3.900 \(\int \frac{x^3}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx\)

Optimal. Leaf size=259 \[ \frac{\sqrt [4]{a+b x} (c+d x)^{3/4} \left (77 a^2 d^2-4 b d x (11 a d+9 b c)+54 a b c d+45 b^2 c^2\right )}{96 b^3 d^3}-\frac{\left (21 a^2 b c d^2+77 a^3 d^3+15 a b^2 c^2 d+15 b^3 c^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{15/4} d^{13/4}}-\frac{\left (21 a^2 b c d^2+77 a^3 d^3+15 a b^2 c^2 d+15 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{15/4} d^{13/4}}+\frac{x^2 \sqrt [4]{a+b x} (c+d x)^{3/4}}{3 b d} \]

[Out]

(x^2*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(3*b*d) + ((a + b*x)^(1/4)*(c + d*x)^(3/4)*(45*b^2*c^2 + 54*a*b*c*d + 77
*a^2*d^2 - 4*b*d*(9*b*c + 11*a*d)*x))/(96*b^3*d^3) - ((15*b^3*c^3 + 15*a*b^2*c^2*d + 21*a^2*b*c*d^2 + 77*a^3*d
^3)*ArcTan[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/(64*b^(15/4)*d^(13/4)) - ((15*b^3*c^3 + 15*a*
b^2*c^2*d + 21*a^2*b*c*d^2 + 77*a^3*d^3)*ArcTanh[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/(64*b^(
15/4)*d^(13/4))

________________________________________________________________________________________

Rubi [A]  time = 0.183745, antiderivative size = 259, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {100, 147, 63, 240, 212, 208, 205} \[ \frac{\sqrt [4]{a+b x} (c+d x)^{3/4} \left (77 a^2 d^2-4 b d x (11 a d+9 b c)+54 a b c d+45 b^2 c^2\right )}{96 b^3 d^3}-\frac{\left (21 a^2 b c d^2+77 a^3 d^3+15 a b^2 c^2 d+15 b^3 c^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{15/4} d^{13/4}}-\frac{\left (21 a^2 b c d^2+77 a^3 d^3+15 a b^2 c^2 d+15 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{15/4} d^{13/4}}+\frac{x^2 \sqrt [4]{a+b x} (c+d x)^{3/4}}{3 b d} \]

Antiderivative was successfully verified.

[In]

Int[x^3/((a + b*x)^(3/4)*(c + d*x)^(1/4)),x]

[Out]

(x^2*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(3*b*d) + ((a + b*x)^(1/4)*(c + d*x)^(3/4)*(45*b^2*c^2 + 54*a*b*c*d + 77
*a^2*d^2 - 4*b*d*(9*b*c + 11*a*d)*x))/(96*b^3*d^3) - ((15*b^3*c^3 + 15*a*b^2*c^2*d + 21*a^2*b*c*d^2 + 77*a^3*d
^3)*ArcTan[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/(64*b^(15/4)*d^(13/4)) - ((15*b^3*c^3 + 15*a*
b^2*c^2*d + 21*a^2*b*c*d^2 + 77*a^3*d^3)*ArcTanh[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/(64*b^(
15/4)*d^(13/4))

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 240

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 1/n + 1), x], x
, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p
+ 1/n]

Rule 212

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

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 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]

