Optimal. Leaf size=490 \[ \frac{d^2 x (7 b-a c) \left (a c+a d x^2+b\right )}{3 (a c+b)^3 \left (c+d x^2\right ) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}+\frac{\sqrt{c} d^{3/2} (3 b-a c) \left (a c+a d x^2+b\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{3 (a c+b)^3 \left (c+d x^2\right ) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}-\frac{\sqrt{c} d^{3/2} (7 b-a c) \left (a c+a d x^2+b\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{3 (a c+b)^3 \left (c+d x^2\right ) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}-\frac{d (7 b-a c) \left (a c+a d x^2+b\right )}{3 x (a c+b)^3 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}+\frac{(3 b-a c) \left (a c+a d x^2+b\right )}{3 a x^3 (a c+b)^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}-\frac{b}{a x^3 (a c+b) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}} \]
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Rubi [A] time = 0.866721, antiderivative size = 567, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {6722, 1975, 468, 583, 531, 418, 492, 411} \[ \frac{d^2 x (7 b-a c) \sqrt{a c+a d x^2+b} \sqrt{a \left (c+d x^2\right )+b}}{3 (a c+b)^3 \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}+\frac{\sqrt{c} d^{3/2} (3 b-a c) \sqrt{a c+a d x^2+b} \sqrt{a \left (c+d x^2\right )+b} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{3 (a c+b)^3 \left (c+d x^2\right ) \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}} \sqrt{a+\frac{b}{c+d x^2}}}-\frac{\sqrt{c} d^{3/2} (7 b-a c) \sqrt{a c+a d x^2+b} \sqrt{a \left (c+d x^2\right )+b} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{3 (a c+b)^3 \left (c+d x^2\right ) \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}} \sqrt{a+\frac{b}{c+d x^2}}}-\frac{d (7 b-a c) \sqrt{a c+a d x^2+b} \sqrt{a \left (c+d x^2\right )+b}}{3 x (a c+b)^3 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{(3 b-a c) \sqrt{a c+a d x^2+b} \sqrt{a \left (c+d x^2\right )+b}}{3 a x^3 (a c+b)^2 \sqrt{a+\frac{b}{c+d x^2}}}-\frac{b \sqrt{a \left (c+d x^2\right )+b}}{a x^3 (a c+b) \sqrt{a c+a d x^2+b} \sqrt{a+\frac{b}{c+d x^2}}} \]
Antiderivative was successfully verified.
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Rule 6722
Rule 1975
Rule 468
Rule 583
Rule 531
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{1}{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{\left (c+d x^2\right )^{3/2}}{x^4 \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{\left (c+d x^2\right )^{3/2}}{x^4 \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{b \sqrt{b+a \left (c+d x^2\right )}}{a (b+a c) x^3 \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{c (3 b-a c) d+(2 b-a c) d^2 x^2}{x^4 \sqrt{c+d x^2} \sqrt{b+a c+a d x^2}} \, dx}{a (b+a c) d \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=-\frac{b \sqrt{b+a \left (c+d x^2\right )}}{a (b+a c) x^3 \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}+\frac{(3 b-a c) \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{3 a (b+a c)^2 x^3 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{\sqrt{b+a \left (c+d x^2\right )} \int \frac{a c^2 (7 b-a c) d^2+a c (3 b-a c) d^3 x^2}{x^2 \sqrt{c+d x^2} \sqrt{b+a c+a d x^2}} \, dx}{3 a c (b+a c)^2 d \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=-\frac{b \sqrt{b+a \left (c+d x^2\right )}}{a (b+a c) x^3 \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}+\frac{(3 b-a c) \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{3 a (b+a c)^2 x^3 \sqrt{a+\frac{b}{c+d x^2}}}-\frac{(7 b-a c) d \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{3 (b+a c)^3 x \sqrt{a+\frac{b}{c+d x^2}}}-\frac{\sqrt{b+a \left (c+d x^2\right )} \int \frac{-a c^2 (3 b-a c) (b+a c) d^3-a^2 c^2 (7 b-a c) d^4 x^2}{\sqrt{c+d x^2} \sqrt{b+a c+a d x^2}} \, dx}{3 a c^2 (b+a c)^3 d \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=-\frac{b \sqrt{b+a \left (c+d x^2\right )}}{a (b+a c) x^3 \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}+\frac{(3 b-a c) \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{3 a (b+a c)^2 x^3 \sqrt{a+\frac{b}{c+d x^2}}}-\frac{(7 b-a c) d \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{3 (b+a c)^3 x \sqrt{a+\frac{b}{c+d x^2}}}+\frac{\left ((3 b-a c) d^2 \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}{3 (b+a c)^2 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}+\frac{\left (a (7 b-a c) d^3 \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}{3 (b+a c)^3 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=-\frac{b \sqrt{b+a \left (c+d x^2\right )}}{a (b+a c) x^3 \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}+\frac{(3 b-a c) \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{3 a (b+a c)^2 x^3 \sqrt{a+\frac{b}{c+d x^2}}}-\frac{(7 b-a c) d \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{3 (b+a c)^3 x \sqrt{a+\frac{b}{c+d x^2}}}+\frac{(7 b-a c) d^2 x \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{3 (b+a c)^3 \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}+\frac{\sqrt{c} (3 b-a c) d^{3/2} \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 )}{3 (b+a c)^3 \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 (7 b-a c) d^2 \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}{3 (b+a c)^3 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=-\frac{b \sqrt{b+a \left (c+d x^2\right )}}{a (b+a c) x^3 \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}+\frac{(3 b-a c) \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{3 a (b+a c)^2 x^3 \sqrt{a+\frac{b}{c+d x^2}}}-\frac{(7 b-a c) d \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{3 (b+a c)^3 x \sqrt{a+\frac{b}{c+d x^2}}}+\frac{(7 b-a c) d^2 x \sqrt{b+a c+a d x^2} \sqrt{b+a \left (c+d x^2\right )}}{3 (b+a c)^3 \left (c+d x^2\right ) \sqrt{a+\frac{b}{c+d x^2}}}-\frac{\sqrt{c} (7 b-a c) d^{3/2} \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 )}{3 (b+a c)^3 \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{\sqrt{c} (3 b-a c) d^{3/2} \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 )}{3 (b+a c)^3 \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 = 0.89798, size = 319, normalized size = 0.65 \[ -\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left (\left (c+d x^2\right ) \sqrt{\frac{a d}{a c+b}} \left (a^2 c \left (c^2-d^2 x^4\right )+a b \left (2 c^2+4 c d x^2+7 d^2 x^4\right )+b^2 \left (c+4 d x^2\right )\right )+i b d^2 x^3 (3 b-5 a c) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} F\left (i \sinh ^{-1}\left (\sqrt{\frac{a d}{b+a c}} x\right )|\frac{b}{a c}+1\right )-i a c d^2 x^3 (a c-7 b) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} E\left (i \sinh ^{-1}\left (\sqrt{\frac{a d}{b+a c}} x\right )|\frac{b}{a c}+1\right )\right )}{3 x^3 (a c+b)^3 \sqrt{\frac{a d}{a c+b}} \left (a \left (c+d x^2\right )+b\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.017, size = 1082, normalized size = 2.2 \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 (a + \frac{b}{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{a d x^{2} + a c + b}{d x^{2} + c}}}{a^{2} d^{2} x^{8} + 2 \,{\left (a^{2} c + a b\right )} d x^{6} +{\left (a^{2} c^{2} + 2 \, a b c + b^{2}\right )} x^{4}}, 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{1}{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{1}{{\left (a + \frac{b}{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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