Optimal. Leaf size=87 \[ \frac{x \sqrt [4]{\frac{c \left (a+b x^3\right )}{a \left (c+d x^3\right )}} \, _2F_1\left (\frac{1}{4},\frac{1}{3};\frac{4}{3};-\frac{(b c-a d) x^3}{a \left (d x^3+c\right )}\right )}{c \sqrt [4]{a+b x^3} \sqrt [12]{c+d x^3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0168134, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {380} \[ \frac{x \sqrt [4]{\frac{c \left (a+b x^3\right )}{a \left (c+d x^3\right )}} \, _2F_1\left (\frac{1}{4},\frac{1}{3};\frac{4}{3};-\frac{(b c-a d) x^3}{a \left (d x^3+c\right )}\right )}{c \sqrt [4]{a+b x^3} \sqrt [12]{c+d x^3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 380
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [4]{a+b x^3} \left (c+d x^3\right )^{13/12}} \, dx &=\frac{x \sqrt [4]{\frac{c \left (a+b x^3\right )}{a \left (c+d x^3\right )}} \, _2F_1\left (\frac{1}{4},\frac{1}{3};\frac{4}{3};-\frac{(b c-a d) x^3}{a \left (c+d x^3\right )}\right )}{c \sqrt [4]{a+b x^3} \sqrt [12]{c+d x^3}}\\ \end{align*}
Mathematica [A] time = 0.0287124, size = 86, normalized size = 0.99 \[ \frac{x \sqrt [4]{\frac{b x^3}{a}+1} \left (\frac{d x^3}{c}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{1}{3};\frac{4}{3};\frac{(a d-b c) x^3}{a \left (d x^3+c\right )}\right )}{\sqrt [4]{a+b x^3} \left (c+d x^3\right )^{13/12}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.446, size = 0, normalized size = 0. \begin{align*} \int{{\frac{1}{\sqrt [4]{b{x}^{3}+a}}} \left ( d{x}^{3}+c \right ) ^{-{\frac{13}{12}}}}\, 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{1}{{\left (b x^{3} + a\right )}^{\frac{1}{4}}{\left (d x^{3} + c\right )}^{\frac{13}{12}}}\,{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 (b x^{3} + a\right )}^{\frac{3}{4}}{\left (d x^{3} + c\right )}^{\frac{11}{12}}}{b d^{2} x^{9} +{\left (2 \, b c d + a d^{2}\right )} x^{6} +{\left (b c^{2} + 2 \, a c d\right )} x^{3} + a c^{2}}, 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{1}{{\left (b x^{3} + a\right )}^{\frac{1}{4}}{\left (d x^{3} + c\right )}^{\frac{13}{12}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]