3.30.85 \(\int \frac {b+d x}{x^4 \sqrt [4]{\frac {b+a x}{d+c x}}} \, dx\)

Optimal. Leaf size=388 \[ \frac {\left (\frac {a x+b}{c x+d}\right )^{3/4} \left (-45 a^2 c d^2 x^3-45 a^2 d^3 x^2+6 a b c^2 d x^3+42 a b c d^2 x^2+36 a b d^3 x+60 a c d^3 x^3+60 a d^4 x^2+7 b^2 c^3 x^3+3 b^2 c^2 d x^2-36 b^2 c d^2 x-32 b^2 d^3-12 b c^2 d^2 x^3-60 b c d^3 x^2-48 b d^4 x\right )}{96 b^2 d^2 x^3}+\frac {\left (-15 a^3 d^3+5 a^2 b c d^2+20 a^2 d^4+3 a b^2 c^2 d-8 a b c d^3+7 b^3 c^3-12 b^2 c^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{\frac {a x+b}{c x+d}}}{\sqrt [4]{b}}\right )}{64 b^{9/4} d^{11/4}}+\frac {\left (15 a^3 d^3-5 a^2 b c d^2-20 a^2 d^4-3 a b^2 c^2 d+8 a b c d^3-7 b^3 c^3+12 b^2 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{\frac {a x+b}{c x+d}}}{\sqrt [4]{b}}\right )}{64 b^{9/4} d^{11/4}} \]

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Rubi [A]  time = 0.58, antiderivative size = 397, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1962, 577, 457, 290, 298, 205, 208} \begin {gather*} -\frac {\left (2 b c d (5 a-6 d)+5 a d^2 (3 a-4 d)+7 b^2 c^2\right ) (b c-a d) \left (\frac {a x+b}{c x+d}\right )^{3/4}}{32 b^2 d^2 \left (b-\frac {d (a x+b)}{c x+d}\right )}+\frac {\left (2 b c d (5 a-6 d)+5 a d^2 (3 a-4 d)+7 b^2 c^2\right ) (b c-a d) \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{\frac {a x+b}{c x+d}}}{\sqrt [4]{b}}\right )}{64 b^{9/4} d^{11/4}}-\frac {\left (2 b c d (5 a-6 d)+5 a d^2 (3 a-4 d)+7 b^2 c^2\right ) (b c-a d) \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{\frac {a x+b}{c x+d}}}{\sqrt [4]{b}}\right )}{64 b^{9/4} d^{11/4}}+\frac {(d (9 a-4 d)+7 b c) (b c-a d)^2 \left (\frac {a x+b}{c x+d}\right )^{3/4}}{24 b d^2 \left (b-\frac {d (a x+b)}{c x+d}\right )^2}-\frac {(b+d x) (b c-a d)^3 \left (\frac {a x+b}{c x+d}\right )^{3/4}}{3 b d (c x+d) \left (b-\frac {d (a x+b)}{c x+d}\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b + d*x)/(x^4*((b + a*x)/(d + c*x))^(1/4)),x]

[Out]

-1/3*((b*c - a*d)^3*((b + a*x)/(d + c*x))^(3/4)*(b + d*x))/(b*d*(d + c*x)*(b - (d*(b + a*x))/(d + c*x))^3) + (
(b*c - a*d)^2*(7*b*c + (9*a - 4*d)*d)*((b + a*x)/(d + c*x))^(3/4))/(24*b*d^2*(b - (d*(b + a*x))/(d + c*x))^2)
- ((b*c - a*d)*(7*b^2*c^2 + 2*b*c*(5*a - 6*d)*d + 5*a*(3*a - 4*d)*d^2)*((b + a*x)/(d + c*x))^(3/4))/(32*b^2*d^
2*(b - (d*(b + a*x))/(d + c*x))) + ((b*c - a*d)*(7*b^2*c^2 + 2*b*c*(5*a - 6*d)*d + 5*a*(3*a - 4*d)*d^2)*ArcTan
[(d^(1/4)*((b + a*x)/(d + c*x))^(1/4))/b^(1/4)])/(64*b^(9/4)*d^(11/4)) - ((b*c - a*d)*(7*b^2*c^2 + 2*b*c*(5*a
- 6*d)*d + 5*a*(3*a - 4*d)*d^2)*ArcTanh[(d^(1/4)*((b + a*x)/(d + c*x))^(1/4))/b^(1/4)])/(64*b^(9/4)*d^(11/4))

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 290

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 0]

Rule 457

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d
)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*b*e*n*(p + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b
*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] &&
LeQ[-1, m, -(n*(p + 1))]))

Rule 577

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> -Simp[((b*e - a*f)*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(a*b*g*n*(p + 1)), x] + Dist[
1/(a*b*n*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(b*e*n*(p + 1) + (b*e - a*f)*(m
+ 1)) + d*(b*e*n*(p + 1) + (b*e - a*f)*(m + n*q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] &&
IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 0] &&  !(EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f])

Rule 1962

Int[(u_)^(r_.)*(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> Wi
th[{q = Denominator[p]}, Dist[(q*e*(b*c - a*d))/n, Subst[Int[SimplifyIntegrand[(x^(q*(p + 1) - 1)*(-(a*e) + c*
x^q)^((m + 1)/n - 1)*(u /. x -> (-(a*e) + c*x^q)^(1/n)/(b*e - d*x^q)^(1/n))^r)/(b*e - d*x^q)^((m + 1)/n + 1),
x], x], x, ((e*(a + b*x^n))/(c + d*x^n))^(1/q)], x]] /; FreeQ[{a, b, c, d, e}, x] && PolynomialQ[u, x] && Frac
tionQ[p] && IntegerQ[1/n] && IntegersQ[m, r]

Rubi steps

\begin {align*} \int \frac {b+d x}{x^4 \sqrt [4]{\frac {b+a x}{d+c x}}} \, dx &=-\left ((4 (b c-a d)) \operatorname {Subst}\left (\int \frac {x^2 \left (a-c x^4\right ) \left (b (a-d)+\left (-b c+d^2\right ) x^4\right )}{\left (b-d x^4\right )^4} \, dx,x,\sqrt [4]{\frac {b+a x}{d+c x}}\right )\right )\\ &=-\frac {(b c-a d)^3 \left (\frac {b+a x}{d+c x}\right )^{3/4} (b+d x)}{3 b d (d+c x) \left (b-\frac {d (b+a x)}{d+c x}\right )^3}-\frac {(b c-a d) \operatorname {Subst}\left (\int \frac {x^2 \left (3 b (a-d) (b c+3 a d)-(7 b c+5 a d) \left (b c-d^2\right ) x^4\right )}{\left (b-d x^4\right )^3} \, dx,x,\sqrt [4]{\frac {b+a x}{d+c x}}\right )}{3 b d}\\ &=-\frac {(b c-a d)^3 \left (\frac {b+a x}{d+c x}\right )^{3/4} (b+d x)}{3 b d (d+c x) \left (b-\frac {d (b+a x)}{d+c x}\right )^3}+\frac {(b c-a d)^2 (7 b c+(9 a-4 d) d) \left (\frac {b+a x}{d+c x}\right )^{3/4}}{24 b d^2 \left (b-\frac {d (b+a x)}{d+c x}\right )^2}-\frac {\left ((b c-a d) \left (7 b^2 c^2+2 b c (5 a-6 d) d+5 a (3 a-4 d) d^2\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b-d x^4\right )^2} \, dx,x,\sqrt [4]{\frac {b+a x}{d+c x}}\right )}{8 b d^2}\\ &=-\frac {(b c-a d)^3 \left (\frac {b+a x}{d+c x}\right )^{3/4} (b+d x)}{3 b d (d+c x) \left (b-\frac {d (b+a x)}{d+c x}\right )^3}+\frac {(b c-a d)^2 (7 b c+(9 a-4 d) d) \left (\frac {b+a x}{d+c x}\right )^{3/4}}{24 b d^2 \left (b-\frac {d (b+a x)}{d+c x}\right )^2}-\frac {(b c-a d) \left (7 b^2 c^2+2 b c (5 a-6 d) d+5 a (3 a-4 d) d^2\right ) \left (\frac {b+a x}{d+c x}\right )^{3/4}}{32 b^2 d^2 \left (b-\frac {d (b+a x)}{d+c x}\right )}-\frac {\left ((b c-a d) \left (7 b^2 c^2+2 b c (5 a-6 d) d+5 a (3 a-4 d) d^2\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{b-d x^4} \, dx,x,\sqrt [4]{\frac {b+a x}{d+c x}}\right )}{32 b^2 d^2}\\ &=-\frac {(b c-a d)^3 \left (\frac {b+a x}{d+c x}\right )^{3/4} (b+d x)}{3 b d (d+c x) \left (b-\frac {d (b+a x)}{d+c x}\right )^3}+\frac {(b c-a d)^2 (7 b c+(9 a-4 d) d) \left (\frac {b+a x}{d+c x}\right )^{3/4}}{24 b d^2 \left (b-\frac {d (b+a x)}{d+c x}\right )^2}-\frac {(b c-a d) \left (7 b^2 c^2+2 b c (5 a-6 d) d+5 a (3 a-4 d) d^2\right ) \left (\frac {b+a x}{d+c x}\right )^{3/4}}{32 b^2 d^2 \left (b-\frac {d (b+a x)}{d+c x}\right )}-\frac {\left ((b c-a d) \left (7 b^2 c^2+2 b c (5 a-6 d) d+5 a (3 a-4 d) d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {d} x^2} \, dx,x,\sqrt [4]{\frac {b+a x}{d+c x}}\right )}{64 b^2 d^{5/2}}+\frac {\left ((b c-a d) \left (7 b^2 c^2+2 b c (5 a-6 d) d+5 a (3 a-4 d) d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {d} x^2} \, dx,x,\sqrt [4]{\frac {b+a x}{d+c x}}\right )}{64 b^2 d^{5/2}}\\ &=-\frac {(b c-a d)^3 \left (\frac {b+a x}{d+c x}\right )^{3/4} (b+d x)}{3 b d (d+c x) \left (b-\frac {d (b+a x)}{d+c x}\right )^3}+\frac {(b c-a d)^2 (7 b c+(9 a-4 d) d) \left (\frac {b+a x}{d+c x}\right )^{3/4}}{24 b d^2 \left (b-\frac {d (b+a x)}{d+c x}\right )^2}-\frac {(b c-a d) \left (7 b^2 c^2+2 b c (5 a-6 d) d+5 a (3 a-4 d) d^2\right ) \left (\frac {b+a x}{d+c x}\right )^{3/4}}{32 b^2 d^2 \left (b-\frac {d (b+a x)}{d+c x}\right )}+\frac {(b c-a d) \left (7 b^2 c^2+2 b c (5 a-6 d) d+5 a (3 a-4 d) d^2\right ) \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{\frac {b+a x}{d+c x}}}{\sqrt [4]{b}}\right )}{64 b^{9/4} d^{11/4}}-\frac {(b c-a d) \left (7 b^2 c^2+2 b c (5 a-6 d) d+5 a (3 a-4 d) d^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{\frac {b+a x}{d+c x}}}{\sqrt [4]{b}}\right )}{64 b^{9/4} d^{11/4}}\\ \end {align*}

