Optimal. Leaf size=67 \[ -\frac{2 \sqrt{e x} (4 b c-3 a d)}{3 a^2 e^3 \sqrt [4]{a+b x^2}}-\frac{2 c}{3 a e (e x)^{3/2} \sqrt [4]{a+b x^2}} \]
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Rubi [A] time = 0.0305918, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {453, 264} \[ -\frac{2 \sqrt{e x} (4 b c-3 a d)}{3 a^2 e^3 \sqrt [4]{a+b x^2}}-\frac{2 c}{3 a e (e x)^{3/2} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 453
Rule 264
Rubi steps
\begin{align*} \int \frac{c+d x^2}{(e x)^{5/2} \left (a+b x^2\right )^{5/4}} \, dx &=-\frac{2 c}{3 a e (e x)^{3/2} \sqrt [4]{a+b x^2}}-\frac{(4 b c-3 a d) \int \frac{1}{\sqrt{e x} \left (a+b x^2\right )^{5/4}} \, dx}{3 a e^2}\\ &=-\frac{2 c}{3 a e (e x)^{3/2} \sqrt [4]{a+b x^2}}-\frac{2 (4 b c-3 a d) \sqrt{e x}}{3 a^2 e^3 \sqrt [4]{a+b x^2}}\\ \end{align*}
Mathematica [A] time = 0.0213481, size = 45, normalized size = 0.67 \[ \frac{x \left (-2 a c+6 a d x^2-8 b c x^2\right )}{3 a^2 (e x)^{5/2} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 39, normalized size = 0.6 \begin{align*} -{\frac{2\,x \left ( -3\,ad{x}^{2}+4\,bc{x}^{2}+ac \right ) }{3\,{a}^{2}}{\frac{1}{\sqrt [4]{b{x}^{2}+a}}} \left ( ex \right ) ^{-{\frac{5}{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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87557, size = 124, normalized size = 1.85 \begin{align*} -\frac{2 \,{\left ({\left (4 \, b c - 3 \, a d\right )} x^{2} + a c\right )}{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{e x}}{3 \,{\left (a^{2} b e^{3} x^{4} + a^{3} e^{3} x^{2}\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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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