3.30.14 \(\int \frac {-1+x^2}{\sqrt [4]{\frac {b+a x}{d+c x}}} \, dx\)

Optimal. Leaf size=329 \[ \frac {\left (\frac {a x+b}{c x+d}\right )^{3/4} \left (32 a^2 c^3 x^3-96 a^2 c^3 x+36 a^2 c^2 d x^2-96 a^2 c^2 d-3 a^2 c d^2 x-7 a^2 d^3-36 a b c^3 x^2-42 a b c^2 d x-6 a b c d^2+45 b^2 c^3 x+45 b^2 c^2 d\right )}{96 a^3 c^2}+\frac {\left (32 a^3 c^2 d-7 a^3 d^3-32 a^2 b c^3-3 a^2 b c d^2-5 a b^2 c^2 d+15 b^3 c^3\right ) \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {a x+b}{c x+d}}}{\sqrt [4]{a}}\right )}{64 a^{13/4} c^{11/4}}+\frac {\left (-32 a^3 c^2 d+7 a^3 d^3+32 a^2 b c^3+3 a^2 b c d^2+5 a b^2 c^2 d-15 b^3 c^3\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {a x+b}{c x+d}}}{\sqrt [4]{a}}\right )}{64 a^{13/4} c^{11/4}} \]

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Rubi [A]  time = 0.77, antiderivative size = 624, normalized size of antiderivative = 1.90, number of steps used = 18, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {1961, 6742, 290, 298, 205, 208} \begin {gather*} \frac {15 (b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {a x+b}{c x+d}}}{\sqrt [4]{a}}\right )}{64 a^{13/4} c^{11/4}}+\frac {5 d (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {a x+b}{c x+d}}}{\sqrt [4]{a}}\right )}{8 a^{9/4} c^{11/4}}-\frac {15 (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {a x+b}{c x+d}}}{\sqrt [4]{a}}\right )}{64 a^{13/4} c^{11/4}}-\frac {5 d (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {a x+b}{c x+d}}}{\sqrt [4]{a}}\right )}{8 a^{9/4} c^{11/4}}-\frac {\left (c^2-d^2\right ) (b c-a d) \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {a x+b}{c x+d}}}{\sqrt [4]{a}}\right )}{2 a^{5/4} c^{11/4}}+\frac {\left (c^2-d^2\right ) (b c-a d) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {a x+b}{c x+d}}}{\sqrt [4]{a}}\right )}{2 a^{5/4} c^{11/4}}+\frac {15 (c x+d) (b c-a d)^2 \left (\frac {a x+b}{c x+d}\right )^{3/4}}{32 a^3 c^2}-\frac {3 (b c-a d)^3 \left (\frac {a x+b}{c x+d}\right )^{3/4}}{8 a^2 c^2 \left (a-\frac {c (a x+b)}{c x+d}\right )^2}+\frac {5 d (c x+d) (b c-a d) \left (\frac {a x+b}{c x+d}\right )^{3/4}}{4 a^2 c^2}-\frac {\left (c^2-d^2\right ) (c x+d) \left (\frac {a x+b}{c x+d}\right )^{3/4}}{a c^2}-\frac {(b c-a d)^3 \left (\frac {a x+b}{c x+d}\right )^{3/4}}{3 a c^2 \left (a-\frac {c (a x+b)}{c x+d}\right )^3}-\frac {d (b c-a d)^2 \left (\frac {a x+b}{c x+d}\right )^{3/4}}{a c^2 \left (a-\frac {c (a x+b)}{c x+d}\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + x^2)/((b + a*x)/(d + c*x))^(1/4),x]

[Out]

(5*d*(b*c - a*d)*((b + a*x)/(d + c*x))^(3/4)*(d + c*x))/(4*a^2*c^2) + (15*(b*c - a*d)^2*((b + a*x)/(d + c*x))^
(3/4)*(d + c*x))/(32*a^3*c^2) - ((c^2 - d^2)*((b + a*x)/(d + c*x))^(3/4)*(d + c*x))/(a*c^2) - ((b*c - a*d)^3*(
(b + a*x)/(d + c*x))^(3/4))/(3*a*c^2*(a - (c*(b + a*x))/(d + c*x))^3) - (d*(b*c - a*d)^2*((b + a*x)/(d + c*x))
^(3/4))/(a*c^2*(a - (c*(b + a*x))/(d + c*x))^2) - (3*(b*c - a*d)^3*((b + a*x)/(d + c*x))^(3/4))/(8*a^2*c^2*(a
- (c*(b + a*x))/(d + c*x))^2) + (5*d*(b*c - a*d)^2*ArcTan[(c^(1/4)*((b + a*x)/(d + c*x))^(1/4))/a^(1/4)])/(8*a
^(9/4)*c^(11/4)) + (15*(b*c - a*d)^3*ArcTan[(c^(1/4)*((b + a*x)/(d + c*x))^(1/4))/a^(1/4)])/(64*a^(13/4)*c^(11
/4)) - ((b*c - a*d)*(c^2 - d^2)*ArcTan[(c^(1/4)*((b + a*x)/(d + c*x))^(1/4))/a^(1/4)])/(2*a^(5/4)*c^(11/4)) -
(5*d*(b*c - a*d)^2*ArcTanh[(c^(1/4)*((b + a*x)/(d + c*x))^(1/4))/a^(1/4)])/(8*a^(9/4)*c^(11/4)) - (15*(b*c - a
*d)^3*ArcTanh[(c^(1/4)*((b + a*x)/(d + c*x))^(1/4))/a^(1/4)])/(64*a^(13/4)*c^(11/4)) + ((b*c - a*d)*(c^2 - d^2
)*ArcTanh[(c^(1/4)*((b + a*x)/(d + c*x))^(1/4))/a^(1/4)])/(2*a^(5/4)*c^(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 1961