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Mathematica [C]  time = 0.34, size = 155, normalized size = 0.40 \begin {gather*} \frac {\left (\frac {a x+b}{c x+d}\right )^{3/4} \left (x \left (4 b^2 (c x+d)^2 (3 d (3 a-4 d)+7 b c)-x \left (2 b c d (5 a-6 d)+5 a d^2 (3 a-4 d)+7 b^2 c^2\right ) \left (x (b c-a d) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {d (b+a x)}{b (d+c x)}\right )+3 b (c x+d)\right )\right )-32 b^3 d (c x+d)^2\right )}{96 b^3 d^2 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b + d*x)/(x^4*((b + a*x)/(d + c*x))^(1/4)),x]

[Out]

(((b + a*x)/(d + c*x))^(3/4)*(-32*b^3*d*(d + c*x)^2 + x*(4*b^2*(7*b*c + 3*(3*a - 4*d)*d)*(d + c*x)^2 - (7*b^2*
c^2 + 2*b*c*(5*a - 6*d)*d + 5*a*(3*a - 4*d)*d^2)*x*(3*b*(d + c*x) + (b*c - a*d)*x*Hypergeometric2F1[3/4, 1, 7/
4, (d*(b + a*x))/(b*(d + c*x))]))))/(96*b^3*d^2*x^3)

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IntegrateAlgebraic [B]  time = 1.06, size = 815, normalized size = 2.10 \begin {gather*} \frac {7 b^5 c^3 \left (\frac {b+a x}{d+c x}\right )^{3/4}+3 a b^4 c^2 d \left (\frac {b+a x}{d+c x}\right )^{3/4}-123 a^2 b^3 c d^2 \left (\frac {b+a x}{d+c x}\right )^{3/4}-12 b^4 c^2 d^2 \left (\frac {b+a x}{d+c x}\right )^{3/4}+113 a^3 b^2 d^3 \left (\frac {b+a x}{d+c x}\right )^{3/4}+120 a b^3 c d^3 \left (\frac {b+a x}{d+c x}\right )^{3/4}-108 a^2 b^2 d^4 \left (\frac {b+a x}{d+c x}\right )^{3/4}-18 b^4 c^3 d \left (\frac {b+a x}{d+c x}\right )^{7/4}+102 a b^3 c^2 d^2 \left (\frac {b+a x}{d+c x}\right )^{7/4}+42 a^2 b^2 c d^3 \left (\frac {b+a x}{d+c x}\right )^{7/4}-24 b^3 c^2 d^3 \left (\frac {b+a x}{d+c x}\right )^{7/4}-126 a^3 b d^4 \left (\frac {b+a x}{d+c x}\right )^{7/4}-144 a b^2 c d^4 \left (\frac {b+a x}{d+c x}\right )^{7/4}+168 a^2 b d^5 \left (\frac {b+a x}{d+c x}\right )^{7/4}-21 b^3 c^3 d^2 \left (\frac {b+a x}{d+c x}\right )^{11/4}-9 a b^2 c^2 d^3 \left (\frac {b+a x}{d+c x}\right )^{11/4}-15 a^2 b c d^4 \left (\frac {b+a x}{d+c x}\right )^{11/4}+36 b^2 c^2 d^4 \left (\frac {b+a x}{d+c x}\right )^{11/4}+45 a^3 d^5 \left (\frac {b+a x}{d+c x}\right )^{11/4}+24 a b c d^5 \left (\frac {b+a x}{d+c x}\right )^{11/4}-60 a^2 d^6 \left (\frac {b+a x}{d+c x}\right )^{11/4}}{96 b^2 d^2 \left (b-\frac {d (b+a x)}{d+c x}\right )^3}+\frac {\left (7 b^3 c^3+3 a b^2 c^2 d+5 a^2 b c d^2-12 b^2 c^2 d^2-15 a^3 d^3-8 a b c d^3+20 a^2 d^4\right ) \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{\frac {b+a x}{d+c x}}}{\sqrt [4]{b}}\right )}{64 b^{9/4} d^{11/4}}+\frac {\left (-7 b^3 c^3-3 a b^2 c^2 d-5 a^2 b c d^2+12 b^2 c^2 d^2+15 a^3 d^3+8 a b c d^3-20 a^2 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{\frac {b+a x}{d+c x}}}{\sqrt [4]{b}}\right )}{64 b^{9/4} d^{11/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(b + d*x)/(x^4*((b + a*x)/(d + c*x))^(1/4)),x]

[Out]

(7*b^5*c^3*((b + a*x)/(d + c*x))^(3/4) + 3*a*b^4*c^2*d*((b + a*x)/(d + c*x))^(3/4) - 123*a^2*b^3*c*d^2*((b + a
*x)/(d + c*x))^(3/4) - 12*b^4*c^2*d^2*((b + a*x)/(d + c*x))^(3/4) + 113*a^3*b^2*d^3*((b + a*x)/(d + c*x))^(3/4
) + 120*a*b^3*c*d^3*((b + a*x)/(d + c*x))^(3/4) - 108*a^2*b^2*d^4*((b + a*x)/(d + c*x))^(3/4) - 18*b^4*c^3*d*(
(b + a*x)/(d + c*x))^(7/4) + 102*a*b^3*c^2*d^2*((b + a*x)/(d + c*x))^(7/4) + 42*a^2*b^2*c*d^3*((b + a*x)/(d +
c*x))^(7/4) - 24*b^3*c^2*d^3*((b + a*x)/(d + c*x))^(7/4) - 126*a^3*b*d^4*((b + a*x)/(d + c*x))^(7/4) - 144*a*b
^2*c*d^4*((b + a*x)/(d + c*x))^(7/4) + 168*a^2*b*d^5*((b + a*x)/(d + c*x))^(7/4) - 21*b^3*c^3*d^2*((b + a*x)/(
d + c*x))^(11/4) - 9*a*b^2*c^2*d^3*((b + a*x)/(d + c*x))^(11/4) - 15*a^2*b*c*d^4*((b + a*x)/(d + c*x))^(11/4)
+ 36*b^2*c^2*d^4*((b + a*x)/(d + c*x))^(11/4) + 45*a^3*d^5*((b + a*x)/(d + c*x))^(11/4) + 24*a*b*c*d^5*((b + a
*x)/(d + c*x))^(11/4) - 60*a^2*d^6*((b + a*x)/(d + c*x))^(11/4))/(96*b^2*d^2*(b - (d*(b + a*x))/(d + c*x))^3)
+ ((7*b^3*c^3 + 3*a*b^2*c^2*d + 5*a^2*b*c*d^2 - 12*b^2*c^2*d^2 - 15*a^3*d^3 - 8*a*b*c*d^3 + 20*a^2*d^4)*ArcTan
[(d^(1/4)*((b + a*x)/(d + c*x))^(1/4))/b^(1/4)])/(64*b^(9/4)*d^(11/4)) + ((-7*b^3*c^3 - 3*a*b^2*c^2*d - 5*a^2*
b*c*d^2 + 12*b^2*c^2*d^2 + 15*a^3*d^3 + 8*a*b*c*d^3 - 20*a^2*d^4)*ArcTanh[(d^(1/4)*((b + a*x)/(d + c*x))^(1/4)
)/b^(1/4)])/(64*b^(9/4)*d^(11/4))

