Optimal. Leaf size=67 \[ \frac{2 \sqrt [4]{a+b x^2} (4 b c-5 a d)}{5 a^2 e^3 \sqrt{e x}}-\frac{2 c \sqrt [4]{a+b x^2}}{5 a e (e x)^{5/2}} \]
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Rubi [A] time = 0.0371541, 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} \[ \frac{2 \sqrt [4]{a+b x^2} (4 b c-5 a d)}{5 a^2 e^3 \sqrt{e x}}-\frac{2 c \sqrt [4]{a+b x^2}}{5 a e (e x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 453
Rule 264
Rubi steps
\begin{align*} \int \frac{c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/4}} \, dx &=-\frac{2 c \sqrt [4]{a+b x^2}}{5 a e (e x)^{5/2}}-\frac{(4 b c-5 a d) \int \frac{1}{(e x)^{3/2} \left (a+b x^2\right )^{3/4}} \, dx}{5 a e^2}\\ &=-\frac{2 c \sqrt [4]{a+b x^2}}{5 a e (e x)^{5/2}}+\frac{2 (4 b c-5 a d) \sqrt [4]{a+b x^2}}{5 a^2 e^3 \sqrt{e x}}\\ \end{align*}
Mathematica [A] time = 0.0196833, size = 44, normalized size = 0.66 \[ -\frac{2 x \sqrt [4]{a+b x^2} \left (a \left (c+5 d x^2\right )-4 b c x^2\right )}{5 a^2 (e x)^{7/2}} \]
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 ( 5\,ad{x}^{2}-4\,bc{x}^{2}+ac \right ) }{5\,{a}^{2}}\sqrt [4]{b{x}^{2}+a} \left ( ex \right ) ^{-{\frac{7}{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{3}{4}} \left (e x\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.07496, size = 101, normalized size = 1.51 \begin{align*} \frac{2 \,{\left ({\left (4 \, b c - 5 \, a d\right )} x^{2} - a c\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{e x}}{5 \, a^{2} e^{4} x^{3}} \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{3}{4}} \left (e x\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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