Optimal. Leaf size=124 \[ \frac {1}{6} d x \left (d^2-e^2 x^2\right )^{5/2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e}+\frac {5 d^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e}+\frac {5}{16} d^5 x \sqrt {d^2-e^2 x^2}+\frac {5}{24} d^3 x \left (d^2-e^2 x^2\right )^{3/2} \]
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Rubi [A] time = 0.04, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {665, 195, 217, 203} \begin {gather*} \frac {5}{16} d^5 x \sqrt {d^2-e^2 x^2}+\frac {5}{24} d^3 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{6} d x \left (d^2-e^2 x^2\right )^{5/2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e}+\frac {5 d^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 195
Rule 203
Rule 217
Rule 665
Rubi steps
\begin {align*} \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{d+e x} \, dx &=\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e}+d \int \left (d^2-e^2 x^2\right )^{5/2} \, dx\\ &=\frac {1}{6} d x \left (d^2-e^2 x^2\right )^{5/2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e}+\frac {1}{6} \left (5 d^3\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx\\ &=\frac {5}{24} d^3 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{6} d x \left (d^2-e^2 x^2\right )^{5/2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e}+\frac {1}{8} \left (5 d^5\right ) \int \sqrt {d^2-e^2 x^2} \, dx\\ &=\frac {5}{16} d^5 x \sqrt {d^2-e^2 x^2}+\frac {5}{24} d^3 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{6} d x \left (d^2-e^2 x^2\right )^{5/2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e}+\frac {1}{16} \left (5 d^7\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {5}{16} d^5 x \sqrt {d^2-e^2 x^2}+\frac {5}{24} d^3 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{6} d x \left (d^2-e^2 x^2\right )^{5/2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e}+\frac {1}{16} \left (5 d^7\right ) \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )\\ &=\frac {5}{16} d^5 x \sqrt {d^2-e^2 x^2}+\frac {5}{24} d^3 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{6} d x \left (d^2-e^2 x^2\right )^{5/2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e}+\frac {5 d^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 113, normalized size = 0.91 \begin {gather*} \frac {105 d^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\sqrt {d^2-e^2 x^2} \left (48 d^6+231 d^5 e x-144 d^4 e^2 x^2-182 d^3 e^3 x^3+144 d^2 e^4 x^4+56 d e^5 x^5-48 e^6 x^6\right )}{336 e} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.44, size = 136, normalized size = 1.10 \begin {gather*} \frac {5 d^7 \sqrt {-e^2} \log \left (\sqrt {d^2-e^2 x^2}-\sqrt {-e^2} x\right )}{16 e^2}+\frac {\sqrt {d^2-e^2 x^2} \left (48 d^6+231 d^5 e x-144 d^4 e^2 x^2-182 d^3 e^3 x^3+144 d^2 e^4 x^4+56 d e^5 x^5-48 e^6 x^6\right )}{336 e} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 116, normalized size = 0.94 \begin {gather*} -\frac {210 \, d^{7} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (48 \, e^{6} x^{6} - 56 \, d e^{5} x^{5} - 144 \, d^{2} e^{4} x^{4} + 182 \, d^{3} e^{3} x^{3} + 144 \, d^{4} e^{2} x^{2} - 231 \, d^{5} e x - 48 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{336 \, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 181, normalized size = 1.46 \begin {gather*} \frac {5 d^{7} \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 )}{16 \sqrt {e^{2}}}+\frac {5 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{5} x}{16}+\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^{3} x}{24}+\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}} d x}{6}+\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}}}{7 e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 2.94, size = 129, normalized size = 1.04 \begin {gather*} -\frac {5 i \, d^{7} \arcsin \left (\frac {e x}{d} + 2\right )}{16 \, e} + \frac {5}{16} \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{5} x + \frac {5 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{6}}{8 \, e} + \frac {5}{24} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} x + \frac {1}{6} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d x + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}{7 \, e} \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}}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 16.13, size = 813, normalized size = 6.56 \begin {gather*} d^{5} \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^{4} 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^{3} 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 ) + 2 d^{2} 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 ) + d 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 ) - 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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