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fricas [B]  time = 4.96, size = 8984, normalized size = 23.15

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+b)/x^4/((a*x+b)/(c*x+d))^(1/4),x, algorithm="fricas")

[Out]

-1/384*(12*b^2*d^2*x^3*((2401*b^12*c^12 + 4116*a*b^11*c^11*d + 160000*a^8*d^16 - 32000*(15*a^9 + 8*a^7*b*c)*d^
15 + 800*(675*a^10 + 920*a^8*b*c - 288*a^6*b^2*c^2)*d^14 - 80*(3375*a^11 + 9900*a^9*b*c - 6720*a^7*b^2*c^2 - 5
248*a^5*b^3*c^3)*d^13 + (50625*a^12 + 378000*a^10*b*c - 429600*a^8*b^2*c^2 - 762880*a^6*b^3*c^3 + 165376*a^4*b
^4*c^4)*d^12 - 4*(16875*a^11*b*c - 31500*a^9*b^2*c^2 - 81600*a^7*b^3*c^3 + 138880*a^5*b^4*c^4 + 62976*a^3*b^5*
c^5)*d^11 - 2*(3375*a^10*b^2*c^2 - 44600*a^8*b^3*c^3 - 359520*a^6*b^4*c^4 - 85248*a^4*b^5*c^5 + 41472*a^2*b^6*
c^6)*d^10 - 4*(15375*a^9*b^3*c^3 + 105400*a^7*b^4*c^4 - 48480*a^5*b^5*c^5 - 104704*a^3*b^6*c^6 - 13824*a*b^7*c
^7)*d^9 + (93775*a^8*b^4*c^4 - 159840*a^6*b^5*c^5 - 423744*a^4*b^6*c^6 + 101376*a^2*b^7*c^7 + 20736*b^8*c^8)*d
^8 + 24*(775*a^7*b^5*c^5 + 3140*a^5*b^6*c^6 - 10976*a^3*b^7*c^7 - 4896*a*b^8*c^8)*d^7 + 4*(7895*a^6*b^6*c^6 +
45624*a^4*b^7*c^7 - 1416*a^2*b^8*c^8 - 12096*b^9*c^9)*d^6 - 24*(2025*a^5*b^7*c^7 - 3334*a^3*b^8*c^8 - 3864*a*b
^9*c^9)*d^5 - (15249*a^4*b^8*c^8 + 31024*a^2*b^9*c^9 - 42336*b^10*c^10)*d^4 - 28*(393*a^3*b^9*c^9 + 1148*a*b^1
0*c^10)*d^3 + 98*(97*a^2*b^10*c^10 - 168*b^11*c^11)*d^2)/(b^9*d^11))^(1/4)*arctan((sqrt((117649*b^18*c^18 + 30
2526*a*b^17*c^17*d + 64000000*a^12*d^24 - 19200000*(15*a^13 + 8*a^11*b*c)*d^23 + 2400000*(225*a^14 + 280*a^12*
b*c - 32*a^10*b^2*c^2)*d^22 - 160000*(3375*a^15 + 7650*a^13*b*c - 1680*a^11*b^2*c^2 - 2368*a^9*b^3*c^3)*d^21 +
6000*(50625*a^16 + 198000*a^14*b*c - 59600*a^12*b^2*c^2 - 218880*a^10*b^3*c^3 + 256*a^8*b^4*c^4)*d^20 - 120*(
759375*a^17 + 5400000*a^15*b*c - 1740000*a^13*b^2*c^2 - 14608000*a^11*b^3*c^3 + 1619200*a^9*b^4*c^4 + 3411968*
a^7*b^5*c^5)*d^19 + (11390625*a^18 + 188325000*a^16*b*c - 32400000*a^14*b^2*c^2 - 1068640000*a^12*b^3*c^3 + 71
8944000*a^10*b^4*c^4 + 1158789120*a^8*b^5*c^5 + 26066944*a^6*b^6*c^6)*d^18 - 6*(3796875*a^17*b*c + 2700000*a^1
5*b^2*c^2 - 38400000*a^13*b^3*c^3 + 179920000*a^11*b^4*c^4 + 202016000*a^9*b^5*c^5 - 68751360*a^7*b^6*c^6 - 40
943616*a^5*b^7*c^7)*d^17 + 3*(1771875*a^16*b^2*c^2 + 12600000*a^14*b^3*c^3 + 271640000*a^12*b^4*c^4 + 18748800
0*a^10*b^5*c^5 - 441555200*a^8*b^6*c^6 - 190455808*a^6*b^7*c^7 + 184320*a^4*b^8*c^8)*d^16 - 16*(1096875*a^15*b
^3*c^3 + 19293750*a^13*b^4*c^4 + 7500000*a^11*b^5*c^5 - 91234000*a^9*b^6*c^6 - 21033600*a^7*b^7*c^7 + 28212096
*a^5*b^8*c^8 + 5114880*a^3*b^9*c^9)*d^15 + 60*(781875*a^14*b^4*c^4 + 433000*a^12*b^5*c^5 - 11277600*a^10*b^6*c
^6 + 3939520*a^8*b^7*c^7 + 16628416*a^6*b^8*c^8 + 2025984*a^4*b^9*c^9 - 165888*a^2*b^10*c^10)*d^14 - 24*(32812
5*a^13*b^5*c^5 - 4055000*a^11*b^6*c^6 + 17584000*a^9*b^7*c^7 + 31071200*a^7*b^8*c^8 - 9901760*a^5*b^9*c^9 - 90
08640*a^3*b^10*c^10 - 497664*a*b^11*c^11)*d^13 + 4*(2100625*a^12*b^6*c^6 + 55086000*a^10*b^7*c^7 + 50212200*a^
8*b^8*c^8 - 153101120*a^6*b^9*c^9 - 76238400*a^4*b^10*c^10 + 4561920*a^2*b^11*c^11 + 746496*b^12*c^12)*d^12 -
48*(879375*a^11*b^7*c^7 + 277625*a^9*b^8*c^8 - 9199200*a^7*b^9*c^9 - 1007760*a^5*b^10*c^10 + 4747200*a^3*b^11*
c^11 + 819072*a*b^12*c^12)*d^11 + 6*(492125*a^10*b^8*c^8 - 17593400*a^8*b^9*c^9 + 22208960*a^6*b^10*c^10 + 485
40800*a^4*b^11*c^11 - 259200*a^2*b^12*c^12 - 1741824*b^13*c^13)*d^10 - 20*(15975*a^9*b^9*c^9 + 4653552*a^7*b^1
0*c^10 + 5058048*a^5*b^11*c^11 - 6051520*a^3*b^12*c^12 - 2685312*a*b^13*c^13)*d^9 + 6*(3566605*a^8*b^10*c^10 -
379744*a^6*b^11*c^11 - 21549600*a^4*b^12*c^12 - 3158400*a^2*b^13*c^13 + 2540160*b^14*c^14)*d^8 + 48*(28305*a^
7*b^11*c^11 + 869438*a^5*b^12*c^12 - 700000*a^3*b^13*c^13 - 811440*a*b^14*c^14)*d^7 - 4*(103199*a^6*b^12*c^12
- 6265560*a^4*b^13*c^13 - 4351200*a^2*b^14*c^14 + 2963520*b^15*c^15)*d^6 - 168*(37083*a^5*b^13*c^13 - 27160*a^
3*b^14*c^14 - 94080*a*b^15*c^15)*d^5 - 2940*(461*a^4*b^14*c^14 + 2128*a^2*b^15*c^15 - 1764*b^16*c^16)*d^4 - 82
32*(30*a^3*b^15*c^15 + 413*a*b^16*c^16)*d^3 + 7203*(115*a^2*b^16*c^16 - 168*b^17*c^17)*d^2)*sqrt((a*x + b)/(c*
x + d)) + (2401*b^17*c^12*d^5 + 4116*a*b^16*c^11*d^6 + 160000*a^8*b^5*d^21 - 32000*(15*a^9*b^5 + 8*a^7*b^6*c)*
