Integrand size = 26, antiderivative size = 149 \[ \int \frac {(e x)^{3/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\frac {2 (b c-a d) (e x)^{5/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {2 d e \sqrt {e x}}{b^2 \sqrt [4]{a+b x^2}}+\frac {d e^{3/2} \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{9/4}}+\frac {d e^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{9/4}} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {463, 294, 335, 246, 218, 214, 211} \[ \int \frac {(e x)^{3/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\frac {d e^{3/2} \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{9/4}}+\frac {d e^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{9/4}}-\frac {2 d e \sqrt {e x}}{b^2 \sqrt [4]{a+b x^2}}+\frac {2 (e x)^{5/2} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}} \]
[In]
[Out]
Rule 211
Rule 214
Rule 218
Rule 246
Rule 294
Rule 335
Rule 463
Rubi steps \begin{align*} \text {integral}& = \frac {2 (b c-a d) (e x)^{5/2}}{5 a b e \left (a+b x^2\right )^{5/4}}+\frac {d \int \frac {(e x)^{3/2}}{\left (a+b x^2\right )^{5/4}} \, dx}{b} \\ & = \frac {2 (b c-a d) (e x)^{5/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {2 d e \sqrt {e x}}{b^2 \sqrt [4]{a+b x^2}}+\frac {\left (d e^2\right ) \int \frac {1}{\sqrt {e x} \sqrt [4]{a+b x^2}} \, dx}{b^2} \\ & = \frac {2 (b c-a d) (e x)^{5/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {2 d e \sqrt {e x}}{b^2 \sqrt [4]{a+b x^2}}+\frac {(2 d e) \text {Subst}\left (\int \frac {1}{\sqrt [4]{a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{b^2} \\ & = \frac {2 (b c-a d) (e x)^{5/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {2 d e \sqrt {e x}}{b^2 \sqrt [4]{a+b x^2}}+\frac {(2 d e) \text {Subst}\left (\int \frac {1}{1-\frac {b x^4}{e^2}} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{a+b x^2}}\right )}{b^2} \\ & = \frac {2 (b c-a d) (e x)^{5/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {2 d e \sqrt {e x}}{b^2 \sqrt [4]{a+b x^2}}+\frac {\left (d e^2\right ) \text {Subst}\left (\int \frac {1}{e-\sqrt {b} x^2} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{a+b x^2}}\right )}{b^2}+\frac {\left (d e^2\right ) \text {Subst}\left (\int \frac {1}{e+\sqrt {b} x^2} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{a+b x^2}}\right )}{b^2} \\ & = \frac {2 (b c-a d) (e x)^{5/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac {2 d e \sqrt {e x}}{b^2 \sqrt [4]{a+b x^2}}+\frac {d e^{3/2} \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{9/4}}+\frac {d e^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{9/4}} \\ \end{align*}
Time = 1.32 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.83 \[ \int \frac {(e x)^{3/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\frac {e \sqrt {e x} \left (\frac {2 \sqrt [4]{b} \left (-5 a^2 d+b^2 c x^2-6 a b d x^2\right )}{a \left (a+b x^2\right )^{5/4}}+\frac {5 d \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a+b x^2}}\right )}{\sqrt {x}}+\frac {5 d \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a+b x^2}}\right )}{\sqrt {x}}\right )}{5 b^{9/4}} \]
[In]
[Out]
\[\int \frac {\left (e x \right )^{\frac {3}{2}} \left (d \,x^{2}+c \right )}{\left (b \,x^{2}+a \right )^{\frac {9}{4}}}d x\]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 468, normalized size of antiderivative = 3.14 \[ \int \frac {(e x)^{3/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=-\frac {4 \, {\left (5 \, a^{2} d e - {\left (b^{2} c - 6 \, a b d\right )} e x^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x} - 5 \, {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \left (\frac {d^{4} e^{6}}{b^{9}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x} d e + {\left (b^{3} x^{2} + a b^{2}\right )} \left (\frac {d^{4} e^{6}}{b^{9}}\right )^{\frac {1}{4}}}{b x^{2} + a}\right ) + 5 \, {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \left (\frac {d^{4} e^{6}}{b^{9}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x} d e - {\left (b^{3} x^{2} + a b^{2}\right )} \left (\frac {d^{4} e^{6}}{b^{9}}\right )^{\frac {1}{4}}}{b x^{2} + a}\right ) + 5 \, {\left (i \, a b^{4} x^{4} + 2 i \, a^{2} b^{3} x^{2} + i \, a^{3} b^{2}\right )} \left (\frac {d^{4} e^{6}}{b^{9}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x} d e - {\left (i \, b^{3} x^{2} + i \, a b^{2}\right )} \left (\frac {d^{4} e^{6}}{b^{9}}\right )^{\frac {1}{4}}}{b x^{2} + a}\right ) + 5 \, {\left (-i \, a b^{4} x^{4} - 2 i \, a^{2} b^{3} x^{2} - i \, a^{3} b^{2}\right )} \left (\frac {d^{4} e^{6}}{b^{9}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x} d e - {\left (-i \, b^{3} x^{2} - i \, a b^{2}\right )} \left (\frac {d^{4} e^{6}}{b^{9}}\right )^{\frac {1}{4}}}{b x^{2} + a}\right )}{10 \, {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 67.63 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.78 \[ \int \frac {(e x)^{3/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\frac {c e^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right )}{2 a^{\frac {9}{4}} \sqrt [4]{1 + \frac {b x^{2}}{a}} \Gamma \left (\frac {9}{4}\right ) + 2 a^{\frac {5}{4}} b x^{2} \sqrt [4]{1 + \frac {b x^{2}}{a}} \Gamma \left (\frac {9}{4}\right )} + \frac {d e^{\frac {3}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {9}{4}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {9}{4}} \Gamma \left (\frac {13}{4}\right )} \]
[In]
[Out]
\[ \int \frac {(e x)^{3/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )} \left (e x\right )^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{\frac {9}{4}}} \,d x } \]
[In]
[Out]
\[ \int \frac {(e x)^{3/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )} \left (e x\right )^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{\frac {9}{4}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(e x)^{3/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx=\int \frac {{\left (e\,x\right )}^{3/2}\,\left (d\,x^2+c\right )}{{\left (b\,x^2+a\right )}^{9/4}} \,d x \]
[In]
[Out]