3.759 \(\int \frac{1}{x^6 (a+b x^3)^{4/3} (c+d x^3)} \, dx\)

Optimal. Leaf size=287 \[ \frac{\left (a+b x^3\right )^{2/3} \left (-5 a^2 d^2-3 a b c d+18 b^2 c^2\right )}{10 a^3 c^2 x^2 (b c-a d)}-\frac{\left (a+b x^3\right )^{2/3} (6 b c-a d)}{5 a^2 c x^5 (b c-a d)}-\frac{d^3 \log \left (c+d x^3\right )}{6 c^{8/3} (b c-a d)^{4/3}}+\frac{d^3 \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{8/3} (b c-a d)^{4/3}}-\frac{d^3 \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} c^{8/3} (b c-a d)^{4/3}}+\frac{b}{a x^5 \sqrt [3]{a+b x^3} (b c-a d)} \]

[Out]

b/(a*(b*c - a*d)*x^5*(a + b*x^3)^(1/3)) - ((6*b*c - a*d)*(a + b*x^3)^(2/3))/(5*a^2*c*(b*c - a*d)*x^5) + ((18*b
^2*c^2 - 3*a*b*c*d - 5*a^2*d^2)*(a + b*x^3)^(2/3))/(10*a^3*c^2*(b*c - a*d)*x^2) - (d^3*ArcTan[(1 + (2*(b*c - a
*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/Sqrt[3]])/(Sqrt[3]*c^(8/3)*(b*c - a*d)^(4/3)) - (d^3*Log[c + d*x^3])
/(6*c^(8/3)*(b*c - a*d)^(4/3)) + (d^3*Log[((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)])/(2*c^(8/3)*(b*c
- a*d)^(4/3))

________________________________________________________________________________________

Rubi [C]  time = 3.85749, antiderivative size = 950, normalized size of antiderivative = 3.31, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {511, 510} \[ \frac{297 d^3 (b c-a d)^2 \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{15}+162 d^3 (b c-a d)^2 \, _3F_2\left (2,2,\frac{7}{3};1,\frac{10}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{15}+27 d^3 (b c-a d)^2 \, _4F_3\left (2,2,2,\frac{7}{3};1,1,\frac{10}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{15}+486 c d^2 (b c-a d)^2 \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}+351 c d^2 (b c-a d)^2 \, _3F_2\left (2,2,\frac{7}{3};1,\frac{10}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}+81 c d^2 (b c-a d)^2 \, _4F_3\left (2,2,2,\frac{7}{3};1,1,\frac{10}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}-1134 c^2 d^3 \left (b x^3+a\right )^2 x^9+1134 c^2 d^3 \left (b x^3+a\right )^2 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9+171 c^2 d (b c-a d)^2 \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9+216 c^2 d (b c-a d)^2 \, _3F_2\left (2,2,\frac{7}{3};1,\frac{10}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9+81 c^2 d (b c-a d)^2 \, _4F_3\left (2,2,2,\frac{7}{3};1,1,\frac{10}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9-1512 c^3 d^2 \left (b x^3+a\right )^2 x^6+1512 c^3 d^2 \left (b x^3+a\right )^2 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6-18 c^3 (b c-a d)^2 \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6+27 c^3 (b c-a d)^2 \, _3F_2\left (2,2,\frac{7}{3};1,\frac{10}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6+27 c^3 (b c-a d)^2 \, _4F_3\left (2,2,2,\frac{7}{3};1,1,\frac{10}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6-252 c^4 d \left (b x^3+a\right )^2 x^3+252 c^4 d \left (b x^3+a\right )^2 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+56 c^5 \left (b x^3+a\right )^2-56 c^5 \left (b x^3+a\right )^2 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{70 c^5 (b c-a d) x^8 \left (b x^3+a\right )^{7/3}} \]

Warning: Unable to verify antiderivative.

[In]

Int[1/(x^6*(a + b*x^3)^(4/3)*(c + d*x^3)),x]

[Out]