d^20 + 800*(675*a^10*b^5 + 920*a^8*b^6*c - 288*a^6*b^7*c^2)*d^19 - 80*(3375*a^11*b^5 + 9900*a^9*b^6*c - 6720*a
^7*b^7*c^2 - 5248*a^5*b^8*c^3)*d^18 + (50625*a^12*b^5 + 378000*a^10*b^6*c - 429600*a^8*b^7*c^2 - 762880*a^6*b^
8*c^3 + 165376*a^4*b^9*c^4)*d^17 - 4*(16875*a^11*b^6*c - 31500*a^9*b^7*c^2 - 81600*a^7*b^8*c^3 + 138880*a^5*b^
9*c^4 + 62976*a^3*b^10*c^5)*d^16 - 2*(3375*a^10*b^7*c^2 - 44600*a^8*b^8*c^3 - 359520*a^6*b^9*c^4 - 85248*a^4*b
^10*c^5 + 41472*a^2*b^11*c^6)*d^15 - 4*(15375*a^9*b^8*c^3 + 105400*a^7*b^9*c^4 - 48480*a^5*b^10*c^5 - 104704*a
^3*b^11*c^6 - 13824*a*b^12*c^7)*d^14 + (93775*a^8*b^9*c^4 - 159840*a^6*b^10*c^5 - 423744*a^4*b^11*c^6 + 101376
*a^2*b^12*c^7 + 20736*b^13*c^8)*d^13 + 24*(775*a^7*b^10*c^5 + 3140*a^5*b^11*c^6 - 10976*a^3*b^12*c^7 - 4896*a*
b^13*c^8)*d^12 + 4*(7895*a^6*b^11*c^6 + 45624*a^4*b^12*c^7 - 1416*a^2*b^13*c^8 - 12096*b^14*c^9)*d^11 - 24*(20
25*a^5*b^12*c^7 - 3334*a^3*b^13*c^8 - 3864*a*b^14*c^9)*d^10 - (15249*a^4*b^13*c^8 + 31024*a^2*b^14*c^9 - 42336
*b^15*c^10)*d^9 - 28*(393*a^3*b^14*c^9 + 1148*a*b^15*c^10)*d^8 + 98*(97*a^2*b^15*c^10 - 168*b^16*c^11)*d^7)*sq
rt((2401*b^12*c^12 + 4116*a*b^11*c^11*d + 160000*a^8*d^16 - 32000*(15*a^9 + 8*a^7*b*c)*d^15 + 800*(675*a^10 +
920*a^8*b*c - 288*a^6*b^2*c^2)*d^14 - 80*(3375*a^11 + 9900*a^9*b*c - 6720*a^7*b^2*c^2 - 5248*a^5*b^3*c^3)*d^13
+ (50625*a^12 + 378000*a^10*b*c - 429600*a^8*b^2*c^2 - 762880*a^6*b^3*c^3 + 165376*a^4*b^4*c^4)*d^12 - 4*(168
75*a^11*b*c - 31500*a^9*b^2*c^2 - 81600*a^7*b^3*c^3 + 138880*a^5*b^4*c^4 + 62976*a^3*b^5*c^5)*d^11 - 2*(3375*a
^10*b^2*c^2 - 44600*a^8*b^3*c^3 - 359520*a^6*b^4*c^4 - 85248*a^4*b^5*c^5 + 41472*a^2*b^6*c^6)*d^10 - 4*(15375*
a^9*b^3*c^3 + 105400*a^7*b^4*c^4 - 48480*a^5*b^5*c^5 - 104704*a^3*b^6*c^6 - 13824*a*b^7*c^7)*d^9 + (93775*a^8*
b^4*c^4 - 159840*a^6*b^5*c^5 - 423744*a^4*b^6*c^6 + 101376*a^2*b^7*c^7 + 20736*b^8*c^8)*d^8 + 24*(775*a^7*b^5*
c^5 + 3140*a^5*b^6*c^6 - 10976*a^3*b^7*c^7 - 4896*a*b^8*c^8)*d^7 + 4*(7895*a^6*b^6*c^6 + 45624*a^4*b^7*c^7 - 1
416*a^2*b^8*c^8 - 12096*b^9*c^9)*d^6 - 24*(2025*a^5*b^7*c^7 - 3334*a^3*b^8*c^8 - 3864*a*b^9*c^9)*d^5 - (15249*
a^4*b^8*c^8 + 31024*a^2*b^9*c^9 - 42336*b^10*c^10)*d^4 - 28*(393*a^3*b^9*c^9 + 1148*a*b^10*c^10)*d^3 + 98*(97*
a^2*b^10*c^10 - 168*b^11*c^11)*d^2)/(b^9*d^11)))*b^2*d^3*((2401*b^12*c^12 + 4116*a*b^11*c^11*d + 160000*a^8*d^
16 - 32000*(15*a^9 + 8*a^7*b*c)*d^15 + 800*(675*a^10 + 920*a^8*b*c - 288*a^6*b^2*c^2)*d^14 - 80*(3375*a^11 + 9
900*a^9*b*c - 6720*a^7*b^2*c^2 - 5248*a^5*b^3*c^3)*d^13 + (50625*a^12 + 378000*a^10*b*c - 429600*a^8*b^2*c^2 -
762880*a^6*b^3*c^3 + 165376*a^4*b^4*c^4)*d^12 - 4*(16875*a^11*b*c - 31500*a^9*b^2*c^2 - 81600*a^7*b^3*c^3 + 1
38880*a^5*b^4*c^4 + 62976*a^3*b^5*c^5)*d^11 - 2*(3375*a^10*b^2*c^2 - 44600*a^8*b^3*c^3 - 359520*a^6*b^4*c^4 -
85248*a^4*b^5*c^5 + 41472*a^2*b^6*c^6)*d^10 - 4*(15375*a^9*b^3*c^3 + 105400*a^7*b^4*c^4 - 48480*a^5*b^5*c^5 -
104704*a^3*b^6*c^6 - 13824*a*b^7*c^7)*d^9 + (93775*a^8*b^4*c^4 - 159840*a^6*b^5*c^5 - 423744*a^4*b^6*c^6 + 101
376*a^2*b^7*c^7 + 20736*b^8*c^8)*d^8 + 24*(775*a^7*b^5*c^5 + 3140*a^5*b^6*c^6 - 10976*a^3*b^7*c^7 - 4896*a*b^8
*c^8)*d^7 + 4*(7895*a^6*b^6*c^6 + 45624*a^4*b^7*c^7 - 1416*a^2*b^8*c^8 - 12096*b^9*c^9)*d^6 - 24*(2025*a^5*b^7
*c^7 - 3334*a^3*b^8*c^8 - 3864*a*b^9*c^9)*d^5 - (15249*a^4*b^8*c^8 + 31024*a^2*b^9*c^9 - 42336*b^10*c^10)*d^4
- 28*(393*a^3*b^9*c^9 + 1148*a*b^10*c^10)*d^3 + 98*(97*a^2*b^10*c^10 - 168*b^11*c^11)*d^2)/(b^9*d^11))^(1/4) -
(343*b^11*c^9*d^3 + 441*a*b^10*c^8*d^4 + 8000*a^6*b^2*d^15 - 1200*(15*a^7*b^2 + 8*a^5*b^3*c)*d^14 + 60*(225*a
^8*b^2 + 340*a^6*b^3*c - 176*a^4*b^4*c^2)*d^13 - (3375*a^9*b^2 + 14400*a^7*b^3*c - 17520*a^5*b^4*c^2 - 11008*a
^3*b^5*c^3)*d^12 + 3*(1125*a^8*b^3*c - 2800*a^6*b^4*c^2 - 3120*a^4*b^5*c^3 + 2112*a^2*b^6*c^4)*d^11 + 12*(75*a
^7*b^4*c^2 - 320*a^5*b^5*c^3 - 1172*a^3*b^6*c^4 - 288*a*b^7*c^5)*d^10 + 4*(875*a^6*b^5*c^3 + 2850*a^4*b^6*c^4
- 1212*a^2*b^7*c^5 - 432*b^8*c^6)*d^9 - 222*(15*a^5*b^6*c^4 - 32*a^3*b^7*c^5 - 24*a*b^8*c^6)*d^8 - 6*(205*a^4*
b^7*c^5 + 152*a^2*b^8*c^6 - 504*b^9*c^7)*d^7 - 12*(129*a^3*b^8*c^6 + 224*a*b^9*c^7)*d^6 + 84*(11*a^2*b^9*c^7 -
21*b^10*c^8)*d^5)*((a*x + b)/(c*x + d))^(1/4)*((2401*b^12*c^12 + 4116*a*b^11*c^11*d + 160000*a^8*d^16 - 32000
