Optimal. Leaf size=71 \[ \frac{2 (e x)^{5/2} \sqrt{c+d x^4} F_1\left (\frac{5}{8};1,-\frac{1}{2};\frac{13}{8};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{5 a e \sqrt{\frac{d x^4}{c}+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11706, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {466, 511, 510} \[ \frac{2 (e x)^{5/2} \sqrt{c+d x^4} F_1\left (\frac{5}{8};1,-\frac{1}{2};\frac{13}{8};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{5 a e \sqrt{\frac{d x^4}{c}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 466
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{(e x)^{3/2} \sqrt{c+d x^4}}{a+b x^4} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{x^4 \sqrt{c+\frac{d x^8}{e^4}}}{a+\frac{b x^8}{e^4}} \, dx,x,\sqrt{e x}\right )}{e}\\ &=\frac{\left (2 \sqrt{c+d x^4}\right ) \operatorname{Subst}\left (\int \frac{x^4 \sqrt{1+\frac{d x^8}{c e^4}}}{a+\frac{b x^8}{e^4}} \, dx,x,\sqrt{e x}\right )}{e \sqrt{1+\frac{d x^4}{c}}}\\ &=\frac{2 (e x)^{5/2} \sqrt{c+d x^4} F_1\left (\frac{5}{8};1,-\frac{1}{2};\frac{13}{8};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{5 a e \sqrt{1+\frac{d x^4}{c}}}\\ \end{align*}
Mathematica [A] time = 0.0354186, size = 70, normalized size = 0.99 \[ \frac{2 x (e x)^{3/2} \sqrt{c+d x^4} F_1\left (\frac{5}{8};-\frac{1}{2},1;\frac{13}{8};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{5 a \sqrt{\frac{c+d x^4}{c}}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.05, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{b{x}^{4}+a} \left ( ex \right ) ^{{\frac{3}{2}}}\sqrt{d{x}^{4}+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{\sqrt{d x^{4} + c} \left (e x\right )^{\frac{3}{2}}}{b x^{4} + a}\,{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{\sqrt{d x^{4} + c} \sqrt{e x} e x}{b x^{4} + a}, x\right ) \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{\left (e x\right )^{\frac{3}{2}} \sqrt{c + d x^{4}}}{a + b x^{4}}\, 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{\sqrt{d x^{4} + c} \left (e x\right )^{\frac{3}{2}}}{b x^{4} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]