Optimal. Leaf size=444 \[ \frac{\left (a+b x^2\right ) (8 b c-7 a d)}{3 a^3 e x \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{\left (a+b x^2\right ) (4 b c-3 a d)}{3 a^2 b e x^3 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{d x \left (a+b x^2\right ) (8 b c-7 a d)}{3 a^3 e \left (c+d x^2\right ) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{\sqrt{c} \sqrt{d} \left (a+b x^2\right ) (4 b c-3 a d) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a^3 e \left (c+d x^2\right ) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{\sqrt{c} \sqrt{d} \left (a+b x^2\right ) (8 b c-7 a d) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a^3 e \left (c+d x^2\right ) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{b c-a d}{a b e x^3 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}} \]
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Rubi [A] time = 0.649331, antiderivative size = 444, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{\left (a+b x^2\right ) (8 b c-7 a d)}{3 a^3 e x \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{\left (a+b x^2\right ) (4 b c-3 a d)}{3 a^2 b e x^3 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{d x \left (a+b x^2\right ) (8 b c-7 a d)}{3 a^3 e \left (c+d x^2\right ) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{\sqrt{c} \sqrt{d} \left (a+b x^2\right ) (4 b c-3 a d) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a^3 e \left (c+d x^2\right ) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{\sqrt{c} \sqrt{d} \left (a+b x^2\right ) (8 b c-7 a d) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a^3 e \left (c+d x^2\right ) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{b c-a d}{a b e x^3 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}} \]
Antiderivative was successfully verified.
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Rule 6719
Rule 468
Rule 583
Rule 531
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{1}{x^4 \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^4 \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^3 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{\sqrt{a+b x^2} \int \frac{-c (4 b c-3 a d)-d (3 b c-2 a d) x^2}{x^4 \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^3 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{(4 b c-3 a d) \left (a+b x^2\right )}{3 a^2 b e x^3 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{\sqrt{a+b x^2} \int \frac{-b c^2 (8 b c-7 a d)-b c d (4 b c-3 a d) x^2}{x^2 \sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{3 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^3 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{(4 b c-3 a d) \left (a+b x^2\right )}{3 a^2 b e x^3 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{(8 b c-7 a d) \left (a+b x^2\right )}{3 a^3 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 (4 b c-3 a d)+b^2 c^2 d (8 b c-7 a d) x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{3 a^3 b c^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^3 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{(4 b c-3 a d) \left (a+b x^2\right )}{3 a^2 b e x^3 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{(8 b c-7 a d) \left (a+b x^2\right )}{3 a^3 e x \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{\left (b d (8 b c-7 a d) \sqrt{a+b x^2}\right ) \int \frac{x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{3 a^3 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}}-\frac{\left (d (4 b c-3 a d) \sqrt{a+b x^2}\right ) \int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{3 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^3 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{(4 b c-3 a d) \left (a+b x^2\right )}{3 a^2 b e x^3 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{(8 b c-7 a d) \left (a+b x^2\right )}{3 a^3 e x \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{d (8 b c-7 a d) x \left (a+b x^2\right )}{3 a^3 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} (4 b c-3 a 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 )}{3 a^3 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 (8 b c-7 a d) \sqrt{a+b x^2}\right ) \int \frac{\sqrt{a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{3 a^3 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^3 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{(4 b c-3 a d) \left (a+b x^2\right )}{3 a^2 b e x^3 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{(8 b c-7 a d) \left (a+b x^2\right )}{3 a^3 e x \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{d (8 b c-7 a d) x \left (a+b x^2\right )}{3 a^3 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} (8 b c-7 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 )}{3 a^3 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{\sqrt{c} \sqrt{d} (4 b c-3 a 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 )}{3 a^3 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*}
Mathematica [C] time = 0.465775, size = 266, normalized size = 0.6 \[ \frac{\sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (-i x^3 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (3 a^2 d^2-11 a b c d+8 b^2 c^2\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )-\sqrt{\frac{b}{a}} \left (c+d x^2\right ) \left (a^2 \left (c+4 d x^2\right )+a b \left (7 d x^4-4 c x^2\right )-8 b^2 c x^4\right )-i b c x^3 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (7 a d-8 b c) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )\right )}{3 a^3 e^2 x^3 \sqrt{\frac{b}{a}} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 866, normalized size = 2. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (\frac{{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac{3}{2}} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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^{8} + 2 \, a b e^{2} x^{6} + a^{2} e^{2} x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (\frac{{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac{3}{2}} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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