*(15*a^9 + 8*a^7*b*c)*d^15 + 800*(675*a^10 + 920*a^8*b*c - 288*a^6*b^2*c^2)*d^14 - 80*(3375*a^11 + 9900*a^9*b*
c - 6720*a^7*b^2*c^2 - 5248*a^5*b^3*c^3)*d^13 + (50625*a^12 + 378000*a^10*b*c - 429600*a^8*b^2*c^2 - 762880*a^
6*b^3*c^3 + 165376*a^4*b^4*c^4)*d^12 - 4*(16875*a^11*b*c - 31500*a^9*b^2*c^2 - 81600*a^7*b^3*c^3 + 138880*a^5*
b^4*c^4 + 62976*a^3*b^5*c^5)*d^11 - 2*(3375*a^10*b^2*c^2 - 44600*a^8*b^3*c^3 - 359520*a^6*b^4*c^4 - 85248*a^4*
b^5*c^5 + 41472*a^2*b^6*c^6)*d^10 - 4*(15375*a^9*b^3*c^3 + 105400*a^7*b^4*c^4 - 48480*a^5*b^5*c^5 - 104704*a^3
*b^6*c^6 - 13824*a*b^7*c^7)*d^9 + (93775*a^8*b^4*c^4 - 159840*a^6*b^5*c^5 - 423744*a^4*b^6*c^6 + 101376*a^2*b^
7*c^7 + 20736*b^8*c^8)*d^8 + 24*(775*a^7*b^5*c^5 + 3140*a^5*b^6*c^6 - 10976*a^3*b^7*c^7 - 4896*a*b^8*c^8)*d^7
+ 4*(7895*a^6*b^6*c^6 + 45624*a^4*b^7*c^7 - 1416*a^2*b^8*c^8 - 12096*b^9*c^9)*d^6 - 24*(2025*a^5*b^7*c^7 - 333
4*a^3*b^8*c^8 - 3864*a*b^9*c^9)*d^5 - (15249*a^4*b^8*c^8 + 31024*a^2*b^9*c^9 - 42336*b^10*c^10)*d^4 - 28*(393*
a^3*b^9*c^9 + 1148*a*b^10*c^10)*d^3 + 98*(97*a^2*b^10*c^10 - 168*b^11*c^11)*d^2)/(b^9*d^11))^(1/4))/(2401*b^12
*c^12 + 4116*a*b^11*c^11*d + 160000*a^8*d^16 - 32000*(15*a^9 + 8*a^7*b*c)*d^15 + 800*(675*a^10 + 920*a^8*b*c -
288*a^6*b^2*c^2)*d^14 - 80*(3375*a^11 + 9900*a^9*b*c - 6720*a^7*b^2*c^2 - 5248*a^5*b^3*c^3)*d^13 + (50625*a^1
2 + 378000*a^10*b*c - 429600*a^8*b^2*c^2 - 762880*a^6*b^3*c^3 + 165376*a^4*b^4*c^4)*d^12 - 4*(16875*a^11*b*c -
31500*a^9*b^2*c^2 - 81600*a^7*b^3*c^3 + 138880*a^5*b^4*c^4 + 62976*a^3*b^5*c^5)*d^11 - 2*(3375*a^10*b^2*c^2 -
44600*a^8*b^3*c^3 - 359520*a^6*b^4*c^4 - 85248*a^4*b^5*c^5 + 41472*a^2*b^6*c^6)*d^10 - 4*(15375*a^9*b^3*c^3 +
105400*a^7*b^4*c^4 - 48480*a^5*b^5*c^5 - 104704*a^3*b^6*c^6 - 13824*a*b^7*c^7)*d^9 + (93775*a^8*b^4*c^4 - 159
840*a^6*b^5*c^5 - 423744*a^4*b^6*c^6 + 101376*a^2*b^7*c^7 + 20736*b^8*c^8)*d^8 + 24*(775*a^7*b^5*c^5 + 3140*a^
5*b^6*c^6 - 10976*a^3*b^7*c^7 - 4896*a*b^8*c^8)*d^7 + 4*(7895*a^6*b^6*c^6 + 45624*a^4*b^7*c^7 - 1416*a^2*b^8*c
^8 - 12096*b^9*c^9)*d^6 - 24*(2025*a^5*b^7*c^7 - 3334*a^3*b^8*c^8 - 3864*a*b^9*c^9)*d^5 - (15249*a^4*b^8*c^8 +
31024*a^2*b^9*c^9 - 42336*b^10*c^10)*d^4 - 28*(393*a^3*b^9*c^9 + 1148*a*b^10*c^10)*d^3 + 98*(97*a^2*b^10*c^10
- 168*b^11*c^11)*d^2)) + 3*b^2*d^2*x^3*((2401*b^12*c^12 + 4116*a*b^11*c^11*d + 160000*a^8*d^16 - 32000*(15*a^
9 + 8*a^7*b*c)*d^15 + 800*(675*a^10 + 920*a^8*b*c - 288*a^6*b^2*c^2)*d^14 - 80*(3375*a^11 + 9900*a^9*b*c - 672
0*a^7*b^2*c^2 - 5248*a^5*b^3*c^3)*d^13 + (50625*a^12 + 378000*a^10*b*c - 429600*a^8*b^2*c^2 - 762880*a^6*b^3*c
^3 + 165376*a^4*b^4*c^4)*d^12 - 4*(16875*a^11*b*c - 31500*a^9*b^2*c^2 - 81600*a^7*b^3*c^3 + 138880*a^5*b^4*c^4
+ 62976*a^3*b^5*c^5)*d^11 - 2*(3375*a^10*b^2*c^2 - 44600*a^8*b^3*c^3 - 359520*a^6*b^4*c^4 - 85248*a^4*b^5*c^5
+ 41472*a^2*b^6*c^6)*d^10 - 4*(15375*a^9*b^3*c^3 + 105400*a^7*b^4*c^4 - 48480*a^5*b^5*c^5 - 104704*a^3*b^6*c^
6 - 13824*a*b^7*c^7)*d^9 + (93775*a^8*b^4*c^4 - 159840*a^6*b^5*c^5 - 423744*a^4*b^6*c^6 + 101376*a^2*b^7*c^7 +
20736*b^8*c^8)*d^8 + 24*(775*a^7*b^5*c^5 + 3140*a^5*b^6*c^6 - 10976*a^3*b^7*c^7 - 4896*a*b^8*c^8)*d^7 + 4*(78
95*a^6*b^6*c^6 + 45624*a^4*b^7*c^7 - 1416*a^2*b^8*c^8 - 12096*b^9*c^9)*d^6 - 24*(2025*a^5*b^7*c^7 - 3334*a^3*b
^8*c^8 - 3864*a*b^9*c^9)*d^5 - (15249*a^4*b^8*c^8 + 31024*a^2*b^9*c^9 - 42336*b^10*c^10)*d^4 - 28*(393*a^3*b^9
*c^9 + 1148*a*b^10*c^10)*d^3 + 98*(97*a^2*b^10*c^10 - 168*b^11*c^11)*d^2)/(b^9*d^11))^(1/4)*log(b^7*d^8*((2401
*b^12*c^12 + 4116*a*b^11*c^11*d + 160000*a^8*d^16 - 32000*(15*a^9 + 8*a^7*b*c)*d^15 + 800*(675*a^10 + 920*a^8*
b*c - 288*a^6*b^2*c^2)*d^14 - 80*(3375*a^11 + 9900*a^9*b*c - 6720*a^7*b^2*c^2 - 5248*a^5*b^3*c^3)*d^13 + (5062
5*a^12 + 378000*a^10*b*c - 429600*a^8*b^2*c^2 - 762880*a^6*b^3*c^3 + 165376*a^4*b^4*c^4)*d^12 - 4*(16875*a^11*
b*c - 31500*a^9*b^2*c^2 - 81600*a^7*b^3*c^3 + 138880*a^5*b^4*c^4 + 62976*a^3*b^5*c^5)*d^11 - 2*(3375*a^10*b^2*
c^2 - 44600*a^8*b^3*c^3 - 359520*a^6*b^4*c^4 - 85248*a^4*b^5*c^5 + 41472*a^2*b^6*c^6)*d^10 - 4*(15375*a^9*b^3*
c^3 + 105400*a^7*b^4*c^4 - 48480*a^5*b^5*c^5 - 104704*a^3*b^6*c^6 - 13824*a*b^7*c^7)*d^9 + (93775*a^8*b^4*c^4
- 159840*a^6*b^5*c^5 - 423744*a^4*b^6*c^6 + 101376*a^2*b^7*c^7 + 20736*b^8*c^8)*d^8 + 24*(775*a^7*b^5*c^5 + 31
40*a^5*b^6*c^6 - 10976*a^3*b^7*c^7 - 4896*a*b^8*c^8)*d^7 + 4*(7895*a^6*b^6*c^6 + 45624*a^4*b^7*c^7 - 1416*a^2*
