Optimal. Leaf size=453 \[ \frac{x \left (a+b x^2\right ) \left (16 a^2 d^2-16 a b c d+b^2 c^2\right )}{5 b^4 e \left (c+d x^2\right ) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{\sqrt{c} \left (a+b x^2\right ) \left (16 a^2 d^2-16 a b c d+b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{5 b^4 \sqrt{d} e \left (c+d x^2\right ) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{c^{3/2} \left (a+b x^2\right ) (7 b c-8 a d) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{5 b^3 \sqrt{d} e \left (c+d x^2\right ) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{6 d x^3 \left (a+b x^2\right )}{5 b^2 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{x \left (a+b x^2\right ) (7 b c-8 a d)}{5 b^3 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{x^3 \left (c+d x^2\right )}{b e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}} \]
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Rubi [A] time = 0.675178, antiderivative size = 453, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {6719, 467, 581, 582, 531, 418, 492, 411} \[ \frac{x \left (a+b x^2\right ) \left (16 a^2 d^2-16 a b c d+b^2 c^2\right )}{5 b^4 e \left (c+d x^2\right ) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{\sqrt{c} \left (a+b x^2\right ) \left (16 a^2 d^2-16 a b c d+b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{5 b^4 \sqrt{d} e \left (c+d x^2\right ) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{c^{3/2} \left (a+b x^2\right ) (7 b c-8 a d) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{5 b^3 \sqrt{d} e \left (c+d x^2\right ) \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{6 d x^3 \left (a+b x^2\right )}{5 b^2 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{x \left (a+b x^2\right ) (7 b c-8 a d)}{5 b^3 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{x^3 \left (c+d x^2\right )}{b e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}} \]
Antiderivative was successfully verified.
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Rule 6719
Rule 467
Rule 581
Rule 582
Rule 531
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{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{x^4 \left (c+d x^2\right )^{3/2}}{\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{x^3 \left (c+d x^2\right )}{b e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{\sqrt{a+b x^2} \int \frac{x^2 \sqrt{c+d x^2} \left (3 c+6 d x^2\right )}{\sqrt{a+b x^2}} \, dx}{b e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}}\\ &=\frac{6 d x^3 \left (a+b x^2\right )}{5 b^2 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{x^3 \left (c+d x^2\right )}{b e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{\sqrt{a+b x^2} \int \frac{x^2 \left (3 c (5 b c-6 a d)+3 d (7 b c-8 a d) x^2\right )}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{5 b^2 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}}\\ &=\frac{(7 b c-8 a d) x \left (a+b x^2\right )}{5 b^3 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{6 d x^3 \left (a+b x^2\right )}{5 b^2 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{x^3 \left (c+d x^2\right )}{b e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{\sqrt{a+b x^2} \int \frac{3 a c d (7 b c-8 a d)-3 d \left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{15 b^3 d e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}}\\ &=\frac{(7 b c-8 a d) x \left (a+b x^2\right )}{5 b^3 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{6 d x^3 \left (a+b x^2\right )}{5 b^2 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{x^3 \left (c+d x^2\right )}{b e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{\left (a c (7 b c-8 a d) \sqrt{a+b x^2}\right ) \int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{5 b^3 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}}+\frac{\left (\left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) \sqrt{a+b x^2}\right ) \int \frac{x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{5 b^3 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}}\\ &=\frac{(7 b c-8 a d) x \left (a+b x^2\right )}{5 b^3 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{6 d x^3 \left (a+b x^2\right )}{5 b^2 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{\left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) x \left (a+b x^2\right )}{5 b^4 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}-\frac{x^3 \left (c+d x^2\right )}{b e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{c^{3/2} (7 b c-8 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 )}{5 b^3 \sqrt{d} 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 \left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) \sqrt{a+b x^2}\right ) \int \frac{\sqrt{a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{5 b^4 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}}\\ &=\frac{(7 b c-8 a d) x \left (a+b x^2\right )}{5 b^3 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{6 d x^3 \left (a+b x^2\right )}{5 b^2 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}+\frac{\left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) x \left (a+b x^2\right )}{5 b^4 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}-\frac{x^3 \left (c+d x^2\right )}{b e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{\sqrt{c} \left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) \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 )}{5 b^4 \sqrt{d} 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} (7 b c-8 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 )}{5 b^3 \sqrt{d} 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 = 0.504485, size = 271, normalized size = 0.6 \[ \frac{\sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (8 a^2 d^2-9 a b c d+b^2 c^2\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (16 a^2 d^2-16 a b c d+b^2 c^2\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+d x \sqrt{\frac{b}{a}} \left (c+d x^2\right ) \left (-8 a^2 d+a b \left (7 c-2 d x^2\right )+b^2 x^2 \left (2 c+d x^2\right )\right )\right )}{5 b^3 d e^2 \sqrt{\frac{b}{a}} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 935, normalized size = 2.1 \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{x^{4}}{\left (\frac{{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac{3}{2}}}\,{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^{8} + 2 \, c d x^{6} + c^{2} x^{4}\right )} \sqrt{\frac{b e x^{2} + a e}{d x^{2} + c}}}{b^{2} e^{2} x^{4} + 2 \, a b e^{2} x^{2} + a^{2} e^{2}}, 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{x^{4}}{\left (\frac{{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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