(56*c^5*(a + b*x^3)^2 - 252*c^4*d*x^3*(a + b*x^3)^2 - 1512*c^3*d^2*x^6*(a + b*x^3)^2 - 1134*c^2*d^3*x^9*(a + b
*x^3)^2 - 56*c^5*(a + b*x^3)^2*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 252*c^4*d*x
^3*(a + b*x^3)^2*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 1512*c^3*d^2*x^6*(a + b*x
^3)^2*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 1134*c^2*d^3*x^9*(a + b*x^3)^2*Hyper
geometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 18*c^3*(b*c - a*d)^2*x^6*Hypergeometric2F1[2, 7
/3, 10/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 171*c^2*d*(b*c - a*d)^2*x^9*Hypergeometric2F1[2, 7/3, 10/3, ((b
*c - a*d)*x^3)/(c*(a + b*x^3))] + 486*c*d^2*(b*c - a*d)^2*x^12*Hypergeometric2F1[2, 7/3, 10/3, ((b*c - a*d)*x^
3)/(c*(a + b*x^3))] + 297*d^3*(b*c - a*d)^2*x^15*Hypergeometric2F1[2, 7/3, 10/3, ((b*c - a*d)*x^3)/(c*(a + b*x
^3))] + 27*c^3*(b*c - a*d)^2*x^6*HypergeometricPFQ[{2, 2, 7/3}, {1, 10/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))]
+ 216*c^2*d*(b*c - a*d)^2*x^9*HypergeometricPFQ[{2, 2, 7/3}, {1, 10/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 3
51*c*d^2*(b*c - a*d)^2*x^12*HypergeometricPFQ[{2, 2, 7/3}, {1, 10/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 162
*d^3*(b*c - a*d)^2*x^15*HypergeometricPFQ[{2, 2, 7/3}, {1, 10/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 27*c^3*
(b*c - a*d)^2*x^6*HypergeometricPFQ[{2, 2, 2, 7/3}, {1, 1, 10/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 81*c^2*
d*(b*c - a*d)^2*x^9*HypergeometricPFQ[{2, 2, 2, 7/3}, {1, 1, 10/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 81*c*
d^2*(b*c - a*d)^2*x^12*HypergeometricPFQ[{2, 2, 2, 7/3}, {1, 1, 10/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 27
*d^3*(b*c - a*d)^2*x^15*HypergeometricPFQ[{2, 2, 2, 7/3}, {1, 1, 10/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(7
0*c^5*(b*c - a*d)*x^8*(a + b*x^3)^(7/3))

Rule 511

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPa
rt[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(e*x)^m*(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rule 510

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^6 \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx &=\frac{\sqrt [3]{1+\frac{b x^3}{a}} \int \frac{1}{x^6 \left (1+\frac{b x^3}{a}\right )^{4/3} \left (c+d x^3\right )} \, dx}{a \sqrt [3]{a+b x^3}}\\ &=\frac{56 c^5 \left (a+b x^3\right )^2-252 c^4 d x^3 \left (a+b x^3\right )^2-1512 c^3 d^2 x^6 \left (a+b x^3\right )^2-1134 c^2 d^3 x^9 \left (a+b x^3\right )^2-56 c^5 \left (a+b x^3\right )^2 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+252 c^4 d x^3 \left (a+b x^3\right )^2 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+1512 c^3 d^2 x^6 \left (a+b x^3\right )^2 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+1134 c^2 d^3 x^9 \left (a+b x^3\right )^2 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-18 c^3 (b c-a d)^2 x^6 \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+171 c^2 d (b c-a d)^2 x^9 \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+486 c d^2 (b c-a d)^2 x^{12} \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+297 d^3 (b c-a d)^2 x^{15} \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+27 c^3 (b c-a d)^2 x^6 \, _3F_2\left (2,2,\frac{7}{3};1,\frac{10}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+216 c^2 d (b c-a d)^2 x^9 \, _3F_2\left (2,2,\frac{7}{3};1,\frac{10}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+351 c d^2 (b c-a d)^2 x^{12} \, _3F_2\left (2,2,\frac{7}{3};1,\frac{10}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+162 d^3 (b c-a d)^2 x^{15} \, _3F_2\left (2,2,\frac{7}{3};1,\frac{10}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+27 c^3 (b c-a d)^2 x^6 \, _4F_3\left (2,2,2,\frac{7}{3};1,1,\frac{10}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+81 c^2 d (b c-a d)^2 x^9 \, _4F_3\left (2,2,2,\frac{7}{3};1,1,\frac{10}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+81 c d^2 (b c-a d)^2 x^{12} \, _4F_3\left (2,2,2,\frac{7}{3};1,1,\frac{10}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+27 d^3 (b c-a d)^2 x^{15} \, _4F_3\left (2,2,2,\frac{7}{3};1,1,\frac{10}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{70 c^5 (b c-a d) x^8 \left (a+b x^3\right )^{7/3}}\\ \end{align*}

Mathematica [C]  time = 1.78511, size = 950, normalized size = 3.31 \[ -\frac{297 d^3 (b c-a d)^2 \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{15}+162 d^3 (b c-a d)^2 \text{HypergeometricPFQ}\left (\left \{2,2,\frac{7}{3}\right \},\left \{1,\frac{10}{3}\right \},\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{15}+27 d^3 (b c-a d)^2 \text{HypergeometricPFQ}\left (\left \{2,2,2,\frac{7}{3}\right \},\left \{1,1,\frac{10}{3}\right \},\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{15}+486 c d^2 (b c-a d)^2 \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}+351 c d^2 (b c-a d)^2 \text{HypergeometricPFQ}\left (\left \{2,2,\frac{7}{3}\right \},\left \{1,\frac{10}{3}\right \},\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}+81 c d^2 (b c-a d)^2 \text{HypergeometricPFQ}\left (\left \{2,2,2,\frac{7}{3}\right \},\left \{1,1,\frac{10}{3}\right \},\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}-1134 c^2 d^3 \left (b x^3+a\right )^2 x^9+1134 c^2 d^3 \left (b x^3+a\right )^2 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9+171 c^2 d (b c-a d)^2 \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9+216 c^2 d (b c-a d)^2 \text{HypergeometricPFQ}\left (\left \{2,2,\frac{7}{3}\right \},\left \{1,\frac{10}{3}\right \},\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9+81 c^2 d (b c-a d)^2 \text{HypergeometricPFQ}\left (\left \{2,2,2,\frac{7}{3}\right \},\left \{1,1,\frac{10}{3}\right \},\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9-1512 c^3 d^2 \left (b x^3+a\right )^2 x^6+1512 c^3 d^2 \left (b x^3+a\right )^2 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6-18 c^3 (b c-a d)^2 \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6+27 c^3 (b c-a d)^2 \text{HypergeometricPFQ}\left (\left \{2,2,\frac{7}{3}\right \},\left \{1,\frac{10}{3}\right \},\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6+27 c^3 (b c-a d)^2 \text{HypergeometricPFQ}\left (\left \{2,2,2,\frac{7}{3}\right \},\left \{1,1,\frac{10}{3}\right \},\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6-252 c^4 d \left (b x^3+a\right )^2 x^3+252 c^4 d \left (b x^3+a\right )^2 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+56 c^5 \left (b x^3+a\right )^2-56 c^5 \left (b x^3+a\right )^2 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{70 c^5 (a d-b c) x^8 \left (b x^3+a\right )^{7/3}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(x^6*(a + b*x^3)^(4/3)*(c + d*x^3)),x]

