Optimal. Leaf size=148 \[ \frac{d^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e^4 (d+e x)^4}-\frac{14 d \left (d^2-e^2 x^2\right )^{3/2}}{15 e^4 (d+e x)^3}+\frac{8 d \sqrt{d^2-e^2 x^2}}{e^4 (d+e x)}-\frac{\left (d^2-e^2 x^2\right )^{3/2}}{e^4 (d+e x)^2}+\frac{4 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^4} \]
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Rubi [A] time = 0.246061, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {1639, 1637, 659, 651, 663, 217, 203} \[ \frac{d^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e^4 (d+e x)^4}-\frac{14 d \left (d^2-e^2 x^2\right )^{3/2}}{15 e^4 (d+e x)^3}+\frac{8 d \sqrt{d^2-e^2 x^2}}{e^4 (d+e x)}-\frac{\left (d^2-e^2 x^2\right )^{3/2}}{e^4 (d+e x)^2}+\frac{4 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^4} \]
Antiderivative was successfully verified.
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Rule 1639
Rule 1637
Rule 659
Rule 651
Rule 663
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{x^3 \sqrt{d^2-e^2 x^2}}{(d+e x)^4} \, dx &=-\frac{\left (d^2-e^2 x^2\right )^{3/2}}{e^4 (d+e x)^2}-\frac{\int \frac{\sqrt{d^2-e^2 x^2} \left (2 d^3 e^2+5 d^2 e^3 x+4 d e^4 x^2\right )}{(d+e x)^4} \, dx}{e^5}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{3/2}}{e^4 (d+e x)^2}-\frac{\int \left (\frac{d^3 e^2 \sqrt{d^2-e^2 x^2}}{(d+e x)^4}-\frac{3 d^2 e^2 \sqrt{d^2-e^2 x^2}}{(d+e x)^3}+\frac{4 d e^2 \sqrt{d^2-e^2 x^2}}{(d+e x)^2}\right ) \, dx}{e^5}\\ &=-\frac{\left (d^2-e^2 x^2\right )^{3/2}}{e^4 (d+e x)^2}-\frac{(4 d) \int \frac{\sqrt{d^2-e^2 x^2}}{(d+e x)^2} \, dx}{e^3}+\frac{\left (3 d^2\right ) \int \frac{\sqrt{d^2-e^2 x^2}}{(d+e x)^3} \, dx}{e^3}-\frac{d^3 \int \frac{\sqrt{d^2-e^2 x^2}}{(d+e x)^4} \, dx}{e^3}\\ &=\frac{8 d \sqrt{d^2-e^2 x^2}}{e^4 (d+e x)}+\frac{d^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e^4 (d+e x)^4}-\frac{d \left (d^2-e^2 x^2\right )^{3/2}}{e^4 (d+e x)^3}-\frac{\left (d^2-e^2 x^2\right )^{3/2}}{e^4 (d+e x)^2}+\frac{(4 d) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{e^3}-\frac{d^2 \int \frac{\sqrt{d^2-e^2 x^2}}{(d+e x)^3} \, dx}{5 e^3}\\ &=\frac{8 d \sqrt{d^2-e^2 x^2}}{e^4 (d+e x)}+\frac{d^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e^4 (d+e x)^4}-\frac{14 d \left (d^2-e^2 x^2\right )^{3/2}}{15 e^4 (d+e x)^3}-\frac{\left (d^2-e^2 x^2\right )^{3/2}}{e^4 (d+e x)^2}+\frac{(4 d) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )}{e^3}\\ &=\frac{8 d \sqrt{d^2-e^2 x^2}}{e^4 (d+e x)}+\frac{d^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e^4 (d+e x)^4}-\frac{14 d \left (d^2-e^2 x^2\right )^{3/2}}{15 e^4 (d+e x)^3}-\frac{\left (d^2-e^2 x^2\right )^{3/2}}{e^4 (d+e x)^2}+\frac{4 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^4}\\ \end{align*}
Mathematica [A] time = 0.144645, size = 85, normalized size = 0.57 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (222 d^2 e x+94 d^3+149 d e^2 x^2+15 e^3 x^3\right )}{(d+e x)^3}+60 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{15 e^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.063, size = 212, normalized size = 1.4 \begin{align*}{\frac{{d}^{2}}{5\,{e}^{8}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-4}}-{\frac{14\,d}{15\,{e}^{7}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{{\frac{3}{2}}} \left ({\frac{d}{e}}+x \right ) ^{-3}}+4\,{\frac{1}{{e}^{4}}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}+4\,{\frac{d}{{e}^{3}\sqrt{{e}^{2}}}\arctan \left ({\sqrt{{e}^{2}}x{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}} \right ) }+3\,{\frac{1}{{e}^{6}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{3/2} \left ({\frac{d}{e}}+x \right ) ^{-2}} \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 = 1.78005, size = 377, normalized size = 2.55 \begin{align*} \frac{94 \, d e^{3} x^{3} + 282 \, d^{2} e^{2} x^{2} + 282 \, d^{3} e x + 94 \, d^{4} - 120 \,{\left (d e^{3} x^{3} + 3 \, d^{2} e^{2} x^{2} + 3 \, d^{3} e x + d^{4}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (15 \, e^{3} x^{3} + 149 \, d e^{2} x^{2} + 222 \, d^{2} e x + 94 \, d^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\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{x^{3} \sqrt{- \left (- d + e x\right ) \left (d + e x\right )}}{\left (d + e x\right )^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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