Optimal. Leaf size=104 \[ \frac{8 \left (a+b x^2\right )^{3/4} (8 b c-7 a d)}{21 a^3 e^3 (e x)^{3/2}}-\frac{2 (8 b c-7 a d)}{7 a^2 e^3 (e x)^{3/2} \sqrt [4]{a+b x^2}}-\frac{2 c}{7 a e (e x)^{7/2} \sqrt [4]{a+b x^2}} \]
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Rubi [A] time = 0.047096, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {453, 273, 264} \[ \frac{8 \left (a+b x^2\right )^{3/4} (8 b c-7 a d)}{21 a^3 e^3 (e x)^{3/2}}-\frac{2 (8 b c-7 a d)}{7 a^2 e^3 (e x)^{3/2} \sqrt [4]{a+b x^2}}-\frac{2 c}{7 a e (e x)^{7/2} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 453
Rule 273
Rule 264
Rubi steps
\begin{align*} \int \frac{c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{5/4}} \, dx &=-\frac{2 c}{7 a e (e x)^{7/2} \sqrt [4]{a+b x^2}}-\frac{(8 b c-7 a d) \int \frac{1}{(e x)^{5/2} \left (a+b x^2\right )^{5/4}} \, dx}{7 a e^2}\\ &=-\frac{2 c}{7 a e (e x)^{7/2} \sqrt [4]{a+b x^2}}-\frac{2 (8 b c-7 a d)}{7 a^2 e^3 (e x)^{3/2} \sqrt [4]{a+b x^2}}-\frac{(4 (8 b c-7 a d)) \int \frac{1}{(e x)^{5/2} \sqrt [4]{a+b x^2}} \, dx}{7 a^2 e^2}\\ &=-\frac{2 c}{7 a e (e x)^{7/2} \sqrt [4]{a+b x^2}}-\frac{2 (8 b c-7 a d)}{7 a^2 e^3 (e x)^{3/2} \sqrt [4]{a+b x^2}}+\frac{8 (8 b c-7 a d) \left (a+b x^2\right )^{3/4}}{21 a^3 e^3 (e x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0295544, size = 71, normalized size = 0.68 \[ -\frac{2 \sqrt{e x} \left (a^2 \left (3 c+7 d x^2\right )+a b \left (28 d x^4-8 c x^2\right )-32 b^2 c x^4\right )}{21 a^3 e^5 x^4 \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 62, normalized size = 0.6 \begin{align*} -{\frac{2\,x \left ( 28\,abd{x}^{4}-32\,{b}^{2}c{x}^{4}+7\,{a}^{2}d{x}^{2}-8\,abc{x}^{2}+3\,{a}^{2}c \right ) }{21\,{a}^{3}}{\frac{1}{\sqrt [4]{b{x}^{2}+a}}} \left ( ex \right ) ^{-{\frac{9}{2}}}} \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{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60484, size = 173, normalized size = 1.66 \begin{align*} \frac{2 \,{\left (4 \,{\left (8 \, b^{2} c - 7 \, a b d\right )} x^{4} - 3 \, a^{2} c +{\left (8 \, a b c - 7 \, a^{2} d\right )} x^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{e x}}{21 \,{\left (a^{3} b e^{5} x^{6} + a^{4} e^{5} x^{4}\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{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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