Optimal. Leaf size=90 \[ \frac{x^2 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{x}{5 d^2 e^3 \sqrt{d^2-e^2 x^2}}-\frac{2 d+3 e x}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0420948, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {819, 778, 191} \[ \frac{x^2 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{x}{5 d^2 e^3 \sqrt{d^2-e^2 x^2}}-\frac{2 d+3 e x}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 819
Rule 778
Rule 191
Rubi steps
\begin{align*} \int \frac{x^3 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{x^2 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{\int \frac{x \left (2 d^3+3 d^2 e x\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2}\\ &=\frac{x^2 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2 d+3 e x}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{\int \frac{1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{5 e^3}\\ &=\frac{x^2 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{2 d+3 e x}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{x}{5 d^2 e^3 \sqrt{d^2-e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0257207, size = 82, normalized size = 0.91 \[ \frac{3 d^2 e^2 x^2+2 d^3 e x-2 d^4-3 d e^3 x^3+3 e^4 x^4}{15 d^2 e^4 (d-e x)^2 (d+e x) \sqrt{d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.046, size = 77, normalized size = 0.9 \begin{align*} -{\frac{ \left ( -ex+d \right ) \left ( ex+d \right ) ^{2} \left ( -3\,{x}^{4}{e}^{4}+3\,{x}^{3}d{e}^{3}-3\,{x}^{2}{d}^{2}{e}^{2}-2\,{d}^{3}xe+2\,{d}^{4} \right ) }{15\,{d}^{2}{e}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.985996, size = 181, normalized size = 2.01 \begin{align*} \frac{x^{3}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{d x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{3 \, d^{2} x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{3}} - \frac{2 \, d^{3}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} + \frac{x}{10 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{3}} + \frac{x}{5 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{2} e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.96432, size = 340, normalized size = 3.78 \begin{align*} -\frac{2 \, e^{5} x^{5} - 2 \, d e^{4} x^{4} - 4 \, d^{2} e^{3} x^{3} + 4 \, d^{3} e^{2} x^{2} + 2 \, d^{4} e x - 2 \, d^{5} +{\left (3 \, e^{4} x^{4} - 3 \, d e^{3} x^{3} + 3 \, d^{2} e^{2} x^{2} + 2 \, d^{3} e x - 2 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{2} e^{9} x^{5} - d^{3} e^{8} x^{4} - 2 \, d^{4} e^{7} x^{3} + 2 \, d^{5} e^{6} x^{2} + d^{6} e^{5} x - d^{7} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 13.4521, size = 338, normalized size = 3.76 \begin{align*} d \left (\begin{cases} - \frac{2 d^{2}}{15 d^{4} e^{4} \sqrt{d^{2} - e^{2} x^{2}} - 30 d^{2} e^{6} x^{2} \sqrt{d^{2} - e^{2} x^{2}} + 15 e^{8} x^{4} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{5 e^{2} x^{2}}{15 d^{4} e^{4} \sqrt{d^{2} - e^{2} x^{2}} - 30 d^{2} e^{6} x^{2} \sqrt{d^{2} - e^{2} x^{2}} + 15 e^{8} x^{4} \sqrt{d^{2} - e^{2} x^{2}}} & \text{for}\: e \neq 0 \\\frac{x^{4}}{4 \left (d^{2}\right )^{\frac{7}{2}}} & \text{otherwise} \end{cases}\right ) + e \left (\begin{cases} - \frac{i x^{5}}{5 d^{7} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} - 10 d^{5} e^{2} x^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}} + 5 d^{3} e^{4} x^{4} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \frac{\left |{e^{2} x^{2}}\right |}{\left |{d^{2}}\right |} > 1 \\\frac{x^{5}}{5 d^{7} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} - 10 d^{5} e^{2} x^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}} + 5 d^{3} e^{4} x^{4} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.20696, size = 78, normalized size = 0.87 \begin{align*} \frac{{\left (2 \, d^{3} e^{\left (-4\right )} -{\left (\frac{3 \, x^{3} e}{d^{2}} + 5 \, d e^{\left (-2\right )}\right )} x^{2}\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]