Optimal. Leaf size=65 \[ -\frac{2 (e x)^{3/2} (4 b c-a d)}{3 a^2 e^3 \left (a+b x^2\right )^{3/4}}-\frac{2 c}{a e \sqrt{e x} \left (a+b x^2\right )^{3/4}} \]
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Rubi [A] time = 0.0311342, antiderivative size = 65, 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} \[ -\frac{2 (e x)^{3/2} (4 b c-a d)}{3 a^2 e^3 \left (a+b x^2\right )^{3/4}}-\frac{2 c}{a e \sqrt{e x} \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Rule 453
Rule 264
Rubi steps
\begin{align*} \int \frac{c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{7/4}} \, dx &=-\frac{2 c}{a e \sqrt{e x} \left (a+b x^2\right )^{3/4}}-\frac{(4 b c-a d) \int \frac{\sqrt{e x}}{\left (a+b x^2\right )^{7/4}} \, dx}{a e^2}\\ &=-\frac{2 c}{a e \sqrt{e x} \left (a+b x^2\right )^{3/4}}-\frac{2 (4 b c-a d) (e x)^{3/2}}{3 a^2 e^3 \left (a+b x^2\right )^{3/4}}\\ \end{align*}
Mathematica [A] time = 0.0185852, size = 44, normalized size = 0.68 \[ \frac{2 x \left (-3 a c+a d x^2-4 b c x^2\right )}{3 a^2 (e x)^{3/2} \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 40, normalized size = 0.6 \begin{align*} -{\frac{2\,x \left ( -ad{x}^{2}+4\,bc{x}^{2}+3\,ac \right ) }{3\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}} \left ( ex \right ) ^{-{\frac{3}{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{7}{4}} \left (e x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.16218, size = 122, normalized size = 1.88 \begin{align*} -\frac{2 \,{\left ({\left (4 \, b c - a d\right )} x^{2} + 3 \, a c\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{e x}}{3 \,{\left (a^{2} b e^{2} x^{3} + a^{3} e^{2} 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{7}{4}} \left (e x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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