Rubi steps

\begin{align*} \int \frac{x^3}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx &=\frac{x^2 \sqrt [4]{a+b x} (c+d x)^{3/4}}{3 b d}+\frac{\int \frac{x \left (-2 a c+\frac{1}{4} (-9 b c-11 a d) x\right )}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{3 b d}\\ &=\frac{x^2 \sqrt [4]{a+b x} (c+d x)^{3/4}}{3 b d}+\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} \left (45 b^2 c^2+54 a b c d+77 a^2 d^2-4 b d (9 b c+11 a d) x\right )}{96 b^3 d^3}-\frac{\left (15 b^3 c^3+15 a b^2 c^2 d+21 a^2 b c d^2+77 a^3 d^3\right ) \int \frac{1}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{128 b^3 d^3}\\ &=\frac{x^2 \sqrt [4]{a+b x} (c+d x)^{3/4}}{3 b d}+\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} \left (45 b^2 c^2+54 a b c d+77 a^2 d^2-4 b d (9 b c+11 a d) x\right )}{96 b^3 d^3}-\frac{\left (15 b^3 c^3+15 a b^2 c^2 d+21 a^2 b c d^2+77 a^3 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{c-\frac{a d}{b}+\frac{d x^4}{b}}} \, dx,x,\sqrt [4]{a+b x}\right )}{32 b^4 d^3}\\ &=\frac{x^2 \sqrt [4]{a+b x} (c+d x)^{3/4}}{3 b d}+\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} \left (45 b^2 c^2+54 a b c d+77 a^2 d^2-4 b d (9 b c+11 a d) x\right )}{96 b^3 d^3}-\frac{\left (15 b^3 c^3+15 a b^2 c^2 d+21 a^2 b c d^2+77 a^3 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^4}{b}} \, dx,x,\frac{\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{32 b^4 d^3}\\ &=\frac{x^2 \sqrt [4]{a+b x} (c+d x)^{3/4}}{3 b d}+\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} \left (45 b^2 c^2+54 a b c d+77 a^2 d^2-4 b d (9 b c+11 a d) x\right )}{96 b^3 d^3}-\frac{\left (15 b^3 c^3+15 a b^2 c^2 d+21 a^2 b c d^2+77 a^3 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b}-\sqrt{d} x^2} \, dx,x,\frac{\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{64 b^{7/2} d^3}-\frac{\left (15 b^3 c^3+15 a b^2 c^2 d+21 a^2 b c d^2+77 a^3 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b}+\sqrt{d} x^2} \, dx,x,\frac{\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{64 b^{7/2} d^3}\\ &=\frac{x^2 \sqrt [4]{a+b x} (c+d x)^{3/4}}{3 b d}+\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} \left (45 b^2 c^2+54 a b c d+77 a^2 d^2-4 b d (9 b c+11 a d) x\right )}{96 b^3 d^3}-\frac{\left (15 b^3 c^3+15 a b^2 c^2 d+21 a^2 b c d^2+77 a^3 d^3\right ) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{15/4} d^{13/4}}-\frac{\left (15 b^3 c^3+15 a b^2 c^2 d+21 a^2 b c d^2+77 a^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{15/4} d^{13/4}}\\ \end{align*}

Mathematica [C]  time = 0.598293, size = 231, normalized size = 0.89 \[ \frac{\sqrt [4]{a+b x} \sqrt [4]{\frac{b (c+d x)}{b c-a d}} \left (b \left (2 c \left (-7 a^2 d^2-2 a b c d+9 b^2 c^2\right ) \, _2F_1\left (-\frac{3}{4},\frac{1}{4};\frac{5}{4};\frac{d (a+b x)}{a d-b c}\right )+b \left (\frac{b d^2 x^2 (c+d x)}{\sqrt [4]{\frac{b (c+d x)}{b c-a d}}}-3 c^2 (a d+3 b c) \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{d (a+b x)}{a d-b c}\right )\right )\right )-(b c-a d)^2 (11 a d+9 b c) \, _2F_1\left (-\frac{7}{4},\frac{1}{4};\frac{5}{4};\frac{d (a+b x)}{a d-b c}\right )\right )}{3 b^4 d^3 \sqrt [4]{c+d x}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/((a + b*x)^(3/4)*(c + d*x)^(1/4)),x]

[Out]

((a + b*x)^(1/4)*((b*(c + d*x))/(b*c - a*d))^(1/4)*(-((b*c - a*d)^2*(9*b*c + 11*a*d)*Hypergeometric2F1[-7/4, 1
/4, 5/4, (d*(a + b*x))/(-(b*c) + a*d)]) + b*(2*c*(9*b^2*c^2 - 2*a*b*c*d - 7*a^2*d^2)*Hypergeometric2F1[-3/4, 1
/4, 5/4, (d*(a + b*x))/(-(b*c) + a*d)] + b*((b*d^2*x^2*(c + d*x))/((b*(c + d*x))/(b*c - a*d))^(1/4) - 3*c^2*(3
*b*c + a*d)*Hypergeometric2F1[1/4, 1/4, 5/4, (d*(a + b*x))/(-(b*c) + a*d)]))))/(3*b^4*d^3*(c + d*x)^(1/4))

________________________________________________________________________________________

Maple [F]  time = 0.024, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \left ( bx+a \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{dx+c}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(b*x+a)^(3/4)/(d*x+c)^(1/4),x)

[Out]

int(x^3/(b*x+a)^(3/4)/(d*x+c)^(1/4),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{{\left (b x + a\right )}^{\frac{3}{4}}{\left (d x + c\right )}^{\frac{1}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x+a)^(3/4)/(d*x+c)^(1/4),x, algorithm="maxima")

[Out]

integrate(x^3/((b*x + a)^(3/4)*(d*x + c)^(1/4)), x)

________________________________________________________________________________________

Fricas [B]  time = 3.57695, size = 5762, normalized size = 22.25 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x+a)^(3/4)/(d*x+c)^(1/4),x, algorithm="fricas")