b^8*c^8 - 12096*b^9*c^9)*d^6 - 24*(2025*a^5*b^7*c^7 - 3334*a^3*b^8*c^8 - 3864*a*b^9*c^9)*d^5 - (15249*a^4*b^8*
c^8 + 31024*a^2*b^9*c^9 - 42336*b^10*c^10)*d^4 - 28*(393*a^3*b^9*c^9 + 1148*a*b^10*c^10)*d^3 + 98*(97*a^2*b^10
*c^10 - 168*b^11*c^11)*d^2)/(b^9*d^11))^(3/4) + (343*b^9*c^9 + 441*a*b^8*c^8*d + 8000*a^6*d^12 - 1200*(15*a^7
+ 8*a^5*b*c)*d^11 + 60*(225*a^8 + 340*a^6*b*c - 176*a^4*b^2*c^2)*d^10 - (3375*a^9 + 14400*a^7*b*c - 17520*a^5*
b^2*c^2 - 11008*a^3*b^3*c^3)*d^9 + 3*(1125*a^8*b*c - 2800*a^6*b^2*c^2 - 3120*a^4*b^3*c^3 + 2112*a^2*b^4*c^4)*d
^8 + 12*(75*a^7*b^2*c^2 - 320*a^5*b^3*c^3 - 1172*a^3*b^4*c^4 - 288*a*b^5*c^5)*d^7 + 4*(875*a^6*b^3*c^3 + 2850*
a^4*b^4*c^4 - 1212*a^2*b^5*c^5 - 432*b^6*c^6)*d^6 - 222*(15*a^5*b^4*c^4 - 32*a^3*b^5*c^5 - 24*a*b^6*c^6)*d^5 -
6*(205*a^4*b^5*c^5 + 152*a^2*b^6*c^6 - 504*b^7*c^7)*d^4 - 12*(129*a^3*b^6*c^6 + 224*a*b^7*c^7)*d^3 + 84*(11*a
^2*b^7*c^7 - 21*b^8*c^8)*d^2)*((a*x + b)/(c*x + d))^(1/4)) - 3*b^2*d^2*x^3*((2401*b^12*c^12 + 4116*a*b^11*c^11
*d + 160000*a^8*d^16 - 32000*(15*a^9 + 8*a^7*b*c)*d^15 + 800*(675*a^10 + 920*a^8*b*c - 288*a^6*b^2*c^2)*d^14 -
80*(3375*a^11 + 9900*a^9*b*c - 6720*a^7*b^2*c^2 - 5248*a^5*b^3*c^3)*d^13 + (50625*a^12 + 378000*a^10*b*c - 42
9600*a^8*b^2*c^2 - 762880*a^6*b^3*c^3 + 165376*a^4*b^4*c^4)*d^12 - 4*(16875*a^11*b*c - 31500*a^9*b^2*c^2 - 816
00*a^7*b^3*c^3 + 138880*a^5*b^4*c^4 + 62976*a^3*b^5*c^5)*d^11 - 2*(3375*a^10*b^2*c^2 - 44600*a^8*b^3*c^3 - 359
520*a^6*b^4*c^4 - 85248*a^4*b^5*c^5 + 41472*a^2*b^6*c^6)*d^10 - 4*(15375*a^9*b^3*c^3 + 105400*a^7*b^4*c^4 - 48
480*a^5*b^5*c^5 - 104704*a^3*b^6*c^6 - 13824*a*b^7*c^7)*d^9 + (93775*a^8*b^4*c^4 - 159840*a^6*b^5*c^5 - 423744
*a^4*b^6*c^6 + 101376*a^2*b^7*c^7 + 20736*b^8*c^8)*d^8 + 24*(775*a^7*b^5*c^5 + 3140*a^5*b^6*c^6 - 10976*a^3*b^
7*c^7 - 4896*a*b^8*c^8)*d^7 + 4*(7895*a^6*b^6*c^6 + 45624*a^4*b^7*c^7 - 1416*a^2*b^8*c^8 - 12096*b^9*c^9)*d^6
- 24*(2025*a^5*b^7*c^7 - 3334*a^3*b^8*c^8 - 3864*a*b^9*c^9)*d^5 - (15249*a^4*b^8*c^8 + 31024*a^2*b^9*c^9 - 423
36*b^10*c^10)*d^4 - 28*(393*a^3*b^9*c^9 + 1148*a*b^10*c^10)*d^3 + 98*(97*a^2*b^10*c^10 - 168*b^11*c^11)*d^2)/(
b^9*d^11))^(1/4)*log(-b^7*d^8*((2401*b^12*c^12 + 4116*a*b^11*c^11*d + 160000*a^8*d^16 - 32000*(15*a^9 + 8*a^7*
b*c)*d^15 + 800*(675*a^10 + 920*a^8*b*c - 288*a^6*b^2*c^2)*d^14 - 80*(3375*a^11 + 9900*a^9*b*c - 6720*a^7*b^2*
c^2 - 5248*a^5*b^3*c^3)*d^13 + (50625*a^12 + 378000*a^10*b*c - 429600*a^8*b^2*c^2 - 762880*a^6*b^3*c^3 + 16537
6*a^4*b^4*c^4)*d^12 - 4*(16875*a^11*b*c - 31500*a^9*b^2*c^2 - 81600*a^7*b^3*c^3 + 138880*a^5*b^4*c^4 + 62976*a
^3*b^5*c^5)*d^11 - 2*(3375*a^10*b^2*c^2 - 44600*a^8*b^3*c^3 - 359520*a^6*b^4*c^4 - 85248*a^4*b^5*c^5 + 41472*a
^2*b^6*c^6)*d^10 - 4*(15375*a^9*b^3*c^3 + 105400*a^7*b^4*c^4 - 48480*a^5*b^5*c^5 - 104704*a^3*b^6*c^6 - 13824*
a*b^7*c^7)*d^9 + (93775*a^8*b^4*c^4 - 159840*a^6*b^5*c^5 - 423744*a^4*b^6*c^6 + 101376*a^2*b^7*c^7 + 20736*b^8
*c^8)*d^8 + 24*(775*a^7*b^5*c^5 + 3140*a^5*b^6*c^6 - 10976*a^3*b^7*c^7 - 4896*a*b^8*c^8)*d^7 + 4*(7895*a^6*b^6
*c^6 + 45624*a^4*b^7*c^7 - 1416*a^2*b^8*c^8 - 12096*b^9*c^9)*d^6 - 24*(2025*a^5*b^7*c^7 - 3334*a^3*b^8*c^8 - 3
864*a*b^9*c^9)*d^5 - (15249*a^4*b^8*c^8 + 31024*a^2*b^9*c^9 - 42336*b^10*c^10)*d^4 - 28*(393*a^3*b^9*c^9 + 114
8*a*b^10*c^10)*d^3 + 98*(97*a^2*b^10*c^10 - 168*b^11*c^11)*d^2)/(b^9*d^11))^(3/4) + (343*b^9*c^9 + 441*a*b^8*c
^8*d + 8000*a^6*d^12 - 1200*(15*a^7 + 8*a^5*b*c)*d^11 + 60*(225*a^8 + 340*a^6*b*c - 176*a^4*b^2*c^2)*d^10 - (3
375*a^9 + 14400*a^7*b*c - 17520*a^5*b^2*c^2 - 11008*a^3*b^3*c^3)*d^9 + 3*(1125*a^8*b*c - 2800*a^6*b^2*c^2 - 31
20*a^4*b^3*c^3 + 2112*a^2*b^4*c^4)*d^8 + 12*(75*a^7*b^2*c^2 - 320*a^5*b^3*c^3 - 1172*a^3*b^4*c^4 - 288*a*b^5*c
^5)*d^7 + 4*(875*a^6*b^3*c^3 + 2850*a^4*b^4*c^4 - 1212*a^2*b^5*c^5 - 432*b^6*c^6)*d^6 - 222*(15*a^5*b^4*c^4 -
32*a^3*b^5*c^5 - 24*a*b^6*c^6)*d^5 - 6*(205*a^4*b^5*c^5 + 152*a^2*b^6*c^6 - 504*b^7*c^7)*d^4 - 12*(129*a^3*b^6
*c^6 + 224*a*b^7*c^7)*d^3 + 84*(11*a^2*b^7*c^7 - 21*b^8*c^8)*d^2)*((a*x + b)/(c*x + d))^(1/4)) + 4*(32*b^2*d^3
- (7*b^2*c^3 + 6*a*b*c^2*d + 60*a*c*d^3 - 3*(15*a^2*c + 4*b*c^2)*d^2)*x^3 - 3*(b^2*c^2*d + 14*a*b*c*d^2 + 20*
a*d^4 - 5*(3*a^2 + 4*b*c)*d^3)*x^2 + 12*(3*b^2*c*d^2 - 3*a*b*d^3 + 4*b*d^4)*x)*((a*x + b)/(c*x + d))^(3/4))/(b
^2*d^2*x^3)

