Integrand size = 28, antiderivative size = 184 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{5/2} \sqrt {c+d x^2}} \, dx=-\frac {2 a^2 \sqrt {c+d x^2}}{3 c e (e x)^{3/2}}+\frac {2 b^2 \sqrt {e x} \sqrt {c+d x^2}}{3 d e^3}-\frac {\left (b^2 c^2-6 a b c d+a^2 d^2\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{3 c^{5/4} d^{5/4} e^{5/2} \sqrt {c+d x^2}} \]
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Time = 0.10 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {473, 470, 335, 226} \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{5/2} \sqrt {c+d x^2}} \, dx=-\frac {\left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (a^2 d^2-6 a b c d+b^2 c^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{3 c^{5/4} d^{5/4} e^{5/2} \sqrt {c+d x^2}}-\frac {2 a^2 \sqrt {c+d x^2}}{3 c e (e x)^{3/2}}+\frac {2 b^2 \sqrt {e x} \sqrt {c+d x^2}}{3 d e^3} \]
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Rule 226
Rule 335
Rule 470
Rule 473
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a^2 \sqrt {c+d x^2}}{3 c e (e x)^{3/2}}+\frac {2 \int \frac {\frac {1}{2} a (6 b c-a d)+\frac {3}{2} b^2 c x^2}{\sqrt {e x} \sqrt {c+d x^2}} \, dx}{3 c e^2} \\ & = -\frac {2 a^2 \sqrt {c+d x^2}}{3 c e (e x)^{3/2}}+\frac {2 b^2 \sqrt {e x} \sqrt {c+d x^2}}{3 d e^3}-\frac {\left (b^2 c^2-6 a b c d+a^2 d^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {c+d x^2}} \, dx}{3 c d e^2} \\ & = -\frac {2 a^2 \sqrt {c+d x^2}}{3 c e (e x)^{3/2}}+\frac {2 b^2 \sqrt {e x} \sqrt {c+d x^2}}{3 d e^3}-\frac {\left (2 \left (b^2 c^2-6 a b c d+a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{3 c d e^3} \\ & = -\frac {2 a^2 \sqrt {c+d x^2}}{3 c e (e x)^{3/2}}+\frac {2 b^2 \sqrt {e x} \sqrt {c+d x^2}}{3 d e^3}-\frac {\left (b^2 c^2-6 a b c d+a^2 d^2\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{3 c^{5/4} d^{5/4} e^{5/2} \sqrt {c+d x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 11.15 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{5/2} \sqrt {c+d x^2}} \, dx=\frac {x \left (2 \sqrt {\frac {i \sqrt {c}}{\sqrt {d}}} \left (-a^2 d+b^2 c x^2\right ) \left (c+d x^2\right )-2 i \left (b^2 c^2-6 a b c d+a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x^{5/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right ),-1\right )\right )}{3 c \sqrt {\frac {i \sqrt {c}}{\sqrt {d}}} d (e x)^{5/2} \sqrt {c+d x^2}} \]
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Time = 3.18 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.13
method | result | size |
risch | \(-\frac {2 \sqrt {d \,x^{2}+c}\, \left (-b^{2} c \,x^{2}+a^{2} d \right )}{3 d c x \,e^{2} \sqrt {e x}}-\frac {\left (a^{2} d^{2}-6 a b c d +b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {e x \left (d \,x^{2}+c \right )}}{3 c \,d^{2} \sqrt {d e \,x^{3}+c e x}\, e^{2} \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(208\) |
elliptic | \(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \left (-\frac {2 a^{2} \sqrt {d e \,x^{3}+c e x}}{3 e^{3} c \,x^{2}}+\frac {2 b^{2} \sqrt {d e \,x^{3}+c e x}}{3 e^{3} d}+\frac {\left (\frac {2 a b}{e^{2}}-\frac {d \,a^{2}}{3 c \,e^{2}}-\frac {b^{2} c}{3 e^{2} d}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{\sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(221\) |
default | \(-\frac {\sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-c d}\, a^{2} d^{2} x -6 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-c d}\, a b c d x +\sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-c d}\, b^{2} c^{2} x -2 b^{2} c \,d^{2} x^{4}+2 a^{2} d^{3} x^{2}-2 b^{2} c^{2} d \,x^{2}+2 c \,a^{2} d^{2}}{3 \sqrt {d \,x^{2}+c}\, x c \,e^{2} \sqrt {e x}\, d^{2}}\) | \(352\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.48 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{5/2} \sqrt {c+d x^2}} \, dx=-\frac {2 \, {\left ({\left (b^{2} c^{2} - 6 \, a b c d + a^{2} d^{2}\right )} \sqrt {d e} x^{2} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) - {\left (b^{2} c d x^{2} - a^{2} d^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}\right )}}{3 \, c d^{2} e^{3} x^{2}} \]
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Result contains complex when optimal does not.
Time = 7.71 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{5/2} \sqrt {c+d x^2}} \, dx=\frac {a^{2} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {c} e^{\frac {5}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} + \frac {a b \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{\sqrt {c} e^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right )} + \frac {b^{2} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {c} e^{\frac {5}{2}} \Gamma \left (\frac {9}{4}\right )} \]
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\[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{5/2} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2}}{\sqrt {d x^{2} + c} \left (e x\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{5/2} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2}}{\sqrt {d x^{2} + c} \left (e x\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{5/2} \sqrt {c+d x^2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2}{{\left (e\,x\right )}^{5/2}\,\sqrt {d\,x^2+c}} \,d x \]
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