Optimal. Leaf size=119 \[ \frac{4 a^{5/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{15 b^{3/2} \sqrt [4]{a+b x^2}}-\frac{4 a^2 x}{15 b \sqrt [4]{a+b x^2}}+\frac{2}{9} x^3 \left (a+b x^2\right )^{3/4}+\frac{2 a x \left (a+b x^2\right )^{3/4}}{15 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0377391, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {279, 321, 229, 227, 196} \[ \frac{4 a^{5/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{15 b^{3/2} \sqrt [4]{a+b x^2}}-\frac{4 a^2 x}{15 b \sqrt [4]{a+b x^2}}+\frac{2}{9} x^3 \left (a+b x^2\right )^{3/4}+\frac{2 a x \left (a+b x^2\right )^{3/4}}{15 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 279
Rule 321
Rule 229
Rule 227
Rule 196
Rubi steps
\begin{align*} \int x^2 \left (a+b x^2\right )^{3/4} \, dx &=\frac{2}{9} x^3 \left (a+b x^2\right )^{3/4}+\frac{1}{3} a \int \frac{x^2}{\sqrt [4]{a+b x^2}} \, dx\\ &=\frac{2 a x \left (a+b x^2\right )^{3/4}}{15 b}+\frac{2}{9} x^3 \left (a+b x^2\right )^{3/4}-\frac{\left (2 a^2\right ) \int \frac{1}{\sqrt [4]{a+b x^2}} \, dx}{15 b}\\ &=\frac{2 a x \left (a+b x^2\right )^{3/4}}{15 b}+\frac{2}{9} x^3 \left (a+b x^2\right )^{3/4}-\frac{\left (2 a^2 \sqrt [4]{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\sqrt [4]{1+\frac{b x^2}{a}}} \, dx}{15 b \sqrt [4]{a+b x^2}}\\ &=-\frac{4 a^2 x}{15 b \sqrt [4]{a+b x^2}}+\frac{2 a x \left (a+b x^2\right )^{3/4}}{15 b}+\frac{2}{9} x^3 \left (a+b x^2\right )^{3/4}+\frac{\left (2 a^2 \sqrt [4]{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{5/4}} \, dx}{15 b \sqrt [4]{a+b x^2}}\\ &=-\frac{4 a^2 x}{15 b \sqrt [4]{a+b x^2}}+\frac{2 a x \left (a+b x^2\right )^{3/4}}{15 b}+\frac{2}{9} x^3 \left (a+b x^2\right )^{3/4}+\frac{4 a^{5/2} \sqrt [4]{1+\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{15 b^{3/2} \sqrt [4]{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0505306, size = 62, normalized size = 0.52 \[ \frac{2 x \left (a+b x^2\right )^{3/4} \left (-\frac{a \, _2F_1\left (-\frac{3}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )}{\left (\frac{b x^2}{a}+1\right )^{3/4}}+a+b x^2\right )}{9 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.024, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{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{\left (b x^{2} + a\right )}^{\frac{3}{4}} x^{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 ({\left (b x^{2} + a\right )}^{\frac{3}{4}} x^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 1.09418, size = 29, normalized size = 0.24 \begin{align*} \frac{a^{\frac{3}{4}} x^{3}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]