Integrand size = 28, antiderivative size = 403 \[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {(b c-a d)^2 (e x)^{3/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}-\frac {(b c-a d) (3 b c+a d) (e x)^{3/2}}{2 c^2 d^2 e \sqrt {c+d x^2}}+\frac {\left (7 b^2 c^2-2 a b c d-a^2 d^2\right ) \sqrt {e x} \sqrt {c+d x^2}}{2 c^2 d^{5/2} \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {\left (7 b^2 c^2-2 a b c d-a^2 d^2\right ) \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{2 c^{7/4} d^{11/4} \sqrt {c+d x^2}}+\frac {\left (7 b^2 c^2-2 a b c d-a^2 d^2\right ) \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 c^{7/4} d^{11/4} \sqrt {c+d x^2}} \]
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Time = 0.25 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {474, 468, 335, 311, 226, 1210} \[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {\sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (-a^2 d^2-2 a b c d+7 b^2 c^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 c^{7/4} d^{11/4} \sqrt {c+d x^2}}-\frac {\sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (-a^2 d^2-2 a b c d+7 b^2 c^2\right ) E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{2 c^{7/4} d^{11/4} \sqrt {c+d x^2}}+\frac {\sqrt {e x} \sqrt {c+d x^2} \left (-a^2 d^2-2 a b c d+7 b^2 c^2\right )}{2 c^2 d^{5/2} \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {(e x)^{3/2} (a d+3 b c) (b c-a d)}{2 c^2 d^2 e \sqrt {c+d x^2}}+\frac {(e x)^{3/2} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}} \]
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Rule 226
Rule 311
Rule 335
Rule 468
Rule 474
Rule 1210
Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d)^2 (e x)^{3/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}-\frac {\int \frac {\sqrt {e x} \left (-\frac {3}{2} \left (2 a^2 d^2-(b c-a d)^2\right )-3 b^2 c d x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx}{3 c d^2} \\ & = \frac {(b c-a d)^2 (e x)^{3/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}-\frac {(b c-a d) (3 b c+a d) (e x)^{3/2}}{2 c^2 d^2 e \sqrt {c+d x^2}}+\frac {\left (7 b^2 c^2-2 a b c d-a^2 d^2\right ) \int \frac {\sqrt {e x}}{\sqrt {c+d x^2}} \, dx}{4 c^2 d^2} \\ & = \frac {(b c-a d)^2 (e x)^{3/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}-\frac {(b c-a d) (3 b c+a d) (e x)^{3/2}}{2 c^2 d^2 e \sqrt {c+d x^2}}+\frac {\left (7 b^2 c^2-2 a b c d-a^2 d^2\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 c^2 d^2 e} \\ & = \frac {(b c-a d)^2 (e x)^{3/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}-\frac {(b c-a d) (3 b c+a d) (e x)^{3/2}}{2 c^2 d^2 e \sqrt {c+d x^2}}+\frac {\left (7 b^2 c^2-2 a b c d-a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 c^{3/2} d^{5/2}}-\frac {\left (7 b^2 c^2-2 a b c d-a^2 d^2\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {d} x^2}{\sqrt {c} e}}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 c^{3/2} d^{5/2}} \\ & = \frac {(b c-a d)^2 (e x)^{3/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}-\frac {(b c-a d) (3 b c+a d) (e x)^{3/2}}{2 c^2 d^2 e \sqrt {c+d x^2}}+\frac {\left (7 b^2 c^2-2 a b c d-a^2 d^2\right ) \sqrt {e x} \sqrt {c+d x^2}}{2 c^2 d^{5/2} \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {\left (7 b^2 c^2-2 a b c d-a^2 d^2\right ) \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{2 c^{7/4} d^{11/4} \sqrt {c+d x^2}}+\frac {\left (7 b^2 c^2-2 a b c d-a^2 d^2\right ) \sqrt {e} \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 )}{4 c^{7/4} d^{11/4} \sqrt {c+d x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 11.13 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.33 \[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {\sqrt {e x} \left (-\left ((b c-a d) x \left (a d \left (5 c+3 d x^2\right )+b c \left (7 c+9 d x^2\right )\right )\right )+3 \left (7 b^2 c^2-2 a b c d-a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x \left (c+d x^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-\frac {c}{d x^2}\right )\right )}{6 c^2 d^2 \left (c+d x^2\right )^{3/2}} \]
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Time = 3.16 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.84
method | result | size |
elliptic | \(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \sqrt {e x}\, \left (\frac {x \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {d e \,x^{3}+c e x}}{3 c \,d^{4} \left (x^{2}+\frac {c}{d}\right )^{2}}+\frac {e \,x^{2} \left (a^{2} d^{2}+2 a b c d -3 b^{2} c^{2}\right )}{2 d^{2} c^{2} \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}+\frac {\left (\frac {b^{2} e}{d^{2}}-\frac {e \left (a^{2} d^{2}+2 a b c d -3 b^{2} c^{2}\right )}{4 d^{2} c^{2}}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{e x \sqrt {d \,x^{2}+c}}\) | \(337\) |
default | \(\text {Expression too large to display}\) | \(1176\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.58 \[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=-\frac {3 \, {\left (7 \, b^{2} c^{4} - 2 \, a b c^{3} d - a^{2} c^{2} d^{2} + {\left (7 \, b^{2} c^{2} d^{2} - 2 \, a b c d^{3} - a^{2} d^{4}\right )} x^{4} + 2 \, {\left (7 \, b^{2} c^{3} d - 2 \, a b c^{2} d^{2} - a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {d e} {\rm weierstrassZeta}\left (-\frac {4 \, c}{d}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right )\right ) + {\left (3 \, {\left (3 \, b^{2} c^{2} d^{2} - 2 \, a b c d^{3} - a^{2} d^{4}\right )} x^{3} + {\left (7 \, b^{2} c^{3} d - 2 \, a b c^{2} d^{2} - 5 \, a^{2} c d^{3}\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{6 \, {\left (c^{2} d^{5} x^{4} + 2 \, c^{3} d^{4} x^{2} + c^{4} d^{3}\right )}} \]
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\[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {e x} \left (a + b x^{2}\right )^{2}}{\left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \sqrt {e x}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \sqrt {e x}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {e\,x}\,{\left (b\,x^2+a\right )}^2}{{\left (d\,x^2+c\right )}^{5/2}} \,d x \]
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