Optimal. Leaf size=365 \[ \frac{12 \sqrt [4]{a} \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (a B+A b) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right ),\frac{1}{2}\right )}{5 e^{7/2} \sqrt{a+b x^2}}-\frac{2 \left (a+b x^2\right )^{3/2} (a B+A b)}{a e^3 \sqrt{e x}}+\frac{12 b (e x)^{3/2} \sqrt{a+b x^2} (a B+A b)}{5 a e^5}+\frac{24 \sqrt{b} \sqrt{e x} \sqrt{a+b x^2} (a B+A b)}{5 e^4 \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{24 \sqrt [4]{a} \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (a B+A b) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 e^{7/2} \sqrt{a+b x^2}}-\frac{2 A \left (a+b x^2\right )^{5/2}}{5 a e (e x)^{5/2}} \]
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Rubi [A] time = 0.288736, antiderivative size = 365, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {453, 277, 279, 329, 305, 220, 1196} \[ -\frac{2 \left (a+b x^2\right )^{3/2} (a B+A b)}{a e^3 \sqrt{e x}}+\frac{12 b (e x)^{3/2} \sqrt{a+b x^2} (a B+A b)}{5 a e^5}+\frac{24 \sqrt{b} \sqrt{e x} \sqrt{a+b x^2} (a B+A b)}{5 e^4 \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{12 \sqrt [4]{a} \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (a B+A b) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 e^{7/2} \sqrt{a+b x^2}}-\frac{24 \sqrt [4]{a} \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (a B+A b) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 e^{7/2} \sqrt{a+b x^2}}-\frac{2 A \left (a+b x^2\right )^{5/2}}{5 a e (e x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 453
Rule 277
Rule 279
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{7/2}} \, dx &=-\frac{2 A \left (a+b x^2\right )^{5/2}}{5 a e (e x)^{5/2}}+\frac{(A b+a B) \int \frac{\left (a+b x^2\right )^{3/2}}{(e x)^{3/2}} \, dx}{a e^2}\\ &=-\frac{2 (A b+a B) \left (a+b x^2\right )^{3/2}}{a e^3 \sqrt{e x}}-\frac{2 A \left (a+b x^2\right )^{5/2}}{5 a e (e x)^{5/2}}+\frac{(6 b (A b+a B)) \int \sqrt{e x} \sqrt{a+b x^2} \, dx}{a e^4}\\ &=\frac{12 b (A b+a B) (e x)^{3/2} \sqrt{a+b x^2}}{5 a e^5}-\frac{2 (A b+a B) \left (a+b x^2\right )^{3/2}}{a e^3 \sqrt{e x}}-\frac{2 A \left (a+b x^2\right )^{5/2}}{5 a e (e x)^{5/2}}+\frac{(12 b (A b+a B)) \int \frac{\sqrt{e x}}{\sqrt{a+b x^2}} \, dx}{5 e^4}\\ &=\frac{12 b (A b+a B) (e x)^{3/2} \sqrt{a+b x^2}}{5 a e^5}-\frac{2 (A b+a B) \left (a+b x^2\right )^{3/2}}{a e^3 \sqrt{e x}}-\frac{2 A \left (a+b x^2\right )^{5/2}}{5 a e (e x)^{5/2}}+\frac{(24 b (A b+a B)) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+\frac{b x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{5 e^5}\\ &=\frac{12 b (A b+a B) (e x)^{3/2} \sqrt{a+b x^2}}{5 a e^5}-\frac{2 (A b+a B) \left (a+b x^2\right )^{3/2}}{a e^3 \sqrt{e x}}-\frac{2 A \left (a+b x^2\right )^{5/2}}{5 a e (e x)^{5/2}}+\frac{\left (24 \sqrt{a} \sqrt{b} (A b+a B)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{5 e^4}-\frac{\left (24 \sqrt{a} \sqrt{b} (A b+a B)\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a} e}}{\sqrt{a+\frac{b x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{5 e^4}\\ &=\frac{12 b (A b+a B) (e x)^{3/2} \sqrt{a+b x^2}}{5 a e^5}+\frac{24 \sqrt{b} (A b+a B) \sqrt{e x} \sqrt{a+b x^2}}{5 e^4 \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{2 (A b+a B) \left (a+b x^2\right )^{3/2}}{a e^3 \sqrt{e x}}-\frac{2 A \left (a+b x^2\right )^{5/2}}{5 a e (e x)^{5/2}}-\frac{24 \sqrt [4]{a} \sqrt [4]{b} (A b+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{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 e^{7/2} \sqrt{a+b x^2}}+\frac{12 \sqrt [4]{a} \sqrt [4]{b} (A b+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{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 e^{7/2} \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0682737, size = 84, normalized size = 0.23 \[ \frac{2 x \sqrt{a+b x^2} \left (-\frac{5 x^2 (a B+A b) \, _2F_1\left (-\frac{3}{2},-\frac{1}{4};\frac{3}{4};-\frac{b x^2}{a}\right )}{\sqrt{\frac{b x^2}{a}+1}}-\frac{A \left (a+b x^2\right )^2}{a}\right )}{5 (e x)^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 422, normalized size = 1.2 \begin{align*}{\frac{2}{5\,{x}^{2}{e}^{3}} \left ( 12\,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}^{2}ab-6\,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}^{2}ab+12\,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}^{2}{a}^{2}-6\,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}^{2}{a}^{2}+B{b}^{2}{x}^{6}-7\,A{b}^{2}{x}^{4}-4\,B{x}^{4}ab-8\,aAb{x}^{2}-5\,B{x}^{2}{a}^{2}-A{a}^{2} \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \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 )}{\left (b x^{2} + a\right )}^{\frac{3}{2}}}{\left (e x\right )^{\frac{7}{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 b x^{4} +{\left (B a + A b\right )} x^{2} + A a\right )} \sqrt{b x^{2} + a} \sqrt{e x}}{e^{4} x^{4}}, 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 )}{\left (b x^{2} + a\right )}^{\frac{3}{2}}}{\left (e x\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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