Optimal. Leaf size=356 \[ \frac{4 b^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (9 A c+b B) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{15 c^{3/4} \sqrt{b x^2+c x^4}}-\frac{8 b^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (9 A c+b B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 c^{3/4} \sqrt{b x^2+c x^4}}+\frac{2 \left (b x^2+c x^4\right )^{3/2} (9 A c+b B)}{9 b x^{3/2}}+\frac{4}{15} \sqrt{x} \sqrt{b x^2+c x^4} (9 A c+b B)+\frac{8 b x^{3/2} \left (b+c x^2\right ) (9 A c+b B)}{15 \sqrt{c} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{2 A \left (b x^2+c x^4\right )^{5/2}}{b x^{11/2}} \]
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Rubi [A] time = 0.445831, antiderivative size = 356, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2038, 2021, 2032, 329, 305, 220, 1196} \[ \frac{4 b^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (9 A c+b B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 c^{3/4} \sqrt{b x^2+c x^4}}-\frac{8 b^{5/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (9 A c+b B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 c^{3/4} \sqrt{b x^2+c x^4}}+\frac{2 \left (b x^2+c x^4\right )^{3/2} (9 A c+b B)}{9 b x^{3/2}}+\frac{4}{15} \sqrt{x} \sqrt{b x^2+c x^4} (9 A c+b B)+\frac{8 b x^{3/2} \left (b+c x^2\right ) (9 A c+b B)}{15 \sqrt{c} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{2 A \left (b x^2+c x^4\right )^{5/2}}{b x^{11/2}} \]
Antiderivative was successfully verified.
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Rule 2038
Rule 2021
Rule 2032
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{x^{9/2}} \, dx &=-\frac{2 A \left (b x^2+c x^4\right )^{5/2}}{b x^{11/2}}-\frac{\left (2 \left (-\frac{b B}{2}-\frac{9 A c}{2}\right )\right ) \int \frac{\left (b x^2+c x^4\right )^{3/2}}{x^{5/2}} \, dx}{b}\\ &=\frac{2 (b B+9 A c) \left (b x^2+c x^4\right )^{3/2}}{9 b x^{3/2}}-\frac{2 A \left (b x^2+c x^4\right )^{5/2}}{b x^{11/2}}+\frac{1}{3} (2 (b B+9 A c)) \int \frac{\sqrt{b x^2+c x^4}}{\sqrt{x}} \, dx\\ &=\frac{4}{15} (b B+9 A c) \sqrt{x} \sqrt{b x^2+c x^4}+\frac{2 (b B+9 A c) \left (b x^2+c x^4\right )^{3/2}}{9 b x^{3/2}}-\frac{2 A \left (b x^2+c x^4\right )^{5/2}}{b x^{11/2}}+\frac{1}{15} (4 b (b B+9 A c)) \int \frac{x^{3/2}}{\sqrt{b x^2+c x^4}} \, dx\\ &=\frac{4}{15} (b B+9 A c) \sqrt{x} \sqrt{b x^2+c x^4}+\frac{2 (b B+9 A c) \left (b x^2+c x^4\right )^{3/2}}{9 b x^{3/2}}-\frac{2 A \left (b x^2+c x^4\right )^{5/2}}{b x^{11/2}}+\frac{\left (4 b (b B+9 A c) x \sqrt{b+c x^2}\right ) \int \frac{\sqrt{x}}{\sqrt{b+c x^2}} \, dx}{15 \sqrt{b x^2+c x^4}}\\ &=\frac{4}{15} (b B+9 A c) \sqrt{x} \sqrt{b x^2+c x^4}+\frac{2 (b B+9 A c) \left (b x^2+c x^4\right )^{3/2}}{9 b x^{3/2}}-\frac{2 A \left (b x^2+c x^4\right )^{5/2}}{b x^{11/2}}+\frac{\left (8 b (b B+9 A c) x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{15 \sqrt{b x^2+c x^4}}\\ &=\frac{4}{15} (b B+9 A c) \sqrt{x} \sqrt{b x^2+c x^4}+\frac{2 (b B+9 A c) \left (b x^2+c x^4\right )^{3/2}}{9 b x^{3/2}}-\frac{2 A \left (b x^2+c x^4\right )^{5/2}}{b x^{11/2}}+\frac{\left (8 b^{3/2} (b B+9 A c) x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{15 \sqrt{c} \sqrt{b x^2+c x^4}}-\frac{\left (8 b^{3/2} (b B+9 A c) x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{b}}}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{15 \sqrt{c} \sqrt{b x^2+c x^4}}\\ &=\frac{8 b (b B+9 A c) x^{3/2} \left (b+c x^2\right )}{15 \sqrt{c} \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}+\frac{4}{15} (b B+9 A c) \sqrt{x} \sqrt{b x^2+c x^4}+\frac{2 (b B+9 A c) \left (b x^2+c x^4\right )^{3/2}}{9 b x^{3/2}}-\frac{2 A \left (b x^2+c x^4\right )^{5/2}}{b x^{11/2}}-\frac{8 b^{5/4} (b B+9 A c) x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 c^{3/4} \sqrt{b x^2+c x^4}}+\frac{4 b^{5/4} (b B+9 A c) x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 c^{3/4} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.0737532, size = 85, normalized size = 0.24 \[ \frac{2 \sqrt{x^2 \left (b+c x^2\right )} \left (\frac{x^2 (9 A c+b B) \, _2F_1\left (-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{b}\right )}{\sqrt{\frac{c x^2}{b}+1}}-\frac{3 A \left (b+c x^2\right )^2}{b}\right )}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 429, normalized size = 1.2 \begin{align*}{\frac{2}{45\, \left ( c{x}^{2}+b \right ) ^{2}c} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 5\,B{c}^{3}{x}^{6}+108\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){b}^{2}c-54\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){b}^{2}c+12\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){b}^{3}-6\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){b}^{3}+9\,A{x}^{4}{c}^{3}+16\,B{x}^{4}b{c}^{2}-36\,A{x}^{2}b{c}^{2}+11\,B{x}^{2}{b}^{2}c-45\,A{b}^{2}c \right ){x}^{-{\frac{7}{2}}}} \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 (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}{\left (B x^{2} + A\right )}}{x^{\frac{9}{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 c x^{4} +{\left (B b + A c\right )} x^{2} + A b\right )} \sqrt{c x^{4} + b x^{2}}}{x^{\frac{5}{2}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}{\left (B x^{2} + A\right )}}{x^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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