Integrand size = 26, antiderivative size = 180 \[ \int \frac {(e x)^{9/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{5/4}} \, dx=-\frac {7 a (10 b c-11 a d) e^3 (e x)^{3/2}}{60 b^3 \sqrt [4]{a+b x^2}}+\frac {(10 b c-11 a d) e (e x)^{7/2}}{30 b^2 \sqrt [4]{a+b x^2}}+\frac {d (e x)^{11/2}}{5 b e \sqrt [4]{a+b x^2}}-\frac {7 a^{3/2} (10 b c-11 a d) e^4 \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {e x} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{20 b^{7/2} \sqrt [4]{a+b x^2}} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {470, 291, 290, 342, 202} \[ \int \frac {(e x)^{9/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{5/4}} \, dx=-\frac {7 a^{3/2} e^4 \sqrt {e x} \sqrt [4]{\frac {a}{b x^2}+1} (10 b c-11 a d) E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{20 b^{7/2} \sqrt [4]{a+b x^2}}-\frac {7 a e^3 (e x)^{3/2} (10 b c-11 a d)}{60 b^3 \sqrt [4]{a+b x^2}}+\frac {e (e x)^{7/2} (10 b c-11 a d)}{30 b^2 \sqrt [4]{a+b x^2}}+\frac {d (e x)^{11/2}}{5 b e \sqrt [4]{a+b x^2}} \]
[In]
[Out]
Rule 202
Rule 290
Rule 291
Rule 342
Rule 470
Rubi steps \begin{align*} \text {integral}& = \frac {d (e x)^{11/2}}{5 b e \sqrt [4]{a+b x^2}}-\frac {\left (-5 b c+\frac {11 a d}{2}\right ) \int \frac {(e x)^{9/2}}{\left (a+b x^2\right )^{5/4}} \, dx}{5 b} \\ & = \frac {(10 b c-11 a d) e (e x)^{7/2}}{30 b^2 \sqrt [4]{a+b x^2}}+\frac {d (e x)^{11/2}}{5 b e \sqrt [4]{a+b x^2}}-\frac {\left (7 a (10 b c-11 a d) e^2\right ) \int \frac {(e x)^{5/2}}{\left (a+b x^2\right )^{5/4}} \, dx}{60 b^2} \\ & = -\frac {7 a (10 b c-11 a d) e^3 (e x)^{3/2}}{60 b^3 \sqrt [4]{a+b x^2}}+\frac {(10 b c-11 a d) e (e x)^{7/2}}{30 b^2 \sqrt [4]{a+b x^2}}+\frac {d (e x)^{11/2}}{5 b e \sqrt [4]{a+b x^2}}+\frac {\left (7 a^2 (10 b c-11 a d) e^4\right ) \int \frac {\sqrt {e x}}{\left (a+b x^2\right )^{5/4}} \, dx}{40 b^3} \\ & = -\frac {7 a (10 b c-11 a d) e^3 (e x)^{3/2}}{60 b^3 \sqrt [4]{a+b x^2}}+\frac {(10 b c-11 a d) e (e x)^{7/2}}{30 b^2 \sqrt [4]{a+b x^2}}+\frac {d (e x)^{11/2}}{5 b e \sqrt [4]{a+b x^2}}+\frac {\left (7 a^2 (10 b c-11 a d) e^4 \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}{40 b^4 \sqrt [4]{a+b x^2}} \\ & = -\frac {7 a (10 b c-11 a d) e^3 (e x)^{3/2}}{60 b^3 \sqrt [4]{a+b x^2}}+\frac {(10 b c-11 a d) e (e x)^{7/2}}{30 b^2 \sqrt [4]{a+b x^2}}+\frac {d (e x)^{11/2}}{5 b e \sqrt [4]{a+b x^2}}-\frac {\left (7 a^2 (10 b c-11 a d) e^4 \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {e x}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{b}\right )^{5/4}} \, dx,x,\frac {1}{x}\right )}{40 b^4 \sqrt [4]{a+b x^2}} \\ & = -\frac {7 a (10 b c-11 a d) e^3 (e x)^{3/2}}{60 b^3 \sqrt [4]{a+b x^2}}+\frac {(10 b c-11 a d) e (e x)^{7/2}}{30 b^2 \sqrt [4]{a+b x^2}}+\frac {d (e x)^{11/2}}{5 b e \sqrt [4]{a+b x^2}}-\frac {7 a^{3/2} (10 b c-11 a d) e^4 \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 )}{20 b^{7/2} \sqrt [4]{a+b x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.10 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.62 \[ \int \frac {(e x)^{9/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{5/4}} \, dx=\frac {e^3 (e x)^{3/2} \left (77 a^2 d+4 b^2 x^2 \left (5 c+3 d x^2\right )-2 a b \left (35 c+11 d x^2\right )+7 a (10 b c-11 a d) \sqrt [4]{1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{4},\frac {7}{4},-\frac {b x^2}{a}\right )\right )}{60 b^3 \sqrt [4]{a+b x^2}} \]
[In]
[Out]
\[\int \frac {\left (e x \right )^{\frac {9}{2}} \left (d \,x^{2}+c \right )}{\left (b \,x^{2}+a \right )^{\frac {5}{4}}}d x\]
[In]
[Out]
\[ \int \frac {(e x)^{9/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{5/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )} \left (e x\right )^{\frac {9}{2}}}{{\left (b x^{2} + a\right )}^{\frac {5}{4}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(e x)^{9/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{5/4}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {(e x)^{9/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{5/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )} \left (e x\right )^{\frac {9}{2}}}{{\left (b x^{2} + a\right )}^{\frac {5}{4}}} \,d x } \]
[In]
[Out]
\[ \int \frac {(e x)^{9/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{5/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )} \left (e x\right )^{\frac {9}{2}}}{{\left (b x^{2} + a\right )}^{\frac {5}{4}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(e x)^{9/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{5/4}} \, dx=\int \frac {{\left (e\,x\right )}^{9/2}\,\left (d\,x^2+c\right )}{{\left (b\,x^2+a\right )}^{5/4}} \,d x \]
[In]
[Out]