Optimal. Leaf size=58 \[ \frac{x (d+e x)^2}{3 a \left (a+c x^2\right )^{3/2}}-\frac{2 d (a e-c d x)}{3 a^2 c \sqrt{a+c x^2}} \]
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Rubi [A] time = 0.0202496, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {729, 637} \[ \frac{x (d+e x)^2}{3 a \left (a+c x^2\right )^{3/2}}-\frac{2 d (a e-c d x)}{3 a^2 c \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Rule 729
Rule 637
Rubi steps
\begin{align*} \int \frac{(d+e x)^2}{\left (a+c x^2\right )^{5/2}} \, dx &=\frac{x (d+e x)^2}{3 a \left (a+c x^2\right )^{3/2}}+\frac{(2 d) \int \frac{d+e x}{\left (a+c x^2\right )^{3/2}} \, dx}{3 a}\\ &=\frac{x (d+e x)^2}{3 a \left (a+c x^2\right )^{3/2}}-\frac{2 d (a e-c d x)}{3 a^2 c \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0801214, size = 57, normalized size = 0.98 \[ \frac{-2 a^2 d e+a c x \left (3 d^2+e^2 x^2\right )+2 c^2 d^2 x^3}{3 a^2 c \left (a+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 55, normalized size = 1. \begin{align*} -{\frac{-ac{e}^{2}{x}^{3}-2\,{c}^{2}{d}^{2}{x}^{3}-3\,{d}^{2}xac+2\,de{a}^{2}}{3\,{a}^{2}c} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17244, size = 124, normalized size = 2.14 \begin{align*} \frac{2 \, d^{2} x}{3 \, \sqrt{c x^{2} + a} a^{2}} + \frac{d^{2} x}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} a} - \frac{e^{2} x}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} c} + \frac{e^{2} x}{3 \, \sqrt{c x^{2} + a} a c} - \frac{2 \, d e}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.16358, size = 153, normalized size = 2.64 \begin{align*} \frac{{\left (3 \, a c d^{2} x - 2 \, a^{2} d e +{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{3}\right )} \sqrt{c x^{2} + a}}{3 \,{\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{2}}{\left (a + c x^{2}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3501, size = 74, normalized size = 1.28 \begin{align*} \frac{{\left (\frac{3 \, d^{2}}{a} + \frac{{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2}}{a^{2} c}\right )} x - \frac{2 \, d e}{c}}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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