Optimal. Leaf size=64 \[ -\frac{\sqrt{\frac{d x^6}{c}+1} F_1\left (-\frac{1}{3};1,\frac{1}{2};\frac{2}{3};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{2 a x^2 \sqrt{c+d x^6}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0868468, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {465, 511, 510} \[ -\frac{\sqrt{\frac{d x^6}{c}+1} F_1\left (-\frac{1}{3};1,\frac{1}{2};\frac{2}{3};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{2 a x^2 \sqrt{c+d x^6}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 465
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (a+b x^6\right ) \sqrt{c+d x^6}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x^3\right ) \sqrt{c+d x^3}} \, dx,x,x^2\right )\\ &=\frac{\sqrt{1+\frac{d x^6}{c}} \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x^3\right ) \sqrt{1+\frac{d x^3}{c}}} \, dx,x,x^2\right )}{2 \sqrt{c+d x^6}}\\ &=-\frac{\sqrt{1+\frac{d x^6}{c}} F_1\left (-\frac{1}{3};1,\frac{1}{2};\frac{2}{3};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{2 a x^2 \sqrt{c+d x^6}}\\ \end{align*}
Mathematica [B] time = 0.109684, size = 141, normalized size = 2.2 \[ \frac{2 b d x^{12} \sqrt{\frac{d x^6}{c}+1} F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )+5 x^6 \sqrt{\frac{d x^6}{c}+1} (a d-2 b c) F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )-20 a \left (c+d x^6\right )}{40 a^2 c x^2 \sqrt{c+d x^6}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \left ( b{x}^{6}+a \right ) }{\frac{1}{\sqrt{d{x}^{6}+c}}}}\, 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^{6} + a\right )} \sqrt{d x^{6} + c} x^{3}}\,{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^{3} \left (a + b x^{6}\right ) \sqrt{c + d x^{6}}}\, 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^{6} + a\right )} \sqrt{d x^{6} + c} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]