Optimal. Leaf size=148 \[ \frac {8 d \sqrt {d^2-e^2 x^2}}{e^4 (d+e x)}+\frac {d^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e^4 (d+e x)^4}-\frac {14 d \left (d^2-e^2 x^2\right )^{3/2}}{15 e^4 (d+e x)^3}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{e^4 (d+e x)^2}+\frac {4 d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^4} \]
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Rubi [A]
time = 0.16, antiderivative size = 148, normalized size of antiderivative = 1.00, 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 = {1653, 1651,
673, 665, 677, 223, 209} \begin {gather*} \frac {4 d \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^4}+\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 223
Rule 665
Rule 673
Rule 677
Rule 1651
Rule 1653
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) \text {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*}
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Mathematica [A]
time = 0.35, size = 106, normalized size = 0.72 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (94 d^3+222 d^2 e x+149 d e^2 x^2+15 e^3 x^3\right )}{15 e^4 (d+e x)^3}+\frac {4 d \sqrt {-e^2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{e^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(348\) vs.
\(2(134)=268\).
time = 0.08, size = 349, normalized size = 2.36
method | result | size |
risch | \(\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{e^{4}}+\frac {4 d \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{3} \sqrt {e^{2}}}+\frac {104 d \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{15 e^{5} \left (x +\frac {d}{e}\right )}-\frac {31 d^{2} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{15 e^{6} \left (x +\frac {d}{e}\right )^{2}}+\frac {2 d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{5 e^{7} \left (x +\frac {d}{e}\right )^{3}}\) | \(186\) |
default | \(\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}+\frac {d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{\sqrt {e^{2}}}}{e^{4}}-\frac {d \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{e^{7} \left (x +\frac {d}{e}\right )^{3}}-\frac {d^{3} \left (-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{5 d e \left (x +\frac {d}{e}\right )^{4}}-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{15 d^{2} \left (x +\frac {d}{e}\right )^{3}}\right )}{e^{7}}-\frac {3 d \left (-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{d e \left (x +\frac {d}{e}\right )^{2}}-\frac {e \left (\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}+\frac {d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{\sqrt {e^{2}}}\right )}{d}\right )}{e^{5}}\) | \(349\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.94, size = 164, normalized size = 1.11 \begin {gather*} \frac {94 \, d x^{3} e^{3} + 282 \, d^{2} x^{2} e^{2} + 282 \, d^{3} x e + 94 \, d^{4} - 120 \, {\left (d x^{3} e^{3} + 3 \, d^{2} x^{2} e^{2} + 3 \, d^{3} x e + d^{4}\right )} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) + {\left (15 \, x^{3} e^{3} + 149 \, d x^{2} e^{2} + 222 \, d^{2} x e + 94 \, d^{3}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (x^{3} e^{7} + 3 \, d x^{2} e^{6} + 3 \, d^{2} x e^{5} + d^{3} e^{4}\right )}} \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^{3} \sqrt {- \left (- d + e x\right ) \left (d + e x\right )}}{\left (d + e x\right )^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.51, size = 193, normalized size = 1.30 \begin {gather*} 4 \, d \arcsin \left (\frac {x e}{d}\right ) e^{\left (-4\right )} \mathrm {sgn}\left (d\right ) + \sqrt {-x^{2} e^{2} + d^{2}} e^{\left (-4\right )} - \frac {2 \, {\left (\frac {335 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d e^{\left (-2\right )}}{x} + \frac {505 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d e^{\left (-4\right )}}{x^{2}} + \frac {285 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d e^{\left (-6\right )}}{x^{3}} + \frac {60 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d e^{\left (-8\right )}}{x^{4}} + 79 \, d\right )} e^{\left (-4\right )}}{15 \, {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} + 1\right )}^{5}} \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^3\,\sqrt {d^2-e^2\,x^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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