Optimal. Leaf size=230 \[ -\frac{77 a^{3/2} e^6 \sqrt{e x} \sqrt [4]{\frac{a}{b x^2}+1} (2 b c-3 a d) E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{20 b^{9/2} \sqrt [4]{a+b x^2}}+\frac{11 e^3 (e x)^{7/2} (2 b c-3 a d)}{30 b^3 \sqrt [4]{a+b x^2}}-\frac{77 a e^5 (e x)^{3/2} (2 b c-3 a d)}{60 b^4 \sqrt [4]{a+b x^2}}-\frac{e (e x)^{11/2} (2 b c-3 a d)}{5 a b^2 \sqrt [4]{a+b x^2}}+\frac{2 (e x)^{15/2} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}} \]
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Rubi [A] time = 0.127523, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {457, 285, 284, 335, 196} \[ -\frac{77 a^{3/2} e^6 \sqrt{e x} \sqrt [4]{\frac{a}{b x^2}+1} (2 b c-3 a d) E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{20 b^{9/2} \sqrt [4]{a+b x^2}}+\frac{11 e^3 (e x)^{7/2} (2 b c-3 a d)}{30 b^3 \sqrt [4]{a+b x^2}}-\frac{77 a e^5 (e x)^{3/2} (2 b c-3 a d)}{60 b^4 \sqrt [4]{a+b x^2}}-\frac{e (e x)^{11/2} (2 b c-3 a d)}{5 a b^2 \sqrt [4]{a+b x^2}}+\frac{2 (e x)^{15/2} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}} \]
Antiderivative was successfully verified.
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Rule 457
Rule 285
Rule 284
Rule 335
Rule 196
Rubi steps
\begin{align*} \int \frac{(e x)^{13/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx &=\frac{2 (b c-a d) (e x)^{15/2}}{5 a b e \left (a+b x^2\right )^{5/4}}+\frac{\left (2 \left (-5 b c+\frac{15 a d}{2}\right )\right ) \int \frac{(e x)^{13/2}}{\left (a+b x^2\right )^{5/4}} \, dx}{5 a b}\\ &=\frac{2 (b c-a d) (e x)^{15/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac{(2 b c-3 a d) e (e x)^{11/2}}{5 a b^2 \sqrt [4]{a+b x^2}}+\frac{\left (11 (2 b c-3 a d) e^2\right ) \int \frac{(e x)^{9/2}}{\left (a+b x^2\right )^{5/4}} \, dx}{10 b^2}\\ &=\frac{2 (b c-a d) (e x)^{15/2}}{5 a b e \left (a+b x^2\right )^{5/4}}+\frac{11 (2 b c-3 a d) e^3 (e x)^{7/2}}{30 b^3 \sqrt [4]{a+b x^2}}-\frac{(2 b c-3 a d) e (e x)^{11/2}}{5 a b^2 \sqrt [4]{a+b x^2}}-\frac{\left (77 a (2 b c-3 a d) e^4\right ) \int \frac{(e x)^{5/2}}{\left (a+b x^2\right )^{5/4}} \, dx}{60 b^3}\\ &=\frac{2 (b c-a d) (e x)^{15/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac{77 a (2 b c-3 a d) e^5 (e x)^{3/2}}{60 b^4 \sqrt [4]{a+b x^2}}+\frac{11 (2 b c-3 a d) e^3 (e x)^{7/2}}{30 b^3 \sqrt [4]{a+b x^2}}-\frac{(2 b c-3 a d) e (e x)^{11/2}}{5 a b^2 \sqrt [4]{a+b x^2}}+\frac{\left (77 a^2 (2 b c-3 a d) e^6\right ) \int \frac{\sqrt{e x}}{\left (a+b x^2\right )^{5/4}} \, dx}{40 b^4}\\ &=\frac{2 (b c-a d) (e x)^{15/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac{77 a (2 b c-3 a d) e^5 (e x)^{3/2}}{60 b^4 \sqrt [4]{a+b x^2}}+\frac{11 (2 b c-3 a d) e^3 (e x)^{7/2}}{30 b^3 \sqrt [4]{a+b x^2}}-\frac{(2 b c-3 a d) e (e x)^{11/2}}{5 a b^2 \sqrt [4]{a+b x^2}}+\frac{\left (77 a^2 (2 b c-3 a d) e^6 \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}{40 b^5 \sqrt [4]{a+b x^2}}\\ &=\frac{2 (b c-a d) (e x)^{15/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac{77 a (2 b c-3 a d) e^5 (e x)^{3/2}}{60 b^4 \sqrt [4]{a+b x^2}}+\frac{11 (2 b c-3 a d) e^3 (e x)^{7/2}}{30 b^3 \sqrt [4]{a+b x^2}}-\frac{(2 b c-3 a d) e (e x)^{11/2}}{5 a b^2 \sqrt [4]{a+b x^2}}-\frac{\left (77 a^2 (2 b c-3 a d) e^6 \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 )}{40 b^5 \sqrt [4]{a+b x^2}}\\ &=\frac{2 (b c-a d) (e x)^{15/2}}{5 a b e \left (a+b x^2\right )^{5/4}}-\frac{77 a (2 b c-3 a d) e^5 (e x)^{3/2}}{60 b^4 \sqrt [4]{a+b x^2}}+\frac{11 (2 b c-3 a d) e^3 (e x)^{7/2}}{30 b^3 \sqrt [4]{a+b x^2}}-\frac{(2 b c-3 a d) e (e x)^{11/2}}{5 a b^2 \sqrt [4]{a+b x^2}}-\frac{77 a^{3/2} (2 b c-3 a d) e^6 \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 )}{20 b^{9/2} \sqrt [4]{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.174434, size = 140, normalized size = 0.61 \[ \frac{e^5 (e x)^{3/2} \left (-110 a^2 b \left (7 c-3 d x^2\right )+1155 a^3 d-20 a b^2 x^2 \left (11 c+3 d x^2\right )-385 a \left (a+b x^2\right ) \sqrt [4]{\frac{b x^2}{a}+1} (3 a d-2 b c) \, _2F_1\left (\frac{3}{4},\frac{9}{4};\frac{7}{4};-\frac{b x^2}{a}\right )+8 b^3 x^4 \left (5 c+3 d x^2\right )\right )}{120 b^4 \left (a+b x^2\right )^{5/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.063, size = 0, normalized size = 0. \begin{align*} \int{(d{x}^{2}+c) \left ( ex \right ) ^{{\frac{13}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{9}{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{13}{2}}}{{\left (b x^{2} + a\right )}^{\frac{9}{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^{6} x^{8} + c e^{6} x^{6}\right )}{\left (b x^{2} + a\right )}^{\frac{3}{4}} \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 (d x^{2} + c\right )} \left (e x\right )^{\frac{13}{2}}}{{\left (b x^{2} + a\right )}^{\frac{9}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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