Int[(u_)^(r_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> With[{q = Den
ominator[p]}, Dist[(q*e*(b*c - a*d))/n, Subst[Int[SimplifyIntegrand[(x^(q*(p + 1) - 1)*(-(a*e) + c*x^q)^(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)^(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] && FractionQ[p] && IntegerQ[1/
n] && IntegerQ[r]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {-1+x^2}{\sqrt [4]{\frac {b+a x}{d+c x}}} \, dx &=-\left ((4 (b c-a d)) \operatorname {Subst}\left (\int \frac {x^2 \left (-1+\frac {\left (b-d x^4\right )^2}{\left (a-c x^4\right )^2}\right )}{\left (a-c x^4\right )^2} \, dx,x,\sqrt [4]{\frac {b+a x}{d+c x}}\right )\right )\\ &=-\left ((4 (b c-a d)) \operatorname {Subst}\left (\int \left (\frac {(b c-a d)^2 x^2}{c^2 \left (-a+c x^4\right )^4}-\frac {2 d (b c-a d) x^2}{c^2 \left (-a+c x^4\right )^3}-\frac {\left (c^2-d^2\right ) x^2}{c^2 \left (-a+c x^4\right )^2}\right ) \, dx,x,\sqrt [4]{\frac {b+a x}{d+c x}}\right )\right )\\ &=\frac {\left (8 d (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-a+c x^4\right )^3} \, dx,x,\sqrt [4]{\frac {b+a x}{d+c x}}\right )}{c^2}-\frac {\left (4 (b c-a d)^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-a+c x^4\right )^4} \, dx,x,\sqrt [4]{\frac {b+a x}{d+c x}}\right )}{c^2}+\left (4 (b c-a d) \left (1-\frac {d^2}{c^2}\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-a+c x^4\right )^2} \, dx,x,\sqrt [4]{\frac {b+a x}{d+c x}}\right )\\ &=-\frac {\left (c^2-d^2\right ) \left (\frac {b+a x}{d+c x}\right )^{3/4} (d+c x)}{a c^2}-\frac {(b c-a d)^3 \left (\frac {b+a x}{d+c x}\right )^{3/4}}{3 a c^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^3}-\frac {d (b c-a d)^2 \left (\frac {b+a x}{d+c x}\right )^{3/4}}{a c^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^2}-\frac {\left (5 d (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-a+c x^4\right )^2} \, dx,x,\sqrt [4]{\frac {b+a x}{d+c x}}\right )}{a c^2}+\frac {\left (3 (b c-a d)^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-a+c x^4\right )^3} \, dx,x,\sqrt [4]{\frac {b+a x}{d+c x}}\right )}{a c^2}-\frac {\left ((b c-a d) \left (1-\frac {d^2}{c^2}\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{-a+c x^4} \, dx,x,\sqrt [4]{\frac {b+a x}{d+c x}}\right )}{a}\\ &=\frac {5 d (b c-a d) \left (\frac {b+a x}{d+c x}\right )^{3/4} (d+c x)}{4 a^2 c^2}-\frac {\left (c^2-d^2\right ) \left (\frac {b+a x}{d+c x}\right )^{3/4} (d+c x)}{a c^2}-\frac {(b c-a d)^3 \left (\frac {b+a x}{d+c x}\right )^{3/4}}{3 a c^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^3}-\frac {d (b c-a d)^2 \left (\frac {b+a x}{d+c x}\right )^{3/4}}{a c^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^2}-\frac {3 (b c-a d)^3 \left (\frac {b+a x}{d+c x}\right )^{3/4}}{8 a^2 c^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^2}+\frac {\left (5 d (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{-a+c x^4} \, dx,x,\sqrt [4]{\frac {b+a x}{d+c x}}\right )}{4 a^2 c^2}-\frac {\left (15 (b c-a d)^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-a+c x^4\right )^2} \, dx,x,\sqrt [4]{\frac {b+a x}{d+c x}}\right )}{8 a^2 c^2}+\frac {\left ((b c-a d) \left (c^2-d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a}-\sqrt {c} x^2} \, dx,x,\sqrt [4]{\frac {b+a x}{d+c x}}\right )}{2 a c^{5/2}}-\frac {\left ((b c-a d) \left (c^2-d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a}+\sqrt {c} x^2} \, dx,x,\sqrt [4]{\frac {b+a x}{d+c x}}\right )}{2 a c^{5/2}}\\ &=\frac {5 d (b c-a d) \left (\frac {b+a x}{d+c x}\right )^{3/4} (d+c x)}{4 a^2 c^2}+\frac {15 (b c-a d)^2 \left (\frac {b+a x}{d+c x}\right )^{3/4} (d+c x)}{32 a^3 c^2}-\frac {\left (c^2-d^2\right ) \left (\frac {b+a x}{d+c x}\right )^{3/4} (d+c x)}{a c^2}-\frac {(b c-a d)^3 \left (\frac {b+a x}{d+c x}\right )^{3/4}}{3 a c^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^3}-\frac {d (b c-a d)^2 \left (\frac {b+a x}{d+c x}\right )^{3/4}}{a c^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^2}-\frac {3 (b c-a d)^3 \left (\frac {b+a x}{d+c x}\right )^{3/4}}{8 a^2 c^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^2}-\frac {(b c-a d) \left (c^2-d^2\right ) \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {b+a x}{d+c x}}}{\sqrt [4]{a}}\right )}{2 a^{5/4} c^{11/4}}+\frac {(b c-a d) \left (c^2-d^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {b+a x}{d+c x}}}{\sqrt [4]{a}}\right )}{2 a^{5/4} c^{11/4}}-\frac {\left (5 d (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a}-\sqrt {c} x^2} \, dx,x,\sqrt [4]{\frac {b+a x}{d+c x}}\right )}{8 a^2 c^{5/2}}+\frac {\left (5 d (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a}+\sqrt {c} x^2} \, dx,x,\sqrt [4]{\frac {b+a x}{d+c x}}\right )}{8 a^2 c^{5/2}}+\frac {\left (15 (b c-a d)^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{-a+c x^4} \, dx,x,\sqrt [4]{\frac {b+a x}{d+c x}}\right )}{32 a^3 c^2}\\ &=\frac {5 d (b c-a d) \left (\frac {b+a x}{d+c x}\right )^{3/4} (d+c x)}{4 a^2 c^2}+\frac {15 (b c-a d)^2 \left (\frac {b+a x}{d+c x}\right )^{3/4} (d+c x)}{32 a^3 c^2}-\frac {\left (c^2-d^2\right ) \left (\frac {b+a x}{d+c x}\right )^{3/4} (d+c x)}{a c^2}-\frac {(b c-a d)^3 \left (\frac {b+a x}{d+c x}\right )^{3/4}}{3 a c^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^3}-\frac {d (b c-a d)^2 \left (\frac {b+a x}{d+c x}\right )^{3/4}}{a c^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^2}-\frac {3 (b c-a d)^3 \left (\frac {b+a x}{d+c x}\right )^{3/4}}{8 a^2 c^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^2}+\frac {5 d (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {b+a x}{d+c x}}}{\sqrt [4]{a}}\right )}{8 a^{9/4} c^{11/4}}-\frac {(b c-a d) \left (c^2-d^2\right ) \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {b+a x}{d+c x}}}{\sqrt [4]{a}}\right )}{2 a^{5/4} c^{11/4}}-\frac {5 d (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {b+a x}{d+c x}}}{\sqrt [4]{a}}\right )}{8 a^{9/4} c^{11/4}}+\frac {(b c-a d) \left (c^2-d^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {b+a x}{d+c x}}}{\sqrt [4]{a}}\right )}{2 a^{5/4} c^{11/4}}-\frac {\left (15 (b c-a d)^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a}-\sqrt {c} x^2} \, dx,x,\sqrt [4]{\frac {b+a x}{d+c x}}\right )}{64 a^3 c^{5/2}}+\frac {\left (15 (b c-a d)^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a}+\sqrt {c} x^2} \, dx,x,\sqrt [4]{\frac {b+a x}{d+c x}}\right )}{64 a^3 c^{5/2}}\\ &=\frac {5 d (b c-a d) \left (\frac {b+a x}{d+c x}\right )^{3/4} (d+c x)}{4 a^2 c^2}+\frac {15 (b c-a d)^2 \left (\frac {b+a x}{d+c x}\right )^{3/4} (d+c x)}{32 a^3 c^2}-\frac {\left (c^2-d^2\right ) \left (\frac {b+a x}{d+c x}\right )^{3/4} (d+c x)}{a c^2}-\frac {(b c-a d)^3 \left (\frac {b+a x}{d+c x}\right )^{3/4}}{3 a c^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^3}-\frac {d (b c-a d)^2 \left (\frac {b+a x}{d+c x}\right )^{3/4}}{a c^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^2}-\frac {3 (b c-a d)^3 \left (\frac {b+a x}{d+c x}\right )^{3/4}}{8 a^2 c^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^2}+\frac {5 d (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {b+a x}{d+c x}}}{\sqrt [4]{a}}\right )}{8 a^{9/4} c^{11/4}}+\frac {15 (b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {b+a x}{d+c x}}}{\sqrt [4]{a}}\right )}{64 a^{13/4} c^{11/4}}-\frac {(b c-a d) \left (c^2-d^2\right ) \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {b+a x}{d+c x}}}{\sqrt [4]{a}}\right )}{2 a^{5/4} c^{11/4}}-\frac {5 d (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {b+a x}{d+c x}}}{\sqrt [4]{a}}\right )}{8 a^{9/4} c^{11/4}}-\frac {15 (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {b+a x}{d+c x}}}{\sqrt [4]{a}}\right )}{64 a^{13/4} c^{11/4}}+\frac {(b c-a d) \left (c^2-d^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {b+a x}{d+c x}}}{\sqrt [4]{a}}\right )}{2 a^{5/4} c^{11/4}}\\ \end {align*}

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Mathematica [C]  time = 0.17, size = 171, normalized size = 0.52 \begin {gather*} \frac {4 (c x+d) \left (\frac {a x+b}{c x+d}\right )^{3/4} \left (a \left (a \left (d^2-c^2\right ) \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {c (b+a x)}{b c-a d}\right )+2 d (b c-a d) \, _2F_1\left (-\frac {5}{4},\frac {3}{4};\frac {7}{4};\frac {c (b+a x)}{b c-a d}\right )\right )+(b c-a d)^2 \, _2F_1\left (-\frac {9}{4},\frac {3}{4};\frac {7}{4};\frac {c (b+a x)}{b c-a d}\right )\right )}{3 a^3 c^2 \sqrt [4]{\frac {a (c x+d)}{a d-b c}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x^2)/((b + a*x)/(d + c*x))^(1/4),x]

