Optimal. Leaf size=144 \[ \frac{2 (6 b c-5 a d)}{5 a^2 e^3 \sqrt{e x} \sqrt [4]{a+b x^2}}-\frac{4 \sqrt{b} \sqrt{e x} \sqrt [4]{\frac{a}{b x^2}+1} (6 b c-5 a d) E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 a^{5/2} e^4 \sqrt [4]{a+b x^2}}-\frac{2 c}{5 a e (e x)^{5/2} \sqrt [4]{a+b x^2}} \]
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Rubi [A] time = 0.0748738, antiderivative size = 144, 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 = {453, 286, 284, 335, 196} \[ \frac{2 (6 b c-5 a d)}{5 a^2 e^3 \sqrt{e x} \sqrt [4]{a+b x^2}}-\frac{4 \sqrt{b} \sqrt{e x} \sqrt [4]{\frac{a}{b x^2}+1} (6 b c-5 a d) E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 a^{5/2} e^4 \sqrt [4]{a+b x^2}}-\frac{2 c}{5 a e (e x)^{5/2} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 453
Rule 286
Rule 284
Rule 335
Rule 196
Rubi steps
\begin{align*} \int \frac{c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{5/4}} \, dx &=-\frac{2 c}{5 a e (e x)^{5/2} \sqrt [4]{a+b x^2}}-\frac{(6 b c-5 a d) \int \frac{1}{(e x)^{3/2} \left (a+b x^2\right )^{5/4}} \, dx}{5 a e^2}\\ &=-\frac{2 c}{5 a e (e x)^{5/2} \sqrt [4]{a+b x^2}}+\frac{2 (6 b c-5 a d)}{5 a^2 e^3 \sqrt{e x} \sqrt [4]{a+b x^2}}+\frac{(2 b (6 b c-5 a d)) \int \frac{\sqrt{e x}}{\left (a+b x^2\right )^{5/4}} \, dx}{5 a^2 e^4}\\ &=-\frac{2 c}{5 a e (e x)^{5/2} \sqrt [4]{a+b x^2}}+\frac{2 (6 b c-5 a d)}{5 a^2 e^3 \sqrt{e x} \sqrt [4]{a+b x^2}}+\frac{\left (2 (6 b c-5 a d) \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}{5 a^2 e^4 \sqrt [4]{a+b x^2}}\\ &=-\frac{2 c}{5 a e (e x)^{5/2} \sqrt [4]{a+b x^2}}+\frac{2 (6 b c-5 a d)}{5 a^2 e^3 \sqrt{e x} \sqrt [4]{a+b x^2}}-\frac{\left (2 (6 b c-5 a d) \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 )}{5 a^2 e^4 \sqrt [4]{a+b x^2}}\\ &=-\frac{2 c}{5 a e (e x)^{5/2} \sqrt [4]{a+b x^2}}+\frac{2 (6 b c-5 a d)}{5 a^2 e^3 \sqrt{e x} \sqrt [4]{a+b x^2}}-\frac{4 \sqrt{b} (6 b c-5 a d) \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 )}{5 a^{5/2} e^4 \sqrt [4]{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0443797, size = 78, normalized size = 0.54 \[ \frac{x \left (2 x^2 \sqrt [4]{\frac{b x^2}{a}+1} (6 b c-5 a d) \, _2F_1\left (-\frac{1}{4},\frac{5}{4};\frac{3}{4};-\frac{b x^2}{a}\right )-2 a c\right )}{5 a^2 (e x)^{7/2} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.065, size = 0, normalized size = 0. \begin{align*} \int{(d{x}^{2}+c) \left ( ex \right ) ^{-{\frac{7}{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{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} \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 x^{2} + a\right )}^{\frac{3}{4}}{\left (d x^{2} + c\right )} \sqrt{e x}}{b^{2} e^{4} x^{8} + 2 \, a b e^{4} x^{6} + a^{2} 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{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} \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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