[Out]

-(56*c^5*(a + b*x^3)^2 - 252*c^4*d*x^3*(a + b*x^3)^2 - 1512*c^3*d^2*x^6*(a + b*x^3)^2 - 1134*c^2*d^3*x^9*(a +
b*x^3)^2 - 56*c^5*(a + b*x^3)^2*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 252*c^4*d*
x^3*(a + b*x^3)^2*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 1512*c^3*d^2*x^6*(a + b*
x^3)^2*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 1134*c^2*d^3*x^9*(a + b*x^3)^2*Hype
rgeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 18*c^3*(b*c - a*d)^2*x^6*Hypergeometric2F1[2,
7/3, 10/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 171*c^2*d*(b*c - a*d)^2*x^9*Hypergeometric2F1[2, 7/3, 10/3, ((
b*c - a*d)*x^3)/(c*(a + b*x^3))] + 486*c*d^2*(b*c - a*d)^2*x^12*Hypergeometric2F1[2, 7/3, 10/3, ((b*c - a*d)*x
^3)/(c*(a + b*x^3))] + 297*d^3*(b*c - a*d)^2*x^15*Hypergeometric2F1[2, 7/3, 10/3, ((b*c - a*d)*x^3)/(c*(a + b*
x^3))] + 27*c^3*(b*c - a*d)^2*x^6*HypergeometricPFQ[{2, 2, 7/3}, {1, 10/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))]
+ 216*c^2*d*(b*c - a*d)^2*x^9*HypergeometricPFQ[{2, 2, 7/3}, {1, 10/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] +
351*c*d^2*(b*c - a*d)^2*x^12*HypergeometricPFQ[{2, 2, 7/3}, {1, 10/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 16
2*d^3*(b*c - a*d)^2*x^15*HypergeometricPFQ[{2, 2, 7/3}, {1, 10/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 27*c^3
*(b*c - a*d)^2*x^6*HypergeometricPFQ[{2, 2, 2, 7/3}, {1, 1, 10/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 81*c^2
*d*(b*c - a*d)^2*x^9*HypergeometricPFQ[{2, 2, 2, 7/3}, {1, 1, 10/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 81*c
*d^2*(b*c - a*d)^2*x^12*HypergeometricPFQ[{2, 2, 2, 7/3}, {1, 1, 10/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 2
7*d^3*(b*c - a*d)^2*x^15*HypergeometricPFQ[{2, 2, 2, 7/3}, {1, 1, 10/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(
70*c^5*(-(b*c) + a*d)*x^8*(a + b*x^3)^(7/3))

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Maple [F]  time = 0.054, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{6} \left ( d{x}^{3}+c \right ) } \left ( b{x}^{3}+a \right ) ^{-{\frac{4}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^6/(b*x^3+a)^(4/3)/(d*x^3+c),x)

[Out]

int(1/x^6/(b*x^3+a)^(4/3)/(d*x^3+c),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{4}{3}}{\left (d x^{3} + c\right )} x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(b*x^3+a)^(4/3)/(d*x^3+c),x, algorithm="maxima")

[Out]

integrate(1/((b*x^3 + a)^(4/3)*(d*x^3 + c)*x^6), x)

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Fricas [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(1/x^6/(b*x^3+a)^(4/3)/(d*x^3+c),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{6} \left (a + b x^{3}\right )^{\frac{4}{3}} \left (c + d x^{3}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**6/(b*x**3+a)**(4/3)/(d*x**3+c),x)

[Out]

Integral(1/(x**6*(a + b*x**3)**(4/3)*(c + d*x**3)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{4}{3}}{\left (d x^{3} + c\right )} x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(b*x^3+a)^(4/3)/(d*x^3+c),x, algorithm="giac")

[Out]

integrate(1/((b*x^3 + a)^(4/3)*(d*x^3 + c)*x^6), x)