Optimal. Leaf size=179 \[ \frac {77}{256} d^8 x \sqrt {d^2-e^2 x^2}+\frac {77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac {77 d^{10} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{256 e} \]
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Rubi [A]
time = 0.05, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {685, 655, 201,
223, 209} \begin {gather*} \frac {77 d^{10} \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{256 e}+\frac {11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac {77}{256} d^8 x \sqrt {d^2-e^2 x^2}+\frac {77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {77}{480} d^4 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
Rule 685
Rubi steps
\begin {align*} \int (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2} \, dx &=-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac {1}{10} (11 d) \int (d+e x) \left (d^2-e^2 x^2\right )^{7/2} \, dx\\ &=-\frac {11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac {1}{10} \left (11 d^2\right ) \int \left (d^2-e^2 x^2\right )^{7/2} \, dx\\ &=\frac {11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac {1}{80} \left (77 d^4\right ) \int \left (d^2-e^2 x^2\right )^{5/2} \, dx\\ &=\frac {77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac {1}{96} \left (77 d^6\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx\\ &=\frac {77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac {1}{128} \left (77 d^8\right ) \int \sqrt {d^2-e^2 x^2} \, dx\\ &=\frac {77}{256} d^8 x \sqrt {d^2-e^2 x^2}+\frac {77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac {1}{256} \left (77 d^{10}\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {77}{256} d^8 x \sqrt {d^2-e^2 x^2}+\frac {77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac {1}{256} \left (77 d^{10}\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 {77}{256} d^8 x \sqrt {d^2-e^2 x^2}+\frac {77}{384} d^6 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {77}{480} d^4 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {11}{80} d^2 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {11 d \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {(d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{10 e}+\frac {77 d^{10} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{256 e}\\ \end {align*}
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Mathematica [A]
time = 0.39, size = 166, normalized size = 0.93 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-2560 d^9+8055 d^8 e x+10240 d^7 e^2 x^2-6150 d^6 e^3 x^3-15360 d^5 e^4 x^4-312 d^4 e^5 x^5+10240 d^3 e^6 x^6+3024 d^2 e^7 x^7-2560 d e^8 x^8-1152 e^9 x^9\right )}{11520 e}-\frac {77 d^{10} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{256 \sqrt {-e^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.49, size = 297, normalized size = 1.66
method | result | size |
risch | \(-\frac {\left (1152 e^{9} x^{9}+2560 d \,e^{8} x^{8}-3024 d^{2} e^{7} x^{7}-10240 d^{3} e^{6} x^{6}+312 d^{4} e^{5} x^{5}+15360 d^{5} e^{4} x^{4}+6150 d^{6} e^{3} x^{3}-10240 d^{7} e^{2} x^{2}-8055 d^{8} e x +2560 d^{9}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{11520 e}+\frac {77 d^{10} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{256 \sqrt {e^{2}}}\) | \(149\) |
default | \(e^{2} \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}{10 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8}+\frac {7 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8}\right )}{10 e^{2}}\right )-\frac {2 d \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}}}{9 e}+d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8}+\frac {7 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8}\right )\) | \(297\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 126, normalized size = 0.70 \begin {gather*} \frac {77}{256} \, d^{10} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} + \frac {77}{256} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{8} x + \frac {77}{384} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{6} x + \frac {77}{480} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} x + \frac {11}{80} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d^{2} x - \frac {2}{9} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {9}{2}} d e^{\left (-1\right )} - \frac {1}{10} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {9}{2}} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.93, size = 138, normalized size = 0.77 \begin {gather*} -\frac {1}{11520} \, {\left (6930 \, d^{10} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) + {\left (1152 \, x^{9} e^{9} + 2560 \, d x^{8} e^{8} - 3024 \, d^{2} x^{7} e^{7} - 10240 \, d^{3} x^{6} e^{6} + 312 \, d^{4} x^{5} e^{5} + 15360 \, d^{5} x^{4} e^{4} + 6150 \, d^{6} x^{3} e^{3} - 10240 \, d^{7} x^{2} e^{2} - 8055 \, d^{8} x e + 2560 \, d^{9}\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 = 138.30, size = 1413, normalized size = 7.89 \begin {gather*} d^{8} \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 ) + 2 d^{7} 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 ) - 2 d^{6} 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 ) - 6 d^{5} 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 ) + 6 d^{3} 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 ) + 2 d^{2} 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 ) - 2 d 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 ) - e^{8} \left (\begin {cases} - \frac {7 i d^{10} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{256 e^{9}} + \frac {7 i d^{9} x}{256 e^{8} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {7 i d^{7} x^{3}}{768 e^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {7 i d^{5} x^{5}}{1920 e^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {i d^{3} x^{7}}{480 e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {9 i d x^{9}}{80 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{11}}{10 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {7 d^{10} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{256 e^{9}} - \frac {7 d^{9} x}{256 e^{8} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {7 d^{7} x^{3}}{768 e^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {7 d^{5} x^{5}}{1920 e^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {d^{3} x^{7}}{480 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {9 d x^{9}}{80 \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {e^{2} x^{11}}{10 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 = 0.94, size = 128, normalized size = 0.72 \begin {gather*} \frac {77}{256} \, d^{10} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} \mathrm {sgn}\left (d\right ) - \frac {1}{11520} \, {\left (2560 \, d^{9} e^{\left (-1\right )} - {\left (8055 \, d^{8} + 2 \, {\left (5120 \, d^{7} e - {\left (3075 \, d^{6} e^{2} + 4 \, {\left (1920 \, d^{5} e^{3} + {\left (39 \, d^{4} e^{4} - 2 \, {\left (640 \, d^{3} e^{5} + {\left (189 \, d^{2} e^{6} - 8 \, {\left (9 \, x e^{8} + 20 \, d e^{7}\right )} x\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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (d^2-e^2\,x^2\right )}^{7/2}\,{\left (d+e\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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