[Out]

1/384*(12*b^3*d^3*((50625*b^12*c^12 + 202500*a*b^11*c^11*d + 587250*a^2*b^10*c^10*d^2 + 2092500*a^3*b^9*c^9*d^
3 + 4614975*a^4*b^8*c^8*d^4 + 8958600*a^5*b^7*c^7*d^5 + 18926460*a^6*b^6*c^6*d^6 + 27042120*a^7*b^5*c^5*d^7 +
36722511*a^8*b^4*c^4*d^8 + 52655988*a^9*b^3*c^3*d^9 + 43080114*a^10*b^2*c^2*d^10 + 38348772*a^11*b*c*d^11 + 35
153041*a^12*d^12)/(b^15*d^13))^(1/4)*arctan(-((15*b^14*c^3*d^10 + 15*a*b^13*c^2*d^11 + 21*a^2*b^12*c*d^12 + 77
*a^3*b^11*d^13)*(b*x + a)^(1/4)*(d*x + c)^(3/4)*((50625*b^12*c^12 + 202500*a*b^11*c^11*d + 587250*a^2*b^10*c^1
0*d^2 + 2092500*a^3*b^9*c^9*d^3 + 4614975*a^4*b^8*c^8*d^4 + 8958600*a^5*b^7*c^7*d^5 + 18926460*a^6*b^6*c^6*d^6
+ 27042120*a^7*b^5*c^5*d^7 + 36722511*a^8*b^4*c^4*d^8 + 52655988*a^9*b^3*c^3*d^9 + 43080114*a^10*b^2*c^2*d^10
+ 38348772*a^11*b*c*d^11 + 35153041*a^12*d^12)/(b^15*d^13))^(3/4) - (b^11*d^11*x + b^11*c*d^10)*sqrt(((225*b^
6*c^6 + 450*a*b^5*c^5*d + 855*a^2*b^4*c^4*d^2 + 2940*a^3*b^3*c^3*d^3 + 2751*a^4*b^2*c^2*d^4 + 3234*a^5*b*c*d^5
+ 5929*a^6*d^6)*sqrt(b*x + a)*sqrt(d*x + c) + (b^8*d^7*x + b^8*c*d^6)*sqrt((50625*b^12*c^12 + 202500*a*b^11*c
^11*d + 587250*a^2*b^10*c^10*d^2 + 2092500*a^3*b^9*c^9*d^3 + 4614975*a^4*b^8*c^8*d^4 + 8958600*a^5*b^7*c^7*d^5
+ 18926460*a^6*b^6*c^6*d^6 + 27042120*a^7*b^5*c^5*d^7 + 36722511*a^8*b^4*c^4*d^8 + 52655988*a^9*b^3*c^3*d^9 +
43080114*a^10*b^2*c^2*d^10 + 38348772*a^11*b*c*d^11 + 35153041*a^12*d^12)/(b^15*d^13)))/(d*x + c))*((50625*b^
12*c^12 + 202500*a*b^11*c^11*d + 587250*a^2*b^10*c^10*d^2 + 2092500*a^3*b^9*c^9*d^3 + 4614975*a^4*b^8*c^8*d^4
+ 8958600*a^5*b^7*c^7*d^5 + 18926460*a^6*b^6*c^6*d^6 + 27042120*a^7*b^5*c^5*d^7 + 36722511*a^8*b^4*c^4*d^8 + 5
2655988*a^9*b^3*c^3*d^9 + 43080114*a^10*b^2*c^2*d^10 + 38348772*a^11*b*c*d^11 + 35153041*a^12*d^12)/(b^15*d^13
))^(3/4))/(50625*b^12*c^13 + 202500*a*b^11*c^12*d + 587250*a^2*b^10*c^11*d^2 + 2092500*a^3*b^9*c^10*d^3 + 4614
975*a^4*b^8*c^9*d^4 + 8958600*a^5*b^7*c^8*d^5 + 18926460*a^6*b^6*c^7*d^6 + 27042120*a^7*b^5*c^6*d^7 + 36722511
*a^8*b^4*c^5*d^8 + 52655988*a^9*b^3*c^4*d^9 + 43080114*a^10*b^2*c^3*d^10 + 38348772*a^11*b*c^2*d^11 + 35153041
*a^12*c*d^12 + (50625*b^12*c^12*d + 202500*a*b^11*c^11*d^2 + 587250*a^2*b^10*c^10*d^3 + 2092500*a^3*b^9*c^9*d^
4 + 4614975*a^4*b^8*c^8*d^5 + 8958600*a^5*b^7*c^7*d^6 + 18926460*a^6*b^6*c^6*d^7 + 27042120*a^7*b^5*c^5*d^8 +
