3.92 \(\int \frac{1}{(a+b x^3)^{10/3} (c+d x^3)} \, dx\)

Optimal. Leaf size=280 \[ \frac{b x \left (67 a^2 d^2-57 a b c d+18 b^2 c^2\right )}{28 a^3 \sqrt [3]{a+b x^3} (b c-a d)^3}+\frac{b x (6 b c-13 a d)}{28 a^2 \left (a+b x^3\right )^{4/3} (b c-a d)^2}-\frac{d^3 \log \left (c+d x^3\right )}{6 c^{2/3} (b c-a d)^{10/3}}+\frac{d^3 \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{2/3} (b c-a d)^{10/3}}-\frac{d^3 \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} c^{2/3} (b c-a d)^{10/3}}+\frac{b x}{7 a \left (a+b x^3\right )^{7/3} (b c-a d)} \]

[Out]

(b*x)/(7*a*(b*c - a*d)*(a + b*x^3)^(7/3)) + (b*(6*b*c - 13*a*d)*x)/(28*a^2*(b*c - a*d)^2*(a + b*x^3)^(4/3)) +
(b*(18*b^2*c^2 - 57*a*b*c*d + 67*a^2*d^2)*x)/(28*a^3*(b*c - a*d)^3*(a + b*x^3)^(1/3)) - (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^(2/3)*(b*c - a*d)^(10/3)) - (d^3*Log[c +
d*x^3])/(6*c^(2/3)*(b*c - a*d)^(10/3)) + (d^3*Log[((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)])/(2*c^(2/
3)*(b*c - a*d)^(10/3))

________________________________________________________________________________________

Rubi [C]  time = 6.64171, antiderivative size = 1172, normalized size of antiderivative = 4.19, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {430, 429} \[ -\frac{-4158 d^3 (b c-a d)^4 \, _2F_1\left (2,\frac{13}{3};\frac{16}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{21}-2268 d^3 (b c-a d)^4 \, _3F_2\left (2,2,\frac{13}{3};1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{21}-378 d^3 (b c-a d)^4 \, _4F_3\left (2,2,2,\frac{13}{3};1,1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{21}-13608 c d^2 (b c-a d)^4 \, _2F_1\left (2,\frac{13}{3};\frac{16}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{18}-7182 c d^2 (b c-a d)^4 \, _3F_2\left (2,2,\frac{13}{3};1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{18}-1134 c d^2 (b c-a d)^4 \, _4F_3\left (2,2,2,\frac{13}{3};1,1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{18}+4212 c^2 d^3 (b c-a d)^2 \left (b x^3+a\right )^2 x^{15}-15246 c^2 d (b c-a d)^4 \, _2F_1\left (2,\frac{13}{3};\frac{16}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{15}-7560 c^2 d (b c-a d)^4 \, _3F_2\left (2,2,\frac{13}{3};1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{15}-1134 c^2 d (b c-a d)^4 \, _4F_3\left (2,2,2,\frac{13}{3};1,1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{15}+7371 c^3 d^3 (b c-a d) \left (b x^3+a\right )^3 x^{12}+14040 c^3 d^2 (b c-a d)^2 \left (b x^3+a\right )^2 x^{12}-5796 c^3 (b c-a d)^4 \, _2F_1\left (2,\frac{13}{3};\frac{16}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}-2646 c^3 (b c-a d)^4 \, _3F_2\left (2,2,\frac{13}{3};1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}-378 c^3 (b c-a d)^4 \, _4F_3\left (2,2,2,\frac{13}{3};1,1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}+29484 c^4 d^3 \left (b x^3+a\right )^4 x^9+24570 c^4 d^2 (b c-a d) \left (b x^3+a\right )^3 x^9+16380 c^4 d (b c-a d)^2 \left (b x^3+a\right )^2 x^9-29484 c^4 d^3 \left (b x^3+a\right )^4 \, _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+98280 c^5 d^2 \left (b x^3+a\right )^4 x^6+28665 c^5 d (b c-a d) \left (b x^3+a\right )^3 x^6+7280 c^5 (b c-a d)^2 \left (b x^3+a\right )^2 x^6-98280 c^5 d^2 \left (b x^3+a\right )^4 \, _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+114660 c^6 d \left (b x^3+a\right )^4 x^3+12740 c^6 (b c-a d) \left (b x^3+a\right )^3 x^3-114660 c^6 d \left (b x^3+a\right )^4 \, _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+50960 c^7 \left (b x^3+a\right )^4-50960 c^7 \left (b x^3+a\right )^4 \, _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 )}{5096 c^5 (b c-a d)^3 x^8 \left (b x^3+a\right )^{13/3}} \]

