Optimal. Leaf size=482 \[ \frac{x \left (a^2 c^2+16 a b c+16 b^2\right ) \left (a c+a d x^2+b\right )}{5 a^4 d^2 \left (c+d x^2\right ) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}-\frac{\sqrt{c} \left (a^2 c^2+16 a b c+16 b^2\right ) \left (a c+a d x^2+b\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{5 a^4 d^{5/2} \left (c+d x^2\right ) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac{c^{3/2} (a c+8 b) \left (a c+a d x^2+b\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{5 a^3 d^{5/2} \left (c+d x^2\right ) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}-\frac{x (a c+8 b) \left (a c+a d x^2+b\right )}{5 a^3 d^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}+\frac{6 x^3 \left (a c+a d x^2+b\right )}{5 a^2 d \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}-\frac{x^3 \left (c+d x^2\right )}{a d \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}} \]
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Rubi [A] time = 0.906294, antiderivative size = 559, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {6722, 1975, 467, 581, 582, 531, 418, 492, 411} \[ \frac{x \left (a^2 c^2+16 a b c+16 b^2\right ) \sqrt{a c+a d x^2+b} \sqrt{a \left (c+d x^2\right )+b}}{5 a^4 d^2 \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}-\frac{\sqrt{c} \left (a^2 c^2+16 a b c+16 b^2\right ) \sqrt{a c+a d x^2+b} \sqrt{a \left (c+d x^2\right )+b} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{5 a^4 d^{5/2} \left (c+d x^2\right ) \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}} \sqrt{a+\frac{b}{c+d x^2}}}+\frac{c^{3/2} (a c+8 b) \sqrt{a c+a d x^2+b} \sqrt{a \left (c+d x^2\right )+b} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{5 a^3 d^{5/2} \left (c+d x^2\right ) \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}} \sqrt{a+\frac{b}{c+d x^2}}}-\frac{x (a c+8 b) \sqrt{a c+a d x^2+b} \sqrt{a \left (c+d x^2\right )+b}}{5 a^3 d^2 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{6 x^3 \sqrt{a c+a d x^2+b} \sqrt{a \left (c+d x^2\right )+b}}{5 a^2 d \sqrt{a+\frac{b}{c+d x^2}}}-\frac{x^3 \left (c+d x^2\right ) \sqrt{a \left (c+d x^2\right )+b}}{a d \sqrt{a c+a d x^2+b} \sqrt{a+\frac{b}{c+d x^2}}} \]
Antiderivative was successfully verified.
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Rule 6722
Rule 1975
Rule 467
Rule 581
Rule 582
Rule 531
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{x^4}{\left (a+\frac{b}{c+d x^2}\right )^{3/2}} \, dx &=\frac{\sqrt{b+a \left (c+d x^2\right )} \int \frac{x^4 \left (c+d x^2\right )^{3/2}}{\left (b+a \left (c+d x^2\right )\right )^{3/2}} \, dx}{\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=\frac{\sqrt{b+a \left (c+d x^2\right )} \int \frac{x^4 \left (c+d x^2\right )^{3/2}}{\left (b+a c+a d x^2\right )^{3/2}} \, dx}{\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=-\frac{x^3 \left (c+d x^2\right ) \sqrt{b+a \left (c+d x^2\right )}}{a d \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}+\frac{\sqrt{b+a \left (c+d x^2\right )} \int \frac{x^2 \sqrt{c+d x^2} \left (3 c+6 d x^2\right )}{\sqrt{b+a c+a d x^2}} \, dx}{a d \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=-\frac{x^3 \left (c+d x^2\right ) \sqrt{b+a \left (c+d x^2\right )}}{a d \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}+\frac{6 x^3 \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{5 a^2 d \sqrt{a+\frac{b}{c+d x^2}}}+\frac{\sqrt{b+a \left (c+d x^2\right )} \int \frac{x^2 \left (-3 c (6 b+a c) d-3 (8 b+a c) d^2 x^2\right )}{\sqrt{c+d x^2} \sqrt{b+a c+a d x^2}} \, dx}{5 a^2 d^2 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=-\frac{x^3 \left (c+d x^2\right ) \sqrt{b+a \left (c+d x^2\right )}}{a d \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}-\frac{(8 b+a c) x \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{5 a^3 d^2 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{6 x^3 \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{5 a^2 d \sqrt{a+\frac{b}{c+d x^2}}}-\frac{\sqrt{b+a \left (c+d x^2\right )} \int \frac{-3 c (b+a c) (8 b+a c) d^2-3 \left (16 b^2+16 a b c+a^2 c^2\right ) d^3 x^2}{\sqrt{c+d x^2} \sqrt{b+a c+a d x^2}} \, dx}{15 a^3 d^4 