Optimal. Leaf size=103 \[ \frac{2 \sqrt{e x} \sqrt [4]{\frac{a}{b x^2}+1} (2 b c-a d) E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{a^{3/2} \sqrt{b} e^2 \sqrt [4]{a+b x^2}}-\frac{2 c}{a e \sqrt{e x} \sqrt [4]{a+b x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0592124, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {453, 284, 335, 196} \[ \frac{2 \sqrt{e x} \sqrt [4]{\frac{a}{b x^2}+1} (2 b c-a d) E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{a^{3/2} \sqrt{b} e^2 \sqrt [4]{a+b x^2}}-\frac{2 c}{a e \sqrt{e x} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 453
Rule 284
Rule 335
Rule 196
Rubi steps
\begin{align*} \int \frac{c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{5/4}} \, dx &=-\frac{2 c}{a e \sqrt{e x} \sqrt [4]{a+b x^2}}-\frac{(2 b c-a d) \int \frac{\sqrt{e x}}{\left (a+b x^2\right )^{5/4}} \, dx}{a e^2}\\ &=-\frac{2 c}{a e \sqrt{e x} \sqrt [4]{a+b x^2}}-\frac{\left ((2 b c-a d) \sqrt [4]{1+\frac{a}{b x^2}} \sqrt{e x}\right ) \int \frac{1}{\left (1+\frac{a}{b x^2}\right )^{5/4} x^2} \, dx}{a b e^2 \sqrt [4]{a+b x^2}}\\ &=-\frac{2 c}{a e \sqrt{e x} \sqrt [4]{a+b x^2}}+\frac{\left ((2 b c-a d) \sqrt [4]{1+\frac{a}{b x^2}} \sqrt{e x}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{a x^2}{b}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )}{a b e^2 \sqrt [4]{a+b x^2}}\\ &=-\frac{2 c}{a e \sqrt{e x} \sqrt [4]{a+b x^2}}+\frac{2 (2 b c-a d) \sqrt [4]{1+\frac{a}{b x^2}} \sqrt{e x} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{a^{3/2} \sqrt{b} e^2 \sqrt [4]{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0412828, size = 77, normalized size = 0.75 \[ \frac{x \left (2 x^2 \sqrt [4]{\frac{b x^2}{a}+1} (a d-2 b c) \, _2F_1\left (\frac{3}{4},\frac{5}{4};\frac{7}{4};-\frac{b x^2}{a}\right )-6 a c\right )}{3 a^2 (e x)^{3/2} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.061, size = 0, normalized size = 0. \begin{align*} \int{(d{x}^{2}+c) \left ( ex \right ) ^{-{\frac{3}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}}}\, 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{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} \left (e x\right )^{\frac{3}{2}}}\,{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{{\left (b x^{2} + a\right )}^{\frac{3}{4}}{\left (d x^{2} + c\right )} \sqrt{e x}}{b^{2} e^{2} x^{6} + 2 \, a b e^{2} x^{4} + a^{2} e^{2} x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 104.662, size = 82, normalized size = 0.8 \begin{align*} - \frac{d{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{4} \\ \frac{3}{2} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{2}}} \right )}}{b^{\frac{5}{4}} e^{\frac{3}{2}} x} + \frac{c \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{5}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{5}{4}} e^{\frac{3}{2}} \sqrt{x} \Gamma \left (\frac{3}{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{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} \left (e x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]