Optimal. Leaf size=187 \[ \frac{2 b^{7/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (5 A b-11 a B) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{231 a^{9/4} \sqrt{a+b x^2}}+\frac{4 b \sqrt{a+b x^2} (5 A b-11 a B)}{231 a^2 x^{3/2}}+\frac{2 \sqrt{a+b x^2} (5 A b-11 a B)}{77 a x^{7/2}}-\frac{2 A \left (a+b x^2\right )^{3/2}}{11 a x^{11/2}} \]
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Rubi [A] time = 0.112352, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {453, 277, 325, 329, 220} \[ \frac{2 b^{7/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (5 A b-11 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{231 a^{9/4} \sqrt{a+b x^2}}+\frac{4 b \sqrt{a+b x^2} (5 A b-11 a B)}{231 a^2 x^{3/2}}+\frac{2 \sqrt{a+b x^2} (5 A b-11 a B)}{77 a x^{7/2}}-\frac{2 A \left (a+b x^2\right )^{3/2}}{11 a x^{11/2}} \]
Antiderivative was successfully verified.
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Rule 453
Rule 277
Rule 325
Rule 329
Rule 220
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x^2} \left (A+B x^2\right )}{x^{13/2}} \, dx &=-\frac{2 A \left (a+b x^2\right )^{3/2}}{11 a x^{11/2}}-\frac{\left (2 \left (\frac{5 A b}{2}-\frac{11 a B}{2}\right )\right ) \int \frac{\sqrt{a+b x^2}}{x^{9/2}} \, dx}{11 a}\\ &=\frac{2 (5 A b-11 a B) \sqrt{a+b x^2}}{77 a x^{7/2}}-\frac{2 A \left (a+b x^2\right )^{3/2}}{11 a x^{11/2}}-\frac{(2 b (5 A b-11 a B)) \int \frac{1}{x^{5/2} \sqrt{a+b x^2}} \, dx}{77 a}\\ &=\frac{2 (5 A b-11 a B) \sqrt{a+b x^2}}{77 a x^{7/2}}+\frac{4 b (5 A b-11 a B) \sqrt{a+b x^2}}{231 a^2 x^{3/2}}-\frac{2 A \left (a+b x^2\right )^{3/2}}{11 a x^{11/2}}+\frac{\left (2 b^2 (5 A b-11 a B)\right ) \int \frac{1}{\sqrt{x} \sqrt{a+b x^2}} \, dx}{231 a^2}\\ &=\frac{2 (5 A b-11 a B) \sqrt{a+b x^2}}{77 a x^{7/2}}+\frac{4 b (5 A b-11 a B) \sqrt{a+b x^2}}{231 a^2 x^{3/2}}-\frac{2 A \left (a+b x^2\right )^{3/2}}{11 a x^{11/2}}+\frac{\left (4 b^2 (5 A b-11 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^4}} \, dx,x,\sqrt{x}\right )}{231 a^2}\\ &=\frac{2 (5 A b-11 a B) \sqrt{a+b x^2}}{77 a x^{7/2}}+\frac{4 b (5 A b-11 a B) \sqrt{a+b x^2}}{231 a^2 x^{3/2}}-\frac{2 A \left (a+b x^2\right )^{3/2}}{11 a x^{11/2}}+\frac{2 b^{7/4} (5 A b-11 a B) \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{231 a^{9/4} \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.100848, size = 80, normalized size = 0.43 \[ \frac{2 \sqrt{a+b x^2} \left (\frac{x^2 (5 A b-11 a B) \, _2F_1\left (-\frac{7}{4},-\frac{1}{2};-\frac{3}{4};-\frac{b x^2}{a}\right )}{\sqrt{\frac{b x^2}{a}+1}}-7 A \left (a+b x^2\right )\right )}{77 a x^{11/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 270, normalized size = 1.4 \begin{align*}{\frac{2}{231\,{a}^{2}} \left ( 5\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}{x}^{5}{b}^{2}-11\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}{x}^{5}ab+10\,A{x}^{6}{b}^{3}-22\,B{x}^{6}a{b}^{2}+4\,A{x}^{4}a{b}^{2}-55\,B{x}^{4}{a}^{2}b-27\,A{x}^{2}{a}^{2}b-33\,B{x}^{2}{a}^{3}-21\,A{a}^{3} \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}{x}^{-{\frac{11}{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 )} \sqrt{b x^{2} + a}}{x^{\frac{13}{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^{2} + A\right )} \sqrt{b x^{2} + a}}{x^{\frac{13}{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 )} \sqrt{b x^{2} + a}}{x^{\frac{13}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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