Integrand size = 26, antiderivative size = 336 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^4 \sqrt {c+d x^2}} \, dx=\frac {\left (3 b^2 c^2+7 a b c d-2 a^2 d^2\right ) x \sqrt {a+b x^2}}{3 c^2 \sqrt {c+d x^2}}-\frac {2 a (3 b c-a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c^2 x}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 c x^3}-\frac {\left (3 b^2 c^2+7 a b c d-2 a^2 d^2\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 c^{3/2} \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {b (9 b c-a d) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 \sqrt {c} \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]
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Time = 0.20 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {485, 594, 545, 429, 506, 422} \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^4 \sqrt {c+d x^2}} \, dx=-\frac {\sqrt {a+b x^2} \left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 c^{3/2} \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {x \sqrt {a+b x^2} \left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right )}{3 c^2 \sqrt {c+d x^2}}+\frac {b \sqrt {a+b x^2} (9 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 \sqrt {c} \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {2 a \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-a d)}{3 c^2 x}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 c x^3} \]
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Rule 422
Rule 429
Rule 485
Rule 506
Rule 545
Rule 594
Rubi steps \begin{align*} \text {integral}& = -\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 c x^3}+\frac {\int \frac {\sqrt {a+b x^2} \left (2 a (3 b c-a d)+b (3 b c+a d) x^2\right )}{x^2 \sqrt {c+d x^2}} \, dx}{3 c} \\ & = -\frac {2 a (3 b c-a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c^2 x}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 c x^3}+\frac {\int \frac {a b c (9 b c-a d)+b \left (3 b^2 c^2+7 a b c d-2 a^2 d^2\right ) x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 c^2} \\ & = -\frac {2 a (3 b c-a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c^2 x}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 c x^3}+\frac {(a b (9 b c-a d)) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 c}+\frac {\left (b \left (3 b^2 c^2+7 a b c d-2 a^2 d^2\right )\right ) \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 c^2} \\ & = \frac {\left (3 b^2 c^2+7 a b c d-2 a^2 d^2\right ) x \sqrt {a+b x^2}}{3 c^2 \sqrt {c+d x^2}}-\frac {2 a (3 b c-a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c^2 x}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 c x^3}+\frac {b (9 b c-a d) \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 \sqrt {c} \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {\left (3 b^2 c^2+7 a b c d-2 a^2 d^2\right ) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{3 c} \\ & = \frac {\left (3 b^2 c^2+7 a b c d-2 a^2 d^2\right ) x \sqrt {a+b x^2}}{3 c^2 \sqrt {c+d x^2}}-\frac {2 a (3 b c-a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c^2 x}-\frac {a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{3 c x^3}-\frac {\left (3 b^2 c^2+7 a b c d-2 a^2 d^2\right ) \sqrt {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 c^{3/2} \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {b (9 b c-a d) \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 \sqrt {c} \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 3.04 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^4 \sqrt {c+d x^2}} \, dx=\frac {a \sqrt {\frac {b}{a}} d \left (a+b x^2\right ) \left (c+d x^2\right ) \left (-a c-7 b c x^2+2 a d x^2\right )+i b c \left (-3 b^2 c^2-7 a b c d+2 a^2 d^2\right ) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i b c \left (-3 b^2 c^2+2 a b c d+a^2 d^2\right ) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 \sqrt {\frac {b}{a}} c^2 d x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \]
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Time = 5.73 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.08
method | result | size |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (-\frac {a^{2} \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{3 c \,x^{3}}+\frac {a \left (2 a d -7 b c \right ) \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{3 c^{2} x}+\frac {\left (3 a \,b^{2}-\frac {b d \,a^{2}}{3 c}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {\left (b^{3}-\frac {a b d \left (2 a d -7 b c \right )}{3 c^{2}}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(362\) |
risch | \(-\frac {a \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (-2 a d \,x^{2}+7 c b \,x^{2}+a c \right )}{3 c^{2} x^{3}}-\frac {b \left (\frac {a^{2} c d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {9 b \,c^{2} a \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {\left (2 a^{2} d^{2}-7 a b c d -3 b^{2} c^{2}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{3 c^{2} \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(418\) |
default | \(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (2 \sqrt {-\frac {b}{a}}\, a^{2} b \,d^{3} x^{6}-7 \sqrt {-\frac {b}{a}}\, a \,b^{2} c \,d^{2} x^{6}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} b c \,d^{2} x^{3}+2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{2} c^{2} d \,x^{3}-3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{3} c^{3} x^{3}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} b c \,d^{2} x^{3}+7 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{2} c^{2} d \,x^{3}+3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{3} c^{3} x^{3}+2 \sqrt {-\frac {b}{a}}\, a^{3} d^{3} x^{4}-6 \sqrt {-\frac {b}{a}}\, a^{2} b c \,d^{2} x^{4}-7 \sqrt {-\frac {b}{a}}\, a \,b^{2} c^{2} d \,x^{4}+\sqrt {-\frac {b}{a}}\, a^{3} c \,d^{2} x^{2}-8 \sqrt {-\frac {b}{a}}\, a^{2} b \,c^{2} d \,x^{2}-\sqrt {-\frac {b}{a}}\, a^{3} c^{2} d \right )}{3 \left (b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c \right ) c^{2} x^{3} \sqrt {-\frac {b}{a}}\, d}\) | \(583\) |
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\[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^4 \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{\sqrt {d x^{2} + c} x^{4}} \,d x } \]
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\[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^4 \sqrt {c+d x^2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {5}{2}}}{x^{4} \sqrt {c + d x^{2}}}\, dx \]
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\[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^4 \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{\sqrt {d x^{2} + c} x^{4}} \,d x } \]
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\[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^4 \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{\sqrt {d x^{2} + c} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^4 \sqrt {c+d x^2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}}{x^4\,\sqrt {d\,x^2+c}} \,d x \]
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