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

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

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Rubi [C]  time = 3.86, 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} \begin {gather*} \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}} \end {gather*}

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 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])

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])

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*}

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Mathematica [C]  time = 2.31, size = 950, normalized size = 3.31 \begin {gather*} -\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 (a d-b c) x^8 \left (b x^3+a\right )^{7/3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/70*(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
*Hypergeometric2F1[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*Hypergeometric2F
1[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))]
+ 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))] + 2
7*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))] + 8
1*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)
)])/(c^5*(-(b*c) + a*d)*x^8*(a + b*x^3)^(7/3))

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IntegrateAlgebraic [C]  time = 4.82, size = 452, normalized size = 1.57 \begin {gather*} \frac {-2 a^3 c d+5 a^3 d^2 x^3+2 a^2 b c^2+a^2 b c d x^3+5 a^2 b d^2 x^6-6 a b^2 c^2 x^3+3 a b^2 c d x^6-18 b^3 c^2 x^6}{10 a^3 c^2 x^5 \sqrt [3]{a+b x^3} (a d-b c)}-\frac {i \left (\sqrt {3} d^3-i d^3\right ) \log \left (2 x \sqrt [3]{b c-a d}+\left (1+i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}\right )}{6 c^{8/3} (b c-a d)^{4/3}}+\frac {\sqrt {\frac {1}{6} \left (-1+i \sqrt {3}\right )} d^3 \tan ^{-1}\left (\frac {3 x \sqrt [3]{b c-a d}}{\sqrt {3} x \sqrt [3]{b c-a d}-\sqrt {3} \sqrt [3]{c} \sqrt [3]{a+b x^3}-3 i \sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )}{c^{8/3} (b c-a d)^{4/3}}+\frac {\left (d^3+i \sqrt {3} d^3\right ) \log \left (\left (\sqrt {3}+i\right ) c^{2/3} \left (a+b x^3\right )^{2/3}+\sqrt [3]{c} \left (-\sqrt {3} x+i x\right ) \sqrt [3]{a+b x^3} \sqrt [3]{b c-a d}-2 i x^2 (b c-a d)^{2/3}\right )}{12 c^{8/3} (b c-a d)^{4/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^2*b*c^2 - 2*a^3*c*d - 6*a*b^2*c^2*x^3 + a^2*b*c*d*x^3 + 5*a^3*d^2*x^3 - 18*b^3*c^2*x^6 + 3*a*b^2*c*d*x^6
+ 5*a^2*b*d^2*x^6)/(10*a^3*c^2*(-(b*c) + a*d)*x^5*(a + b*x^3)^(1/3)) + (Sqrt[(-1 + I*Sqrt[3])/6]*d^3*ArcTan[(3
*(b*c - a*d)^(1/3)*x)/(Sqrt[3]*(b*c - a*d)^(1/3)*x - (3*I)*c^(1/3)*(a + b*x^3)^(1/3) - Sqrt[3]*c^(1/3)*(a + b*
x^3)^(1/3))])/(c^(8/3)*(b*c - a*d)^(4/3)) - ((I/6)*((-I)*d^3 + Sqrt[3]*d^3)*Log[2*(b*c - a*d)^(1/3)*x + (1 + I
*Sqrt[3])*c^(1/3)*(a + b*x^3)^(1/3)])/(c^(8/3)*(b*c - a*d)^(4/3)) + ((d^3 + I*Sqrt[3]*d^3)*Log[(-2*I)*(b*c - a
*d)^(2/3)*x^2 + c^(1/3)*(b*c - a*d)^(1/3)*(I*x - Sqrt[3]*x)*(a + b*x^3)^(1/3) + (I + Sqrt[3])*c^(2/3)*(a + b*x
^3)^(2/3)])/(12*c^(8/3)*(b*c - a*d)^(4/3))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (d x^{3} + c\right )} x^{6}}\,{d x} \end {gather*}

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)

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maple [F]  time = 0.45, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {4}{3}} \left (d \,x^{3}+c \right ) x^{6}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (d x^{3} + c\right )} x^{6}}\,{d x} \end {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x^6\,{\left (b\,x^3+a\right )}^{4/3}\,\left (d\,x^3+c\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{6} \left (a + b x^{3}\right )^{\frac {4}{3}} \left (c + d x^{3}\right )}\, dx \end {gather*}

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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