Optimal. Leaf size=132 \[ -\frac {35}{2} d x \sqrt {d^2-e^2 x^2}-\frac {35 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{e (d+e x)^2}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{e (d+e x)^4}-\frac {35 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e} \]
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Rubi [A]
time = 0.04, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {677, 679, 201,
223, 209} \begin {gather*} -\frac {35 d^3 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{e (d+e x)^4}-\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{e (d+e x)^2}-\frac {35 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {35}{2} d x \sqrt {d^2-e^2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 209
Rule 223
Rule 677
Rule 679
Rubi steps
\begin {align*} \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^5} \, dx &=-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{e (d+e x)^4}-7 \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx\\ &=-\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{e (d+e x)^2}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{e (d+e x)^4}-35 \int \frac {\left (d^2-e^2 x^2\right )^{3/2}}{d+e x} \, dx\\ &=-\frac {35 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{e (d+e x)^2}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{e (d+e x)^4}-(35 d) \int \sqrt {d^2-e^2 x^2} \, dx\\ &=-\frac {35}{2} d x \sqrt {d^2-e^2 x^2}-\frac {35 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{e (d+e x)^2}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{e (d+e x)^4}-\frac {1}{2} \left (35 d^3\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=-\frac {35}{2} d x \sqrt {d^2-e^2 x^2}-\frac {35 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{e (d+e x)^2}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{e (d+e x)^4}-\frac {1}{2} \left (35 d^3\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )\\ &=-\frac {35}{2} d x \sqrt {d^2-e^2 x^2}-\frac {35 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{e (d+e x)^2}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{e (d+e x)^4}-\frac {35 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e}\\ \end {align*}
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Mathematica [A]
time = 0.35, size = 107, normalized size = 0.81 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-166 d^3-55 d^2 e x+13 d e^2 x^2-2 e^3 x^3\right )}{6 e (d+e x)}+\frac {35 d^3 \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{2 \sqrt {-e^2}} \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. \(454\) vs.
\(2(116)=232\).
time = 0.50, size = 455, normalized size = 3.45
method | result | size |
risch | \(-\frac {\left (2 e^{2} x^{2}-15 d x e +70 d^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{6 e}-\frac {35 d^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}-\frac {16 d^{3} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{e^{2} \left (x +\frac {d}{e}\right )}\) | \(117\) |
default | \(\frac {-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{d e \left (x +\frac {d}{e}\right )^{5}}-\frac {4 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{d e \left (x +\frac {d}{e}\right )^{4}}+\frac {5 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{3 d e \left (x +\frac {d}{e}\right )^{3}}+\frac {2 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{5 d e \left (x +\frac {d}{e}\right )^{2}}+\frac {7 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{7}+d e \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{12 e^{2}}+\frac {5 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )\right )}{5 d}\right )}{d}\right )}{d}\right )}{d}}{e^{5}}\) | \(455\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.57, size = 187, normalized size = 1.42 \begin {gather*} -\frac {35}{2} \, d^{3} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} + \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}}}{3 \, {\left (x^{4} e^{5} + 4 \, d x^{3} e^{4} + 6 \, d^{2} x^{2} e^{3} + 4 \, d^{3} x e^{2} + d^{4} e\right )}} + \frac {7 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d}{6 \, {\left (x^{3} e^{4} + 3 \, d x^{2} e^{3} + 3 \, d^{2} x e^{2} + d^{3} e\right )}} + \frac {35 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{6 \, {\left (x^{2} e^{3} + 2 \, d x e^{2} + d^{2} e\right )}} - \frac {35 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{3}}{x e^{2} + d e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.17, size = 108, normalized size = 0.82 \begin {gather*} -\frac {166 \, d^{3} x e + 166 \, d^{4} - 210 \, {\left (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 (2 \, x^{3} e^{3} - 13 \, d x^{2} e^{2} + 55 \, d^{2} x e + 166 \, d^{3}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{6 \, {\left (x e^{2} + d e\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 {\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}{\left (d + e x\right )^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.15, size = 178, normalized size = 1.35 \begin {gather*} \frac {{\left (840 \, d^{4} \arctan \left (\sqrt {\frac {2 \, d}{x e + d} - 1}\right ) e^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 384 \, d^{4} \sqrt {\frac {2 \, d}{x e + d} - 1} e^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - \frac {{\left (87 \, d^{4} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} e^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 136 \, d^{4} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} e^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 57 \, d^{4} \sqrt {\frac {2 \, d}{x e + d} - 1} e^{4} \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )} {\left (x e + d\right )}^{3}}{d^{3}}\right )} e^{\left (-5\right )}}{24 \, d} \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 {{\left (d^2-e^2\,x^2\right )}^{7/2}}{{\left (d+e\,x\right )}^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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