Optimal. Leaf size=64 \[ -\frac{\sqrt{\frac{d x^8}{c}+1} F_1\left (-\frac{3}{8};1,\frac{1}{2};\frac{5}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right )}{3 a x^3 \sqrt{c+d x^8}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05565, 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} \[ -\frac{\sqrt{\frac{d x^8}{c}+1} F_1\left (-\frac{3}{8};1,\frac{1}{2};\frac{5}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right )}{3 a x^3 \sqrt{c+d x^8}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{1}{x^4 \left (a+b x^8\right ) \sqrt{c+d x^8}} \, dx &=\frac{\sqrt{1+\frac{d x^8}{c}} \int \frac{1}{x^4 \left (a+b x^8\right ) \sqrt{1+\frac{d x^8}{c}}} \, dx}{\sqrt{c+d x^8}}\\ &=-\frac{\sqrt{1+\frac{d x^8}{c}} F_1\left (-\frac{3}{8};1,\frac{1}{2};\frac{5}{8};-\frac{b x^8}{a},-\frac{d x^8}{c}\right )}{3 a x^3 \sqrt{c+d x^8}}\\ \end{align*}
Mathematica [B] time = 0.113915, size = 141, normalized size = 2.2 \[ \frac{5 b d x^{16} \sqrt{\frac{d x^8}{c}+1} F_1\left (\frac{13}{8};\frac{1}{2},1;\frac{21}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )+13 x^8 \sqrt{\frac{d x^8}{c}+1} (a d-3 b c) F_1\left (\frac{5}{8};\frac{1}{2},1;\frac{13}{8};-\frac{d x^8}{c},-\frac{b x^8}{a}\right )-65 a \left (c+d x^8\right )}{195 a^2 c x^3 \sqrt{c+d x^8}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4} \left ( b{x}^{8}+a \right ) }{\frac{1}{\sqrt{d{x}^{8}+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^{8} + a\right )} \sqrt{d x^{8} + c} x^{4}}\,{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^{4} \left (a + b x^{8}\right ) \sqrt{c + d x^{8}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [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]