Optimal. Leaf size=148 \[ \frac {35}{128} d^7 x \sqrt {d^2-e^2 x^2}+\frac {35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac {35 d^9 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e} \]
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Rubi [A]
time = 0.03, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {655, 201, 223,
209} \begin {gather*} \frac {35 d^9 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e}+\frac {1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac {35}{128} d^7 x \sqrt {d^2-e^2 x^2}+\frac {35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 209
Rule 223
Rule 655
Rubi steps
\begin {align*} \int (d+e x) \left (d^2-e^2 x^2\right )^{7/2} \, dx &=-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+d \int \left (d^2-e^2 x^2\right )^{7/2} \, dx\\ &=\frac {1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac {1}{8} \left (7 d^3\right ) \int \left (d^2-e^2 x^2\right )^{5/2} \, dx\\ &=\frac {7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac {1}{48} \left (35 d^5\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx\\ &=\frac {35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac {1}{64} \left (35 d^7\right ) \int \sqrt {d^2-e^2 x^2} \, dx\\ &=\frac {35}{128} d^7 x \sqrt {d^2-e^2 x^2}+\frac {35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac {1}{128} \left (35 d^9\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {35}{128} d^7 x \sqrt {d^2-e^2 x^2}+\frac {35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac {1}{128} \left (35 d^9\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}{128} d^7 x \sqrt {d^2-e^2 x^2}+\frac {35}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {7}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {1}{8} d x \left (d^2-e^2 x^2\right )^{7/2}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{9 e}+\frac {35 d^9 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e}\\ \end {align*}
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Mathematica [A]
time = 0.35, size = 155, normalized size = 1.05 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-128 d^8+837 d^7 e x+512 d^6 e^2 x^2-978 d^5 e^3 x^3-768 d^4 e^4 x^4+600 d^3 e^5 x^5+512 d^2 e^6 x^6-144 d e^7 x^7-128 e^8 x^8\right )}{1152 e}-\frac {35 d^9 \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{128 \sqrt {-e^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 142, normalized size = 0.96
method | result | size |
risch | \(-\frac {\left (128 e^{8} x^{8}+144 d \,e^{7} x^{7}-512 d^{2} e^{6} x^{6}-600 d^{3} e^{5} x^{5}+768 d^{4} e^{4} x^{4}+978 d^{5} e^{3} x^{3}-512 d^{6} e^{2} x^{2}-837 d^{7} e x +128 d^{8}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{1152 e}+\frac {35 d^{9} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{128 \sqrt {e^{2}}}\) | \(138\) |
default | \(-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}{9 e}+d \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8}+\frac {7 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8}\right )\) | \(142\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 107, normalized size = 0.72 \begin {gather*} \frac {35}{128} \, d^{9} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} + \frac {35}{128} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{7} x + \frac {35}{192} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} x + \frac {7}{48} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} x + \frac {1}{8} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d x - \frac {1}{9} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {9}{2}} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.01, size = 128, normalized size = 0.86 \begin {gather*} -\frac {1}{1152} \, {\left (630 \, d^{9} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) + {\left (128 \, x^{8} e^{8} + 144 \, d x^{7} e^{7} - 512 \, d^{2} x^{6} e^{6} - 600 \, d^{3} x^{5} e^{5} + 768 \, d^{4} x^{4} e^{4} + 978 \, d^{5} x^{3} e^{3} - 512 \, d^{6} x^{2} e^{2} - 837 \, d^{7} x e + 128 \, d^{8}\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 [C] Result contains complex when optimal does not.
time = 30.20, size = 1284, normalized size = 8.68 \begin {gather*} d^{7} \left (\begin {cases} - \frac {i d^{2} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{2 e} - \frac {i d x}{2 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{3}}{2 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {d^{2} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{2 e} + \frac {d x \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{2} & \text {otherwise} \end {cases}\right ) + d^{6} e \left (\begin {cases} \frac {x^{2} \sqrt {d^{2}}}{2} & \text {for}\: e^{2} = 0 \\- \frac {\left (d^{2} - e^{2} x^{2}\right )^{\frac {3}{2}}}{3 e^{2}} & \text {otherwise} \end {cases}\right ) - 3 d^{5} e^{2} \left (\begin {cases} - \frac {i d^{4} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{8 e^{3}} + \frac {i d^{3} x}{8 e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {3 i d x^{3}}{8 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{5}}{4 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {d^{4} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{8 e^{3}} - \frac {d^{3} x}{8 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {3 d x^{3}}{8 \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {e^{2} x^{5}}{4 d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) - 3 d^{4} e^{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 ) + 3 d^{3} e^{4} \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 ) + 3 d^{2} e^{5} \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 ) - d e^{6} \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 ) - e^{7} \left (\begin {cases} - \frac {16 d^{8} \sqrt {d^{2} - e^{2} x^{2}}}{315 e^{8}} - \frac {8 d^{6} x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{315 e^{6}} - \frac {2 d^{4} x^{4} \sqrt {d^{2} - e^{2} x^{2}}}{105 e^{4}} - \frac {d^{2} x^{6} \sqrt {d^{2} - e^{2} x^{2}}}{63 e^{2}} + \frac {x^{8} \sqrt {d^{2} - e^{2} x^{2}}}{9} & \text {for}\: e \neq 0 \\\frac {x^{8} \sqrt {d^{2}}}{8} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.85, size = 119, normalized size = 0.80 \begin {gather*} \frac {35}{128} \, d^{9} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} \mathrm {sgn}\left (d\right ) - \frac {1}{1152} \, {\left (128 \, d^{8} e^{\left (-1\right )} - {\left (837 \, d^{7} + 2 \, {\left (256 \, d^{6} e - {\left (489 \, d^{5} e^{2} + 4 \, {\left (96 \, d^{4} e^{3} - {\left (75 \, d^{3} e^{4} + 2 \, {\left (32 \, d^{2} e^{5} - {\left (8 \, x e^{7} + 9 \, d e^{6}\right )} x\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 [B]
time = 0.85, size = 67, normalized size = 0.45 \begin {gather*} \frac {d\,x\,{\left (d^2-e^2\,x^2\right )}^{7/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {7}{2},\frac {1}{2};\ \frac {3}{2};\ \frac {e^2\,x^2}{d^2}\right )}{{\left (1-\frac {e^2\,x^2}{d^2}\right )}^{7/2}}-\frac {{\left (d^2-e^2\,x^2\right )}^{9/2}}{9\,e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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