Optimal. Leaf size=136 \[ \frac {35}{8} d^2 x \sqrt {d^2-e^2 x^2}+\frac {35 d \left (d^2-e^2 x^2\right )^{3/2}}{12 e}+\frac {7 (d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{4 e}+\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{e (d+e x)^3}+\frac {35 d^4 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e} \]
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Rubi [A]
time = 0.04, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {677, 669, 685,
655, 201, 223, 209} \begin {gather*} \frac {35 d^4 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e}+\frac {35}{8} d^2 x \sqrt {d^2-e^2 x^2}+\frac {35 d \left (d^2-e^2 x^2\right )^{3/2}}{12 e}+\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{e (d+e x)^3}+\frac {7 (d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{4 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 209
Rule 223
Rule 655
Rule 669
Rule 677
Rule 685
Rubi steps
\begin {align*} \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^4} \, dx &=\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{e (d+e x)^3}+7 \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx\\ &=\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{e (d+e x)^3}+7 \int (d-e x)^2 \sqrt {d^2-e^2 x^2} \, dx\\ &=\frac {7 (d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{4 e}+\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{e (d+e x)^3}+\frac {1}{4} (35 d) \int (d-e x) \sqrt {d^2-e^2 x^2} \, dx\\ &=\frac {35 d \left (d^2-e^2 x^2\right )^{3/2}}{12 e}+\frac {7 (d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{4 e}+\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{e (d+e x)^3}+\frac {1}{4} \left (35 d^2\right ) \int \sqrt {d^2-e^2 x^2} \, dx\\ &=\frac {35}{8} d^2 x \sqrt {d^2-e^2 x^2}+\frac {35 d \left (d^2-e^2 x^2\right )^{3/2}}{12 e}+\frac {7 (d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{4 e}+\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{e (d+e x)^3}+\frac {1}{8} \left (35 d^4\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {35}{8} d^2 x \sqrt {d^2-e^2 x^2}+\frac {35 d \left (d^2-e^2 x^2\right )^{3/2}}{12 e}+\frac {7 (d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{4 e}+\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{e (d+e x)^3}+\frac {1}{8} \left (35 d^4\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}{8} d^2 x \sqrt {d^2-e^2 x^2}+\frac {35 d \left (d^2-e^2 x^2\right )^{3/2}}{12 e}+\frac {7 (d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{4 e}+\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{e (d+e x)^3}+\frac {35 d^4 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e}\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 100, normalized size = 0.74 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (160 d^3-81 d^2 e x+32 d e^2 x^2-6 e^3 x^3\right )}{24 e}-\frac {35 d^4 \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{8 \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. \(402\) vs.
\(2(118)=236\).
time = 0.48, size = 403, normalized size = 2.96
method | result | size |
risch | \(\frac {\left (-6 e^{3} x^{3}+32 d \,e^{2} x^{2}-81 d^{2} e x +160 d^{3}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{24 e}+\frac {35 d^{4} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 \sqrt {e^{2}}}\) | \(83\) |
default | \(\frac {\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{d e \left (x +\frac {d}{e}\right )^{4}}+\frac {5 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{3 d e \left (x +\frac {d}{e}\right )^{3}}+\frac {2 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{5 d e \left (x +\frac {d}{e}\right )^{2}}+\frac {7 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{7}+d e \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{12 e^{2}}+\frac {5 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )\right )}{5 d}\right )}{d}\right )}{d}}{e^{4}}\) | \(403\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 148, normalized size = 1.09 \begin {gather*} \frac {35}{8} \, d^{4} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} + \frac {35}{8} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{3} e^{\left (-1\right )} + \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}}}{4 \, {\left (x^{3} e^{4} + 3 \, d x^{2} e^{3} + 3 \, d^{2} x e^{2} + d^{3} e\right )}} + \frac {7 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d}{12 \, {\left (x^{2} e^{3} + 2 \, d x e^{2} + d^{2} e\right )}} + \frac {35 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{24 \, {\left (x e^{2} + d e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.86, size = 78, normalized size = 0.57 \begin {gather*} -\frac {1}{24} \, {\left (210 \, d^{4} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) + {\left (6 \, x^{3} e^{3} - 32 \, d x^{2} e^{2} + 81 \, d^{2} x e - 160 \, d^{3}\right )} \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\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 )^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.91, size = 64, normalized size = 0.47 \begin {gather*} \frac {35}{8} \, d^{4} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} \mathrm {sgn}\left (d\right ) + \frac {1}{24} \, {\left (160 \, d^{3} e^{\left (-1\right )} - {\left (81 \, d^{2} + 2 \, {\left (3 \, x e^{2} - 16 \, d e\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{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 {{\left (d^2-e^2\,x^2\right )}^{7/2}}{{\left (d+e\,x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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