Integrand size = 28, antiderivative size = 288 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{5/2}} \, dx=-\frac {4 \left (3 b^2 c^2-11 a d (6 b c+7 a d)\right ) \sqrt {e x} \sqrt {c+d x^2}}{231 d e^3}-\frac {2 \left (3 b^2 c^2-11 a d (6 b c+7 a d)\right ) \sqrt {e x} \left (c+d x^2\right )^{3/2}}{231 c d e^3}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{3 c e (e x)^{3/2}}+\frac {2 b^2 \sqrt {e x} \left (c+d x^2\right )^{5/2}}{11 d e^3}-\frac {4 c^{3/4} \left (3 b^2 c^2-11 a d (6 b c+7 a d)\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{231 d^{5/4} e^{5/2} \sqrt {c+d x^2}} \]
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Time = 0.18 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {473, 470, 285, 335, 226} \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{5/2}} \, dx=-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{3 c e (e x)^{3/2}}-\frac {4 c^{3/4} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (3 b^2 c^2-11 a d (7 a d+6 b c)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{231 d^{5/4} e^{5/2} \sqrt {c+d x^2}}-\frac {2 \sqrt {e x} \left (c+d x^2\right )^{3/2} \left (3 b^2 c^2-11 a d (7 a d+6 b c)\right )}{231 c d e^3}-\frac {4 \sqrt {e x} \sqrt {c+d x^2} \left (3 b^2 c^2-11 a d (7 a d+6 b c)\right )}{231 d e^3}+\frac {2 b^2 \sqrt {e x} \left (c+d x^2\right )^{5/2}}{11 d e^3} \]
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Rule 226
Rule 285
Rule 335
Rule 470
Rule 473
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{3 c e (e x)^{3/2}}+\frac {2 \int \frac {\left (\frac {1}{2} a (6 b c+7 a d)+\frac {3}{2} b^2 c x^2\right ) \left (c+d x^2\right )^{3/2}}{\sqrt {e x}} \, dx}{3 c e^2} \\ & = -\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{3 c e (e x)^{3/2}}+\frac {2 b^2 \sqrt {e x} \left (c+d x^2\right )^{5/2}}{11 d e^3}-\frac {\left (3 b^2 c^2-11 a d (6 b c+7 a d)\right ) \int \frac {\left (c+d x^2\right )^{3/2}}{\sqrt {e x}} \, dx}{33 c d e^2} \\ & = -\frac {2 \left (3 b^2 c^2-11 a d (6 b c+7 a d)\right ) \sqrt {e x} \left (c+d x^2\right )^{3/2}}{231 c d e^3}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{3 c e (e x)^{3/2}}+\frac {2 b^2 \sqrt {e x} \left (c+d x^2\right )^{5/2}}{11 d e^3}-\frac {\left (2 \left (3 b^2 c^2-11 a d (6 b c+7 a d)\right )\right ) \int \frac {\sqrt {c+d x^2}}{\sqrt {e x}} \, dx}{77 d e^2} \\ & = -\frac {4 \left (3 b^2 c^2-11 a d (6 b c+7 a d)\right ) \sqrt {e x} \sqrt {c+d x^2}}{231 d e^3}-\frac {2 \left (3 b^2 c^2-11 a d (6 b c+7 a d)\right ) \sqrt {e x} \left (c+d x^2\right )^{3/2}}{231 c d e^3}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{3 c e (e x)^{3/2}}+\frac {2 b^2 \sqrt {e x} \left (c+d x^2\right )^{5/2}}{11 d e^3}-\frac {\left (4 c \left (3 b^2 c^2-11 a d (6 b c+7 a d)\right )\right ) \int \frac {1}{\sqrt {e x} \sqrt {c+d x^2}} \, dx}{231 d e^2} \\ & = -\frac {4 \left (3 b^2 c^2-11 a d (6 b c+7 a d)\right ) \sqrt {e x} \sqrt {c+d x^2}}{231 d e^3}-\frac {2 \left (3 b^2 c^2-11 a d (6 b c+7 a d)\right ) \sqrt {e x} \left (c+d x^2\right )^{3/2}}{231 c d e^3}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{3 c e (e x)^{3/2}}+\frac {2 b^2 \sqrt {e x} \left (c+d x^2\right )^{5/2}}{11 d e^3}-\frac {\left (8 c \left (3 b^2 c^2-11 a d (6 b c+7 a d)\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{231 d e^3} \\ & = -\frac {4 \left (3 b^2 c^2-11 a d (6 b c+7 a d)\right ) \sqrt {e x} \sqrt {c+d x^2}}{231 d e^3}-\frac {2 \left (3 b^2 c^2-11 a d (6 b c+7 a d)\right ) \sqrt {e x} \left (c+d x^2\right )^{3/2}}{231 c d e^3}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{3 c e (e x)^{3/2}}+\frac {2 b^2 \sqrt {e x} \left (c+d x^2\right )^{5/2}}{11 d e^3}-\frac {4 c^{3/4} \left (3 b^2 c^2-11 a d (6 b c+7 a d)\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 )}{231 d^{5/4} e^{5/2} \sqrt {c+d x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 11.16 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.70 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{5/2}} \, dx=\frac {x^{5/2} \left (\frac {2 \left (c+d x^2\right ) \left (77 a^2 d \left (-c+d x^2\right )+66 a b d x^2 \left (3 c+d x^2\right )+3 b^2 x^2 \left (4 c^2+13 c d x^2+7 d^2 x^4\right )\right )}{d x^{3/2}}+\frac {8 i c \left (-3 b^2 c^2+66 a b c d+77 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right ),-1\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}} d}\right )}{231 (e x)^{5/2} \sqrt {c+d x^2}} \]