Warning: Unable to verify antiderivative.

[In]

Int[1/((a + b*x^3)^(10/3)*(c + d*x^3)),x]

[Out]

-(7280*c^5*(b*c - a*d)^2*x^6*(a + b*x^3)^2 + 16380*c^4*d*(b*c - a*d)^2*x^9*(a + b*x^3)^2 + 14040*c^3*d^2*(b*c
- a*d)^2*x^12*(a + b*x^3)^2 + 4212*c^2*d^3*(b*c - a*d)^2*x^15*(a + b*x^3)^2 + 12740*c^6*(b*c - a*d)*x^3*(a + b
*x^3)^3 + 28665*c^5*d*(b*c - a*d)*x^6*(a + b*x^3)^3 + 24570*c^4*d^2*(b*c - a*d)*x^9*(a + b*x^3)^3 + 7371*c^3*d
^3*(b*c - a*d)*x^12*(a + b*x^3)^3 + 50960*c^7*(a + b*x^3)^4 + 114660*c^6*d*x^3*(a + b*x^3)^4 + 98280*c^5*d^2*x
^6*(a + b*x^3)^4 + 29484*c^4*d^3*x^9*(a + b*x^3)^4 - 50960*c^7*(a + b*x^3)^4*Hypergeometric2F1[1/3, 1, 4/3, ((
b*c - a*d)*x^3)/(c*(a + b*x^3))] - 114660*c^6*d*x^3*(a + b*x^3)^4*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*
x^3)/(c*(a + b*x^3))] - 98280*c^5*d^2*x^6*(a + b*x^3)^4*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a
+ b*x^3))] - 29484*c^4*d^3*x^9*(a + b*x^3)^4*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))
] - 5796*c^3*(b*c - a*d)^4*x^12*Hypergeometric2F1[2, 13/3, 16/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 15246*c^
2*d*(b*c - a*d)^4*x^15*Hypergeometric2F1[2, 13/3, 16/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 13608*c*d^2*(b*c
- a*d)^4*x^18*Hypergeometric2F1[2, 13/3, 16/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 4158*d^3*(b*c - a*d)^4*x^2
1*Hypergeometric2F1[2, 13/3, 16/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 2646*c^3*(b*c - a*d)^4*x^12*Hypergeome
tricPFQ[{2, 2, 13/3}, {1, 16/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 7560*c^2*d*(b*c - a*d)^4*x^15*Hypergeome
tricPFQ[{2, 2, 13/3}, {1, 16/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 7182*c*d^2*(b*c - a*d)^4*x^18*Hypergeome
tricPFQ[{2, 2, 13/3}, {1, 16/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 2268*d^3*(b*c - a*d)^4*x^21*Hypergeometr
icPFQ[{2, 2, 13/3}, {1, 16/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 378*c^3*(b*c - a*d)^4*x^12*HypergeometricP
FQ[{2, 2, 2, 13/3}, {1, 1, 16/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 1134*c^2*d*(b*c - a*d)^4*x^15*Hypergeom
etricPFQ[{2, 2, 2, 13/3}, {1, 1, 16/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 1134*c*d^2*(b*c - a*d)^4*x^18*Hyp
ergeometricPFQ[{2, 2, 2, 13/3}, {1, 1, 16/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 378*d^3*(b*c - a*d)^4*x^21*
HypergeometricPFQ[{2, 2, 2, 13/3}, {1, 1, 16/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(5096*c^5*(b*c - a*d)^3*x
^8*(a + b*x^3)^(13/3))