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=-\frac{x^3 \left (c+d x^2\right ) \sqrt{b+a \left (c+d x^2\right )}}{a d \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}-\frac{(8 b+a c) x \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{5 a^3 d^2 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{6 x^3 \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{5 a^2 d \sqrt{a+\frac{b}{c+d x^2}}}+\frac{\left (c (b+a c) (8 b+a c) \sqrt{b+a \left (c+d x^2\right )}\right ) \int \frac{1}{\sqrt{c+d x^2} \sqrt{b+a c+a d x^2}} \, dx}{5 a^3 d^2 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}+\frac{\left (\left (16 b^2+16 a b c+a^2 c^2\right ) \sqrt{b+a \left (c+d x^2\right )}\right ) \int \frac{x^2}{\sqrt{c+d x^2} \sqrt{b+a c+a d x^2}} \, dx}{5 a^3 d \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=-\frac{x^3 \left (c+d x^2\right ) \sqrt{b+a \left (c+d x^2\right )}}{a d \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}-\frac{(8 b+a c) x \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{5 a^3 d^2 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{6 x^3 \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{5 a^2 d \sqrt{a+\frac{b}{c+d x^2}}}+\frac{\left (16 b^2+16 a b c+a^2 c^2\right ) x \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{5 a^4 d^2 \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}+\frac{c^{3/2} (8 b+a c) \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{5 a^3 d^{5/2} \left (c+d x^2\right ) \sqrt{\frac{c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt{a+\frac{b}{c+d x^2}}}-\frac{\left (c \left (16 b^2+16 a b c+a^2 c^2\right ) \sqrt{b+a \left (c+d x^2\right )}\right ) \int \frac{\sqrt{b+a c+a d x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{5 a^4 d^2 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=-\frac{x^3 \left (c+d x^2\right ) \sqrt{b+a \left (c+d x^2\right )}}{a d \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}-\frac{(8 b+a c) x \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{5 a^3 d^2 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{6 x^3 \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{5 a^2 d \sqrt{a+\frac{b}{c+d x^2}}}+\frac{\left (16 b^2+16 a b c+a^2 c^2\right ) x \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{5 a^4 d^2 \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}-\frac{\sqrt{c} \left (16 b^2+16 a b c+a^2 c^2\right ) \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{5 a^4 d^{5/2} \left (c+d x^2\right ) \sqrt{\frac{c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt{a+\frac{b}{c+d x^2}}}+\frac{c^{3/2} (8 b+a c) \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{5 a^3 d^{5/2} \left (c+d x^2\right ) \sqrt{\frac{c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt{a+\frac{b}{c+d x^2}}}\\ \end{align*}
Mathematica [C] time = 0.813272, size = 296, normalized size = 0.61 \[ -\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left (x \left (c+d x^2\right ) \sqrt{\frac{a d}{a c+b}} \left (a^2 \left (c^2-d^2 x^4\right )+a b \left (9 c+2 d x^2\right )+8 b^2\right )+i c \left (a^2 c^2+16 a b c+16 b^2\right ) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} E\left (i \sinh ^{-1}\left (\sqrt{\frac{a d}{b+a c}} x\right )|\frac{b}{a c}+1\right )-i b c (7 a c+8 b) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} F\left (i \sinh ^{-1}\left (\sqrt{\frac{a d}{b+a c}} x\right )|\frac{b}{a c}+1\right )\right )}{5 a^3 d^2 \sqrt{\frac{a d}{a c+b}} \left (a \left (c+d x^2\right )+b\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.037, size = 1158, normalized size = 2.4 \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 (a + \frac{b}{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{a d x^{2} + a c + b}{d x^{2} + c}}}{a^{2} d^{2} x^{4} + a^{2} c^{2} + 2 \,{\left (a^{2} c + a b\right )} d x^{2} + 2 \, a b c + b^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\left (\frac{a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac{3}{2}}}\, dx \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 (a + \frac{b}{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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