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Time = 3.08 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.90
method | result | size |
risch | \(-\frac {2 \sqrt {d \,x^{2}+c}\, \left (-21 b^{2} d^{2} x^{6}-66 a b \,d^{2} x^{4}-39 b^{2} c d \,x^{4}-77 a^{2} d^{2} x^{2}-198 a b c d \,x^{2}-12 b^{2} c^{2} x^{2}+77 a^{2} c d \right )}{231 d x \,e^{2} \sqrt {e x}}+\frac {4 c \left (77 a^{2} d^{2}+66 a b c d -3 b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {e x \left (d \,x^{2}+c \right )}}{231 d^{2} \sqrt {d e \,x^{3}+c e x}\, e^{2} \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(260\) |
elliptic | \(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \left (-\frac {2 c \,a^{2} \sqrt {d e \,x^{3}+c e x}}{3 e^{3} x^{2}}+\frac {2 b^{2} d \,x^{4} \sqrt {d e \,x^{3}+c e x}}{11 e^{3}}+\frac {2 \left (\frac {2 b d \left (a d +b c \right )}{e^{2}}-\frac {9 b^{2} d c}{11 e^{2}}\right ) x^{2} \sqrt {d e \,x^{3}+c e x}}{7 d e}+\frac {2 \left (\frac {a^{2} d^{2}+4 a b c d +b^{2} c^{2}}{e^{2}}-\frac {5 \left (\frac {2 b d \left (a d +b c \right )}{e^{2}}-\frac {9 b^{2} d c}{11 e^{2}}\right ) c}{7 d}\right ) \sqrt {d e \,x^{3}+c e x}}{3 d e}+\frac {\left (\frac {2 a c \left (a d +b c \right )}{e^{2}}-\frac {d c \,a^{2}}{3 e^{2}}-\frac {\left (\frac {a^{2} d^{2}+4 a b c d +b^{2} c^{2}}{e^{2}}-\frac {5 \left (\frac {2 b d \left (a d +b c \right )}{e^{2}}-\frac {9 b^{2} d c}{11 e^{2}}\right ) c}{7 d}\right ) c}{3 d}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{\sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(403\) |
default | \(\frac {\frac {2 b^{2} d^{4} x^{8}}{11}+\frac {4 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2} x}{3}+\frac {8 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d x}{7}-\frac {4 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3} x}{77}+\frac {4 a b \,d^{4} x^{6}}{7}+\frac {40 b^{2} c \,d^{3} x^{6}}{77}+\frac {2 a^{2} d^{4} x^{4}}{3}+\frac {16 c a b \,x^{4} d^{3}}{7}+\frac {34 b^{2} c^{2} d^{2} x^{4}}{77}+\frac {12 a b \,c^{2} d^{2} x^{2}}{7}+\frac {8 b^{2} c^{3} d \,x^{2}}{77}-\frac {2 a^{2} c^{2} d^{2}}{3}}{\sqrt {d \,x^{2}+c}\, x \,d^{2} e^{2} \sqrt {e x}}\) | \(415\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.51 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{5/2}} \, dx=-\frac {2 \, {\left (4 \, {\left (3 \, b^{2} c^{3} - 66 \, a b c^{2} d - 77 \, a^{2} c d^{2}\right )} \sqrt {d e} x^{2} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) - {\left (21 \, b^{2} d^{3} x^{6} - 77 \, a^{2} c d^{2} + 3 \, {\left (13 \, b^{2} c d^{2} + 22 \, a b d^{3}\right )} x^{4} + {\left (12 \, b^{2} c^{2} d + 198 \, a b c d^{2} + 77 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}\right )}}{231 \, d^{2} e^{3} x^{2}} \]
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Result contains complex when optimal does not.
Time = 16.30 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{5/2}} \, dx=\frac {a^{2} c^{\frac {3}{2}} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac {5}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} + \frac {a^{2} \sqrt {c} d \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right )} + \frac {a b c^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{e^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right )} + \frac {a b \sqrt {c} d x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{e^{\frac {5}{2}} \Gamma \left (\frac {9}{4}\right )} + \frac {b^{2} c^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac {5}{2}} \Gamma \left (\frac {9}{4}\right )} + \frac {b^{2} \sqrt {c} d x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac {5}{2}} \Gamma \left (\frac {13}{4}\right )} \]
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\[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}{\left (e x\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}{\left (e x\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{5/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2\,{\left (d\,x^2+c\right )}^{3/2}}{{\left (e\,x\right )}^{5/2}} \,d x \]
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