Rule 430

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

Rule 429

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

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b x^3\right )^{10/3} \left (c+d x^3\right )} \, dx &=\frac{\sqrt [3]{1+\frac{b x^3}{a}} \int \frac{1}{\left (1+\frac{b x^3}{a}\right )^{10/3} \left (c+d x^3\right )} \, dx}{a^3 \sqrt [3]{a+b x^3}}\\ &=-\frac{7280 c^5 (b c-a d)^2 x^6 \left (a+b x^3\right )^2+16380 c^4 d (b c-a d)^2 x^9 \left (a+b x^3\right )^2+14040 c^3 d^2 (b c-a d)^2 x^{12} \left (a+b x^3\right )^2+4212 c^2 d^3 (b c-a d)^2 x^{15} \left (a+b x^3\right )^2+12740 c^6 (b c-a d) x^3 \left (a+b x^3\right )^3+28665 c^5 d (b c-a d) x^6 \left (a+b x^3\right )^3+24570 c^4 d^2 (b c-a d) x^9 \left (a+b x^3\right )^3+7371 c^3 d^3 (b c-a d) x^{12} \left (a+b x^3\right )^3+50960 c^7 \left (a+b x^3\right )^4+114660 c^6 d x^3 \left (a+b x^3\right )^4+98280 c^5 d^2 x^6 \left (a+b x^3\right )^4+29484 c^4 d^3 x^9 \left (a+b x^3\right )^4-50960 c^7 \left (a+b x^3\right )^4 \, _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 )-114660 c^6 d x^3 \left (a+b x^3\right )^4 \, _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 )-98280 c^5 d^2 x^6 \left (a+b x^3\right )^4 \, _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 )-29484 c^4 d^3 x^9 \left (a+b x^3\right )^4 \, _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 )-5796 c^3 (b c-a d)^4 x^{12} \, _2F_1\left (2,\frac{13}{3};\frac{16}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-15246 c^2 d (b c-a d)^4 x^{15} \, _2F_1\left (2,\frac{13}{3};\frac{16}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-13608 c d^2 (b c-a d)^4 x^{18} \, _2F_1\left (2,\frac{13}{3};\frac{16}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-4158 d^3 (b c-a d)^4 x^{21} \, _2F_1\left (2,\frac{13}{3};\frac{16}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-2646 c^3 (b c-a d)^4 x^{12} \, _3F_2\left (2,2,\frac{13}{3};1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-7560 c^2 d (b c-a d)^4 x^{15} \, _3F_2\left (2,2,\frac{13}{3};1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-7182 c d^2 (b c-a d)^4 x^{18} \, _3F_2\left (2,2,\frac{13}{3};1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-2268 d^3 (b c-a d)^4 x^{21} \, _3F_2\left (2,2,\frac{13}{3};1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-378 c^3 (b c-a d)^4 x^{12} \, _4F_3\left (2,2,2,\frac{13}{3};1,1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-1134 c^2 d (b c-a d)^4 x^{15} \, _4F_3\left (2,2,2,\frac{13}{3};1,1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-1134 c d^2 (b c-a d)^4 x^{18} \, _4F_3\left (2,2,2,\frac{13}{3};1,1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-378 d^3 (b c-a d)^4 x^{21} \, _4F_3\left (2,2,2,\frac{13}{3};1,1,\frac{16}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{5096 c^5 (b c-a d)^3 x^8 \left (a+b x^3\right )^{13/3}}\\ \end{align*}

Mathematica [C]  time = 4.35543, size = 1172, normalized size = 4.19 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((a + b*x^3)^(10/3)*(c + d*x^3)),x]

[Out]

