Optimal. Leaf size=229 \[ -\frac {1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}+\frac {d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {5 d^2 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^2}-\frac {5 d^9 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{64 e^6}-\frac {5 d^7 x \sqrt {d^2-e^2 x^2}}{64 e^5}-\frac {d^5 (256 d-315 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2016 e^6}-\frac {4 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^4}+\frac {5 d^3 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 e^3} \]
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Rubi [A] time = 0.31, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {852, 1809, 833, 780, 195, 217, 203} \[ -\frac {5 d^7 x \sqrt {d^2-e^2 x^2}}{64 e^5}-\frac {d^5 (256 d-315 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2016 e^6}-\frac {4 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^4}+\frac {5 d^3 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 e^3}-\frac {5 d^2 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^2}+\frac {d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}-\frac {5 d^9 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{64 e^6} \]
Antiderivative was successfully verified.
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Rule 195
Rule 203
Rule 217
Rule 780
Rule 833
Rule 852
Rule 1809
Rubi steps
\begin {align*} \int \frac {x^5 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx &=\int x^5 (d-e x)^2 \sqrt {d^2-e^2 x^2} \, dx\\ &=-\frac {1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}-\frac {\int x^5 \left (-15 d^2 e^2+18 d e^3 x\right ) \sqrt {d^2-e^2 x^2} \, dx}{9 e^2}\\ &=\frac {d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}+\frac {\int x^4 \left (-90 d^3 e^3+120 d^2 e^4 x\right ) \sqrt {d^2-e^2 x^2} \, dx}{72 e^4}\\ &=-\frac {5 d^2 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^2}+\frac {d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}-\frac {\int x^3 \left (-480 d^4 e^4+630 d^3 e^5 x\right ) \sqrt {d^2-e^2 x^2} \, dx}{504 e^6}\\ &=\frac {5 d^3 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 e^3}-\frac {5 d^2 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^2}+\frac {d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}+\frac {\int x^2 \left (-1890 d^5 e^5+2880 d^4 e^6 x\right ) \sqrt {d^2-e^2 x^2} \, dx}{3024 e^8}\\ &=-\frac {4 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^4}+\frac {5 d^3 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 e^3}-\frac {5 d^2 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^2}+\frac {d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}-\frac {\int x \left (-5760 d^6 e^6+9450 d^5 e^7 x\right ) \sqrt {d^2-e^2 x^2} \, dx}{15120 e^{10}}\\ &=-\frac {4 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^4}+\frac {5 d^3 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 e^3}-\frac {5 d^2 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^2}+\frac {d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}-\frac {d^5 (256 d-315 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2016 e^6}-\frac {\left (5 d^7\right ) \int \sqrt {d^2-e^2 x^2} \, dx}{32 e^5}\\ &=-\frac {5 d^7 x \sqrt {d^2-e^2 x^2}}{64 e^5}-\frac {4 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^4}+\frac {5 d^3 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 e^3}-\frac {5 d^2 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^2}+\frac {d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}-\frac {d^5 (256 d-315 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2016 e^6}-\frac {\left (5 d^9\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{64 e^5}\\ &=-\frac {5 d^7 x \sqrt {d^2-e^2 x^2}}{64 e^5}-\frac {4 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^4}+\frac {5 d^3 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 e^3}-\frac {5 d^2 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^2}+\frac {d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}-\frac {d^5 (256 d-315 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2016 e^6}-\frac {\left (5 d^9\right ) \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{64 e^5}\\ &=-\frac {5 d^7 x \sqrt {d^2-e^2 x^2}}{64 e^5}-\frac {4 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^4}+\frac {5 d^3 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 e^3}-\frac {5 d^2 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{21 e^2}+\frac {d x^5 \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {1}{9} x^6 \left (d^2-e^2 x^2\right )^{3/2}-\frac {d^5 (256 d-315 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2016 e^6}-\frac {5 d^9 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{64 e^6}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 135, normalized size = 0.59 \[ \frac {\sqrt {d^2-e^2 x^2} \left (-512 d^8+315 d^7 e x-256 d^6 e^2 x^2+210 d^5 e^3 x^3-192 d^4 e^4 x^4+168 d^3 e^5 x^5+512 d^2 e^6 x^6-1008 d e^7 x^7+448 e^8 x^8\right )-315 d^9 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{4032 e^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 138, normalized size = 0.60 \[ \frac {630 \, d^{9} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (448 \, e^{8} x^{8} - 1008 \, d e^{7} x^{7} + 512 \, d^{2} e^{6} x^{6} + 168 \, d^{3} e^{5} x^{5} - 192 \, d^{4} e^{4} x^{4} + 210 \, d^{5} e^{3} x^{3} - 256 \, d^{6} e^{2} x^{2} + 315 \, d^{7} e x - 512 \, d^{8}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{4032 \, e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 375, normalized size = 1.64 \[ \frac {5 d^{9} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{4 \sqrt {e^{2}}\, e^{5}}-\frac {85 d^{9} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{64 \sqrt {e^{2}}\, e^{5}}-\frac {85 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{7} x}{64 e^{5}}+\frac {5 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{7} x}{4 e^{5}}-\frac {85 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{5} x}{96 e^{5}}+\frac {5 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} d^{5} x}{6 e^{5}}-\frac {17 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d^{3} x}{24 e^{5}}+\frac {2 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}} d^{4}}{3 e^{6}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} x^{2}}{9 e^{4}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d x}{4 e^{5}}-\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}} d^{4}}{3 \left (x +\frac {d}{e}\right )^{2} e^{8}}-\frac {29 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{2}}{63 e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.07, size = 299, normalized size = 1.31 \[ -\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5}}{4 \, {\left (e^{7} x + d e^{6}\right )}} - \frac {5 i \, d^{9} \arcsin \left (\frac {e x}{d} + 2\right )}{4 \, e^{6}} - \frac {85 \, d^{9} \arcsin \left (\frac {e x}{d}\right )}{64 \, e^{6}} + \frac {5 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{7} x}{4 \, e^{5}} - \frac {85 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{7} x}{64 \, e^{5}} + \frac {5 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{8}}{2 \, e^{6}} + \frac {35 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} x}{96 \, e^{5}} - \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{6}}{12 \, e^{6}} - \frac {17 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} x}{24 \, e^{5}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} x^{2}}{9 \, e^{4}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4}}{e^{6}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d x}{4 \, e^{5}} - \frac {29 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{2}}{63 \, e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^5\,{\left (d^2-e^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 17.48, size = 571, normalized size = 2.49 \[ d^{2} \left (\begin {cases} - \frac {8 d^{6} \sqrt {d^{2} - e^{2} x^{2}}}{105 e^{6}} - \frac {4 d^{4} x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{105 e^{4}} - \frac {d^{2} x^{4} \sqrt {d^{2} - e^{2} x^{2}}}{35 e^{2}} + \frac {x^{6} \sqrt {d^{2} - e^{2} x^{2}}}{7} & \text {for}\: e \neq 0 \\\frac {x^{6} \sqrt {d^{2}}}{6} & \text {otherwise} \end {cases}\right ) - 2 d e \left (\begin {cases} - \frac {5 i d^{8} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{128 e^{7}} + \frac {5 i d^{7} x}{128 e^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {5 i d^{5} x^{3}}{384 e^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {i d^{3} x^{5}}{192 e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {7 i d x^{7}}{48 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{9}}{8 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {5 d^{8} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{128 e^{7}} - \frac {5 d^{7} x}{128 e^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {5 d^{5} x^{3}}{384 e^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {d^{3} x^{5}}{192 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {7 d x^{7}}{48 \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {e^{2} x^{9}}{8 d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} - \frac {16 d^{8} \sqrt {d^{2} - e^{2} x^{2}}}{315 e^{8}} - \frac {8 d^{6} x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{315 e^{6}} - \frac {2 d^{4} x^{4} \sqrt {d^{2} - e^{2} x^{2}}}{105 e^{4}} - \frac {d^{2} x^{6} \sqrt {d^{2} - e^{2} x^{2}}}{63 e^{2}} + \frac {x^{8} \sqrt {d^{2} - e^{2} x^{2}}}{9} & \text {for}\: e \neq 0 \\\frac {x^{8} \sqrt {d^{2}}}{8} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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