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giac [B]  time = 6.05, size = 1632, normalized size = 4.21

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+b)/x^4/((a*x+b)/(c*x+d))^(1/4),x, algorithm="giac")

[Out]

1/768*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)*(6*sqrt(2)*(7*b^4*c^4 - 4*a*b^3*c^3*d + 2*a^2*b^2*c^2*d^2 - 12*b
^3*c^3*d^2 - 20*a^3*b*c*d^3 + 4*a*b^2*c^2*d^3 + 15*a^4*d^4 + 28*a^2*b*c*d^4 - 20*a^3*d^5)*arctan(1/2*sqrt(2)*(
sqrt(2)*(-b/d)^(1/4) + 2*((a*x + b)/(c*x + d))^(1/4))/(-b/d)^(1/4))/((-b*d^3)^(1/4)*b^2*d^2) + 6*sqrt(2)*(7*b^
4*c^4 - 4*a*b^3*c^3*d + 2*a^2*b^2*c^2*d^2 - 12*b^3*c^3*d^2 - 20*a^3*b*c*d^3 + 4*a*b^2*c^2*d^3 + 15*a^4*d^4 + 2
8*a^2*b*c*d^4 - 20*a^3*d^5)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-b/d)^(1/4) - 2*((a*x + b)/(c*x + d))^(1/4))/(-b/d)^
(1/4))/((-b*d^3)^(1/4)*b^2*d^2) - 3*sqrt(2)*(7*b^4*c^4 - 4*a*b^3*c^3*d + 2*a^2*b^2*c^2*d^2 - 12*b^3*c^3*d^2 -
20*a^3*b*c*d^3 + 4*a*b^2*c^2*d^3 + 15*a^4*d^4 + 28*a^2*b*c*d^4 - 20*a^3*d^5)*log(sqrt(2)*((a*x + b)/(c*x + d))
^(1/4)*(-b/d)^(1/4) + sqrt((a*x + b)/(c*x + d)) + sqrt(-b/d))/((-b*d^3)^(1/4)*b^2*d^2) + 3*sqrt(2)*(7*b^4*c^4
- 4*a*b^3*c^3*d + 2*a^2*b^2*c^2*d^2 - 12*b^3*c^3*d^2 - 20*a^3*b*c*d^3 + 4*a*b^2*c^2*d^3 + 15*a^4*d^4 + 28*a^2*
b*c*d^4 - 20*a^3*d^5)*log(-sqrt(2)*((a*x + b)/(c*x + d))^(1/4)*(-b/d)^(1/4) + sqrt((a*x + b)/(c*x + d)) + sqrt
(-b/d))/((-b*d^3)^(1/4)*b^2*d^2) + 8*(7*b^6*c^4*((a*x + b)/(c*x + d))^(3/4) - 4*a*b^5*c^3*d*((a*x + b)/(c*x +
d))^(3/4) - 18*(a*x + b)*b^5*c^4*d*((a*x + b)/(c*x + d))^(3/4)/(c*x + d) - 126*a^2*b^4*c^2*d^2*((a*x + b)/(c*x
+ d))^(3/4) + 120*(a*x + b)*a*b^4*c^3*d^2*((a*x + b)/(c*x + d))^(3/4)/(c*x + d) - 12*b^5*c^3*d^2*((a*x + b)/(
c*x + d))^(3/4) - 21*(a*x + b)^2*b^4*c^4*d^2*((a*x + b)/(c*x + d))^(3/4)/(c*x + d)^2 + 236*a^3*b^3*c*d^3*((a*x
+ b)/(c*x + d))^(3/4) - 60*(a*x + b)*a^2*b^3*c^2*d^3*((a*x + b)/(c*x + d))^(3/4)/(c*x + d) + 132*a*b^4*c^2*d^
3*((a*x + b)/(c*x + d))^(3/4) + 12*(a*x + b)^2*a*b^3*c^3*d^3*((a*x + b)/(c*x + d))^(3/4)/(c*x + d)^2 - 24*(a*x
+ b)*b^4*c^3*d^3*((a*x + b)/(c*x + d))^(3/4)/(c*x + d) - 113*a^4*b^2*d^4*((a*x + b)/(c*x + d))^(3/4) - 168*(a
*x + b)*a^3*b^2*c*d^4*((a*x + b)/(c*x + d))^(3/4)/(c*x + d) - 228*a^2*b^3*c*d^4*((a*x + b)/(c*x + d))^(3/4) -
6*(a*x + b)^2*a^2*b^2*c^2*d^4*((a*x + b)/(c*x + d))^(3/4)/(c*x + d)^2 - 120*(a*x + b)*a*b^3*c^2*d^4*((a*x + b)
/(c*x + d))^(3/4)/(c*x + d) + 36*(a*x + b)^2*b^3*c^3*d^4*((a*x + b)/(c*x + d))^(3/4)/(c*x + d)^2 + 126*(a*x +
b)*a^4*b*d^5*((a*x + b)/(c*x + d))^(3/4)/(c*x + d) + 108*a^3*b^2*d^5*((a*x + b)/(c*x + d))^(3/4) + 60*(a*x + b
)^2*a^3*b*c*d^5*((a*x + b)/(c*x + d))^(3/4)/(c*x + d)^2 + 312*(a*x + b)*a^2*b^2*c*d^5*((a*x + b)/(c*x + d))^(3
/4)/(c*x + d) - 12*(a*x + b)^2*a*b^2*c^2*d^5*((a*x + b)/(c*x + d))^(3/4)/(c*x + d)^2 - 45*(a*x + b)^2*a^4*d^6*
((a*x + b)/(c*x + d))^(3/4)/(c*x + d)^2 - 168*(a*x + b)*a^3*b*d^6*((a*x + b)/(c*x + d))^(3/4)/(c*x + d) - 84*(
a*x + b)^2*a^2*b*c*d^6*((a*x + b)/(c*x + d))^(3/4)/(c*x + d)^2 + 60*(a*x + b)^2*a^3*d^7*((a*x + b)/(c*x + d))^
(3/4)/(c*x + d)^2)/((b - (a*x + b)*d/(c*x + d))^3*b^2*d^2))

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {d x +b}{x^{4} \left (\frac {a x +b}{c x +d}\right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+b)/x^4/((a*x+b)/(c*x+d))^(1/4),x)

[Out]

int((d*x+b)/x^4/((a*x+b)/(c*x+d))^(1/4),x)

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maxima [A]  time = 0.43, size = 535, normalized size = 1.38 \begin {gather*} -\frac {3 \, {\left (7 \, b^{3} c^{3} d^{2} + 3 \, a b^{2} c^{2} d^{3} + 20 \, a^{2} d^{6} - {\left (15 \, a^{3} + 8 \, a b c\right )} d^{5} + {\left (5 \, a^{2} b c - 12 \, b^{2} c^{2}\right )} d^{4}\right )} \left (\frac {a x + b}{c x + d}\right )^{\frac {11}{4}} + 6 \, {\left (3 \, b^{4} c^{3} d - 17 \, a b^{3} c^{2} d^{2} - 28 \, a^{2} b d^{5} + 3 \, {\left (7 \, a^{3} b + 8 \, a b^{2} c\right )} d^{4} - {\left (7 \, a^{2} b^{2} c - 4 \, b^{3} c^{2}\right )} d^{3}\right )} \left (\frac {a x + b}{c x + d}\right )^{\frac {7}{4}} - {\left (7 \, b^{5} c^{3} + 3 \, a b^{4} c^{2} d - 108 \, a^{2} b^{2} d^{4} + {\left (113 \, a^{3} b^{2} + 120 \, a b^{3} c\right )} d^{3} - 3 \, {\left (41 \, a^{2} b^{3} c + 4 \, b^{4} c^{2}\right )} d^{2}\right )} \left (\frac {a x + b}{c x + d}\right )^{\frac {3}{4}}}{96 \, {\left (b^{5} d^{2} - \frac {3 \, {\left (a x + b\right )} b^{4} d^{3}}{c x + d} + \frac {3 \, {\left (a x + b\right )}^{2} b^{3} d^{4}}{{\left (c x + d\right )}^{2}} - \frac {{\left (a x + b\right )}^{3} b^{2} d^{5}}{{\left (c x + d\right )}^{3}}\right )}} + \frac {{\left (7 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 20 \, a^{2} d^{4} - {\left (15 \, a^{3} + 8 \, a b c\right )} d^{3} + {\left (5 \, a^{2} b c - 12 \, b^{2} c^{2}\right )} d^{2}\right )} {\left (\frac {2 \, \arctan \left (\frac {\sqrt {d} \left (\frac {a x + b}{c x + d}\right )^{\frac {1}{4}}}{\sqrt {\sqrt {b} \sqrt {d}}}\right )}{\sqrt {\sqrt {b} \sqrt {d}} \sqrt {d}} + \frac {\log \left (\frac {\sqrt {d} \left (\frac {a x + b}{c x + d}\right )^{\frac {1}{4}} - \sqrt {\sqrt {b} \sqrt {d}}}{\sqrt {d} \left (\frac {a x + b}{c x + d}\right )^{\frac {1}{4}} + \sqrt {\sqrt {b} \sqrt {d}}}\right )}{\sqrt {\sqrt {b} \sqrt {d}} \sqrt {d}}\right )}}{128 \, b^{2} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+b)/x^4/((a*x+b)/(c*x+d))^(1/4),x, algorithm="maxima")