36722511*a^8*b^4*c^4*d^9 + 52655988*a^9*b^3*c^3*d^10 + 43080114*a^10*b^2*c^2*d^11 + 38348772*a^11*b*c*d^12 + 3
5153041*a^12*d^13)*x)) - 3*b^3*d^3*((50625*b^12*c^12 + 202500*a*b^11*c^11*d + 587250*a^2*b^10*c^10*d^2 + 20925
00*a^3*b^9*c^9*d^3 + 4614975*a^4*b^8*c^8*d^4 + 8958600*a^5*b^7*c^7*d^5 + 18926460*a^6*b^6*c^6*d^6 + 27042120*a
^7*b^5*c^5*d^7 + 36722511*a^8*b^4*c^4*d^8 + 52655988*a^9*b^3*c^3*d^9 + 43080114*a^10*b^2*c^2*d^10 + 38348772*a
^11*b*c*d^11 + 35153041*a^12*d^12)/(b^15*d^13))^(1/4)*log(((15*b^3*c^3 + 15*a*b^2*c^2*d + 21*a^2*b*c*d^2 + 77*
a^3*d^3)*(b*x + a)^(1/4)*(d*x + c)^(3/4) + (b^4*d^4*x + b^4*c*d^3)*((50625*b^12*c^12 + 202500*a*b^11*c^11*d +
587250*a^2*b^10*c^10*d^2 + 2092500*a^3*b^9*c^9*d^3 + 4614975*a^4*b^8*c^8*d^4 + 8958600*a^5*b^7*c^7*d^5 + 18926
460*a^6*b^6*c^6*d^6 + 27042120*a^7*b^5*c^5*d^7 + 36722511*a^8*b^4*c^4*d^8 + 52655988*a^9*b^3*c^3*d^9 + 4308011
4*a^10*b^2*c^2*d^10 + 38348772*a^11*b*c*d^11 + 35153041*a^12*d^12)/(b^15*d^13))^(1/4))/(d*x + c)) + 3*b^3*d^3*
((50625*b^12*c^12 + 202500*a*b^11*c^11*d + 587250*a^2*b^10*c^10*d^2 + 2092500*a^3*b^9*c^9*d^3 + 4614975*a^4*b^
8*c^8*d^4 + 8958600*a^5*b^7*c^7*d^5 + 18926460*a^6*b^6*c^6*d^6 + 27042120*a^7*b^5*c^5*d^7 + 36722511*a^8*b^4*c
^4*d^8 + 52655988*a^9*b^3*c^3*d^9 + 43080114*a^10*b^2*c^2*d^10 + 38348772*a^11*b*c*d^11 + 35153041*a^12*d^12)/
(b^15*d^13))^(1/4)*log(((15*b^3*c^3 + 15*a*b^2*c^2*d + 21*a^2*b*c*d^2 + 77*a^3*d^3)*(b*x + a)^(1/4)*(d*x + c)^
(3/4) - (b^4*d^4*x + b^4*c*d^3)*((50625*b^12*c^12 + 202500*a*b^11*c^11*d + 587250*a^2*b^10*c^10*d^2 + 2092500*
a^3*b^9*c^9*d^3 + 4614975*a^4*b^8*c^8*d^4 + 8958600*a^5*b^7*c^7*d^5 + 18926460*a^6*b^6*c^6*d^6 + 27042120*a^7*
b^5*c^5*d^7 + 36722511*a^8*b^4*c^4*d^8 + 52655988*a^9*b^3*c^3*d^9 + 43080114*a^10*b^2*c^2*d^10 + 38348772*a^11
*b*c*d^11 + 35153041*a^12*d^12)/(b^15*d^13))^(1/4))/(d*x + c)) + 4*(32*b^2*d^2*x^2 + 45*b^2*c^2 + 54*a*b*c*d +
77*a^2*d^2 - 4*(9*b^2*c*d + 11*a*b*d^2)*x)*(b*x + a)^(1/4)*(d*x + c)^(3/4))/(b^3*d^3)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\left (a + b x\right )^{\frac{3}{4}} \sqrt [4]{c + d x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(b*x+a)**(3/4)/(d*x+c)**(1/4),x)

[Out]

Integral(x**3/((a + b*x)**(3/4)*(c + d*x)**(1/4)), x)

________________________________________________________________________________________

Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x+a)^(3/4)/(d*x+c)^(1/4),x, algorithm="giac")

[Out]

Timed out