Optimal. Leaf size=130 \[ \frac {\left (d^2-e^2 x^2\right )^{7/2}}{e^2 (d+e x)^4}+\frac {8 \left (d^2-e^2 x^2\right )^{5/2}}{e^2 (d+e x)^2}+\frac {20 \left (d^2-e^2 x^2\right )^{3/2}}{3 e^2}+\frac {10 d x \sqrt {d^2-e^2 x^2}}{e}+\frac {10 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^2} \]
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Rubi [A] time = 0.06, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {793, 663, 665, 195, 217, 203} \begin {gather*} \frac {\left (d^2-e^2 x^2\right )^{7/2}}{e^2 (d+e x)^4}+\frac {8 \left (d^2-e^2 x^2\right )^{5/2}}{e^2 (d+e x)^2}+\frac {20 \left (d^2-e^2 x^2\right )^{3/2}}{3 e^2}+\frac {10 d x \sqrt {d^2-e^2 x^2}}{e}+\frac {10 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 195
Rule 203
Rule 217
Rule 663
Rule 665
Rule 793
Rubi steps
\begin {align*} \int \frac {x \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx &=\frac {\left (d^2-e^2 x^2\right )^{7/2}}{e^2 (d+e x)^4}+\frac {4 \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx}{e}\\ &=\frac {8 \left (d^2-e^2 x^2\right )^{5/2}}{e^2 (d+e x)^2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{e^2 (d+e x)^4}+\frac {20 \int \frac {\left (d^2-e^2 x^2\right )^{3/2}}{d+e x} \, dx}{e}\\ &=\frac {20 \left (d^2-e^2 x^2\right )^{3/2}}{3 e^2}+\frac {8 \left (d^2-e^2 x^2\right )^{5/2}}{e^2 (d+e x)^2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{e^2 (d+e x)^4}+\frac {(20 d) \int \sqrt {d^2-e^2 x^2} \, dx}{e}\\ &=\frac {10 d x \sqrt {d^2-e^2 x^2}}{e}+\frac {20 \left (d^2-e^2 x^2\right )^{3/2}}{3 e^2}+\frac {8 \left (d^2-e^2 x^2\right )^{5/2}}{e^2 (d+e x)^2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{e^2 (d+e x)^4}+\frac {\left (10 d^3\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{e}\\ &=\frac {10 d x \sqrt {d^2-e^2 x^2}}{e}+\frac {20 \left (d^2-e^2 x^2\right )^{3/2}}{3 e^2}+\frac {8 \left (d^2-e^2 x^2\right )^{5/2}}{e^2 (d+e x)^2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{e^2 (d+e x)^4}+\frac {\left (10 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 )}{e}\\ &=\frac {10 d x \sqrt {d^2-e^2 x^2}}{e}+\frac {20 \left (d^2-e^2 x^2\right )^{3/2}}{3 e^2}+\frac {8 \left (d^2-e^2 x^2\right )^{5/2}}{e^2 (d+e x)^2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{e^2 (d+e x)^4}+\frac {10 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^2}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 83, normalized size = 0.64 \begin {gather*} \frac {1}{3} \sqrt {d^2-e^2 x^2} \left (\frac {24 d^3}{e^2 (d+e x)}+\frac {23 d^2}{e^2}-\frac {6 d x}{e}+x^2\right )+\frac {10 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.48, size = 107, normalized size = 0.82 \begin {gather*} \frac {10 d^3 \sqrt {-e^2} \log \left (\sqrt {d^2-e^2 x^2}-\sqrt {-e^2} x\right )}{e^3}+\frac {\sqrt {d^2-e^2 x^2} \left (47 d^3+17 d^2 e x-5 d e^2 x^2+e^3 x^3\right )}{3 e^2 (d+e x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 111, normalized size = 0.85 \begin {gather*} \frac {47 \, d^{3} e x + 47 \, d^{4} - 60 \, {\left (d^{3} e x + d^{4}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (e^{3} x^{3} - 5 \, d e^{2} x^{2} + 17 \, d^{2} e x + 47 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{3 \, {\left (e^{3} x + d e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 290, normalized size = 2.23 \begin {gather*} \frac {10 d^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{\sqrt {e^{2}}\, e}+\frac {10 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d x}{e}+\frac {20 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} x}{3 d e}+\frac {16 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}}}{3 d^{2} e^{2}}+\frac {4 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}}}{\left (x +\frac {d}{e}\right )^{3} d \,e^{5}}+\frac {16 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}}}{3 \left (x +\frac {d}{e}\right )^{2} d^{2} e^{4}}+\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}}}{\left (x +\frac {d}{e}\right )^{4} e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 235, normalized size = 1.81 \begin {gather*} -\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d}{2 \, {\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )}} - \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{2 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} + \frac {15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3}}{e^{3} x + d e^{2}} + \frac {10 \, d^{3} \arcsin \left (\frac {e x}{d}\right )}{e^{2}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{3 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} + \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d}{6 \, {\left (e^{3} x + d e^{2}\right )}} + \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{2 \, e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x\,{\left (d^2-e^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}}}{\left (d + e x\right )^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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