[Out]

(4*((b + a*x)/(d + c*x))^(3/4)*(d + c*x)*((b*c - a*d)^2*Hypergeometric2F1[-9/4, 3/4, 7/4, (c*(b + a*x))/(b*c -
a*d)] + a*(2*d*(b*c - a*d)*Hypergeometric2F1[-5/4, 3/4, 7/4, (c*(b + a*x))/(b*c - a*d)] + a*(-c^2 + d^2)*Hype
rgeometric2F1[-1/4, 3/4, 7/4, (c*(b + a*x))/(b*c - a*d)])))/(3*a^3*c^2*((a*(d + c*x))/(-(b*c) + a*d))^(1/4))

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IntegrateAlgebraic [B]  time = 1.01, size = 711, normalized size = 2.16 \begin {gather*} \frac {96 a^4 b c^3 \left (\frac {b+a x}{d+c x}\right )^{3/4}-113 a^2 b^3 c^3 \left (\frac {b+a x}{d+c x}\right )^{3/4}-96 a^5 c^2 d \left (\frac {b+a x}{d+c x}\right )^{3/4}+123 a^3 b^2 c^2 d \left (\frac {b+a x}{d+c x}\right )^{3/4}-3 a^4 b c d^2 \left (\frac {b+a x}{d+c x}\right )^{3/4}-7 a^5 d^3 \left (\frac {b+a x}{d+c x}\right )^{3/4}-192 a^3 b c^4 \left (\frac {b+a x}{d+c x}\right )^{7/4}+126 a b^3 c^4 \left (\frac {b+a x}{d+c x}\right )^{7/4}+192 a^4 c^3 d \left (\frac {b+a x}{d+c x}\right )^{7/4}-42 a^2 b^2 c^3 d \left (\frac {b+a x}{d+c x}\right )^{7/4}-102 a^3 b c^2 d^2 \left (\frac {b+a x}{d+c x}\right )^{7/4}+18 a^4 c d^3 \left (\frac {b+a x}{d+c x}\right )^{7/4}+96 a^2 b c^5 \left (\frac {b+a x}{d+c x}\right )^{11/4}-45 b^3 c^5 \left (\frac {b+a x}{d+c x}\right )^{11/4}-96 a^3 c^4 d \left (\frac {b+a x}{d+c x}\right )^{11/4}+15 a b^2 c^4 d \left (\frac {b+a x}{d+c x}\right )^{11/4}+9 a^2 b c^3 d^2 \left (\frac {b+a x}{d+c x}\right )^{11/4}+21 a^3 c^2 d^3 \left (\frac {b+a x}{d+c x}\right )^{11/4}}{96 a^3 c^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^3}+\frac {\left (-32 a^2 b c^3+15 b^3 c^3+32 a^3 c^2 d-5 a b^2 c^2 d-3 a^2 b c d^2-7 a^3 d^3\right ) \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {b+a x}{d+c x}}}{\sqrt [4]{a}}\right )}{64 a^{13/4} c^{11/4}}+\frac {\left (32 a^2 b c^3-15 b^3 c^3-32 a^3 c^2 d+5 a b^2 c^2 d+3 a^2 b c d^2+7 a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {b+a x}{d+c x}}}{\sqrt [4]{a}}\right )}{64 a^{13/4} c^{11/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 + x^2)/((b + a*x)/(d + c*x))^(1/4),x]

[Out]

(96*a^4*b*c^3*((b + a*x)/(d + c*x))^(3/4) - 113*a^2*b^3*c^3*((b + a*x)/(d + c*x))^(3/4) - 96*a^5*c^2*d*((b + a
*x)/(d + c*x))^(3/4) + 123*a^3*b^2*c^2*d*((b + a*x)/(d + c*x))^(3/4) - 3*a^4*b*c*d^2*((b + a*x)/(d + c*x))^(3/
4) - 7*a^5*d^3*((b + a*x)/(d + c*x))^(3/4) - 192*a^3*b*c^4*((b + a*x)/(d + c*x))^(7/4) + 126*a*b^3*c^4*((b + a
*x)/(d + c*x))^(7/4) + 192*a^4*c^3*d*((b + a*x)/(d + c*x))^(7/4) - 42*a^2*b^2*c^3*d*((b + a*x)/(d + c*x))^(7/4
) - 102*a^3*b*c^2*d^2*((b + a*x)/(d + c*x))^(7/4) + 18*a^4*c*d^3*((b + a*x)/(d + c*x))^(7/4) + 96*a^2*b*c^5*((
b + a*x)/(d + c*x))^(11/4) - 45*b^3*c^5*((b + a*x)/(d + c*x))^(11/4) - 96*a^3*c^4*d*((b + a*x)/(d + c*x))^(11/
4) + 15*a*b^2*c^4*d*((b + a*x)/(d + c*x))^(11/4) + 9*a^2*b*c^3*d^2*((b + a*x)/(d + c*x))^(11/4) + 21*a^3*c^2*d
^3*((b + a*x)/(d + c*x))^(11/4))/(96*a^3*c^2*(a - (c*(b + a*x))/(d + c*x))^3) + ((-32*a^2*b*c^3 + 15*b^3*c^3 +
32*a^3*c^2*d - 5*a*b^2*c^2*d - 3*a^2*b*c*d^2 - 7*a^3*d^3)*ArcTan[(c^(1/4)*((b + a*x)/(d + c*x))^(1/4))/a^(1/4
)])/(64*a^(13/4)*c^(11/4)) + ((32*a^2*b*c^3 - 15*b^3*c^3 - 32*a^3*c^2*d + 5*a*b^2*c^2*d + 3*a^2*b*c*d^2 + 7*a^
3*d^3)*ArcTanh[(c^(1/4)*((b + a*x)/(d + c*x))^(1/4))/a^(1/4)])/(64*a^(13/4)*c^(11/4))

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fricas [B]  time = 2.26, size = 6305, normalized size = 19.16

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-1)/((a*x+b)/(c*x+d))^(1/4),x, algorithm="fricas")

[Out]

