Optimal. Leaf size=104 \[ -\frac{8 \sqrt{e x} (8 b c-3 a d)}{15 a^3 e^3 \sqrt [4]{a+b x^2}}-\frac{2 \sqrt{e x} (8 b c-3 a d)}{15 a^2 e^3 \left (a+b x^2\right )^{5/4}}-\frac{2 c}{3 a e (e x)^{3/2} \left (a+b x^2\right )^{5/4}} \]
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Rubi [A] time = 0.0481076, 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} \[ -\frac{8 \sqrt{e x} (8 b c-3 a d)}{15 a^3 e^3 \sqrt [4]{a+b x^2}}-\frac{2 \sqrt{e x} (8 b c-3 a d)}{15 a^2 e^3 \left (a+b x^2\right )^{5/4}}-\frac{2 c}{3 a e (e x)^{3/2} \left (a+b x^2\right )^{5/4}} \]
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)^{5/2} \left (a+b x^2\right )^{9/4}} \, dx &=-\frac{2 c}{3 a e (e x)^{3/2} \left (a+b x^2\right )^{5/4}}-\frac{(8 b c-3 a d) \int \frac{1}{\sqrt{e x} \left (a+b x^2\right )^{9/4}} \, dx}{3 a e^2}\\ &=-\frac{2 c}{3 a e (e x)^{3/2} \left (a+b x^2\right )^{5/4}}-\frac{2 (8 b c-3 a d) \sqrt{e x}}{15 a^2 e^3 \left (a+b x^2\right )^{5/4}}-\frac{(4 (8 b c-3 a d)) \int \frac{1}{\sqrt{e x} \left (a+b x^2\right )^{5/4}} \, dx}{15 a^2 e^2}\\ &=-\frac{2 c}{3 a e (e x)^{3/2} \left (a+b x^2\right )^{5/4}}-\frac{2 (8 b c-3 a d) \sqrt{e x}}{15 a^2 e^3 \left (a+b x^2\right )^{5/4}}-\frac{8 (8 b c-3 a d) \sqrt{e x}}{15 a^3 e^3 \sqrt [4]{a+b x^2}}\\ \end{align*}
Mathematica [A] time = 0.0347551, size = 65, normalized size = 0.62 \[ \frac{x \left (-10 a^2 \left (c-3 d x^2\right )+a b \left (24 d x^4-80 c x^2\right )-64 b^2 c x^4\right )}{15 a^3 (e x)^{5/2} \left (a+b x^2\right )^{5/4}} \]
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 ( -12\,abd{x}^{4}+32\,{b}^{2}c{x}^{4}-15\,{a}^{2}d{x}^{2}+40\,abc{x}^{2}+5\,{a}^{2}c \right ) }{15\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}} \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{9}{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 = 1.82257, size = 204, normalized size = 1.96 \begin{align*} -\frac{2 \,{\left (4 \,{\left (8 \, b^{2} c - 3 \, a b d\right )} x^{4} + 5 \, a^{2} c + 5 \,{\left (8 \, a b c - 3 \, a^{2} d\right )} x^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{e x}}{15 \,{\left (a^{3} b^{2} e^{3} x^{6} + 2 \, a^{4} b e^{3} x^{4} + a^{5} 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{9}{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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