Optimal. Leaf size=388 \[ -\frac{d^2 x (a c+8 b) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{3 c^3}-\frac{a d^{3/2} (a c+4 b) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{3 c^{3/2} (a c+b) \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac{d^{3/2} (a c+8 b) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{3 c^{5/2} \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac{d (a c+8 b) \left (c+d x^2\right ) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{3 c^3 x}-\frac{(a c+4 b) \left (c+d x^2\right ) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{3 c^2 x^3}+\frac{b \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{c x^3} \]
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Rubi [A] time = 0.868577, antiderivative size = 520, normalized size of antiderivative = 1.34, 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} \[ -\frac{d^2 x (a c+8 b) \sqrt{a c+a d x^2+b} \sqrt{a+\frac{b}{c+d x^2}}}{3 c^3 \sqrt{a \left (c+d x^2\right )+b}}-\frac{a d^{3/2} (a c+4 b) \sqrt{a c+a d x^2+b} \sqrt{a+\frac{b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{3 c^{3/2} (a c+b) \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}} \sqrt{a \left (c+d x^2\right )+b}}+\frac{d^{3/2} (a c+8 b) \sqrt{a c+a d x^2+b} \sqrt{a+\frac{b}{c+d x^2}} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{3 c^{5/2} \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}} \sqrt{a \left (c+d x^2\right )+b}}+\frac{d (a c+8 b) \left (c+d x^2\right ) \sqrt{a c+a d x^2+b} \sqrt{a+\frac{b}{c+d x^2}}}{3 c^3 x \sqrt{a \left (c+d x^2\right )+b}}-\frac{(a c+4 b) \left (c+d x^2\right ) \sqrt{a c+a d x^2+b} \sqrt{a+\frac{b}{c+d x^2}}}{3 c^2 x^3 \sqrt{a \left (c+d x^2\right )+b}}+\frac{b \sqrt{a c+a d x^2+b} \sqrt{a+\frac{b}{c+d x^2}}}{c x^3 \sqrt{a \left (c+d x^2\right )+b}} \]
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^4} \, 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^4 \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^4 \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^3 \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) (4 b+a c) d-a (3 b+a c) d^2 x^2}{x^4 \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^3 \sqrt{b+a \left (c+d x^2\right )}}-\frac{(4 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}}}{3 c^2 x^3 \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 (8 b+a c) d^2-a (b+a c) (4 b+a c) d^3 x^2}{x^2 \sqrt{c+d x^2} \sqrt{b+a c+a d x^2}} \, dx}{3 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^3 \sqrt{b+a \left (c+d x^2\right )}}-\frac{(4 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}}}{3 c^2 x^3 \sqrt{b+a \left (c+d x^2\right )}}+\frac{(8 b+a c) d \left (c+d x^2\right ) \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{3 c^3 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 (4 b+a c) d^3+a (b+a c)^2 (8 b+a c) d^4 x^2}{\sqrt{c+d x^2} \sqrt{b+a c+a d x^2}} \, dx}{3 c^3 (b+a c)^2 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^3 \sqrt{b+a \left (c+d x^2\right )}}-\frac{(4 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}}}{3 c^2 x^3 \sqrt{b+a \left (c+d x^2\right )}}+\frac{(8 b+a c) d \left (c+d x^2\right ) \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{3 c^3 x \sqrt{b+a \left (c+d x^2\right )}}-\frac{\left (a (4 b+a c) d^2 \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}{3 c^2 \sqrt{b+a \left (c+d x^2\right )}}-\frac{\left (a (8 b+a c) d^3 \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}{3 c^3 \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^3 \sqrt{b+a \left (c+d x^2\right )}}-\frac{(8 b+a c) d^2 x \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{3 c^3 \sqrt{b+a \left (c+d x^2\right )}}-\frac{(4 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}}}{3 c^2 x^3 \sqrt{b+a \left (c+d x^2\right )}}+\frac{(8 b+a c) d \left (c+d x^2\right ) \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{3 c^3 x \sqrt{b+a \left (c+d x^2\right )}}-\frac{a (4 b+a c) d^{3/2} \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 )}{3 c^{3/2} (b+a 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 ((8 b+a c) d^2 \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}{3 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^3 \sqrt{b+a \left (c+d x^2\right )}}-\frac{(8 b+a c) d^2 x \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{3 c^3 \sqrt{b+a \left (c+d x^2\right )}}-\frac{(4 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}}}{3 c^2 x^3 \sqrt{b+a \left (c+d x^2\right )}}+\frac{(8 b+a c) d \left (c+d x^2\right ) \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{3 c^3 x \sqrt{b+a \left (c+d x^2\right )}}+\frac{(8 b+a c) d^{3/2} \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 )}{3 c^{5/2} \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 (4 b+a c) d^{3/2} \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 )}{3 c^{3/2} (b+a 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 = 0.833336, size = 329, normalized size = 0.85 \[ -\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left (\sqrt{\frac{a d}{a c+b}} \left (a^2 c \left (c-d x^2\right ) \left (c+d x^2\right )^2+a b \left (-3 c^2 d x^2+2 c^3-13 c d^2 x^4-8 d^3 x^6\right )+b^2 \left (c^2-4 c d x^2-8 d^2 x^4\right )\right )+4 i a b c d^2 x^3 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} F\left (i \sinh ^{-1}\left (\sqrt{\frac{a d}{b+a c}} x\right )|\frac{b}{a c}+1\right )-i a c d^2 x^3 (a c+8 b) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} E\left (i \sinh ^{-1}\left (\sqrt{\frac{a d}{b+a c}} x\right )|\frac{b}{a c}+1\right )\right )}{3 c^3 x^3 \sqrt{\frac{a d}{a c+b}} \left (a \left (c+d x^2\right )+b\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.016, size = 1039, normalized size = 2.7 \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^{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 (a d x^{2} + a c + b\right )} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{d x^{6} + c 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{\left (\frac{a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac{3}{2}}}{x^{4}}\, 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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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