Optimal. Leaf size=64 \[ \frac{x^7 \left (\frac{b x^3}{a}+1\right )^{2/3} F_1\left (\frac{7}{3};\frac{2}{3},1;\frac{10}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{7 c \left (a+b x^3\right )^{2/3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0575557, antiderivative size = 64, normalized size of antiderivative = 1., 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} \[ \frac{x^7 \left (\frac{b x^3}{a}+1\right )^{2/3} F_1\left (\frac{7}{3};\frac{2}{3},1;\frac{10}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{7 c \left (a+b x^3\right )^{2/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{x^6}{\left (a+b x^3\right )^{2/3} \left (c+d x^3\right )} \, dx &=\frac{\left (1+\frac{b x^3}{a}\right )^{2/3} \int \frac{x^6}{\left (1+\frac{b x^3}{a}\right )^{2/3} \left (c+d x^3\right )} \, dx}{\left (a+b x^3\right )^{2/3}}\\ &=\frac{x^7 \left (1+\frac{b x^3}{a}\right )^{2/3} F_1\left (\frac{7}{3};\frac{2}{3},1;\frac{10}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{7 c \left (a+b x^3\right )^{2/3}}\\ \end{align*}
Mathematica [B] time = 0.333766, size = 249, normalized size = 3.89 \[ \frac{x \left (4 \left (\frac{4 a^2 c^2 F_1\left (\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{b \left (c+d x^3\right ) \left (x^3 \left (3 a d F_1\left (\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+2 b c F_1\left (\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )-4 a c F_1\left (\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )}+\frac{a}{b}+x^3\right )-\frac{x^3 \left (\frac{b x^3}{a}+1\right )^{2/3} (a d+2 b c) F_1\left (\frac{4}{3};\frac{2}{3},1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{b c}\right )}{8 d \left (a+b x^3\right )^{2/3}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.047, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{6}}{d{x}^{3}+c} \left ( b{x}^{3}+a \right ) ^{-{\frac{2}{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{x^{6}}{{\left (b x^{3} + a\right )}^{\frac{2}{3}}{\left (d x^{3} + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6}}{\left (a + b x^{3}\right )^{\frac{2}{3}} \left (c + d x^{3}\right )}\, dx \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{x^{6}}{{\left (b x^{3} + a\right )}^{\frac{2}{3}}{\left (d x^{3} + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]