1/384*(12*a^3*c^2*((4116*a^11*b*c*d^11 + 2401*a^12*d^12 + (1048576*a^8*b^4 - 1966080*a^6*b^6 + 1382400*a^4*b^8
- 432000*a^2*b^10 + 50625*b^12)*c^12 - 4*(1048576*a^9*b^3 - 1638400*a^7*b^5 + 921600*a^5*b^7 - 216000*a^3*b^9
+ 16875*a*b^11)*c^11*d + 6*(1048576*a^10*b^2 - 1245184*a^8*b^4 + 471040*a^6*b^6 - 52800*a^4*b^8 - 1125*a^2*b^
10)*c^10*d^2 - 4*(1048576*a^11*b - 917504*a^9*b^3 + 307200*a^7*b^5 - 83200*a^5*b^7 + 15375*a^3*b^9)*c^9*d^3 +
(1048576*a^12 - 2228224*a^10*b^2 + 2297856*a^8*b^4 - 874240*a^6*b^6 + 93775*a^4*b^8)*c^8*d^4 + 24*(98304*a^11*
b - 75776*a^9*b^3 + 11200*a^7*b^5 + 775*a^5*b^7)*c^7*d^5 - 4*(229376*a^12 - 67584*a^10*b^2 + 10176*a^8*b^4 - 7
895*a^6*b^6)*c^6*d^6 - 24*(14336*a^11*b - 13184*a^9*b^3 + 2025*a^7*b^5)*c^5*d^7 + 3*(100352*a^12 - 20608*a^10*
b^2 - 5083*a^8*b^4)*c^4*d^8 - 28*(448*a^11*b + 393*a^9*b^3)*c^3*d^9 - 98*(448*a^12 - 97*a^10*b^2)*c^2*d^10)/(a
^13*c^11))^(1/4)*arctan((sqrt((302526*a^17*b*c*d^17 + 117649*a^18*d^18 + (1073741824*a^12*b^6 - 3019898880*a^1
0*b^8 + 3538944000*a^8*b^10 - 2211840000*a^6*b^12 + 777600000*a^4*b^14 - 145800000*a^2*b^16 + 11390625*b^18)*c
^18 - 6*(1073741824*a^13*b^5 - 2684354560*a^11*b^7 + 2752512000*a^9*b^9 - 1474560000*a^7*b^11 + 432000000*a^5*
b^13 - 64800000*a^3*b^15 + 3796875*a*b^17)*c^17*d + 3*(5368709120*a^14*b^4 - 11542724608*a^12*b^6 + 9882828800
*a^10*b^8 - 4227072000*a^8*b^10 + 915840000*a^6*b^12 - 86400000*a^4*b^14 + 1771875*a^2*b^16)*c^16*d^2 - 16*(13
42177280*a^15*b^3 - 2415919104*a^13*b^5 + 1690828800*a^11*b^7 - 585728000*a^9*b^9 + 110880000*a^7*b^11 - 13500
000*a^5*b^13 + 1096875*a^3*b^15)*c^15*d^3 + 60*(268435456*a^16*b^2 - 436207616*a^14*b^4 + 323485696*a^12*b^6 -
158433280*a^10*b^8 + 53766400*a^8*b^10 - 10392000*a^6*b^12 + 781875*a^4*b^14)*c^14*d^4 - 24*(268435456*a^17*b
- 671088640*a^15*b^3 + 815267840*a^13*b^5 - 501350400*a^11*b^7 + 147488000*a^9*b^9 - 17320000*a^7*b^11 + 3281
25*a^5*b^13)*c^13*d^5 + 4*(268435456*a^18 - 3019898880*a^16*b^2 + 3971481600*a^14*b^4 - 1859256320*a^12*b^6 +
338515200*a^10*b^8 - 23424000*a^8*b^10 + 2100625*a^6*b^12)*c^12*d^6 + 48*(134217728*a^17*b - 144179200*a^15*b^
3 + 71106560*a^13*b^5 - 29811200*a^11*b^7 + 8516000*a^9*b^9 - 879375*a^7*b^11)*c^11*d^7 - 6*(234881024*a^18 -
537395200*a^16*b^2 + 627507200*a^14*b^4 - 290639360*a^12*b^6 + 44430400*a^10*b^8 - 492125*a^8*b^10)*c^10*d^8 -
20*(121110528*a^17*b - 109445120*a^15*b^3 + 22353408*a^13*b^5 + 447360*a^11*b^7 + 15975*a^9*b^9)*c^9*d^9 + 6*
(128450560*a^18 - 68812800*a^16*b^2 + 41638400*a^14*b^4 - 23924224*a^12*b^6 + 3566605*a^10*b^8)*c^8*d^10 + 48*
(8028160*a^17*b - 8288000*a^15*b^3 + 1718368*a^13*b^5 + 28305*a^11*b^7)*c^7*d^11 - 4*(56197120*a^18 - 14112000
*a^16*b^2 - 4320960*a^14*b^4 + 103199*a^12*b^6)*c^6*d^12 - 168*(62720*a^17*b - 176960*a^15*b^3 + 37083*a^13*b^
5)*c^5*d^13 + 2940*(12544*a^18 - 3584*a^16*b^2 - 461*a^14*b^4)*c^4*d^14 - 16464*(224*a^17*b + 15*a^15*b^3)*c^3
*d^15 - 7203*(448*a^18 - 115*a^16*b^2)*c^2*d^16)*sqrt((a*x + b)/(c*x + d)) + (4116*a^18*b*c^6*d^11 + 2401*a^19
*c^5*d^12 + (1048576*a^15*b^4 - 1966080*a^13*b^6 + 1382400*a^11*b^8 - 432000*a^9*b^10 + 50625*a^7*b^12)*c^17 -
4*(1048576*a^16*b^3 - 1638400*a^14*b^5 + 921600*a^12*b^7 - 216000*a^10*b^9 + 16875*a^8*b^11)*c^16*d + 6*(1048
576*a^17*b^2 - 1245184*a^15*b^4 + 471040*a^13*b^6 - 52800*a^11*b^8 - 1125*a^9*b^10)*c^15*d^2 - 4*(1048576*a^18
*b - 917504*a^16*b^3 + 307200*a^14*b^5 - 83200*a^12*b^7 + 15375*a^10*b^9)*c^14*d^3 + (1048576*a^19 - 2228224*a
^17*b^2 + 2297856*a^15*b^4 - 874240*a^13*b^6 + 93775*a^11*b^8)*c^13*d^4 + 24*(98304*a^18*b - 75776*a^16*b^3 +
11200*a^14*b^5 + 775*a^12*b^7)*c^12*d^5 - 4*(229376*a^19 - 67584*a^17*b^2 + 10176*a^15*b^4 - 7895*a^13*b^6)*c^
11*d^6 - 24*(14336*a^18*b - 13184*a^16*b^3 + 2025*a^14*b^5)*c^10*d^7 + 3*(100352*a^19 - 20608*a^17*b^2 - 5083*
a^15*b^4)*c^9*d^8 - 28*(448*a^18*b + 393*a^16*b^3)*c^8*d^9 - 98*(448*a^19 - 97*a^17*b^2)*c^7*d^10)*sqrt((4116*
a^11*b*c*d^11 + 2401*a^12*d^12 + (1048576*a^8*b^4 - 1966080*a^6*b^6 + 1382400*a^4*b^8 - 432000*a^2*b^10 + 5062
5*b^12)*c^12 - 4*(1048576*a^9*b^3 - 1638400*a^7*b^5 + 921600*a^5*b^7 - 216000*a^3*b^9 + 16875*a*b^11)*c^11*d +
6*(1048576*a^10*b^2 - 1245184*a^8*b^4 + 471040*a^6*b^6 - 52800*a^4*b^8 - 1125*a^2*b^10)*c^10*d^2 - 4*(1048576
*a^11*b - 917504*a^9*b^3 + 307200*a^7*b^5 - 83200*a^5*b^7 + 15375*a^3*b^9)*c^9*d^3 + (1048576*a^12 - 2228224*a
^10*b^2 + 2297856*a^8*b^4 - 874240*a^6*b^6 + 93775*a^4*b^8)*c^8*d^4 + 24*(98304*a^11*b - 75776*a^9*b^3 + 11200
*a^7*b^5 + 775*a^5*b^7)*c^7*d^5 - 4*(229376*a^12 - 67584*a^10*b^2 + 10176*a^8*b^4 - 7895*a^6*b^6)*c^6*d^6 - 24
*(14336*a^11*b - 13184*a^9*b^3 + 2025*a^7*b^5)*c^5*d^7 + 3*(100352*a^12 - 20608*a^10*b^2 - 5083*a^8*b^4)*c^4*d
^8 - 28*(448*a^11*b + 393*a^9*b^3)*c^3*d^9 - 98*(448*a^12 - 97*a^10*b^2)*c^2*d^10)/(a^13*c^11)))*a^3*c^3*((411
6*a^11*b*c*d^11 + 2401*a^12*d^12 + (1048576*a^8*b^4 - 1966080*a^6*b^6 + 1382400*a^4*b^8 - 432000*a^2*b^10 + 50
625*b^12)*c^12 - 4*(1048576*a^9*b^3 - 1638400*a^7*b^5 + 921600*a^5*b^7 - 216000*a^3*b^9 + 16875*a*b^11)*c^11*d
+ 6*(1048576*a^10*b^2 - 1245184*a^8*b^4 + 471040*a^6*b^6 - 52800*a^4*b^8 - 1125*a^2*b^10)*c^10*d^2 - 4*(10485
76*a^11*b - 917504*a^9*b^3 + 307200*a^7*b^5 - 83200*a^5*b^7 + 15375*a^3*b^9)*c^9*d^3 + (1048576*a^12 - 2228224
*a^10*b^2 + 2297856*a^8*b^4 - 874240*a^6*b^6 + 93775*a^4*b^8)*c^8*d^4 + 24*(98304*a^11*b - 75776*a^9*b^3 + 112
00*a^7*b^5 + 775*a^5*b^7)*c^7*d^5 - 4*(229376*a^12 - 67584*a^10*b^2 + 10176*a^8*b^4 - 7895*a^6*b^6)*c^6*d^6 -
24*(14336*a^11*b - 13184*a^9*b^3 + 2025*a^7*b^5)*c^5*d^7 + 3*(100352*a^12 - 20608*a^10*b^2 - 5083*a^8*b^4)*c^4
*d^8 - 28*(448*a^11*b + 393*a^9*b^3)*c^3*d^9 - 98*(448*a^12 - 97*a^10*b^2)*c^2*d^10)/(a^13*c^11))^(1/4) - (441
*a^11*b*c^4*d^8 + 343*a^12*c^3*d^9 + (32768*a^9*b^3 - 46080*a^7*b^5 + 21600*a^5*b^7 - 3375*a^3*b^9)*c^12 - 3*(
32768*a^10*b^2 - 35840*a^8*b^4 + 12000*a^6*b^6 - 1125*a^4*b^8)*c^11*d + 12*(8192*a^11*b - 5632*a^9*b^3 + 680*a
^7*b^5 + 75*a^5*b^7)*c^10*d^2 - 4*(8192*a^12 - 4608*a^10*b^2 + 2760*a^8*b^4 - 875*a^6*b^6)*c^9*d^3 - 6*(5632*a
^11*b - 4144*a^9*b^3 + 555*a^7*b^5)*c^8*d^4 + 6*(3584*a^12 - 592*a^10*b^2 - 205*a^8*b^4)*c^7*d^5 + 12*(56*a^11
*b - 129*a^9*b^3)*c^6*d^6 - 84*(56*a^12 - 11*a^10*b^2)*c^5*d^7)*((a*x + b)/(c*x + d))^(1/4)*((4116*a^11*b*c*d^
11 + 2401*a^12*d^12 + (1048576*a^8*b^4 - 1966080*a^6*b^6 + 1382400*a^4*b^8 - 432000*a^2*b^10 + 50625*b^12)*c^1
2 - 4*(1048576*a^9*b^3 - 1638400*a^7*b^5 + 921600*a^5*b^7 - 216000*a^3*b^9 + 16875*a*b^11)*c^11*d + 6*(1048576
