Optimal. Leaf size=344 \[ \frac{\sqrt{c} \sqrt{a+b x^2} \left (15 a^2 d^2-11 a b c d+4 b^2 c^2\right ) \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right ),1-\frac{b c}{a d}\right )}{15 d^{5/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{x \sqrt{a+b x^2} \left (23 a^2 d^2-23 a b c d+8 b^2 c^2\right )}{15 d^2 \sqrt{c+d x^2}}-\frac{\sqrt{c} \sqrt{a+b x^2} \left (23 a^2 d^2-23 a b c d+8 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 d^{5/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{4 b x \sqrt{a+b x^2} \sqrt{c+d x^2} (b c-2 a d)}{15 d^2}+\frac{b x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{5 d} \]
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Rubi [A] time = 0.282602, antiderivative size = 344, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {416, 528, 531, 418, 492, 411} \[ \frac{x \sqrt{a+b x^2} \left (23 a^2 d^2-23 a b c d+8 b^2 c^2\right )}{15 d^2 \sqrt{c+d x^2}}+\frac{\sqrt{c} \sqrt{a+b x^2} \left (15 a^2 d^2-11 a b c d+4 b^2 c^2\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 d^{5/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{\sqrt{c} \sqrt{a+b x^2} \left (23 a^2 d^2-23 a b c d+8 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 d^{5/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{4 b x \sqrt{a+b x^2} \sqrt{c+d x^2} (b c-2 a d)}{15 d^2}+\frac{b x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{5 d} \]
Antiderivative was successfully verified.
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Rule 416
Rule 528
Rule 531
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{5/2}}{\sqrt{c+d x^2}} \, dx &=\frac{b x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{5 d}+\frac{\int \frac{\sqrt{a+b x^2} \left (-a (b c-5 a d)-4 b (b c-2 a d) x^2\right )}{\sqrt{c+d x^2}} \, dx}{5 d}\\ &=-\frac{4 b (b c-2 a d) x \sqrt{a+b x^2} \sqrt{c+d x^2}}{15 d^2}+\frac{b x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{5 d}+\frac{\int \frac{a \left (4 b^2 c^2-11 a b c d+15 a^2 d^2\right )+b \left (8 b^2 c^2-23 a b c d+23 a^2 d^2\right ) x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{15 d^2}\\ &=-\frac{4 b (b c-2 a d) x \sqrt{a+b x^2} \sqrt{c+d x^2}}{15 d^2}+\frac{b x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{5 d}+\frac{\left (a \left (4 b^2 c^2-11 a b c d+15 a^2 d^2\right )\right ) \int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{15 d^2}+\frac{\left (b \left (8 b^2 c^2-23 a b c d+23 a^2 d^2\right )\right ) \int \frac{x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{15 d^2}\\ &=\frac{\left (8 b^2 c^2-23 a b c d+23 a^2 d^2\right ) x \sqrt{a+b x^2}}{15 d^2 \sqrt{c+d x^2}}-\frac{4 b (b c-2 a d) x \sqrt{a+b x^2} \sqrt{c+d x^2}}{15 d^2}+\frac{b x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{5 d}+\frac{\sqrt{c} \left (4 b^2 c^2-11 a b c d+15 a^2 d^2\right ) \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 d^{5/2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}-\frac{\left (c \left (8 b^2 c^2-23 a b c d+23 a^2 d^2\right )\right ) \int \frac{\sqrt{a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{15 d^2}\\ &=\frac{\left (8 b^2 c^2-23 a b c d+23 a^2 d^2\right ) x \sqrt{a+b x^2}}{15 d^2 \sqrt{c+d x^2}}-\frac{4 b (b c-2 a d) x \sqrt{a+b x^2} \sqrt{c+d x^2}}{15 d^2}+\frac{b x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{5 d}-\frac{\sqrt{c} \left (8 b^2 c^2-23 a b c d+23 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 )}{15 d^{5/2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}+\frac{\sqrt{c} \left (4 b^2 c^2-11 a b c d+15 a^2 d^2\right ) \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 d^{5/2} \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 = 0.478191, size = 260, normalized size = 0.76 \[ \frac{-i \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (-34 a^2 b c d^2+15 a^3 d^3+27 a b^2 c^2 d-8 b^3 c^3\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{b}{a}}\right ),\frac{a d}{b c}\right )-i b c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (23 a^2 d^2-23 a b c d+8 b^2 c^2\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+b d x \sqrt{\frac{b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (11 a d-4 b c+3 b d x^2\right )}{15 d^3 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 615, normalized size = 1.8 \begin{align*}{\frac{1}{15\,{d}^{3} \left ( bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac \right ) }\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( 3\,\sqrt{-{\frac{b}{a}}}{x}^{7}{b}^{3}{d}^{3}+14\,\sqrt{-{\frac{b}{a}}}{x}^{5}a{b}^{2}{d}^{3}-\sqrt{-{\frac{b}{a}}}{x}^{5}{b}^{3}c{d}^{2}+11\,\sqrt{-{\frac{b}{a}}}{x}^{3}{a}^{2}b{d}^{3}+10\,\sqrt{-{\frac{b}{a}}}{x}^{3}a{b}^{2}c{d}^{2}-4\,\sqrt{-{\frac{b}{a}}}{x}^{3}{b}^{3}{c}^{2}d+15\,\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 ){a}^{3}{d}^{3}-34\,\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 ){a}^{2}bc{d}^{2}+27\,\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 ) a{b}^{2}{c}^{2}d-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 ){b}^{3}{c}^{3}+23\,\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 ){a}^{2}bc{d}^{2}-23\,\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 ) a{b}^{2}{c}^{2}d+8\,\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 ){b}^{3}{c}^{3}+11\,\sqrt{-{\frac{b}{a}}}x{a}^{2}bc{d}^{2}-4\,\sqrt{-{\frac{b}{a}}}xa{b}^{2}{c}^{2}d \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}}\,{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}}, 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}}}{\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}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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