Integrand size = 17, antiderivative size = 119 \[ \int \left (c \sqrt {a+b x^2}\right )^{3/2} \, dx=\frac {2}{5} x \left (c \sqrt {a+b x^2}\right )^{3/2}+\frac {6 a x \left (c \sqrt {a+b x^2}\right )^{3/2}}{5 \left (a+b x^2\right )}-\frac {6 \sqrt {a} \left (c \sqrt {a+b x^2}\right )^{3/2} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{5 \sqrt {b} \left (1+\frac {b x^2}{a}\right )^{3/4}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {1973, 201, 233, 202} \[ \int \left (c \sqrt {a+b x^2}\right )^{3/2} \, dx=-\frac {6 \sqrt {a} \left (c \sqrt {a+b x^2}\right )^{3/2} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{5 \sqrt {b} \left (\frac {b x^2}{a}+1\right )^{3/4}}+\frac {2}{5} x \left (c \sqrt {a+b x^2}\right )^{3/2}+\frac {6 a x \left (c \sqrt {a+b x^2}\right )^{3/2}}{5 \left (a+b x^2\right )} \]
[In]
[Out]
Rule 201
Rule 202
Rule 233
Rule 1973
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \int \left (1+\frac {b x^2}{a}\right )^{3/4} \, dx}{\left (1+\frac {b x^2}{a}\right )^{3/4}} \\ & = \frac {2}{5} x \left (c \sqrt {a+b x^2}\right )^{3/2}+\frac {\left (3 \left (c \sqrt {a+b x^2}\right )^{3/2}\right ) \int \frac {1}{\sqrt [4]{1+\frac {b x^2}{a}}} \, dx}{5 \left (1+\frac {b x^2}{a}\right )^{3/4}} \\ & = \frac {2}{5} x \left (c \sqrt {a+b x^2}\right )^{3/2}+\frac {6 a x \left (c \sqrt {a+b x^2}\right )^{3/2}}{5 \left (a+b x^2\right )}-\frac {\left (3 \left (c \sqrt {a+b x^2}\right )^{3/2}\right ) \int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{5/4}} \, dx}{5 \left (1+\frac {b x^2}{a}\right )^{3/4}} \\ & = \frac {2}{5} x \left (c \sqrt {a+b x^2}\right )^{3/2}+\frac {6 a x \left (c \sqrt {a+b x^2}\right )^{3/2}}{5 \left (a+b x^2\right )}-\frac {6 \sqrt {a} \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 )}{5 \sqrt {b} \left (1+\frac {b x^2}{a}\right )^{3/4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.44 \[ \int \left (c \sqrt {a+b x^2}\right )^{3/2} \, dx=\frac {x \left (c \sqrt {a+b x^2}\right )^{3/2} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {1}{2},\frac {3}{2},-\frac {b x^2}{a}\right )}{\left (1+\frac {b x^2}{a}\right )^{3/4}} \]
[In]
[Out]
\[\int \left (c \sqrt {b \,x^{2}+a}\right )^{\frac {3}{2}}d x\]
[In]
[Out]
\[ \int \left (c \sqrt {a+b x^2}\right )^{3/2} \, dx=\int { \left (\sqrt {b x^{2} + a} c\right )^{\frac {3}{2}} \,d x } \]
[In]
[Out]
\[ \int \left (c \sqrt {a+b x^2}\right )^{3/2} \, dx=\int \left (c \sqrt {a + b x^{2}}\right )^{\frac {3}{2}}\, dx \]
[In]
[Out]
\[ \int \left (c \sqrt {a+b x^2}\right )^{3/2} \, dx=\int { \left (\sqrt {b x^{2} + a} c\right )^{\frac {3}{2}} \,d x } \]
[In]
[Out]
\[ \int \left (c \sqrt {a+b x^2}\right )^{3/2} \, dx=\int { \left (\sqrt {b x^{2} + a} c\right )^{\frac {3}{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \left (c \sqrt {a+b x^2}\right )^{3/2} \, dx=\int {\left (c\,\sqrt {b\,x^2+a}\right )}^{3/2} \,d x \]
[In]
[Out]