*a^10*b^2 - 1245184*a^8*b^4 + 471040*a^6*b^6 - 52800*a^4*b^8 - 1125*a^2*b^10)*c^10*d^2 - 4*(1048576*a^11*b - 9
17504*a^9*b^3 + 307200*a^7*b^5 - 83200*a^5*b^7 + 15375*a^3*b^9)*c^9*d^3 + (1048576*a^12 - 2228224*a^10*b^2 + 2
297856*a^8*b^4 - 874240*a^6*b^6 + 93775*a^4*b^8)*c^8*d^4 + 24*(98304*a^11*b - 75776*a^9*b^3 + 11200*a^7*b^5 +
775*a^5*b^7)*c^7*d^5 - 4*(229376*a^12 - 67584*a^10*b^2 + 10176*a^8*b^4 - 7895*a^6*b^6)*c^6*d^6 - 24*(14336*a^1
1*b - 13184*a^9*b^3 + 2025*a^7*b^5)*c^5*d^7 + 3*(100352*a^12 - 20608*a^10*b^2 - 5083*a^8*b^4)*c^4*d^8 - 28*(44
8*a^11*b + 393*a^9*b^3)*c^3*d^9 - 98*(448*a^12 - 97*a^10*b^2)*c^2*d^10)/(a^13*c^11))^(1/4))/(4116*a^11*b*c*d^1
1 + 2401*a^12*d^12 + (1048576*a^8*b^4 - 1966080*a^6*b^6 + 1382400*a^4*b^8 - 432000*a^2*b^10 + 50625*b^12)*c^12
- 4*(1048576*a^9*b^3 - 1638400*a^7*b^5 + 921600*a^5*b^7 - 216000*a^3*b^9 + 16875*a*b^11)*c^11*d + 6*(1048576*
a^10*b^2 - 1245184*a^8*b^4 + 471040*a^6*b^6 - 52800*a^4*b^8 - 1125*a^2*b^10)*c^10*d^2 - 4*(1048576*a^11*b - 91
7504*a^9*b^3 + 307200*a^7*b^5 - 83200*a^5*b^7 + 15375*a^3*b^9)*c^9*d^3 + (1048576*a^12 - 2228224*a^10*b^2 + 22
97856*a^8*b^4 - 874240*a^6*b^6 + 93775*a^4*b^8)*c^8*d^4 + 24*(98304*a^11*b - 75776*a^9*b^3 + 11200*a^7*b^5 + 7
75*a^5*b^7)*c^7*d^5 - 4*(229376*a^12 - 67584*a^10*b^2 + 10176*a^8*b^4 - 7895*a^6*b^6)*c^6*d^6 - 24*(14336*a^11
*b - 13184*a^9*b^3 + 2025*a^7*b^5)*c^5*d^7 + 3*(100352*a^12 - 20608*a^10*b^2 - 5083*a^8*b^4)*c^4*d^8 - 28*(448
*a^11*b + 393*a^9*b^3)*c^3*d^9 - 98*(448*a^12 - 97*a^10*b^2)*c^2*d^10)) + 3*a^3*c^2*((4116*a^11*b*c*d^11 + 240
1*a^12*d^12 + (1048576*a^8*b^4 - 1966080*a^6*b^6 + 1382400*a^4*b^8 - 432000*a^2*b^10 + 50625*b^12)*c^12 - 4*(1
048576*a^9*b^3 - 1638400*a^7*b^5 + 921600*a^5*b^7 - 216000*a^3*b^9 + 16875*a*b^11)*c^11*d + 6*(1048576*a^10*b^
2 - 1245184*a^8*b^4 + 471040*a^6*b^6 - 52800*a^4*b^8 - 1125*a^2*b^10)*c^10*d^2 - 4*(1048576*a^11*b - 917504*a^
9*b^3 + 307200*a^7*b^5 - 83200*a^5*b^7 + 15375*a^3*b^9)*c^9*d^3 + (1048576*a^12 - 2228224*a^10*b^2 + 2297856*a
^8*b^4 - 874240*a^6*b^6 + 93775*a^4*b^8)*c^8*d^4 + 24*(98304*a^11*b - 75776*a^9*b^3 + 11200*a^7*b^5 + 775*a^5*
b^7)*c^7*d^5 - 4*(229376*a^12 - 67584*a^10*b^2 + 10176*a^8*b^4 - 7895*a^6*b^6)*c^6*d^6 - 24*(14336*a^11*b - 13
184*a^9*b^3 + 2025*a^7*b^5)*c^5*d^7 + 3*(100352*a^12 - 20608*a^10*b^2 - 5083*a^8*b^4)*c^4*d^8 - 28*(448*a^11*b
+ 393*a^9*b^3)*c^3*d^9 - 98*(448*a^12 - 97*a^10*b^2)*c^2*d^10)/(a^13*c^11))^(1/4)*log(a^10*c^8*((4116*a^11*b*
c*d^11 + 2401*a^12*d^12 + (1048576*a^8*b^4 - 1966080*a^6*b^6 + 1382400*a^4*b^8 - 432000*a^2*b^10 + 50625*b^12)
*c^12 - 4*(1048576*a^9*b^3 - 1638400*a^7*b^5 + 921600*a^5*b^7 - 216000*a^3*b^9 + 16875*a*b^11)*c^11*d + 6*(104
8576*a^10*b^2 - 1245184*a^8*b^4 + 471040*a^6*b^6 - 52800*a^4*b^8 - 1125*a^2*b^10)*c^10*d^2 - 4*(1048576*a^11*b
- 917504*a^9*b^3 + 307200*a^7*b^5 - 83200*a^5*b^7 + 15375*a^3*b^9)*c^9*d^3 + (1048576*a^12 - 2228224*a^10*b^2
+ 2297856*a^8*b^4 - 874240*a^6*b^6 + 93775*a^4*b^8)*c^8*d^4 + 24*(98304*a^11*b - 75776*a^9*b^3 + 11200*a^7*b^
5 + 775*a^5*b^7)*c^7*d^5 - 4*(229376*a^12 - 67584*a^10*b^2 + 10176*a^8*b^4 - 7895*a^6*b^6)*c^6*d^6 - 24*(14336
*a^11*b - 13184*a^9*b^3 + 2025*a^7*b^5)*c^5*d^7 + 3*(100352*a^12 - 20608*a^10*b^2 - 5083*a^8*b^4)*c^4*d^8 - 28
*(448*a^11*b + 393*a^9*b^3)*c^3*d^9 - 98*(448*a^12 - 97*a^10*b^2)*c^2*d^10)/(a^13*c^11))^(3/4) + (441*a^8*b*c*
d^8 + 343*a^9*d^9 + (32768*a^6*b^3 - 46080*a^4*b^5 + 21600*a^2*b^7 - 3375*b^9)*c^9 - 3*(32768*a^7*b^2 - 35840*
a^5*b^4 + 12000*a^3*b^6 - 1125*a*b^8)*c^8*d + 12*(8192*a^8*b - 5632*a^6*b^3 + 680*a^4*b^5 + 75*a^2*b^7)*c^7*d^
2 - 4*(8192*a^9 - 4608*a^7*b^2 + 2760*a^5*b^4 - 875*a^3*b^6)*c^6*d^3 - 6*(5632*a^8*b - 4144*a^6*b^3 + 555*a^4*
b^5)*c^5*d^4 + 6*(3584*a^9 - 592*a^7*b^2 - 205*a^5*b^4)*c^4*d^5 + 12*(56*a^8*b - 129*a^6*b^3)*c^3*d^6 - 84*(56
*a^9 - 11*a^7*b^2)*c^2*d^7)*((a*x + b)/(c*x + d))^(1/4)) - 3*a^3*c^2*((4116*a^11*b*c*d^11 + 2401*a^12*d^12 + (
1048576*a^8*b^4 - 1966080*a^6*b^6 + 1382400*a^4*b^8 - 432000*a^2*b^10 + 50625*b^12)*c^12 - 4*(1048576*a^9*b^3
- 1638400*a^7*b^5 + 921600*a^5*b^7 - 216000*a^3*b^9 + 16875*a*b^11)*c^11*d + 6*(1048576*a^10*b^2 - 1245184*a^8
*b^4 + 471040*a^6*b^6 - 52800*a^4*b^8 - 1125*a^2*b^10)*c^10*d^2 - 4*(1048576*a^11*b - 917504*a^9*b^3 + 307200*
a^7*b^5 - 83200*a^5*b^7 + 15375*a^3*b^9)*c^9*d^3 + (1048576*a^12 - 2228224*a^10*b^2 + 2297856*a^8*b^4 - 874240
*a^6*b^6 + 93775*a^4*b^8)*c^8*d^4 + 24*(98304*a^11*b - 75776*a^9*b^3 + 11200*a^7*b^5 + 775*a^5*b^7)*c^7*d^5 -
4*(229376*a^12 - 67584*a^10*b^2 + 10176*a^8*b^4 - 7895*a^6*b^6)*c^6*d^6 - 24*(14336*a^11*b - 13184*a^9*b^3 + 2
025*a^7*b^5)*c^5*d^7 + 3*(100352*a^12 - 20608*a^10*b^2 - 5083*a^8*b^4)*c^4*d^8 - 28*(448*a^11*b + 393*a^9*b^3)
*c^3*d^9 - 98*(448*a^12 - 97*a^10*b^2)*c^2*d^10)/(a^13*c^11))^(1/4)*log(-a^10*c^8*((4116*a^11*b*c*d^11 + 2401*
a^12*d^12 + (1048576*a^8*b^4 - 1966080*a^6*b^6 + 1382400*a^4*b^8 - 432000*a^2*b^10 + 50625*b^12)*c^12 - 4*(104
8576*a^9*b^3 - 1638400*a^7*b^5 + 921600*a^5*b^7 - 216000*a^3*b^9 + 16875*a*b^11)*c^11*d + 6*(1048576*a^10*b^2
- 1245184*a^8*b^4 + 471040*a^6*b^6 - 52800*a^4*b^8 - 1125*a^2*b^10)*c^10*d^2 - 4*(1048576*a^11*b - 917504*a^9*
b^3 + 307200*a^7*b^5 - 83200*a^5*b^7 + 15375*a^3*b^9)*c^9*d^3 + (1048576*a^12 - 2228224*a^10*b^2 + 2297856*a^8
*b^4 - 874240*a^6*b^6 + 93775*a^4*b^8)*c^8*d^4 + 24*(98304*a^11*b - 75776*a^9*b^3 + 11200*a^7*b^5 + 775*a^5*b^
7)*c^7*d^5 - 4*(229376*a^12 - 67584*a^10*b^2 + 10176*a^8*b^4 - 7895*a^6*b^6)*c^6*d^6 - 24*(14336*a^11*b - 1318
4*a^9*b^3 + 2025*a^7*b^5)*c^5*d^7 + 3*(100352*a^12 - 20608*a^10*b^2 - 5083*a^8*b^4)*c^4*d^8 - 28*(448*a^11*b +
393*a^9*b^3)*c^3*d^9 - 98*(448*a^12 - 97*a^10*b^2)*c^2*d^10)/(a^13*c^11))^(3/4) + (441*a^8*b*c*d^8 + 343*a^9*
d^9 + (32768*a^6*b^3 - 46080*a^4*b^5 + 21600*a^2*b^7 - 3375*b^9)*c^9 - 3*(32768*a^7*b^2 - 35840*a^5*b^4 + 1200
0*a^3*b^6 - 1125*a*b^8)*c^8*d + 12*(8192*a^8*b - 5632*a^6*b^3 + 680*a^4*b^5 + 75*a^2*b^7)*c^7*d^2 - 4*(8192*a^
9 - 4608*a^7*b^2 + 2760*a^5*b^4 - 875*a^3*b^6)*c^6*d^3 - 6*(5632*a^8*b - 4144*a^6*b^3 + 555*a^4*b^5)*c^5*d^4 +
6*(3584*a^9 - 592*a^7*b^2 - 205*a^5*b^4)*c^4*d^5 + 12*(56*a^8*b - 129*a^6*b^3)*c^3*d^6 - 84*(56*a^9 - 11*a^7*
b^2)*c^2*d^7)*((a*x + b)/(c*x + d))^(1/4)) + 4*(32*a^2*c^3*x^3 - 6*a*b*c*d^2 - 7*a^2*d^3 - 3*(32*a^2 - 15*b^2)
*c^2*d - 36*(a*b*c^3 - a^2*c^2*d)*x^2 - 3*(14*a*b*c^2*d + a^2*c*d^2 + (32*a^2 - 15*b^2)*c^3)*x)*((a*x + b)/(c*
x + d))^(3/4))/(a^3*c^2)

