Optimal. Leaf size=83 \[ \frac{e^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2}}-\frac{d e \sqrt{a+c x^2}}{a c}-\frac{(d+e x) (a e-c d x)}{a c \sqrt{a+c x^2}} \]
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Rubi [A] time = 0.0345132, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {739, 641, 217, 206} \[ \frac{e^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2}}-\frac{d e \sqrt{a+c x^2}}{a c}-\frac{(d+e x) (a e-c d x)}{a c \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Rule 739
Rule 641
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{(d+e x)^2}{\left (a+c x^2\right )^{3/2}} \, dx &=-\frac{(a e-c d x) (d+e x)}{a c \sqrt{a+c x^2}}+\frac{\int \frac{a e^2-c d e x}{\sqrt{a+c x^2}} \, dx}{a c}\\ &=-\frac{(a e-c d x) (d+e x)}{a c \sqrt{a+c x^2}}-\frac{d e \sqrt{a+c x^2}}{a c}+\frac{e^2 \int \frac{1}{\sqrt{a+c x^2}} \, dx}{c}\\ &=-\frac{(a e-c d x) (d+e x)}{a c \sqrt{a+c x^2}}-\frac{d e \sqrt{a+c x^2}}{a c}+\frac{e^2 \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{c}\\ &=-\frac{(a e-c d x) (d+e x)}{a c \sqrt{a+c x^2}}-\frac{d e \sqrt{a+c x^2}}{a c}+\frac{e^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0697005, size = 69, normalized size = 0.83 \[ \frac{e^2 \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{c^{3/2}}+\frac{-2 a d e-a e^2 x+c d^2 x}{a c \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 76, normalized size = 0.9 \begin{align*} -{\frac{{e}^{2}x}{c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{{e}^{2}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}-2\,{\frac{de}{c\sqrt{c{x}^{2}+a}}}+{\frac{{d}^{2}x}{a}{\frac{1}{\sqrt{c{x}^{2}+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99505, size = 428, normalized size = 5.16 \begin{align*} \left [\frac{{\left (a c e^{2} x^{2} + a^{2} e^{2}\right )} \sqrt{c} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) - 2 \,{\left (2 \, a c d e -{\left (c^{2} d^{2} - a c e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{2 \,{\left (a c^{3} x^{2} + a^{2} c^{2}\right )}}, -\frac{{\left (a c e^{2} x^{2} + a^{2} e^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) +{\left (2 \, a c d e -{\left (c^{2} d^{2} - a c e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{a c^{3} x^{2} + a^{2} 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 )^{2}}{\left (a + c x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.39254, size = 93, normalized size = 1.12 \begin{align*} -\frac{\frac{2 \, d e}{c} - \frac{{\left (c^{2} d^{2} - a c e^{2}\right )} x}{a c^{2}}}{\sqrt{c x^{2} + a}} - \frac{e^{2} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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