Optimal. Leaf size=172 \[ \frac {3 d^6 x \sqrt {d^2-e^2 x^2}}{128 e^3}+\frac {d^4 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^3}-\frac {d x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e^2}-\frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac {d^2 (32 d+35 e x) \left (d^2-e^2 x^2\right )^{5/2}}{560 e^4}+\frac {3 d^8 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^4} \]
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Rubi [A]
time = 0.07, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {847, 794, 201,
223, 209} \begin {gather*} \frac {3 d^8 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^4}-\frac {d x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e^2}-\frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac {d^2 (32 d+35 e x) \left (d^2-e^2 x^2\right )^{5/2}}{560 e^4}+\frac {3 d^6 x \sqrt {d^2-e^2 x^2}}{128 e^3}+\frac {d^4 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 209
Rule 223
Rule 794
Rule 847
Rubi steps
\begin {align*} \int x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx &=-\frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac {\int x^2 \left (-3 d^2 e-8 d e^2 x\right ) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{8 e^2}\\ &=-\frac {d x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e^2}-\frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}+\frac {\int x \left (16 d^3 e^2+21 d^2 e^3 x\right ) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{56 e^4}\\ &=-\frac {d x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e^2}-\frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac {d^2 (32 d+35 e x) \left (d^2-e^2 x^2\right )^{5/2}}{560 e^4}+\frac {d^4 \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{16 e^3}\\ &=\frac {d^4 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^3}-\frac {d x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e^2}-\frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac {d^2 (32 d+35 e x) \left (d^2-e^2 x^2\right )^{5/2}}{560 e^4}+\frac {\left (3 d^6\right ) \int \sqrt {d^2-e^2 x^2} \, dx}{64 e^3}\\ &=\frac {3 d^6 x \sqrt {d^2-e^2 x^2}}{128 e^3}+\frac {d^4 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^3}-\frac {d x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e^2}-\frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac {d^2 (32 d+35 e x) \left (d^2-e^2 x^2\right )^{5/2}}{560 e^4}+\frac {\left (3 d^8\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{128 e^3}\\ &=\frac {3 d^6 x \sqrt {d^2-e^2 x^2}}{128 e^3}+\frac {d^4 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^3}-\frac {d x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e^2}-\frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac {d^2 (32 d+35 e x) \left (d^2-e^2 x^2\right )^{5/2}}{560 e^4}+\frac {\left (3 d^8\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^3}\\ &=\frac {3 d^6 x \sqrt {d^2-e^2 x^2}}{128 e^3}+\frac {d^4 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^3}-\frac {d x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e^2}-\frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac {d^2 (32 d+35 e x) \left (d^2-e^2 x^2\right )^{5/2}}{560 e^4}+\frac {3 d^8 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^4}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 147, normalized size = 0.85 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-256 d^7-105 d^6 e x-128 d^5 e^2 x^2-70 d^4 e^3 x^3+1024 d^3 e^4 x^4+840 d^2 e^5 x^5-640 d e^6 x^6-560 e^7 x^7\right )}{4480 e^4}+\frac {3 d^8 \sqrt {-e^2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{128 e^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 184, normalized size = 1.07
method | result | size |
risch | \(-\frac {\left (560 e^{7} x^{7}+640 d \,e^{6} x^{6}-840 d^{2} e^{5} x^{5}-1024 d^{3} e^{4} x^{4}+70 d^{4} e^{3} x^{3}+128 d^{5} e^{2} x^{2}+105 d^{6} e x +256 d^{7}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{4480 e^{4}}+\frac {3 d^{8} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{128 e^{3} \sqrt {e^{2}}}\) | \(130\) |
