Optimal. Leaf size=152 \[ \frac{4 a^{3/2} \left (c \sqrt{a+b x^2}\right )^{3/2} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{15 b^{3/2} \left (\frac{b x^2}{a}+1\right )^{3/4}}-\frac{4 a^2 x \left (c \sqrt{a+b x^2}\right )^{3/2}}{15 b \left (a+b x^2\right )}+\frac{2}{9} x^3 \left (c \sqrt{a+b x^2}\right )^{3/2}+\frac{2 a x \left (c \sqrt{a+b x^2}\right )^{3/2}}{15 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.165506, antiderivative size = 191, normalized size of antiderivative = 1.26, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {6720, 279, 321, 229, 227, 196} \[ \frac{4 a^{5/2} c \sqrt [4]{\frac{b x^2}{a}+1} \sqrt{c \sqrt{a+b x^2}} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{15 b^{3/2} \sqrt{a+b x^2}}-\frac{4 a^2 c x \sqrt{c \sqrt{a+b x^2}}}{15 b \sqrt{a+b x^2}}+\frac{2}{9} c x^3 \sqrt{a+b x^2} \sqrt{c \sqrt{a+b x^2}}+\frac{2 a c x \sqrt{a+b x^2} \sqrt{c \sqrt{a+b x^2}}}{15 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6720
Rule 279
Rule 321
Rule 229
Rule 227
Rule 196
Rubi steps
\begin{align*} \int x^2 \left (c \sqrt{a+b x^2}\right )^{3/2} \, dx &=\frac{\left (c \sqrt{c \sqrt{a+b x^2}}\right ) \int x^2 \left (a+b x^2\right )^{3/4} \, dx}{\sqrt [4]{a+b x^2}}\\ &=\frac{2}{9} c x^3 \sqrt{c \sqrt{a+b x^2}} \sqrt{a+b x^2}+\frac{\left (a c \sqrt{c \sqrt{a+b x^2}}\right ) \int \frac{x^2}{\sqrt [4]{a+b x^2}} \, dx}{3 \sqrt [4]{a+b x^2}}\\ &=\frac{2 a c x \sqrt{c \sqrt{a+b x^2}} \sqrt{a+b x^2}}{15 b}+\frac{2}{9} c x^3 \sqrt{c \sqrt{a+b x^2}} \sqrt{a+b x^2}-\frac{\left (2 a^2 c \sqrt{c \sqrt{a+b x^2}}\right ) \int \frac{1}{\sqrt [4]{a+b x^2}} \, dx}{15 b \sqrt [4]{a+b x^2}}\\ &=\frac{2 a c x \sqrt{c \sqrt{a+b x^2}} \sqrt{a+b x^2}}{15 b}+\frac{2}{9} c x^3 \sqrt{c \sqrt{a+b x^2}} \sqrt{a+b x^2}-\frac{\left (2 a^2 c \sqrt{c \sqrt{a+b x^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{a+b x^2}}\\ &=-\frac{4 a^2 c x \sqrt{c \sqrt{a+b x^2}}}{15 b \sqrt{a+b x^2}}+\frac{2 a c x \sqrt{c \sqrt{a+b x^2}} \sqrt{a+b x^2}}{15 b}+\frac{2}{9} c x^3 \sqrt{c \sqrt{a+b x^2}} \sqrt{a+b x^2}+\frac{\left (2 a^2 c \sqrt{c \sqrt{a+b x^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{a+b x^2}}\\ &=-\frac{4 a^2 c x \sqrt{c \sqrt{a+b x^2}}}{15 b \sqrt{a+b x^2}}+\frac{2 a c x \sqrt{c \sqrt{a+b x^2}} \sqrt{a+b x^2}}{15 b}+\frac{2}{9} c x^3 \sqrt{c \sqrt{a+b x^2}} \sqrt{a+b x^2}+\frac{4 a^{5/2} c \sqrt{c \sqrt{a+b x^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{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0583909, size = 68, normalized size = 0.45 \[ \frac{2 x \left (c \sqrt{a+b x^2}\right )^{3/2} \left (-\frac{a \, _2F_1\left (-\frac{3}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )}{\left (\frac{b x^2}{a}+1\right )^{3/4}}+a+b x^2\right )}{9 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.008, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( c\sqrt{b{x}^{2}+a} \right ) ^{{\frac{3}{2}}}\, 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 (\sqrt{b x^{2} + a} c\right )^{\frac{3}{2}} 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 (\sqrt{b x^{2} + a} \sqrt{\sqrt{b x^{2} + a} c} c x^{2}, 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 x^{2} \left (c \sqrt{a + b x^{2}}\right )^{\frac{3}{2}}\, dx \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]