Optimal. Leaf size=214 \[ \frac{5 b^{3/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (9 b B-7 A c) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{42 c^{13/4} \sqrt{b x^2+c x^4}}+\frac{x^{3/2} \sqrt{b x^2+c x^4} (9 b B-7 A c)}{7 b c^2}-\frac{5 \sqrt{b x^2+c x^4} (9 b B-7 A c)}{21 c^3 \sqrt{x}}-\frac{x^{11/2} (b B-A c)}{b c \sqrt{b x^2+c x^4}} \]
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Rubi [A] time = 0.326987, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {2037, 2024, 2032, 329, 220} \[ \frac{5 b^{3/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (9 b B-7 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{42 c^{13/4} \sqrt{b x^2+c x^4}}+\frac{x^{3/2} \sqrt{b x^2+c x^4} (9 b B-7 A c)}{7 b c^2}-\frac{5 \sqrt{b x^2+c x^4} (9 b B-7 A c)}{21 c^3 \sqrt{x}}-\frac{x^{11/2} (b B-A c)}{b c \sqrt{b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 2037
Rule 2024
Rule 2032
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{x^{13/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx &=-\frac{(b B-A c) x^{11/2}}{b c \sqrt{b x^2+c x^4}}+\frac{\left (\frac{9 b B}{2}-\frac{7 A c}{2}\right ) \int \frac{x^{9/2}}{\sqrt{b x^2+c x^4}} \, dx}{b c}\\ &=-\frac{(b B-A c) x^{11/2}}{b c \sqrt{b x^2+c x^4}}+\frac{(9 b B-7 A c) x^{3/2} \sqrt{b x^2+c x^4}}{7 b c^2}-\frac{(5 (9 b B-7 A c)) \int \frac{x^{5/2}}{\sqrt{b x^2+c x^4}} \, dx}{14 c^2}\\ &=-\frac{(b B-A c) x^{11/2}}{b c \sqrt{b x^2+c x^4}}-\frac{5 (9 b B-7 A c) \sqrt{b x^2+c x^4}}{21 c^3 \sqrt{x}}+\frac{(9 b B-7 A c) x^{3/2} \sqrt{b x^2+c x^4}}{7 b c^2}+\frac{(5 b (9 b B-7 A c)) \int \frac{\sqrt{x}}{\sqrt{b x^2+c x^4}} \, dx}{42 c^3}\\ &=-\frac{(b B-A c) x^{11/2}}{b c \sqrt{b x^2+c x^4}}-\frac{5 (9 b B-7 A c) \sqrt{b x^2+c x^4}}{21 c^3 \sqrt{x}}+\frac{(9 b B-7 A c) x^{3/2} \sqrt{b x^2+c x^4}}{7 b c^2}+\frac{\left (5 b (9 b B-7 A c) x \sqrt{b+c x^2}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x^2}} \, dx}{42 c^3 \sqrt{b x^2+c x^4}}\\ &=-\frac{(b B-A c) x^{11/2}}{b c \sqrt{b x^2+c x^4}}-\frac{5 (9 b B-7 A c) \sqrt{b x^2+c x^4}}{21 c^3 \sqrt{x}}+\frac{(9 b B-7 A c) x^{3/2} \sqrt{b x^2+c x^4}}{7 b c^2}+\frac{\left (5 b (9 b B-7 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 )}{21 c^3 \sqrt{b x^2+c x^4}}\\ &=-\frac{(b B-A c) x^{11/2}}{b c \sqrt{b x^2+c x^4}}-\frac{5 (9 b B-7 A c) \sqrt{b x^2+c x^4}}{21 c^3 \sqrt{x}}+\frac{(9 b B-7 A c) x^{3/2} \sqrt{b x^2+c x^4}}{7 b c^2}+\frac{5 b^{3/4} (9 b B-7 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 )}{42 c^{13/4} \sqrt{b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 0.133174, size = 110, normalized size = 0.51 \[ \frac{x^{3/2} \left (5 b \sqrt{\frac{c x^2}{b}+1} (9 b B-7 A c) \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^2}{b}\right )+b c \left (35 A-18 B x^2\right )+2 c^2 x^2 \left (7 A+3 B x^2\right )-45 b^2 B\right )}{21 c^3 \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 255, normalized size = 1.2 \begin{align*} -{\frac{c{x}^{2}+b}{42\,{c}^{4}}{x}^{{\frac{5}{2}}} \left ( 35\,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 ) \sqrt{-bc}bc-45\,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 ) \sqrt{-bc}{b}^{2}-12\,B{c}^{3}{x}^{5}-28\,A{x}^{3}{c}^{3}+36\,B{x}^{3}b{c}^{2}-70\,Ab{c}^{2}x+90\,B{b}^{2}cx \right ) \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{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 (B x^{2} + A\right )} x^{\frac{13}{2}}}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{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 x^{4} + A x^{2}\right )} \sqrt{c x^{4} + b x^{2}} \sqrt{x}}{c^{2} x^{4} + 2 \, b c x^{2} + b^{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 (B x^{2} + A\right )} x^{\frac{13}{2}}}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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