default | \(e \left (-\frac {x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6 e^{2}}\right )}{8 e^{2}}\right )+d \left (-\frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{7 e^{2}}-\frac {2 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{35 e^{4}}\right )\) | \(184\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 140, normalized size = 0.81 \begin {gather*} \frac {3}{128} \, d^{8} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-4\right )} + \frac {3}{128} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{6} x e^{\left (-3\right )} + \frac {1}{64} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4} x e^{\left (-3\right )} - \frac {1}{8} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} x^{3} e^{\left (-1\right )} - \frac {1}{7} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d x^{2} e^{\left (-2\right )} - \frac {1}{16} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} x e^{\left (-3\right )} - \frac {2}{35} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.32, size = 118, normalized size = 0.69 \begin {gather*} -\frac {1}{4480} \, {\left (210 \, d^{8} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) + {\left (560 \, x^{7} e^{7} + 640 \, d x^{6} e^{6} - 840 \, d^{2} x^{5} e^{5} - 1024 \, d^{3} x^{4} e^{4} + 70 \, d^{4} x^{3} e^{3} + 128 \, d^{5} x^{2} e^{2} + 105 \, d^{6} x e + 256 \, d^{7}\right )} \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 25.55, size = 775, normalized size = 4.51 \begin {gather*} d^{3} \left (\begin {cases} - \frac {2 d^{4} \sqrt {d^{2} - e^{2} x^{2}}}{15 e^{4}} - \frac {d^{2} x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{15 e^{2}} + \frac {x^{4} \sqrt {d^{2} - e^{2} x^{2}}}{5} & \text {for}\: e \neq 0 \\\frac {x^{4} \sqrt {d^{2}}}{4} & \text {otherwise} \end {cases}\right ) + d^{2} e \left (\begin {cases} - \frac {i d^{6} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{16 e^{5}} + \frac {i d^{5} x}{16 e^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {i d^{3} x^{3}}{48 e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {5 i d x^{5}}{24 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{7}}{6 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {d^{6} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{16 e^{5}} - \frac {d^{5} x}{16 e^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {d^{3} x^{3}}{48 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {5 d x^{5}}{24 \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {e^{2} x^{7}}{6 d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) - d e^{2} \left (\begin {cases} - \frac {8 d^{6} \sqrt {d^{2} - e^{2} x^{2}}}{105 e^{6}} - \frac {4 d^{4} x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{105 e^{4}} - \frac {d^{2} x^{4} \sqrt {d^{2} - e^{2} x^{2}}}{35 e^{2}} + \frac {x^{6} \sqrt {d^{2} - e^{2} x^{2}}}{7} & \text {for}\: e \neq 0 \\\frac {x^{6} \sqrt {d^{2}}}{6} & \text {otherwise} \end {cases}\right ) - e^{3} \left (\begin {cases} - \frac {5 i d^{8} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{128 e^{7}} + \frac {5 i d^{7} x}{128 e^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {5 i d^{5} x^{3}}{384 e^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {i d^{3} x^{5}}{192 e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {7 i d x^{7}}{48 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{9}}{8 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {5 d^{8} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{128 e^{7}} - \frac {5 d^{7} x}{128 e^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {5 d^{5} x^{3}}{384 e^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {d^{3} x^{5}}{192 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {7 d x^{7}}{48 \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {e^{2} x^{9}}{8 d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.93, size = 106, normalized size = 0.62 \begin {gather*} \frac {3}{128} \, d^{8} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-4\right )} \mathrm {sgn}\left (d\right ) - \frac {1}{4480} \, {\left (256 \, d^{7} e^{\left (-4\right )} + {\left (105 \, d^{6} e^{\left (-3\right )} + 2 \, {\left (64 \, d^{5} e^{\left (-2\right )} + {\left (35 \, d^{4} e^{\left (-1\right )} - 4 \, {\left (128 \, d^{3} + 5 \, {\left (21 \, d^{2} e - 2 \, {\left (7 \, x e^{3} + 8 \, d e^{2}\right )} x\right )} x\right )} x\right )} x\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 x^3\,{\left (d^2-e^2\,x^2\right )}^{3/2}\,\left (d+e\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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