-(-7280*c^5*(b*c - a*d)^2*x^6*(a + b*x^3)^2 - 16380*c^4*d*(b*c - a*d)^2*x^9*(a + b*x^3)^2 - 14040*c^3*d^2*(b*c
- a*d)^2*x^12*(a + b*x^3)^2 - 4212*c^2*d^3*(b*c - a*d)^2*x^15*(a + b*x^3)^2 - 12740*c^6*(b*c - a*d)*x^3*(a +
b*x^3)^3 - 28665*c^5*d*(b*c - a*d)*x^6*(a + b*x^3)^3 - 24570*c^4*d^2*(b*c - a*d)*x^9*(a + b*x^3)^3 - 7371*c^3*
d^3*(b*c - a*d)*x^12*(a + b*x^3)^3 - 50960*c^7*(a + b*x^3)^4 - 114660*c^6*d*x^3*(a + b*x^3)^4 - 98280*c^5*d^2*
x^6*(a + b*x^3)^4 - 29484*c^4*d^3*x^9*(a + b*x^3)^4 + 50960*c^7*(a + b*x^3)^4*Hypergeometric2F1[1/3, 1, 4/3, (
(b*c - a*d)*x^3)/(c*(a + b*x^3))] + 114660*c^6*d*x^3*(a + b*x^3)^4*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)
*x^3)/(c*(a + b*x^3))] + 98280*c^5*d^2*x^6*(a + b*x^3)^4*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(
a + b*x^3))] + 29484*c^4*d^3*x^9*(a + b*x^3)^4*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3)
)] + 5796*c^3*(b*c - a*d)^4*x^12*Hypergeometric2F1[2, 13/3, 16/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 15246*c
^2*d*(b*c - a*d)^4*x^15*Hypergeometric2F1[2, 13/3, 16/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 13608*c*d^2*(b*c
- a*d)^4*x^18*Hypergeometric2F1[2, 13/3, 16/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 4158*d^3*(b*c - a*d)^4*x^
21*Hypergeometric2F1[2, 13/3, 16/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 2646*c^3*(b*c - a*d)^4*x^12*Hypergeom
etricPFQ[{2, 2, 13/3}, {1, 16/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 7560*c^2*d*(b*c - a*d)^4*x^15*Hypergeom
etricPFQ[{2, 2, 13/3}, {1, 16/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 7182*c*d^2*(b*c - a*d)^4*x^18*Hypergeom
etricPFQ[{2, 2, 13/3}, {1, 16/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 2268*d^3*(b*c - a*d)^4*x^21*Hypergeomet
ricPFQ[{2, 2, 13/3}, {1, 16/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 378*c^3*(b*c - a*d)^4*x^12*Hypergeometric
PFQ[{2, 2, 2, 13/3}, {1, 1, 16/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 1134*c^2*d*(b*c - a*d)^4*x^15*Hypergeo
metricPFQ[{2, 2, 2, 13/3}, {1, 1, 16/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 1134*c*d^2*(b*c - a*d)^4*x^18*Hy
pergeometricPFQ[{2, 2, 2, 13/3}, {1, 1, 16/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 378*d^3*(b*c - a*d)^4*x^21
*HypergeometricPFQ[{2, 2, 2, 13/3}, {1, 1, 16/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(5096*c^5*(-(b*c) + a*d)
^3*x^8*(a + b*x^3)^(13/3))

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Maple [F]  time = 0.408, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{d{x}^{3}+c} \left ( b{x}^{3}+a \right ) ^{-{\frac{10}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^3+a)^(10/3)/(d*x^3+c),x)

[Out]

int(1/(b*x^3+a)^(10/3)/(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{10}{3}}{\left (d x^{3} + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^3+a)^(10/3)/(d*x^3+c),x, algorithm="maxima")

[Out]

integrate(1/((b*x^3 + a)^(10/3)*(d*x^3 + c)), 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/(b*x^3+a)^(10/3)/(d*x^3+c),x, algorithm="fricas")

[Out]

Timed out

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Sympy [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/(b*x**3+a)**(10/3)/(d*x**3+c),x)

[Out]

Timed out

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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{10}{3}}{\left (d x^{3} + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^3+a)^(10/3)/(d*x^3+c),x, algorithm="giac")

[Out]

integrate(1/((b*x^3 + a)^(10/3)*(d*x^3 + c)), x)