Optimal. Leaf size=331 \[ -\frac{2 b^{5/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (A b-3 a B) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{15 a^{7/4} \sqrt{a+b x^2}}-\frac{4 b^{3/2} \sqrt{x} \sqrt{a+b x^2} (A b-3 a B)}{15 a^2 \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{4 b^{5/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (A b-3 a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 a^{7/4} \sqrt{a+b x^2}}+\frac{4 b \sqrt{a+b x^2} (A b-3 a B)}{15 a^2 \sqrt{x}}+\frac{2 \sqrt{a+b x^2} (A b-3 a B)}{15 a x^{5/2}}-\frac{2 A \left (a+b x^2\right )^{3/2}}{9 a x^{9/2}} \]
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Rubi [A] time = 0.233339, antiderivative size = 331, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {453, 277, 325, 329, 305, 220, 1196} \[ -\frac{4 b^{3/2} \sqrt{x} \sqrt{a+b x^2} (A b-3 a B)}{15 a^2 \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{2 b^{5/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (A b-3 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 a^{7/4} \sqrt{a+b x^2}}+\frac{4 b^{5/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (A b-3 a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 a^{7/4} \sqrt{a+b x^2}}+\frac{4 b \sqrt{a+b x^2} (A b-3 a B)}{15 a^2 \sqrt{x}}+\frac{2 \sqrt{a+b x^2} (A b-3 a B)}{15 a x^{5/2}}-\frac{2 A \left (a+b x^2\right )^{3/2}}{9 a x^{9/2}} \]
Antiderivative was successfully verified.
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Rule 453
Rule 277
Rule 325
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x^2} \left (A+B x^2\right )}{x^{11/2}} \, dx &=-\frac{2 A \left (a+b x^2\right )^{3/2}}{9 a x^{9/2}}-\frac{\left (2 \left (\frac{3 A b}{2}-\frac{9 a B}{2}\right )\right ) \int \frac{\sqrt{a+b x^2}}{x^{7/2}} \, dx}{9 a}\\ &=\frac{2 (A b-3 a B) \sqrt{a+b x^2}}{15 a x^{5/2}}-\frac{2 A \left (a+b x^2\right )^{3/2}}{9 a x^{9/2}}-\frac{(2 b (A b-3 a B)) \int \frac{1}{x^{3/2} \sqrt{a+b x^2}} \, dx}{15 a}\\ &=\frac{2 (A b-3 a B) \sqrt{a+b x^2}}{15 a x^{5/2}}+\frac{4 b (A b-3 a B) \sqrt{a+b x^2}}{15 a^2 \sqrt{x}}-\frac{2 A \left (a+b x^2\right )^{3/2}}{9 a x^{9/2}}-\frac{\left (2 b^2 (A b-3 a B)\right ) \int \frac{\sqrt{x}}{\sqrt{a+b x^2}} \, dx}{15 a^2}\\ &=\frac{2 (A b-3 a B) \sqrt{a+b x^2}}{15 a x^{5/2}}+\frac{4 b (A b-3 a B) \sqrt{a+b x^2}}{15 a^2 \sqrt{x}}-\frac{2 A \left (a+b x^2\right )^{3/2}}{9 a x^{9/2}}-\frac{\left (4 b^2 (A b-3 a B)\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b x^4}} \, dx,x,\sqrt{x}\right )}{15 a^2}\\ &=\frac{2 (A b-3 a B) \sqrt{a+b x^2}}{15 a x^{5/2}}+\frac{4 b (A b-3 a B) \sqrt{a+b x^2}}{15 a^2 \sqrt{x}}-\frac{2 A \left (a+b x^2\right )^{3/2}}{9 a x^{9/2}}-\frac{\left (4 b^{3/2} (A b-3 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^4}} \, dx,x,\sqrt{x}\right )}{15 a^{3/2}}+\frac{\left (4 b^{3/2} (A b-3 a B)\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a+b x^4}} \, dx,x,\sqrt{x}\right )}{15 a^{3/2}}\\ &=\frac{2 (A b-3 a B) \sqrt{a+b x^2}}{15 a x^{5/2}}+\frac{4 b (A b-3 a B) \sqrt{a+b x^2}}{15 a^2 \sqrt{x}}-\frac{4 b^{3/2} (A b-3 a B) \sqrt{x} \sqrt{a+b x^2}}{15 a^2 \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{2 A \left (a+b x^2\right )^{3/2}}{9 a x^{9/2}}+\frac{4 b^{5/4} (A b-3 a B) \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 a^{7/4} \sqrt{a+b x^2}}-\frac{2 b^{5/4} (A b-3 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 )}{15 a^{7/4} \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0944826, size = 80, normalized size = 0.24 \[ \frac{2 \sqrt{a+b x^2} \left (\frac{3 x^2 (A b-3 a B) \, _2F_1\left (-\frac{5}{4},-\frac{1}{2};-\frac{1}{4};-\frac{b x^2}{a}\right )}{\sqrt{\frac{b x^2}{a}+1}}-5 A \left (a+b x^2\right )\right )}{45 a x^{9/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 439, normalized size = 1.3 \begin{align*} -{\frac{2}{45\,{a}^{2}} \left ( 6\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){x}^{4}a{b}^{2}-3\,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 ){x}^{4}a{b}^{2}-18\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){x}^{4}{a}^{2}b+9\,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 ){x}^{4}{a}^{2}b-6\,A{x}^{6}{b}^{3}+18\,B{x}^{6}a{b}^{2}-4\,A{x}^{4}a{b}^{2}+27\,B{x}^{4}{a}^{2}b+7\,A{x}^{2}{a}^{2}b+9\,B{x}^{2}{a}^{3}+5\,A{a}^{3} \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}{x}^{-{\frac{9}{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{11}{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{11}{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{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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