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giac [B]  time = 7.67, size = 1490, normalized size = 4.53

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-1)/((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)*(32*a^2*b^2*c^4 - 15*b^4*c^4 - 64*a^3*b*c^3*d + 20*a
*b^3*c^3*d + 32*a^4*c^2*d^2 - 2*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 - 7*a^4*d^4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a/c
)^(1/4) + 2*((a*x + b)/(c*x + d))^(1/4))/(-a/c)^(1/4))/((-a*c^3)^(1/4)*a^3*c^2) + 6*sqrt(2)*(32*a^2*b^2*c^4 -
15*b^4*c^4 - 64*a^3*b*c^3*d + 20*a*b^3*c^3*d + 32*a^4*c^2*d^2 - 2*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 - 7*a^4*d^4)
*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a/c)^(1/4) - 2*((a*x + b)/(c*x + d))^(1/4))/(-a/c)^(1/4))/((-a*c^3)^(1/4)*a^3*
c^2) - 3*sqrt(2)*(32*a^2*b^2*c^4 - 15*b^4*c^4 - 64*a^3*b*c^3*d + 20*a*b^3*c^3*d + 32*a^4*c^2*d^2 - 2*a^2*b^2*c
^2*d^2 + 4*a^3*b*c*d^3 - 7*a^4*d^4)*log(sqrt(2)*((a*x + b)/(c*x + d))^(1/4)*(-a/c)^(1/4) + sqrt((a*x + b)/(c*x
+ d)) + sqrt(-a/c))/((-a*c^3)^(1/4)*a^3*c^2) + 3*sqrt(2)*(32*a^2*b^2*c^4 - 15*b^4*c^4 - 64*a^3*b*c^3*d + 20*a
*b^3*c^3*d + 32*a^4*c^2*d^2 - 2*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 - 7*a^4*d^4)*log(-sqrt(2)*((a*x + b)/(c*x + d)
)^(1/4)*(-a/c)^(1/4) + sqrt((a*x + b)/(c*x + d)) + sqrt(-a/c))/((-a*c^3)^(1/4)*a^3*c^2) - 8*(96*a^4*b^2*c^4*((
a*x + b)/(c*x + d))^(3/4) - 113*a^2*b^4*c^4*((a*x + b)/(c*x + d))^(3/4) - 192*(a*x + b)*a^3*b^2*c^5*((a*x + b)
/(c*x + d))^(3/4)/(c*x + d) + 126*(a*x + b)*a*b^4*c^5*((a*x + b)/(c*x + d))^(3/4)/(c*x + d) + 96*(a*x + b)^2*a
^2*b^2*c^6*((a*x + b)/(c*x + d))^(3/4)/(c*x + d)^2 - 45*(a*x + b)^2*b^4*c^6*((a*x + b)/(c*x + d))^(3/4)/(c*x +
d)^2 - 192*a^5*b*c^3*d*((a*x + b)/(c*x + d))^(3/4) + 236*a^3*b^3*c^3*d*((a*x + b)/(c*x + d))^(3/4) + 384*(a*x
+ b)*a^4*b*c^4*d*((a*x + b)/(c*x + d))^(3/4)/(c*x + d) - 168*(a*x + b)*a^2*b^3*c^4*d*((a*x + b)/(c*x + d))^(3
/4)/(c*x + d) - 192*(a*x + b)^2*a^3*b*c^5*d*((a*x + b)/(c*x + d))^(3/4)/(c*x + d)^2 + 60*(a*x + b)^2*a*b^3*c^5
*d*((a*x + b)/(c*x + d))^(3/4)/(c*x + d)^2 + 96*a^6*c^2*d^2*((a*x + b)/(c*x + d))^(3/4) - 126*a^4*b^2*c^2*d^2*
((a*x + b)/(c*x + d))^(3/4) - 192*(a*x + b)*a^5*c^3*d^2*((a*x + b)/(c*x + d))^(3/4)/(c*x + d) - 60*(a*x + b)*a
^3*b^2*c^3*d^2*((a*x + b)/(c*x + d))^(3/4)/(c*x + d) + 96*(a*x + b)^2*a^4*c^4*d^2*((a*x + b)/(c*x + d))^(3/4)/
(c*x + d)^2 - 6*(a*x + b)^2*a^2*b^2*c^4*d^2*((a*x + b)/(c*x + d))^(3/4)/(c*x + d)^2 - 4*a^5*b*c*d^3*((a*x + b)
/(c*x + d))^(3/4) + 120*(a*x + b)*a^4*b*c^2*d^3*((a*x + b)/(c*x + d))^(3/4)/(c*x + d) + 12*(a*x + b)^2*a^3*b*c
^3*d^3*((a*x + b)/(c*x + d))^(3/4)/(c*x + d)^2 + 7*a^6*d^4*((a*x + b)/(c*x + d))^(3/4) - 18*(a*x + b)*a^5*c*d^
4*((a*x + b)/(c*x + d))^(3/4)/(c*x + d) - 21*(a*x + b)^2*a^4*c^2*d^4*((a*x + b)/(c*x + d))^(3/4)/(c*x + d)^2)/
((a - (a*x + b)*c/(c*x + d))^3*a^3*c^2))

