Optimal. Leaf size=79 \[ -\frac{2 \left (a e^2+c d^2\right ) (a e-c d x)}{3 a^2 c^2 \sqrt{a+c x^2}}-\frac{(d+e x)^2 (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0267251, antiderivative size = 79, 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 = {723, 637} \[ -\frac{2 \left (a e^2+c d^2\right ) (a e-c d x)}{3 a^2 c^2 \sqrt{a+c x^2}}-\frac{(d+e x)^2 (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 723
Rule 637
Rubi steps
\begin{align*} \int \frac{(d+e x)^3}{\left (a+c x^2\right )^{5/2}} \, dx &=-\frac{(a e-c d x) (d+e x)^2}{3 a c \left (a+c x^2\right )^{3/2}}+\frac{\left (2 \left (c d^2+a e^2\right )\right ) \int \frac{d+e x}{\left (a+c x^2\right )^{3/2}} \, dx}{3 a c}\\ &=-\frac{(a e-c d x) (d+e x)^2}{3 a c \left (a+c x^2\right )^{3/2}}-\frac{2 \left (c d^2+a e^2\right ) (a e-c d x)}{3 a^2 c^2 \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.104651, size = 78, normalized size = 0.99 \[ \frac{-3 a^2 c e \left (d^2+e^2 x^2\right )-2 a^3 e^3+3 a c^2 d x \left (d^2+e^2 x^2\right )+2 c^3 d^3 x^3}{3 a^2 c^2 \left (a+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 83, normalized size = 1.1 \begin{align*} -{\frac{-3\,a{c}^{2}d{e}^{2}{x}^{3}-2\,{c}^{3}{d}^{3}{x}^{3}+3\,{e}^{3}{x}^{2}{a}^{2}c-3\,{d}^{3}xa{c}^{2}+2\,{a}^{3}{e}^{3}+3\,{a}^{2}c{d}^{2}e}{3\,{a}^{2}{c}^{2}} \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 = 1.70726, size = 180, normalized size = 2.28 \begin{align*} -\frac{e^{3} x^{2}}{{\left (c x^{2} + a\right )}^{\frac{3}{2}} c} + \frac{2 \, d^{3} x}{3 \, \sqrt{c x^{2} + a} a^{2}} + \frac{d^{3} x}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} a} - \frac{d e^{2} x}{{\left (c x^{2} + a\right )}^{\frac{3}{2}} c} + \frac{d e^{2} x}{\sqrt{c x^{2} + a} a c} - \frac{d^{2} e}{{\left (c x^{2} + a\right )}^{\frac{3}{2}} c} - \frac{2 \, a e^{3}}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.25374, size = 213, normalized size = 2.7 \begin{align*} -\frac{{\left (3 \, a^{2} c e^{3} x^{2} - 3 \, a c^{2} d^{3} x + 3 \, a^{2} c d^{2} e + 2 \, a^{3} e^{3} -{\left (2 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} x^{3}\right )} \sqrt{c x^{2} + a}}{3 \,{\left (a^{2} c^{4} x^{4} + 2 \, a^{3} c^{3} x^{2} + a^{4} c^{2}\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 )^{3}}{\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 = 1.27188, size = 119, normalized size = 1.51 \begin{align*} \frac{{\left (\frac{3 \, d^{3}}{a} - x{\left (\frac{3 \, e^{3}}{c} - \frac{{\left (2 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} x}{a^{2} c^{2}}\right )}\right )} x - \frac{3 \, a^{2} c d^{2} e + 2 \, a^{3} e^{3}}{a^{2} c^{2}}}{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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