Optimal. Leaf size=142 \[ \frac{\sqrt{a} e^2 \sqrt{e x} \sqrt [4]{\frac{a}{b x^2}+1} (6 b c-7 a d) E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 b^{5/2} \sqrt [4]{a+b x^2}}+\frac{e (e x)^{3/2} (6 b c-7 a d)}{6 b^2 \sqrt [4]{a+b x^2}}+\frac{d (e x)^{7/2}}{3 b e \sqrt [4]{a+b x^2}} \]
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Rubi [A] time = 0.0697724, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {459, 285, 284, 335, 196} \[ \frac{\sqrt{a} e^2 \sqrt{e x} \sqrt [4]{\frac{a}{b x^2}+1} (6 b c-7 a d) E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 b^{5/2} \sqrt [4]{a+b x^2}}+\frac{e (e x)^{3/2} (6 b c-7 a d)}{6 b^2 \sqrt [4]{a+b x^2}}+\frac{d (e x)^{7/2}}{3 b e \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 459
Rule 285
Rule 284
Rule 335
Rule 196
Rubi steps
\begin{align*} \int \frac{(e x)^{5/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{5/4}} \, dx &=\frac{d (e x)^{7/2}}{3 b e \sqrt [4]{a+b x^2}}-\frac{\left (-3 b c+\frac{7 a d}{2}\right ) \int \frac{(e x)^{5/2}}{\left (a+b x^2\right )^{5/4}} \, dx}{3 b}\\ &=\frac{(6 b c-7 a d) e (e x)^{3/2}}{6 b^2 \sqrt [4]{a+b x^2}}+\frac{d (e x)^{7/2}}{3 b e \sqrt [4]{a+b x^2}}-\frac{\left (a (6 b c-7 a d) e^2\right ) \int \frac{\sqrt{e x}}{\left (a+b x^2\right )^{5/4}} \, dx}{4 b^2}\\ &=\frac{(6 b c-7 a d) e (e x)^{3/2}}{6 b^2 \sqrt [4]{a+b x^2}}+\frac{d (e x)^{7/2}}{3 b e \sqrt [4]{a+b x^2}}-\frac{\left (a (6 b c-7 a d) e^2 \sqrt [4]{1+\frac{a}{b x^2}} \sqrt{e x}\right ) \int \frac{1}{\left (1+\frac{a}{b x^2}\right )^{5/4} x^2} \, dx}{4 b^3 \sqrt [4]{a+b x^2}}\\ &=\frac{(6 b c-7 a d) e (e x)^{3/2}}{6 b^2 \sqrt [4]{a+b x^2}}+\frac{d (e x)^{7/2}}{3 b e \sqrt [4]{a+b x^2}}+\frac{\left (a (6 b c-7 a d) e^2 \sqrt [4]{1+\frac{a}{b x^2}} \sqrt{e x}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{a x^2}{b}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )}{4 b^3 \sqrt [4]{a+b x^2}}\\ &=\frac{(6 b c-7 a d) e (e x)^{3/2}}{6 b^2 \sqrt [4]{a+b x^2}}+\frac{d (e x)^{7/2}}{3 b e \sqrt [4]{a+b x^2}}+\frac{\sqrt{a} (6 b c-7 a d) e^2 \sqrt [4]{1+\frac{a}{b x^2}} \sqrt{e x} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 b^{5/2} \sqrt [4]{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.11123, size = 85, normalized size = 0.6 \[ \frac{e (e x)^{3/2} \left (\sqrt [4]{\frac{b x^2}{a}+1} (7 a d-6 b c) \, _2F_1\left (\frac{3}{4},\frac{5}{4};\frac{7}{4};-\frac{b x^2}{a}\right )-7 a d+6 b c+2 b d x^2\right )}{6 b^2 \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{(d{x}^{2}+c) \left ( ex \right ) ^{{\frac{5}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}}}\, dx \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 (d x^{2} + c\right )} \left (e x\right )^{\frac{5}{2}}}{{\left (b x^{2} + a\right )}^{\frac{5}{4}}}\,{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 (d e^{2} x^{4} + c e^{2} x^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{e x}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{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 (d x^{2} + c\right )} \left (e x\right )^{\frac{5}{2}}}{{\left (b x^{2} + a\right )}^{\frac{5}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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