Optimal. Leaf size=149 \[ -\frac{35 d^3 \sqrt{d^2-e^2 x^2}}{8 e}-\frac{35 d^2 (d+e x) \sqrt{d^2-e^2 x^2}}{24 e}-\frac{7 d (d+e x)^2 \sqrt{d^2-e^2 x^2}}{12 e}-\frac{(d+e x)^3 \sqrt{d^2-e^2 x^2}}{4 e}+\frac{35 d^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e} \]
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Rubi [A] time = 0.0618129, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {671, 641, 217, 203} \[ -\frac{35 d^3 \sqrt{d^2-e^2 x^2}}{8 e}-\frac{35 d^2 (d+e x) \sqrt{d^2-e^2 x^2}}{24 e}-\frac{7 d (d+e x)^2 \sqrt{d^2-e^2 x^2}}{12 e}-\frac{(d+e x)^3 \sqrt{d^2-e^2 x^2}}{4 e}+\frac{35 d^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e} \]
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)^4}{\sqrt{d^2-e^2 x^2}} \, dx &=-\frac{(d+e x)^3 \sqrt{d^2-e^2 x^2}}{4 e}+\frac{1}{4} (7 d) \int \frac{(d+e x)^3}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=-\frac{7 d (d+e x)^2 \sqrt{d^2-e^2 x^2}}{12 e}-\frac{(d+e x)^3 \sqrt{d^2-e^2 x^2}}{4 e}+\frac{1}{12} \left (35 d^2\right ) \int \frac{(d+e x)^2}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=-\frac{35 d^2 (d+e x) \sqrt{d^2-e^2 x^2}}{24 e}-\frac{7 d (d+e x)^2 \sqrt{d^2-e^2 x^2}}{12 e}-\frac{(d+e x)^3 \sqrt{d^2-e^2 x^2}}{4 e}+\frac{1}{8} \left (35 d^3\right ) \int \frac{d+e x}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=-\frac{35 d^3 \sqrt{d^2-e^2 x^2}}{8 e}-\frac{35 d^2 (d+e x) \sqrt{d^2-e^2 x^2}}{24 e}-\frac{7 d (d+e x)^2 \sqrt{d^2-e^2 x^2}}{12 e}-\frac{(d+e x)^3 \sqrt{d^2-e^2 x^2}}{4 e}+\frac{1}{8} \left (35 d^4\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=-\frac{35 d^3 \sqrt{d^2-e^2 x^2}}{8 e}-\frac{35 d^2 (d+e x) \sqrt{d^2-e^2 x^2}}{24 e}-\frac{7 d (d+e x)^2 \sqrt{d^2-e^2 x^2}}{12 e}-\frac{(d+e x)^3 \sqrt{d^2-e^2 x^2}}{4 e}+\frac{1}{8} \left (35 d^4\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{35 d^3 \sqrt{d^2-e^2 x^2}}{8 e}-\frac{35 d^2 (d+e x) \sqrt{d^2-e^2 x^2}}{24 e}-\frac{7 d (d+e x)^2 \sqrt{d^2-e^2 x^2}}{12 e}-\frac{(d+e x)^3 \sqrt{d^2-e^2 x^2}}{4 e}+\frac{35 d^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e}\\ \end{align*}
Mathematica [A] time = 0.0835836, size = 81, normalized size = 0.54 \[ \frac{105 d^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\sqrt{d^2-e^2 x^2} \left (81 d^2 e x+160 d^3+32 d e^2 x^2+6 e^3 x^3\right )}{24 e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 119, normalized size = 0.8 \begin{align*} -{\frac{{e}^{2}{x}^{3}}{4}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{27\,{d}^{2}x}{8}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{35\,{d}^{4}}{8}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{4\,de{x}^{2}}{3}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{20\,{d}^{3}}{3\,e}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.6912, size = 150, normalized size = 1.01 \begin{align*} -\frac{1}{4} \, \sqrt{-e^{2} x^{2} + d^{2}} e^{2} x^{3} - \frac{4}{3} \, \sqrt{-e^{2} x^{2} + d^{2}} d e x^{2} + \frac{35 \, d^{4} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{8 \, \sqrt{e^{2}}} - \frac{27}{8} \, \sqrt{-e^{2} x^{2} + d^{2}} d^{2} x - \frac{20 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{3}}{3 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.19204, size = 181, normalized size = 1.21 \begin{align*} -\frac{210 \, d^{4} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (6 \, e^{3} x^{3} + 32 \, d e^{2} x^{2} + 81 \, d^{2} e x + 160 \, d^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{24 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.19789, size = 549, normalized size = 3.68 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29989, size = 85, normalized size = 0.57 \begin{align*} \frac{35}{8} \, d^{4} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )} \mathrm{sgn}\left (d\right ) - \frac{1}{24} \,{\left (160 \, d^{3} e^{\left (-1\right )} +{\left (81 \, d^{2} + 2 \,{\left (3 \, x e^{2} + 16 \, d e\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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