Optimal. Leaf size=147 \[ \frac{x \left (\frac{b x^3}{a}+1\right )^{2/3} \left (2 a^2 d^2+a b c d+2 b^2 c^2\right ) \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{5 a^2 b^2 \left (a+b x^3\right )^{2/3}}+\frac{2 x \left (\frac{c^2}{a^2}-\frac{d^2}{b^2}\right )}{5 \left (a+b x^3\right )^{2/3}}+\frac{x \left (c+d x^3\right ) (b c-a d)}{5 a b \left (a+b x^3\right )^{5/3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0877303, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {413, 385, 246, 245} \[ \frac{x \left (\frac{b x^3}{a}+1\right )^{2/3} \left (2 a^2 d^2+a b c d+2 b^2 c^2\right ) \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{5 a^2 b^2 \left (a+b x^3\right )^{2/3}}+\frac{2 x \left (\frac{c^2}{a^2}-\frac{d^2}{b^2}\right )}{5 \left (a+b x^3\right )^{2/3}}+\frac{x \left (c+d x^3\right ) (b c-a d)}{5 a b \left (a+b x^3\right )^{5/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 413
Rule 385
Rule 246
Rule 245
Rubi steps
\begin{align*} \int \frac{\left (c+d x^3\right )^2}{\left (a+b x^3\right )^{8/3}} \, dx &=\frac{(b c-a d) x \left (c+d x^3\right )}{5 a b \left (a+b x^3\right )^{5/3}}+\frac{\int \frac{c (4 b c+a d)+d (b c+4 a d) x^3}{\left (a+b x^3\right )^{5/3}} \, dx}{5 a b}\\ &=\frac{2 \left (\frac{c^2}{a^2}-\frac{d^2}{b^2}\right ) x}{5 \left (a+b x^3\right )^{2/3}}+\frac{(b c-a d) x \left (c+d x^3\right )}{5 a b \left (a+b x^3\right )^{5/3}}+\frac{\left (2 b^2 c^2+a b c d+2 a^2 d^2\right ) \int \frac{1}{\left (a+b x^3\right )^{2/3}} \, dx}{5 a^2 b^2}\\ &=\frac{2 \left (\frac{c^2}{a^2}-\frac{d^2}{b^2}\right ) x}{5 \left (a+b x^3\right )^{2/3}}+\frac{(b c-a d) x \left (c+d x^3\right )}{5 a b \left (a+b x^3\right )^{5/3}}+\frac{\left (\left (2 b^2 c^2+a b c d+2 a^2 d^2\right ) \left (1+\frac{b x^3}{a}\right )^{2/3}\right ) \int \frac{1}{\left (1+\frac{b x^3}{a}\right )^{2/3}} \, dx}{5 a^2 b^2 \left (a+b x^3\right )^{2/3}}\\ &=\frac{2 \left (\frac{c^2}{a^2}-\frac{d^2}{b^2}\right ) x}{5 \left (a+b x^3\right )^{2/3}}+\frac{(b c-a d) x \left (c+d x^3\right )}{5 a b \left (a+b x^3\right )^{5/3}}+\frac{\left (2 b^2 c^2+a b c d+2 a^2 d^2\right ) x \left (1+\frac{b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{5 a^2 b^2 \left (a+b x^3\right )^{2/3}}\\ \end{align*}
Mathematica [A] time = 4.47114, size = 171, normalized size = 1.16 \[ \frac{x \text{Gamma}\left (\frac{2}{3}\right ) \left (\frac{b x^3}{a}+1\right )^{2/3} \left (-6 b x^3 \left (c+d x^3\right )^2 \text{HypergeometricPFQ}\left (\left \{\frac{4}{3},2,\frac{11}{3}\right \},\left \{1,\frac{13}{3}\right \},-\frac{b x^3}{a}\right )-2 b x^3 \left (11 c^2+16 c d x^3+5 d^2 x^6\right ) \, _2F_1\left (\frac{4}{3},\frac{11}{3};\frac{13}{3};-\frac{b x^3}{a}\right )+5 a \left (14 c^2+7 c d x^3+2 d^2 x^6\right ) \, _2F_1\left (\frac{1}{3},\frac{8}{3};\frac{10}{3};-\frac{b x^3}{a}\right )\right )}{63 a^3 \text{Gamma}\left (\frac{8}{3}\right ) \left (a+b x^3\right )^{2/3}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.369, size = 0, normalized size = 0. \begin{align*} \int{ \left ( d{x}^{3}+c \right ) ^{2} \left ( b{x}^{3}+a \right ) ^{-{\frac{8}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x^{3} + c\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac{8}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (d^{2} x^{6} + 2 \, c d x^{3} + c^{2}\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{b^{3} x^{9} + 3 \, a b^{2} x^{6} + 3 \, a^{2} b x^{3} + a^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x^{3} + c\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac{8}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]