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{2}-1}{\left (\frac {a x +b}{c x +d}\right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2-1)/((a*x+b)/(c*x+d))^(1/4),x)

[Out]

int((x^2-1)/((a*x+b)/(c*x+d))^(1/4),x)

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maxima [A]  time = 1.65, size = 487, normalized size = 1.48 \begin {gather*} \frac {3 \, {\left (3 \, a^{2} b c^{3} d^{2} + 7 \, a^{3} c^{2} d^{3} + {\left (32 \, a^{2} b - 15 \, b^{3}\right )} c^{5} - {\left (32 \, a^{3} - 5 \, a b^{2}\right )} c^{4} d\right )} \left (\frac {a x + b}{c x + d}\right )^{\frac {11}{4}} - 6 \, {\left (17 \, a^{3} b c^{2} d^{2} - 3 \, a^{4} c d^{3} + {\left (32 \, a^{3} b - 21 \, a b^{3}\right )} c^{4} - {\left (32 \, a^{4} - 7 \, a^{2} b^{2}\right )} c^{3} d\right )} \left (\frac {a x + b}{c x + d}\right )^{\frac {7}{4}} - {\left (3 \, a^{4} b c d^{2} + 7 \, a^{5} d^{3} - {\left (96 \, a^{4} b - 113 \, a^{2} b^{3}\right )} c^{3} + 3 \, {\left (32 \, a^{5} - 41 \, a^{3} b^{2}\right )} c^{2} d\right )} \left (\frac {a x + b}{c x + d}\right )^{\frac {3}{4}}}{96 \, {\left (a^{6} c^{2} - \frac {3 \, {\left (a x + b\right )} a^{5} c^{3}}{c x + d} + \frac {3 \, {\left (a x + b\right )}^{2} a^{4} c^{4}}{{\left (c x + d\right )}^{2}} - \frac {{\left (a x + b\right )}^{3} a^{3} c^{5}}{{\left (c x + d\right )}^{3}}\right )}} - \frac {{\left (3 \, a^{2} b c d^{2} + 7 \, a^{3} d^{3} + {\left (32 \, a^{2} b - 15 \, b^{3}\right )} c^{3} - {\left (32 \, a^{3} - 5 \, a b^{2}\right )} c^{2} d\right )} {\left (\frac {2 \, \arctan \left (\frac {\sqrt {c} \left (\frac {a x + b}{c x + d}\right )^{\frac {1}{4}}}{\sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\log \left (\frac {\sqrt {c} \left (\frac {a x + b}{c x + d}\right )^{\frac {1}{4}} - \sqrt {\sqrt {a} \sqrt {c}}}{\sqrt {c} \left (\frac {a x + b}{c x + d}\right )^{\frac {1}{4}} + \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}}\right )}}{128 \, a^{3} c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-1)/((a*x+b)/(c*x+d))^(1/4),x, algorithm="maxima")

[Out]

1/96*(3*(3*a^2*b*c^3*d^2 + 7*a^3*c^2*d^3 + (32*a^2*b - 15*b^3)*c^5 - (32*a^3 - 5*a*b^2)*c^4*d)*((a*x + b)/(c*x
+ d))^(11/4) - 6*(17*a^3*b*c^2*d^2 - 3*a^4*c*d^3 + (32*a^3*b - 21*a*b^3)*c^4 - (32*a^4 - 7*a^2*b^2)*c^3*d)*((
a*x + b)/(c*x + d))^(7/4) - (3*a^4*b*c*d^2 + 7*a^5*d^3 - (96*a^4*b - 113*a^2*b^3)*c^3 + 3*(32*a^5 - 41*a^3*b^2
)*c^2*d)*((a*x + b)/(c*x + d))^(3/4))/(a^6*c^2 - 3*(a*x + b)*a^5*c^3/(c*x + d) + 3*(a*x + b)^2*a^4*c^4/(c*x +
d)^2 - (a*x + b)^3*a^3*c^5/(c*x + d)^3) - 1/128*(3*a^2*b*c*d^2 + 7*a^3*d^3 + (32*a^2*b - 15*b^3)*c^3 - (32*a^3
- 5*a*b^2)*c^2*d)*(2*arctan(sqrt(c)*((a*x + b)/(c*x + d))^(1/4)/sqrt(sqrt(a)*sqrt(c)))/(sqrt(sqrt(a)*sqrt(c))
*sqrt(c)) + log((sqrt(c)*((a*x + b)/(c*x + d))^(1/4) - sqrt(sqrt(a)*sqrt(c)))/(sqrt(c)*((a*x + b)/(c*x + d))^(
1/4) + sqrt(sqrt(a)*sqrt(c))))/(sqrt(sqrt(a)*sqrt(c))*sqrt(c)))/(a^3*c^2)

