Optimal. Leaf size=330 \[ -\frac{b \sqrt{c} \sqrt{a+b x^2} (b c-9 a d) \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right ),1-\frac{b c}{a d}\right )}{3 d^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{\sqrt{a+b x^2} \left (-3 a^2 d^2-7 a b c d+2 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 \sqrt{c} d^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{x \sqrt{a+b x^2} \left (\frac{3 a^2 d}{c}+7 a b-\frac{2 b^2 c}{d}\right )}{3 \sqrt{c+d x^2}}-\frac{a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{c x}+\frac{b x \sqrt{a+b x^2} \sqrt{c+d x^2} (3 a d+b c)}{3 c d} \]
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Rubi [A] time = 0.290934, antiderivative size = 330, 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, 528, 531, 418, 492, 411} \[ \frac{\sqrt{a+b x^2} \left (-3 a^2 d^2-7 a b c d+2 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 \sqrt{c} d^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{x \sqrt{a+b x^2} \left (\frac{3 a^2 d}{c}+7 a b-\frac{2 b^2 c}{d}\right )}{3 \sqrt{c+d x^2}}-\frac{b \sqrt{c} \sqrt{a+b x^2} (b c-9 a d) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 d^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{c x}+\frac{b x \sqrt{a+b x^2} \sqrt{c+d x^2} (3 a d+b c)}{3 c d} \]
Antiderivative was successfully verified.
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Rule 474
Rule 528
Rule 531
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{5/2}}{x^2 \sqrt{c+d x^2}} \, dx &=-\frac{a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{c x}+\frac{\int \frac{\sqrt{a+b x^2} \left (4 a b c+b (b c+3 a d) x^2\right )}{\sqrt{c+d x^2}} \, dx}{c}\\ &=\frac{b (b c+3 a d) x \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 c d}-\frac{a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{c x}+\frac{\int \frac{-a b c (b c-9 a d)-b \left (2 b^2 c^2-7 a b c d-3 a^2 d^2\right ) x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{3 c d}\\ &=\frac{b (b c+3 a d) x \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 c d}-\frac{a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{c x}-\frac{(a b (b c-9 a d)) \int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{3 d}-\frac{\left (b \left (2 b^2 c^2-7 a b c d-3 a^2 d^2\right )\right ) \int \frac{x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{3 c d}\\ &=\frac{\left (7 a b-\frac{2 b^2 c}{d}+\frac{3 a^2 d}{c}\right ) x \sqrt{a+b x^2}}{3 \sqrt{c+d x^2}}+\frac{b (b c+3 a d) x \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 c d}-\frac{a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{c x}-\frac{b \sqrt{c} (b c-9 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 d^{3/2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}+\frac{\left (2 b^2 c^2-7 a b c d-3 a^2 d^2\right ) \int \frac{\sqrt{a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{3 d}\\ &=\frac{\left (7 a b-\frac{2 b^2 c}{d}+\frac{3 a^2 d}{c}\right ) x \sqrt{a+b x^2}}{3 \sqrt{c+d x^2}}+\frac{b (b c+3 a d) x \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 c d}-\frac{a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{c x}+\frac{\left (2 b^2 c^2-7 a b c d-3 a^2 d^2\right ) \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 \sqrt{c} d^{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 \sqrt{c} (b c-9 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 d^{3/2} \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 = 0.43935, size = 254, normalized size = 0.77 \[ \frac{-2 i b c x \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (3 a^2 d^2-4 a b c d+b^2 c^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{b}{a}}\right ),\frac{a d}{b c}\right )-i b c x \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (3 a^2 d^2+7 a b c d-2 b^2 c^2\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+d \left (-\sqrt{\frac{b}{a}}\right ) \left (a+b x^2\right ) \left (c+d x^2\right ) \left (3 a^2 d-b^2 c x^2\right )}{3 c d^2 x \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 568, normalized size = 1.7 \begin{align*}{\frac{1}{ \left ( 3\,bd{x}^{4}+3\,ad{x}^{2}+3\,bc{x}^{2}+3\,ac \right ){d}^{2}cx}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( \sqrt{-{\frac{b}{a}}}{x}^{6}{b}^{3}c{d}^{2}-3\,\sqrt{-{\frac{b}{a}}}{x}^{4}{a}^{2}b{d}^{3}+\sqrt{-{\frac{b}{a}}}{x}^{4}a{b}^{2}c{d}^{2}+\sqrt{-{\frac{b}{a}}}{x}^{4}{b}^{3}{c}^{2}d+3\,{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) \sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}x{a}^{2}bc{d}^{2}+7\,{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) \sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}xa{b}^{2}{c}^{2}d-2\,{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) \sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}x{b}^{3}{c}^{3}+6\,\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{a}^{2}bc{d}^{2}-8\,\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 ) xa{b}^{2}{c}^{2}d+2\,\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{b}^{3}{c}^{3}-3\,\sqrt{-{\frac{b}{a}}}{x}^{2}{a}^{3}{d}^{3}-3\,\sqrt{-{\frac{b}{a}}}{x}^{2}{a}^{2}bc{d}^{2}+\sqrt{-{\frac{b}{a}}}{x}^{2}a{b}^{2}{c}^{2}d-3\,\sqrt{-{\frac{b}{a}}}{a}^{3}c{d}^{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{5}{2}}}{\sqrt{d x^{2} + c} 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 (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt{b x^{2} + a} \sqrt{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 (a + b x^{2}\right )^{\frac{5}{2}}}{x^{2} \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{5}{2}}}{\sqrt{d x^{2} + c} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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