Optimal. Leaf size=311 \[ \frac{b \sqrt{a+b x^2} (3 b c-a d) \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right ),1-\frac{b c}{a d}\right )}{3 a \sqrt{c} \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{2 \sqrt{a+b x^2} \sqrt{c+d x^2} (2 b c-a d)}{3 c^2 x}+\frac{2 d x \sqrt{a+b x^2} (2 b c-a d)}{3 c^2 \sqrt{c+d x^2}}-\frac{2 \sqrt{d} \sqrt{a+b x^2} (2 b c-a d) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 c^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{a \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 c x^3} \]
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Rubi [A] time = 0.292151, antiderivative size = 311, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {474, 583, 531, 418, 492, 411} \[ -\frac{2 \sqrt{a+b x^2} \sqrt{c+d x^2} (2 b c-a d)}{3 c^2 x}+\frac{2 d x \sqrt{a+b x^2} (2 b c-a d)}{3 c^2 \sqrt{c+d x^2}}-\frac{2 \sqrt{d} \sqrt{a+b x^2} (2 b c-a d) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 c^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{a \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 c x^3}+\frac{b \sqrt{a+b x^2} (3 b c-a d) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a \sqrt{c} \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
Antiderivative was successfully verified.
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Rule 474
Rule 583
Rule 531
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{3/2}}{x^4 \sqrt{c+d x^2}} \, dx &=-\frac{a \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 c x^3}+\frac{\int \frac{2 a (2 b c-a d)+b (3 b c-a d) x^2}{x^2 \sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{3 c}\\ &=-\frac{a \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 c x^3}-\frac{2 (2 b c-a d) \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 c^2 x}-\frac{\int \frac{-a b c (3 b c-a d)-2 a b d (2 b c-a d) x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{3 a c^2}\\ &=-\frac{a \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 c x^3}-\frac{2 (2 b c-a d) \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 c^2 x}+\frac{(2 b d (2 b c-a d)) \int \frac{x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{3 c^2}+\frac{(b (3 b c-a d)) \int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{3 c}\\ &=\frac{2 d (2 b c-a d) x \sqrt{a+b x^2}}{3 c^2 \sqrt{c+d x^2}}-\frac{a \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 c x^3}-\frac{2 (2 b c-a d) \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 c^2 x}+\frac{b (3 b c-a d) \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a \sqrt{c} \sqrt{d} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}-\frac{(2 d (2 b c-a d)) \int \frac{\sqrt{a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{3 c}\\ &=\frac{2 d (2 b c-a d) x \sqrt{a+b x^2}}{3 c^2 \sqrt{c+d x^2}}-\frac{a \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 c x^3}-\frac{2 (2 b c-a d) \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 c^2 x}-\frac{2 \sqrt{d} (2 b c-a d) \sqrt{a+b x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 c^{3/2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}+\frac{b (3 b c-a d) \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a \sqrt{c} \sqrt{d} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}\\ \end{align*}
Mathematica [C] time = 0.353747, size = 227, normalized size = 0.73 \[ \frac{-i b c x^3 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-b c) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{b}{a}}\right ),\frac{a d}{b c}\right )+\sqrt{\frac{b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (-a c+2 a d x^2-4 b c x^2\right )+2 i b c x^3 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-2 b c) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )}{3 c^2 x^3 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 433, normalized size = 1.4 \begin{align*}{\frac{1}{ \left ( 3\,bd{x}^{4}+3\,ad{x}^{2}+3\,bc{x}^{2}+3\,ac \right ){c}^{2}{x}^{3}}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( 2\,\sqrt{-{\frac{b}{a}}}{x}^{6}ab{d}^{2}-4\,\sqrt{-{\frac{b}{a}}}{x}^{6}{b}^{2}cd+bd\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}ac-\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}{b}^{2}{c}^{2}-2\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}abcd+4\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}{b}^{2}{c}^{2}+2\,\sqrt{-{\frac{b}{a}}}{x}^{4}{a}^{2}{d}^{2}-3\,\sqrt{-{\frac{b}{a}}}{x}^{4}abcd-4\,\sqrt{-{\frac{b}{a}}}{x}^{4}{b}^{2}{c}^{2}+\sqrt{-{\frac{b}{a}}}{x}^{2}{a}^{2}cd-5\,\sqrt{-{\frac{b}{a}}}{x}^{2}ab{c}^{2}-\sqrt{-{\frac{b}{a}}}{a}^{2}{c}^{2} \right ){\frac{1}{\sqrt{-{\frac{b}{a}}}}}} \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 (b x^{2} + a\right )}^{\frac{3}{2}}}{\sqrt{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{{\left (b x^{2} + a\right )}^{\frac{3}{2}} \sqrt{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 (a + b x^{2}\right )^{\frac{3}{2}}}{x^{4} \sqrt{c + d 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 (b x^{2} + a\right )}^{\frac{3}{2}}}{\sqrt{d x^{2} + c} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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