Optimal. Leaf size=380 \[ \frac {c^{3/2} \sqrt {d} \left (a+b x^2\right ) F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{a^2 e \left (c+d x^2\right ) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {\left (a+b x^2\right ) (2 b c-a d)}{a^2 b e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {d x \left (a+b x^2\right ) (2 b c-a d)}{a^2 b e \left (c+d x^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {\sqrt {c} \sqrt {d} \left (a+b x^2\right ) (2 b c-a d) E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{a^2 b e \left (c+d x^2\right ) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {b c-a d}{a b e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \]
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Rubi [A] time = 0.47, antiderivative size = 380, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {6719, 468, 583, 531, 418, 492, 411} \[ \frac {c^{3/2} \sqrt {d} \left (a+b x^2\right ) F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{a^2 e \left (c+d x^2\right ) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {\left (a+b x^2\right ) (2 b c-a d)}{a^2 b e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {d x \left (a+b x^2\right ) (2 b c-a d)}{a^2 b e \left (c+d x^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {\sqrt {c} \sqrt {d} \left (a+b x^2\right ) (2 b c-a d) E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{a^2 b e \left (c+d x^2\right ) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {b c-a d}{a b e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \]
Antiderivative was successfully verified.
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Rule 411
Rule 418
Rule 468
Rule 492
Rule 531
Rule 583
Rule 6719
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx &=\frac {\sqrt {a+b x^2} \int \frac {\left (c+d x^2\right )^{3/2}}{x^2 \left (a+b x^2\right )^{3/2}} \, dx}{e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}}\\ &=\frac {b c-a d}{a b e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {\sqrt {a+b x^2} \int \frac {-c (2 b c-a d)-b c d x^2}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{a b e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}}\\ &=\frac {b c-a d}{a b e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {(2 b c-a d) \left (a+b x^2\right )}{a^2 b e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {\sqrt {a+b x^2} \int \frac {a b c^2 d+b c d (2 b c-a d) x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{a^2 b c e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}}\\ &=\frac {b c-a d}{a b e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {(2 b c-a d) \left (a+b x^2\right )}{a^2 b e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {\left (c d \sqrt {a+b x^2}\right ) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{a e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}}+\frac {\left (d (2 b c-a d) \sqrt {a+b x^2}\right ) \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{a^2 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}}\\ &=\frac {b c-a d}{a b e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {(2 b c-a d) \left (a+b x^2\right )}{a^2 b e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {d (2 b c-a d) x \left (a+b x^2\right )}{a^2 b e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}+\frac {c^{3/2} \sqrt {d} \left (a+b x^2\right ) F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{a^2 e \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}-\frac {\left (c d (2 b c-a d) \sqrt {a+b x^2}\right ) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{a^2 b e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}}\\ &=\frac {b c-a d}{a b e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {(2 b c-a d) \left (a+b x^2\right )}{a^2 b e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {d (2 b c-a d) x \left (a+b x^2\right )}{a^2 b e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}-\frac {\sqrt {c} \sqrt {d} (2 b c-a d) \left (a+b x^2\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{a^2 b e \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}+\frac {c^{3/2} \sqrt {d} \left (a+b x^2\right ) F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{a^2 e \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}\\ \end {align*}
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Mathematica [C] time = 0.37, size = 223, normalized size = 0.59 \[ \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (-\sqrt {\frac {b}{a}} \left (c+d x^2\right ) \left (a c-a d x^2+2 b c x^2\right )-2 i c x \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} (a d-b c) F\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i c x \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} (a d-2 b c) E\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )\right )}{a^2 e^2 x \sqrt {\frac {b}{a}} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d^{2} x^{4} + 2 \, c d x^{2} + c^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{b^{2} e^{2} x^{6} + 2 \, a b e^{2} x^{4} + a^{2} e^{2} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 650, normalized size = 1.71 \[ -\frac {\left (b \,x^{2}+a \right ) \left (-\sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, a \,d^{2} x^{4}+\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, b c d \,x^{4}+\sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, b c d \,x^{4}+\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a c d \,x^{2}-\sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, a c d \,x^{2}+\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a c d x \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )-2 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, a c d x \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )+\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, b \,c^{2} x^{2}+\sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, b \,c^{2} x^{2}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, b \,c^{2} x \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )+2 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, b \,c^{2} x \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )+\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a \,c^{2}\right )}{\left (\frac {\left (b \,x^{2}+a \right ) e}{d \,x^{2}+c}\right )^{\frac {3}{2}} \left (d \,x^{2}+c \right )^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^2\,{\left (\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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