Optimal. Leaf size=64 \[ -\frac{\sqrt [3]{\frac{b x^3}{a}+1} F_1\left (-\frac{4}{3};\frac{1}{3},1;-\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{4 c x^4 \sqrt [3]{a+b x^3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0544019, 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} \[ -\frac{\sqrt [3]{\frac{b x^3}{a}+1} F_1\left (-\frac{4}{3};\frac{1}{3},1;-\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{4 c x^4 \sqrt [3]{a+b x^3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{1}{x^5 \sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx &=\frac{\sqrt [3]{1+\frac{b x^3}{a}} \int \frac{1}{x^5 \sqrt [3]{1+\frac{b x^3}{a}} \left (c+d x^3\right )} \, dx}{\sqrt [3]{a+b x^3}}\\ &=-\frac{\sqrt [3]{1+\frac{b x^3}{a}} F_1\left (-\frac{4}{3};\frac{1}{3},1;-\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{4 c x^4 \sqrt [3]{a+b x^3}}\\ \end{align*}
Mathematica [B] time = 0.171355, size = 183, normalized size = 2.86 \[ \frac{5 x^6 \sqrt [3]{\frac{b x^3}{a}+1} \left (2 a^2 d^2-2 a b c d-b^2 c^2\right ) F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )-2 b d x^9 \sqrt [3]{\frac{b x^3}{a}+1} (2 a d+b c) F_1\left (\frac{5}{3};\frac{1}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+5 c \left (a+b x^3\right ) \left (-a c+4 a d x^3+2 b c x^3\right )}{20 a^2 c^3 x^4 \sqrt [3]{a+b x^3}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.062, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5} \left ( d{x}^{3}+c \right ) }{\frac{1}{\sqrt [3]{b{x}^{3}+a}}}}\, 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}{3}}{\left (d x^{3} + c\right )} x^{5}}\,{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{1}{x^{5} \sqrt [3]{a + b x^{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{1}{{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (d x^{3} + c\right )} x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]