Optimal. Leaf size=362 \[ -\frac{d^2 x (a c+2 b) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{3 c^2 (a c+b)}+\frac{d^{3/2} (a c+2 b) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{3 c^{3/2} (a c+b) \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac{d (a c+2 b) \left (c+d x^2\right ) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{3 c^2 x (a c+b)}-\frac{a d^{3/2} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{3 \sqrt{c} (a c+b) \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}-\frac{\left (c+d x^2\right ) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{3 c x^3} \]
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Rubi [A] time = 0.630172, antiderivative size = 472, normalized size of antiderivative = 1.3, 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, 475, 583, 531, 418, 492, 411} \[ -\frac{d^2 x (a c+2 b) \sqrt{a c+a d x^2+b} \sqrt{a+\frac{b}{c+d x^2}}}{3 c^2 (a c+b) \sqrt{a \left (c+d x^2\right )+b}}+\frac{d^{3/2} (a c+2 b) \sqrt{a c+a d x^2+b} \sqrt{a+\frac{b}{c+d x^2}} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{3 c^{3/2} (a c+b) \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}} \sqrt{a \left (c+d x^2\right )+b}}+\frac{d (a c+2 b) \left (c+d x^2\right ) \sqrt{a c+a d x^2+b} \sqrt{a+\frac{b}{c+d x^2}}}{3 c^2 x (a c+b) \sqrt{a \left (c+d x^2\right )+b}}-\frac{a d^{3/2} \sqrt{a c+a d x^2+b} \sqrt{a+\frac{b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b}{b+a c}\right )}{3 \sqrt{c} (a c+b) \sqrt{\frac{c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}} \sqrt{a \left (c+d x^2\right )+b}}-\frac{\left (c+d x^2\right ) \sqrt{a c+a d x^2+b} \sqrt{a+\frac{b}{c+d x^2}}}{3 c x^3 \sqrt{a \left (c+d x^2\right )+b}} \]
Antiderivative was successfully verified.
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Rule 6722
Rule 1975
Rule 475
Rule 583
Rule 531
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{\sqrt{a+\frac{b}{c+d x^2}}}{x^4} \, dx &=\frac{\left (\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}\right ) \int \frac{\sqrt{b+a \left (c+d x^2\right )}}{x^4 \sqrt{c+d x^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{\sqrt{b+a c+a d x^2}}{x^4 \sqrt{c+d x^2}} \, dx}{\sqrt{b+a \left (c+d x^2\right )}}\\ &=-\frac{\left (c+d x^2\right ) \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{3 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{-(2 b+a c) d-a d^2 x^2}{x^2 \sqrt{c+d x^2} \sqrt{b+a c+a d x^2}} \, dx}{3 c \sqrt{b+a \left (c+d x^2\right )}}\\ &=-\frac{\left (c+d x^2\right ) \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{3 c x^3 \sqrt{b+a \left (c+d x^2\right )}}+\frac{(2 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^2 (b+a 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{a c (b+a c) d^2+a (2 b+a c) d^3 x^2}{\sqrt{c+d x^2} \sqrt{b+a c+a d x^2}} \, dx}{3 c^2 (b+a c) \sqrt{b+a \left (c+d x^2\right )}}\\ &=-\frac{\left (c+d x^2\right ) \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{3 c x^3 \sqrt{b+a \left (c+d x^2\right )}}+\frac{(2 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^2 (b+a c) x \sqrt{b+a \left (c+d x^2\right )}}-\frac{\left (a 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 \sqrt{b+a \left (c+d x^2\right )}}-\frac{\left (a (2 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^2 (b+a c) \sqrt{b+a \left (c+d x^2\right )}}\\ &=-\frac{(2 b+a c) d^2 x \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{3 c^2 (b+a c) \sqrt{b+a \left (c+d x^2\right )}}-\frac{\left (c+d x^2\right ) \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{3 c x^3 \sqrt{b+a \left (c+d x^2\right )}}+\frac{(2 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^2 (b+a c) x \sqrt{b+a \left (c+d x^2\right )}}-\frac{a 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 \sqrt{c} (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 ((2 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 (b+a c) \sqrt{b+a \left (c+d x^2\right )}}\\ &=-\frac{(2 b+a c) d^2 x \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{3 c^2 (b+a c) \sqrt{b+a \left (c+d x^2\right )}}-\frac{\left (c+d x^2\right ) \sqrt{b+a c+a d x^2} \sqrt{a+\frac{b}{c+d x^2}}}{3 c x^3 \sqrt{b+a \left (c+d x^2\right )}}+\frac{(2 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^2 (b+a c) x \sqrt{b+a \left (c+d x^2\right )}}+\frac{(2 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^{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{a 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 \sqrt{c} (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 = 0.971735, size = 314, normalized size = 0.87 \[ -\frac{\sqrt{\frac{a d}{a c+b}} \sqrt{\frac{a c+a d x^2+b}{c+d x^2}} \left (\left (c+d x^2\right ) \sqrt{\frac{a d}{a c+b}} \left (a^2 c \left (c^2-d^2 x^4\right )+2 a b \left (c^2-c d x^2-d^2 x^4\right )+b^2 \left (c-2 d x^2\right )\right )+i a b c d^2 x^3 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} F\left (i \sinh ^{-1}\left (\sqrt{\frac{a d}{b+a c}} x\right )|\frac{b}{a c}+1\right )-i a c d^2 x^3 (a c+2 b) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{a c+a d x^2+b}{a c+b}} E\left (i \sinh ^{-1}\left (\sqrt{\frac{a d}{b+a c}} x\right )|\frac{b}{a c}+1\right )\right )}{3 a c^2 d x^3 \left (a \left (c+d x^2\right )+b\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 571, normalized size = 1.6 \begin{align*}{\frac{d{x}^{2}+c}{ \left ( 3\,ac+3\,b \right ){x}^{3}{c}^{2}} \left ( \sqrt{-{\frac{ad}{ac+b}}}{x}^{6}{a}^{2}c{d}^{3}+2\,\sqrt{-{\frac{ad}{ac+b}}}{x}^{6}ab{d}^{3}-\sqrt{{\frac{ad{x}^{2}+ac+b}{ac+b}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{ad}{ac+b}}},\sqrt{{\frac{ac+b}{ac}}} \right ){x}^{3}{a}^{2}{c}^{2}{d}^{2}+\sqrt{-{\frac{ad}{ac+b}}}{x}^{4}{a}^{2}{c}^{2}{d}^{2}+\sqrt{{\frac{ad{x}^{2}+ac+b}{ac+b}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{ad}{ac+b}}},\sqrt{{\frac{ac+b}{ac}}} \right ){x}^{3}abc{d}^{2}-2\,\sqrt{{\frac{ad{x}^{2}+ac+b}{ac+b}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{ad}{ac+b}}},\sqrt{{\frac{ac+b}{ac}}} \right ){x}^{3}abc{d}^{2}+4\,\sqrt{-{\frac{ad}{ac+b}}}{x}^{4}abc{d}^{2}+2\,\sqrt{-{\frac{ad}{ac+b}}}{x}^{4}{b}^{2}{d}^{2}-\sqrt{-{\frac{ad}{ac+b}}}{x}^{2}{a}^{2}{c}^{3}d+\sqrt{-{\frac{ad}{ac+b}}}{x}^{2}{b}^{2}cd-\sqrt{-{\frac{ad}{ac+b}}}{a}^{2}{c}^{4}-2\,\sqrt{-{\frac{ad}{ac+b}}}ab{c}^{3}-\sqrt{-{\frac{ad}{ac+b}}}{b}^{2}{c}^{2} \right ) \sqrt{{\frac{ad{x}^{2}+ac+b}{d{x}^{2}+c}}}{\frac{1}{\sqrt{a{d}^{2}{x}^{4}+2\,acd{x}^{2}+bd{x}^{2}+{c}^{2}a+bc}}}{\frac{1}{\sqrt{-{\frac{ad}{ac+b}}}}}{\frac{1}{\sqrt{ \left ( d{x}^{2}+c \right ) \left ( ad{x}^{2}+ac+b \right ) }}}} \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{\sqrt{a + \frac{b}{d x^{2} + c}}}{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{\sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + 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{\sqrt{\frac{a c + a d x^{2} + b}{c + d x^{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{\sqrt{a + \frac{b}{d x^{2} + c}}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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