Integrand size = 28, antiderivative size = 318 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx=-\frac {2 A \sqrt {x}}{b \sqrt {b x^2+c x^4}}+\frac {(b B-3 A c) x^{5/2}}{b^2 \sqrt {b x^2+c x^4}}-\frac {(b B-3 A c) x^{3/2} \left (b+c x^2\right )}{b^2 \sqrt {c} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}+\frac {(b B-3 A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{b^{7/4} c^{3/4} \sqrt {b x^2+c x^4}}-\frac {(b B-3 A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 b^{7/4} c^{3/4} \sqrt {b x^2+c x^4}} \]
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Time = 0.25 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2063, 2048, 2057, 335, 311, 226, 1210} \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx=-\frac {x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (b B-3 A c) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 b^{7/4} c^{3/4} \sqrt {b x^2+c x^4}}+\frac {x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (b B-3 A c) E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{b^{7/4} c^{3/4} \sqrt {b x^2+c x^4}}+\frac {x^{5/2} (b B-3 A c)}{b^2 \sqrt {b x^2+c x^4}}-\frac {x^{3/2} \left (b+c x^2\right ) (b B-3 A c)}{b^2 \sqrt {c} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {2 A \sqrt {x}}{b \sqrt {b x^2+c x^4}} \]
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Rule 226
Rule 311
Rule 335
Rule 1210
Rule 2048
Rule 2057
Rule 2063
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A \sqrt {x}}{b \sqrt {b x^2+c x^4}}-\frac {\left (2 \left (-\frac {b B}{2}+\frac {3 A c}{2}\right )\right ) \int \frac {x^{7/2}}{\left (b x^2+c x^4\right )^{3/2}} \, dx}{b} \\ & = -\frac {2 A \sqrt {x}}{b \sqrt {b x^2+c x^4}}+\frac {(b B-3 A c) x^{5/2}}{b^2 \sqrt {b x^2+c x^4}}-\frac {(b B-3 A c) \int \frac {x^{3/2}}{\sqrt {b x^2+c x^4}} \, dx}{2 b^2} \\ & = -\frac {2 A \sqrt {x}}{b \sqrt {b x^2+c x^4}}+\frac {(b B-3 A c) x^{5/2}}{b^2 \sqrt {b x^2+c x^4}}-\frac {\left ((b B-3 A c) x \sqrt {b+c x^2}\right ) \int \frac {\sqrt {x}}{\sqrt {b+c x^2}} \, dx}{2 b^2 \sqrt {b x^2+c x^4}} \\ & = -\frac {2 A \sqrt {x}}{b \sqrt {b x^2+c x^4}}+\frac {(b B-3 A c) x^{5/2}}{b^2 \sqrt {b x^2+c x^4}}-\frac {\left ((b B-3 A c) x \sqrt {b+c x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{b^2 \sqrt {b x^2+c x^4}} \\ & = -\frac {2 A \sqrt {x}}{b \sqrt {b x^2+c x^4}}+\frac {(b B-3 A c) x^{5/2}}{b^2 \sqrt {b x^2+c x^4}}-\frac {\left ((b B-3 A c) x \sqrt {b+c x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{b^{3/2} \sqrt {c} \sqrt {b x^2+c x^4}}+\frac {\left ((b B-3 A c) x \sqrt {b+c x^2}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {b}}}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{b^{3/2} \sqrt {c} \sqrt {b x^2+c x^4}} \\ & = -\frac {2 A \sqrt {x}}{b \sqrt {b x^2+c x^4}}+\frac {(b B-3 A c) x^{5/2}}{b^2 \sqrt {b x^2+c x^4}}-\frac {(b B-3 A c) x^{3/2} \left (b+c x^2\right )}{b^2 \sqrt {c} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}+\frac {(b B-3 A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{b^{7/4} c^{3/4} \sqrt {b x^2+c x^4}}-\frac {(b B-3 A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 b^{7/4} c^{3/4} \sqrt {b x^2+c x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.06 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.24 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx=\frac {2 \sqrt {x} \left (-3 A b+(b B-3 A c) x^2 \sqrt {1+\frac {c x^2}{b}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\frac {c x^2}{b}\right )\right )}{3 b^2 \sqrt {x^2 \left (b+c x^2\right )}} \]
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Time = 2.28 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.23
method | result | size |
default | \(\frac {x^{\frac {5}{2}} \left (c \,x^{2}+b \right ) \left (6 A b c \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-3 A b c \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-2 B \,b^{2} \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )+B \,b^{2} \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-6 A \,c^{2} x^{2}+2 B b c \,x^{2}-4 A b c \right )}{2 \left (x^{4} c +b \,x^{2}\right )^{\frac {3}{2}} c \,b^{2}}\) | \(392\) |
risch | \(-\frac {2 A \left (c \,x^{2}+b \right ) \sqrt {x}}{b^{2} \sqrt {x^{2} \left (c \,x^{2}+b \right )}}+\frac {\left (\frac {A \sqrt {-b c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \left (-\frac {2 \sqrt {-b c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-b c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{\sqrt {c \,x^{3}+b x}}-b \left (A c -B b \right ) \left (\frac {x^{2}}{b \sqrt {\left (x^{2}+\frac {b}{c}\right ) c x}}-\frac {\sqrt {-b c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \left (-\frac {2 \sqrt {-b c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-b c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{2 b c \sqrt {c \,x^{3}+b x}}\right )\right ) \sqrt {x}\, \sqrt {x \left (c \,x^{2}+b \right )}}{b^{2} \sqrt {x^{2} \left (c \,x^{2}+b \right )}}\) | \(412\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.36 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx=\frac {{\left ({\left (B b c - 3 \, A c^{2}\right )} x^{4} + {\left (B b^{2} - 3 \, A b c\right )} x^{2}\right )} \sqrt {c} {\rm weierstrassZeta}\left (-\frac {4 \, b}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right )\right ) - \sqrt {c x^{4} + b x^{2}} {\left (2 \, A b c - {\left (B b c - 3 \, A c^{2}\right )} x^{2}\right )} \sqrt {x}}{b^{2} c^{2} x^{4} + b^{3} c x^{2}} \]
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\[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {x^{\frac {3}{2}} \left (A + B x^{2}\right )}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} x^{\frac {3}{2}}}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} x^{\frac {3}{2}}}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {x^{3/2}\,\left (B\,x^2+A\right )}{{\left (c\,x^4+b\,x^2\right )}^{3/2}} \,d x \]
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