Optimal. Leaf size=131 \[ -\frac{5 b^{3/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{12 a^{9/4} \sqrt{a+b x^4}}-\frac{5 \sqrt{a+b x^4}}{6 a^2 x^3}+\frac{1}{2 a x^3 \sqrt{a+b x^4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0353922, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {290, 325, 220} \[ -\frac{5 b^{3/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{12 a^{9/4} \sqrt{a+b x^4}}-\frac{5 \sqrt{a+b x^4}}{6 a^2 x^3}+\frac{1}{2 a x^3 \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 290
Rule 325
Rule 220
Rubi steps
\begin{align*} \int \frac{1}{x^4 \left (a+b x^4\right )^{3/2}} \, dx &=\frac{1}{2 a x^3 \sqrt{a+b x^4}}+\frac{5 \int \frac{1}{x^4 \sqrt{a+b x^4}} \, dx}{2 a}\\ &=\frac{1}{2 a x^3 \sqrt{a+b x^4}}-\frac{5 \sqrt{a+b x^4}}{6 a^2 x^3}-\frac{(5 b) \int \frac{1}{\sqrt{a+b x^4}} \, dx}{6 a^2}\\ &=\frac{1}{2 a x^3 \sqrt{a+b x^4}}-\frac{5 \sqrt{a+b x^4}}{6 a^2 x^3}-\frac{5 b^{3/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{12 a^{9/4} \sqrt{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.0090256, size = 54, normalized size = 0.41 \[ -\frac{\sqrt{\frac{b x^4}{a}+1} \, _2F_1\left (-\frac{3}{4},\frac{3}{2};\frac{1}{4};-\frac{b x^4}{a}\right )}{3 a x^3 \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.014, size = 113, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,{x}^{3}{a}^{2}}\sqrt{b{x}^{4}+a}}-{\frac{bx}{2\,{a}^{2}}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{b}} \right ) b}}}}-{\frac{5\,b}{6\,{a}^{2}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \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^{4} + a\right )}^{\frac{3}{2}} x^{4}}\,{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{b x^{4} + a}}{b^{2} x^{12} + 2 \, a b x^{8} + a^{2} x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 1.40557, size = 41, normalized size = 0.31 \begin{align*} \frac{\Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{3}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{2}} x^{3} \Gamma \left (\frac{1}{4}\right )} \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^{4} + a\right )}^{\frac{3}{2}} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]