[Out]

-1/96*(3*(7*b^3*c^3*d^2 + 3*a*b^2*c^2*d^3 + 20*a^2*d^6 - (15*a^3 + 8*a*b*c)*d^5 + (5*a^2*b*c - 12*b^2*c^2)*d^4
)*((a*x + b)/(c*x + d))^(11/4) + 6*(3*b^4*c^3*d - 17*a*b^3*c^2*d^2 - 28*a^2*b*d^5 + 3*(7*a^3*b + 8*a*b^2*c)*d^
4 - (7*a^2*b^2*c - 4*b^3*c^2)*d^3)*((a*x + b)/(c*x + d))^(7/4) - (7*b^5*c^3 + 3*a*b^4*c^2*d - 108*a^2*b^2*d^4
+ (113*a^3*b^2 + 120*a*b^3*c)*d^3 - 3*(41*a^2*b^3*c + 4*b^4*c^2)*d^2)*((a*x + b)/(c*x + d))^(3/4))/(b^5*d^2 -
3*(a*x + b)*b^4*d^3/(c*x + d) + 3*(a*x + b)^2*b^3*d^4/(c*x + d)^2 - (a*x + b)^3*b^2*d^5/(c*x + d)^3) + 1/128*(
7*b^3*c^3 + 3*a*b^2*c^2*d + 20*a^2*d^4 - (15*a^3 + 8*a*b*c)*d^3 + (5*a^2*b*c - 12*b^2*c^2)*d^2)*(2*arctan(sqrt
(d)*((a*x + b)/(c*x + d))^(1/4)/sqrt(sqrt(b)*sqrt(d)))/(sqrt(sqrt(b)*sqrt(d))*sqrt(d)) + log((sqrt(d)*((a*x +
b)/(c*x + d))^(1/4) - sqrt(sqrt(b)*sqrt(d)))/(sqrt(d)*((a*x + b)/(c*x + d))^(1/4) + sqrt(sqrt(b)*sqrt(d))))/(s
qrt(sqrt(b)*sqrt(d))*sqrt(d)))/(b^2*d^2)

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mupad [B]  time = 4.22, size = 532, normalized size = 1.37 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {d^{1/4}\,{\left (\frac {b+a\,x}{d+c\,x}\right )}^{1/4}}{b^{1/4}}\right )\,\left (a\,d-b\,c\right )\,\left (15\,a^2\,d^2+10\,a\,b\,c\,d-20\,a\,d^3+7\,b^2\,c^2-12\,b\,c\,d^2\right )}{64\,b^{9/4}\,d^{11/4}}-\frac {\mathrm {atan}\left (\frac {d^{1/4}\,{\left (\frac {b+a\,x}{d+c\,x}\right )}^{1/4}}{b^{1/4}}\right )\,\left (a\,d-b\,c\right )\,\left (15\,a^2\,d^2+10\,a\,b\,c\,d-20\,a\,d^3+7\,b^2\,c^2-12\,b\,c\,d^2\right )}{64\,b^{9/4}\,d^{11/4}}-\frac {\frac {c^2\,{\left (\frac {b+a\,x}{d+c\,x}\right )}^{7/4}\,\left (\frac {21\,a^3\,c\,d^3}{16}-\frac {7\,a^2\,b\,c^2\,d^2}{16}-\frac {7\,a^2\,c\,d^4}{4}-\frac {17\,a\,b^2\,c^3\,d}{16}+\frac {3\,a\,b\,c^2\,d^3}{2}+\frac {3\,b^3\,c^4}{16}+\frac {b^2\,c^3\,d^2}{4}\right )}{a^3\,b\,d^4}-\frac {c\,{\left (\frac {b+a\,x}{d+c\,x}\right )}^{3/4}\,\left (\frac {113\,a^3\,c^2\,d^3}{96}-\frac {41\,a^2\,b\,c^3\,d^2}{32}-\frac {9\,a^2\,c^2\,d^4}{8}+\frac {a\,b^2\,c^4\,d}{32}+\frac {5\,a\,b\,c^3\,d^3}{4}+\frac {7\,b^3\,c^5}{96}-\frac {b^2\,c^4\,d^2}{8}\right )}{a^3\,d^5}+\frac {c^3\,{\left (\frac {b+a\,x}{d+c\,x}\right )}^{11/4}\,\left (-\frac {15\,a^3\,d^3}{32}+\frac {5\,a^2\,b\,c\,d^2}{32}+\frac {5\,a^2\,d^4}{8}+\frac {3\,a\,b^2\,c^2\,d}{32}-\frac {a\,b\,c\,d^3}{4}+\frac {7\,b^3\,c^3}{32}-\frac {3\,b^2\,c^2\,d^2}{8}\right )}{a^3\,b^2\,d^3}}{\frac {b^3\,c^3}{a^3\,d^3}-\frac {c^3\,{\left (b+a\,x\right )}^3}{a^3\,{\left (d+c\,x\right )}^3}+\frac {3\,b\,c^3\,{\left (b+a\,x\right )}^2}{a^3\,d\,{\left (d+c\,x\right )}^2}-\frac {3\,b^2\,c^3\,\left (b+a\,x\right )}{a^3\,d^2\,\left (d+c\,x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b + d*x)/(x^4*((b + a*x)/(d + c*x))^(1/4)),x)

[Out]

(atanh((d^(1/4)*((b + a*x)/(d + c*x))^(1/4))/b^(1/4))*(a*d - b*c)*(15*a^2*d^2 - 20*a*d^3 + 7*b^2*c^2 - 12*b*c*
d^2 + 10*a*b*c*d))/(64*b^(9/4)*d^(11/4)) - (atan((d^(1/4)*((b + a*x)/(d + c*x))^(1/4))/b^(1/4))*(a*d - b*c)*(1
5*a^2*d^2 - 20*a*d^3 + 7*b^2*c^2 - 12*b*c*d^2 + 10*a*b*c*d))/(64*b^(9/4)*d^(11/4)) - ((c^2*((b + a*x)/(d + c*x
))^(7/4)*((3*b^3*c^4)/16 - (7*a^2*c*d^4)/4 + (21*a^3*c*d^3)/16 + (b^2*c^3*d^2)/4 - (7*a^2*b*c^2*d^2)/16 + (3*a
*b*c^2*d^3)/2 - (17*a*b^2*c^3*d)/16))/(a^3*b*d^4) - (c*((b + a*x)/(d + c*x))^(3/4)*((7*b^3*c^5)/96 - (9*a^2*c^
2*d^4)/8 + (113*a^3*c^2*d^3)/96 - (b^2*c^4*d^2)/8 - (41*a^2*b*c^3*d^2)/32 + (5*a*b*c^3*d^3)/4 + (a*b^2*c^4*d)/
32))/(a^3*d^5) + (c^3*((b + a*x)/(d + c*x))^(11/4)*((5*a^2*d^4)/8 - (15*a^3*d^3)/32 + (7*b^3*c^3)/32 - (3*b^2*
c^2*d^2)/8 - (a*b*c*d^3)/4 + (3*a*b^2*c^2*d)/32 + (5*a^2*b*c*d^2)/32))/(a^3*b^2*d^3))/((b^3*c^3)/(a^3*d^3) - (
c^3*(b + a*x)^3)/(a^3*(d + c*x)^3) + (3*b*c^3*(b + a*x)^2)/(a^3*d*(d + c*x)^2) - (3*b^2*c^3*(b + a*x))/(a^3*d^
2*(d + c*x)))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {b + d x}{x^{4} \sqrt [4]{\frac {a x + b}{c x + d}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+b)/x**4/((a*x+b)/(c*x+d))**(1/4),x)

[Out]

Integral((b + d*x)/(x**4*((a*x + b)/(c*x + d))**(1/4)), x)

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