Optimal. Leaf size=312 \[ \frac{d x (a c+2 b) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{c^2}-\frac{(a c+2 b) \left (c+d x^2\right ) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{c^2 x}-\frac{\sqrt{d} (a c+2 b) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{c^{3/2} \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac{b \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{c x}+\frac{a \sqrt{d} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{\sqrt{c} \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}} \]
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Rubi [A] time = 0.663576, antiderivative size = 422, normalized size of antiderivative = 1.35, number of steps used = 8, 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} \[ \frac{d x (a c+2 b) \sqrt{a c+a d x^2+b} \sqrt{a+\frac{b}{c+d x^2}}}{c^2 \sqrt{a \left (c+d x^2\right )+b}}-\frac{(a c+2 b) \left (c+d x^2\right ) \sqrt{a c+a d x^2+b} \sqrt{a+\frac{b}{c+d x^2}}}{c^2 x \sqrt{a \left (c+d x^2\right )+b}}-\frac{\sqrt{d} (a c+2 b) \sqrt{a c+a d x^2+b} \sqrt{a+\frac{b}{c+d x^2}} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{c^{3/2} \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}} \sqrt{a \left (c+d x^2\right )+b}}+\frac{b \sqrt{a c+a d x^2+b} \sqrt{a+\frac{b}{c+d x^2}}}{c x \sqrt{a \left (c+d x^2\right )+b}}+\frac{a \sqrt{d} \sqrt{a c+a d x^2+b} \sqrt{a+\frac{b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{\sqrt{c} \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}} \sqrt{a \left (c+d x^2\right )+b}} \]
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{\left (a+\frac{b}{c+d x^2}\right )^{3/2}}{x^2} \, dx &=\frac{\left (\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \int \frac{\left (b+a \left (c+d x^2\right )\right )^{3/2}}{x^2 \left (c+d x^2\right )^{3/2}} \, dx}{\sqrt{b+a \left (c+d x^2\right )}}\\ &=\frac{\left (\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \int \frac{\left (b+a c+a d x^2\right )^{3/2}}{x^2 \left (c+d x^2\right )^{3/2}} \, dx}{\sqrt{b+a \left (c+d x^2\right )}}\\ &=\frac{b \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{c x \sqrt{b+a \left (c+d x^2\right )}}-\frac{\left (\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \int \frac{-(b+a c) (2 b+a c) d-a (b+a c) d^2 x^2}{x^2 \sqrt{c+d x^2} \sqrt{b+a c+a d x^2}} \, dx}{c d \sqrt{b+a \left (c+d x^2\right )}}\\ &=\frac{b \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{c x \sqrt{b+a \left (c+d x^2\right )}}-\frac{(2 b+a c) \left (c+d x^2\right ) \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{c^2 x \sqrt{b+a \left (c+d x^2\right )}}+\frac{\left (\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \int \frac{a c (b+a c)^2 d^2+a (b+a c) (2 b+a c) d^3 x^2}{\sqrt{c+d x^2} \sqrt{b+a c+a d x^2}} \, dx}{c^2 (b+a c) d \sqrt{b+a \left (c+d x^2\right )}}\\ &=\frac{b \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{c x \sqrt{b+a \left (c+d x^2\right )}}-\frac{(2 b+a c) \left (c+d x^2\right ) \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{c^2 x \sqrt{b+a \left (c+d x^2\right )}}+\frac{\left (a (b+a c) d \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \int \frac{1}{\sqrt{c+d x^2} \sqrt{b+a c+a d x^2}} \, dx}{c \sqrt{b+a \left (c+d x^2\right )}}+\frac{\left (a (2 b+a c) d^2 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \int \frac{x^2}{\sqrt{c+d x^2} \sqrt{b+a c+a d x^2}} \, dx}{c^2 \sqrt{b+a \left (c+d x^2\right )}}\\ &=\frac{b \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{c x \sqrt{b+a \left (c+d x^2\right )}}+\frac{(2 b+a c) d x \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{c^2 \sqrt{b+a \left (c+d x^2\right )}}-\frac{(2 b+a c) \left (c+d x^2\right ) \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{c^2 x \sqrt{b+a \left (c+d x^2\right )}}+\frac{a \sqrt{d} \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{\sqrt{c} \sqrt{\frac{c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt{b+a \left (c+d x^2\right )}}-\frac{\left ((2 b+a c) d \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \int \frac{\sqrt{b+a c+a d x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{c \sqrt{b+a \left (c+d x^2\right )}}\\ &=\frac{b \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{c x \sqrt{b+a \left (c+d x^2\right )}}+\frac{(2 b+a c) d x \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{c^2 \sqrt{b+a \left (c+d x^2\right )}}-\frac{(2 b+a c) \left (c+d x^2\right ) \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{c^2 x \sqrt{b+a \left (c+d x^2\right )}}-\frac{(2 b+a c) \sqrt{d} \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{c^{3/2} \sqrt{\frac{c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt{b+a \left (c+d x^2\right )}}+\frac{a \sqrt{d} \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{\sqrt{c} \sqrt{\frac{c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt{b+a \left (c+d x^2\right )}}\\ \end{align*}
Mathematica [C] time = 0.703092, size = 278, normalized size = 0.89 \[ -\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left (\sqrt{\frac{a d}{a c+b}} \left (a^2 c \left (c+d x^2\right )^2+2 a b \left (c+d x^2\right )^2+b^2 \left (c+2 d x^2\right )\right )-i a b c d x \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} F\left (i \sinh ^{-1}\left (\sqrt{\frac{a d}{b+a c}} x\right )|\frac{b}{a c}+1\right )+i a c d x (a c+2 b) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} E\left (i \sinh ^{-1}\left (\sqrt{\frac{a d}{b+a c}} x\right )|\frac{b}{a c}+1\right )\right )}{c^2 x \sqrt{\frac{a d}{a c+b}} \left (a \left (c+d x^2\right )+b\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.016, size = 873, normalized size = 2.8 \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{{\left (a + \frac{b}{d x^{2} + c}\right )}^{\frac{3}{2}}}{x^{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 (a d x^{2} + a c + b\right )} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{d x^{4} + c x^{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{\left (\frac{a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac{3}{2}}}{x^{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{{\left (a + \frac{b}{d x^{2} + c}\right )}^{\frac{3}{2}}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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