Optimal. Leaf size=116 \[ -\frac{5 d^2 \sqrt{d^2-e^2 x^2}}{2 e}-\frac{5 d (d+e x) \sqrt{d^2-e^2 x^2}}{6 e}-\frac{(d+e x)^2 \sqrt{d^2-e^2 x^2}}{3 e}+\frac{5 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]
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Rubi [A] time = 0.0427732, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {671, 641, 217, 203} \[ -\frac{5 d^2 \sqrt{d^2-e^2 x^2}}{2 e}-\frac{5 d (d+e x) \sqrt{d^2-e^2 x^2}}{6 e}-\frac{(d+e x)^2 \sqrt{d^2-e^2 x^2}}{3 e}+\frac{5 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]
Antiderivative was successfully verified.
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Rule 671
Rule 641
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{(d+e x)^3}{\sqrt{d^2-e^2 x^2}} \, dx &=-\frac{(d+e x)^2 \sqrt{d^2-e^2 x^2}}{3 e}+\frac{1}{3} (5 d) \int \frac{(d+e x)^2}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=-\frac{5 d (d+e x) \sqrt{d^2-e^2 x^2}}{6 e}-\frac{(d+e x)^2 \sqrt{d^2-e^2 x^2}}{3 e}+\frac{1}{2} \left (5 d^2\right ) \int \frac{d+e x}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=-\frac{5 d^2 \sqrt{d^2-e^2 x^2}}{2 e}-\frac{5 d (d+e x) \sqrt{d^2-e^2 x^2}}{6 e}-\frac{(d+e x)^2 \sqrt{d^2-e^2 x^2}}{3 e}+\frac{1}{2} \left (5 d^3\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=-\frac{5 d^2 \sqrt{d^2-e^2 x^2}}{2 e}-\frac{5 d (d+e x) \sqrt{d^2-e^2 x^2}}{6 e}-\frac{(d+e x)^2 \sqrt{d^2-e^2 x^2}}{3 e}+\frac{1}{2} \left (5 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )\\ &=-\frac{5 d^2 \sqrt{d^2-e^2 x^2}}{2 e}-\frac{5 d (d+e x) \sqrt{d^2-e^2 x^2}}{6 e}-\frac{(d+e x)^2 \sqrt{d^2-e^2 x^2}}{3 e}+\frac{5 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e}\\ \end{align*}
Mathematica [A] time = 0.0600884, size = 70, normalized size = 0.6 \[ \frac{15 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\sqrt{d^2-e^2 x^2} \left (22 d^2+9 d e x+2 e^2 x^2\right )}{6 e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 94, normalized size = 0.8 \begin{align*} -{\frac{e{x}^{2}}{3}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{11\,{d}^{2}}{3\,e}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{3\,dx}{2}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{5\,{d}^{3}}{2}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.7962, size = 116, normalized size = 1. \begin{align*} -\frac{1}{3} \, \sqrt{-e^{2} x^{2} + d^{2}} e x^{2} + \frac{5 \, d^{3} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{2 \, \sqrt{e^{2}}} - \frac{3}{2} \, \sqrt{-e^{2} x^{2} + d^{2}} d x - \frac{11 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{2}}{3 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.16217, size = 153, normalized size = 1.32 \begin{align*} -\frac{30 \, d^{3} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (2 \, e^{2} x^{2} + 9 \, d e x + 22 \, d^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{6 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.36979, size = 338, normalized size = 2.91 \begin{align*} d^{3} \left (\begin{cases} \frac{\sqrt{\frac{d^{2}}{e^{2}}} \operatorname{asin}{\left (x \sqrt{\frac{e^{2}}{d^{2}}} \right )}}{\sqrt{d^{2}}} & \text{for}\: d^{2} > 0 \wedge e^{2} > 0 \\\frac{\sqrt{- \frac{d^{2}}{e^{2}}} \operatorname{asinh}{\left (x \sqrt{- \frac{e^{2}}{d^{2}}} \right )}}{\sqrt{d^{2}}} & \text{for}\: d^{2} > 0 \wedge e^{2} < 0 \\\frac{\sqrt{\frac{d^{2}}{e^{2}}} \operatorname{acosh}{\left (x \sqrt{\frac{e^{2}}{d^{2}}} \right )}}{\sqrt{- d^{2}}} & \text{for}\: d^{2} < 0 \wedge e^{2} < 0 \end{cases}\right ) + 3 d^{2} e \left (\begin{cases} \frac{x^{2}}{2 \sqrt{d^{2}}} & \text{for}\: e^{2} = 0 \\- \frac{\sqrt{d^{2} - e^{2} x^{2}}}{e^{2}} & \text{otherwise} \end{cases}\right ) + 3 d e^{2} \left (\begin{cases} - \frac{i d^{2} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{2 e^{3}} - \frac{i d x \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}}{2 e^{2}} & \text{for}\: \frac{\left |{e^{2} x^{2}}\right |}{\left |{d^{2}}\right |} > 1 \\\frac{d^{2} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{2 e^{3}} - \frac{d x}{2 e^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{x^{3}}{2 d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) + e^{3} \left (\begin{cases} - \frac{2 d^{2} \sqrt{d^{2} - e^{2} x^{2}}}{3 e^{4}} - \frac{x^{2} \sqrt{d^{2} - e^{2} x^{2}}}{3 e^{2}} & \text{for}\: e \neq 0 \\\frac{x^{4}}{4 \sqrt{d^{2}}} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36994, size = 70, normalized size = 0.6 \begin{align*} \frac{5}{2} \, d^{3} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )} \mathrm{sgn}\left (d\right ) - \frac{1}{6} \, \sqrt{-x^{2} e^{2} + d^{2}}{\left (22 \, d^{2} e^{\left (-1\right )} +{\left (2 \, x e + 9 \, d\right )} x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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