Optimal. Leaf size=349 \[ -\frac{e^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (A b-7 a B) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right ),\frac{1}{2}\right )}{4 a^{3/4} b^{11/4} \sqrt{a+b x^2}}+\frac{e^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (A b-7 a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{2 a^{3/4} b^{11/4} \sqrt{a+b x^2}}-\frac{e^2 \sqrt{e x} \sqrt{a+b x^2} (A b-7 a B)}{2 a b^{5/2} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{e (e x)^{3/2} (A b-7 a B)}{6 a b^2 \sqrt{a+b x^2}}+\frac{(e x)^{7/2} (A b-a B)}{3 a b e \left (a+b x^2\right )^{3/2}} \]
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Rubi [A] time = 0.260169, antiderivative size = 349, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {457, 288, 329, 305, 220, 1196} \[ -\frac{e^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (A b-7 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{4 a^{3/4} b^{11/4} \sqrt{a+b x^2}}+\frac{e^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (A b-7 a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{2 a^{3/4} b^{11/4} \sqrt{a+b x^2}}-\frac{e^2 \sqrt{e x} \sqrt{a+b x^2} (A b-7 a B)}{2 a b^{5/2} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{e (e x)^{3/2} (A b-7 a B)}{6 a b^2 \sqrt{a+b x^2}}+\frac{(e x)^{7/2} (A b-a B)}{3 a b e \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 457
Rule 288
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{(e x)^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx &=\frac{(A b-a B) (e x)^{7/2}}{3 a b e \left (a+b x^2\right )^{3/2}}+\frac{\left (-\frac{A b}{2}+\frac{7 a B}{2}\right ) \int \frac{(e x)^{5/2}}{\left (a+b x^2\right )^{3/2}} \, dx}{3 a b}\\ &=\frac{(A b-a B) (e x)^{7/2}}{3 a b e \left (a+b x^2\right )^{3/2}}+\frac{(A b-7 a B) e (e x)^{3/2}}{6 a b^2 \sqrt{a+b x^2}}-\frac{\left ((A b-7 a B) e^2\right ) \int \frac{\sqrt{e x}}{\sqrt{a+b x^2}} \, dx}{4 a b^2}\\ &=\frac{(A b-a B) (e x)^{7/2}}{3 a b e \left (a+b x^2\right )^{3/2}}+\frac{(A b-7 a B) e (e x)^{3/2}}{6 a b^2 \sqrt{a+b x^2}}-\frac{((A b-7 a B) e) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+\frac{b x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{2 a b^2}\\ &=\frac{(A b-a B) (e x)^{7/2}}{3 a b e \left (a+b x^2\right )^{3/2}}+\frac{(A b-7 a B) e (e x)^{3/2}}{6 a b^2 \sqrt{a+b x^2}}-\frac{\left ((A b-7 a B) e^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{2 \sqrt{a} b^{5/2}}+\frac{\left ((A b-7 a B) e^2\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 )}{2 \sqrt{a} b^{5/2}}\\ &=\frac{(A b-a B) (e x)^{7/2}}{3 a b e \left (a+b x^2\right )^{3/2}}+\frac{(A b-7 a B) e (e x)^{3/2}}{6 a b^2 \sqrt{a+b x^2}}-\frac{(A b-7 a B) e^2 \sqrt{e x} \sqrt{a+b x^2}}{2 a b^{5/2} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{(A b-7 a B) e^{5/2} \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 )}{2 a^{3/4} b^{11/4} \sqrt{a+b x^2}}-\frac{(A b-7 a B) e^{5/2} \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 )}{4 a^{3/4} b^{11/4} \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.127703, size = 97, normalized size = 0.28 \[ -\frac{2 e (e x)^{3/2} \left (\left (a+b x^2\right ) \sqrt{\frac{b x^2}{a}+1} (7 a B-A b) \, _2F_1\left (\frac{3}{4},\frac{5}{2};\frac{7}{4};-\frac{b x^2}{a}\right )+a \left (-7 a B+A b-3 b B x^2\right )\right )}{3 a b^2 \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.037, size = 767, normalized size = 2.2 \begin{align*} \text{result too large to display} \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 (e x\right )^{\frac{5}{2}}}{{\left (b x^{2} + a\right )}^{\frac{5}{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 e^{2} x^{4} + A e^{2} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{e x}}{b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}}, 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 (e x\right )^{\frac{5}{2}}}{{\left (b x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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