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mupad [B]  time = 3.77, size = 1566, normalized size = 4.76

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 - 1)/((b + a*x)/(d + c*x))^(1/4),x)

[Out]

((((b + a*x)/(d + c*x))^(11/4)*((7*a^3*d^3)/32 - (15*b^3*c^3)/32 + a^2*b*c^3 - a^3*c^2*d + (5*a*b^2*c^2*d)/32
+ (3*a^2*b*c*d^2)/32))/a^6 + (((b + a*x)/(d + c*x))^(7/4)*((3*a^3*d^3)/16 + (21*b^3*c^3)/16 - 2*a^2*b*c^3 + 2*
a^3*c^2*d - (7*a*b^2*c^2*d)/16 - (17*a^2*b*c*d^2)/16))/(a^5*c) - (((b + a*x)/(d + c*x))^(3/4)*((7*a^3*d^3)/96
+ (113*b^3*c^3)/96 - a^2*b*c^3 + a^3*c^2*d - (41*a*b^2*c^2*d)/32 + (a^2*b*c*d^2)/32))/(a^4*c^2))/((3*c^2*(b +
a*x)^2)/(a^2*(d + c*x)^2) - (c^3*(b + a*x)^3)/(a^3*(d + c*x)^3) - (3*c*(b + a*x))/(a*(d + c*x)) + 1) - (atan((
c^(1/2)*(a*d - b*c)*((b + a*x)/(d + c*x))^(1/4)*(7*a^2*d^2 - 32*a^2*c^2 + 15*b^2*c^2 + 10*a*b*c*d)*(225*b^6*c^
(13/2) - 960*a^2*b^4*c^(13/2) + 1024*a^4*b^2*c^(13/2) + 49*a^6*c^(1/2)*d^6 - 448*a^6*c^(5/2)*d^4 + 1024*a^6*c^
(9/2)*d^2 + 42*a^5*b*c^(3/2)*d^5 + 256*a^5*b*c^(7/2)*d^3 + 1280*a^3*b^3*c^(11/2)*d + 79*a^4*b^2*c^(5/2)*d^4 -
180*a^3*b^3*c^(7/2)*d^3 - 65*a^2*b^4*c^(9/2)*d^2 - 128*a^4*b^2*c^(9/2)*d^2 - 150*a*b^5*c^(11/2)*d - 2048*a^5*b
*c^(11/2)*d))/(a^(1/4)*(21600*a^2*b^7*c^(39/4) - 3375*b^9*c^(39/4) - 46080*a^4*b^5*c^(39/4) + 32768*a^6*b^3*c^
(39/4) + 343*a^9*c^(3/4)*d^9 - 4704*a^9*c^(11/4)*d^7 + 21504*a^9*c^(19/4)*d^5 - 32768*a^9*c^(27/4)*d^3 + 441*a
^8*b*c^(7/4)*d^8 + 672*a^8*b*c^(15/4)*d^6 - 33792*a^8*b*c^(23/4)*d^4 + 98304*a^8*b*c^(31/4)*d^2 - 36000*a^3*b^
6*c^(35/4)*d + 107520*a^5*b^4*c^(35/4)*d - 98304*a^7*b^2*c^(35/4)*d + 924*a^7*b^2*c^(11/4)*d^7 - 1548*a^6*b^3*
c^(15/4)*d^6 - 1230*a^5*b^4*c^(19/4)*d^5 - 3552*a^7*b^2*c^(19/4)*d^5 - 3330*a^4*b^5*c^(23/4)*d^4 + 24864*a^6*b
^3*c^(23/4)*d^4 + 3500*a^3*b^6*c^(27/4)*d^3 - 11040*a^5*b^4*c^(27/4)*d^3 + 18432*a^7*b^2*c^(27/4)*d^3 + 900*a^
2*b^7*c^(31/4)*d^2 + 8160*a^4*b^5*c^(31/4)*d^2 - 67584*a^6*b^3*c^(31/4)*d^2 + 3375*a*b^8*c^(35/4)*d)))*(a*d -
b*c)*(7*a^2*d^2 - 32*a^2*c^2 + 15*b^2*c^2 + 10*a*b*c*d))/(64*a^(13/4)*c^(11/4)) + (atanh((c^(1/2)*(a*d - b*c)*
((b + a*x)/(d + c*x))^(1/4)*(7*a^2*d^2 - 32*a^2*c^2 + 15*b^2*c^2 + 10*a*b*c*d)*(225*b^6*c^(13/2) - 960*a^2*b^4
*c^(13/2) + 1024*a^4*b^2*c^(13/2) + 49*a^6*c^(1/2)*d^6 - 448*a^6*c^(5/2)*d^4 + 1024*a^6*c^(9/2)*d^2 + 42*a^5*b
*c^(3/2)*d^5 + 256*a^5*b*c^(7/2)*d^3 + 1280*a^3*b^3*c^(11/2)*d + 79*a^4*b^2*c^(5/2)*d^4 - 180*a^3*b^3*c^(7/2)*
d^3 - 65*a^2*b^4*c^(9/2)*d^2 - 128*a^4*b^2*c^(9/2)*d^2 - 150*a*b^5*c^(11/2)*d - 2048*a^5*b*c^(11/2)*d))/(a^(1/
4)*(21600*a^2*b^7*c^(39/4) - 3375*b^9*c^(39/4) - 46080*a^4*b^5*c^(39/4) + 32768*a^6*b^3*c^(39/4) + 343*a^9*c^(
3/4)*d^9 - 4704*a^9*c^(11/4)*d^7 + 21504*a^9*c^(19/4)*d^5 - 32768*a^9*c^(27/4)*d^3 + 441*a^8*b*c^(7/4)*d^8 + 6
72*a^8*b*c^(15/4)*d^6 - 33792*a^8*b*c^(23/4)*d^4 + 98304*a^8*b*c^(31/4)*d^2 - 36000*a^3*b^6*c^(35/4)*d + 10752
0*a^5*b^4*c^(35/4)*d - 98304*a^7*b^2*c^(35/4)*d + 924*a^7*b^2*c^(11/4)*d^7 - 1548*a^6*b^3*c^(15/4)*d^6 - 1230*
a^5*b^4*c^(19/4)*d^5 - 3552*a^7*b^2*c^(19/4)*d^5 - 3330*a^4*b^5*c^(23/4)*d^4 + 24864*a^6*b^3*c^(23/4)*d^4 + 35
00*a^3*b^6*c^(27/4)*d^3 - 11040*a^5*b^4*c^(27/4)*d^3 + 18432*a^7*b^2*c^(27/4)*d^3 + 900*a^2*b^7*c^(31/4)*d^2 +
8160*a^4*b^5*c^(31/4)*d^2 - 67584*a^6*b^3*c^(31/4)*d^2 + 3375*a*b^8*c^(35/4)*d)))*(a*d - b*c)*(7*a^2*d^2 - 32
*a^2*c^2 + 15*b^2*c^2 + 10*a*b*c*d))/(64*a^(13/4)*c^(11/4))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 1\right ) \left (x + 1\right )}{\sqrt [4]{\frac {a x + b}{c x + d}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2-1)/((a*x+b)/(c*x+d))**(1/4),x)

[Out]

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

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