Optimal. Leaf size=162 \[ \frac {x^6 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(32 d-35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}+\frac {7 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^8}+\frac {x^2 (24 d-35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}-\frac {x^4 (6 d-7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.16, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {850, 819, 780, 217, 203} \[ \frac {x^6 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d-7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d-35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {(32 d-35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}+\frac {7 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^8} \]
Antiderivative was successfully verified.
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Rule 203
Rule 217
Rule 780
Rule 819
Rule 850
Rubi steps
\begin {align*} \int \frac {x^7}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx &=\int \frac {x^7 (d-e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\\ &=\frac {x^6 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {x^5 \left (6 d^3-7 d^2 e x\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2}\\ &=\frac {x^6 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d-7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {x^3 \left (24 d^5-35 d^4 e x\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4 e^4}\\ &=\frac {x^6 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d-7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d-35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {x \left (48 d^7-105 d^6 e x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{15 d^6 e^6}\\ &=\frac {x^6 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d-7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d-35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {(32 d-35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}+\frac {\left (7 d^2\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{2 e^7}\\ &=\frac {x^6 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d-7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d-35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {(32 d-35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}+\frac {\left (7 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^7}\\ &=\frac {x^6 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d-7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d-35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {(32 d-35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}+\frac {7 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^8}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 128, normalized size = 0.79 \[ \frac {105 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {\sqrt {d^2-e^2 x^2} \left (96 d^6-9 d^5 e x-249 d^4 e^2 x^2-4 d^3 e^3 x^3+176 d^2 e^4 x^4+15 d e^5 x^5-15 e^6 x^6\right )}{(d-e x)^2 (d+e x)^3}}{30 e^8} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 274, normalized size = 1.69 \[ \frac {96 \, d^{2} e^{5} x^{5} + 96 \, d^{3} e^{4} x^{4} - 192 \, d^{4} e^{3} x^{3} - 192 \, d^{5} e^{2} x^{2} + 96 \, d^{6} e x + 96 \, d^{7} - 210 \, {\left (d^{2} e^{5} x^{5} + d^{3} e^{4} x^{4} - 2 \, d^{4} e^{3} x^{3} - 2 \, d^{5} e^{2} x^{2} + d^{6} e x + d^{7}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (15 \, e^{6} x^{6} - 15 \, d e^{5} x^{5} - 176 \, d^{2} e^{4} x^{4} + 4 \, d^{3} e^{3} x^{3} + 249 \, d^{4} e^{2} x^{2} + 9 \, d^{5} e x - 96 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{30 \, {\left (e^{13} x^{5} + d e^{12} x^{4} - 2 \, d^{2} e^{11} x^{3} - 2 \, d^{3} e^{10} x^{2} + d^{4} e^{9} x + d^{5} e^{8}\right )}} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 318, normalized size = 1.96 \[ -\frac {x^{5}}{2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{3}}+\frac {d \,x^{4}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{4}}+\frac {7 d^{2} x^{3}}{6 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{5}}-\frac {5 d^{3} x^{2}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{6}}+\frac {2 d^{4} x}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{7}}-\frac {4 d^{4} x}{15 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} e^{7}}+\frac {d^{6}}{5 \left (x +\frac {d}{e}\right ) \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} e^{9}}+\frac {3 d^{5}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{8}}-\frac {19 d^{2} x}{6 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{7}}-\frac {8 d^{2} x}{15 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, e^{7}}+\frac {7 d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}\, e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.09, size = 289, normalized size = 1.78 \[ \frac {d^{6}}{5 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{9} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d e^{8}\right )}} - \frac {x^{5}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{3}} + \frac {d x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}} + \frac {25 \, d^{2} x^{3}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{5}} - \frac {65 \, d^{3} x^{2}}{6 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{6}} - \frac {164 \, d^{4} x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{7}} - \frac {7 \, d x^{2}}{6 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{6}} + \frac {53 \, d^{5}}{6 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{8}} + \frac {229 \, d^{2} x}{30 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{7}} + \frac {7 \, d^{2} \arcsin \left (\frac {e x}{d}\right )}{2 \, e^{8}} - \frac {14 \, d^{3}}{3 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{8}} - \frac {7 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{6 \, e^{8}} \]
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 \[ \int \frac {x^7}{{\left (d^2-e^2\,x^2\right )}^{5/2}\,\left (d+e\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{7}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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