Optimal. Leaf size=209 \[ -\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}-\frac {45}{8} d^2 e^4 (d-e x) \sqrt {d^2-e^2 x^2}+\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}+\frac {45}{8} d^4 e^4 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {45}{8} d^4 e^4 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
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Rubi [A] time = 0.32, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1807, 813, 815, 844, 217, 203, 266, 63, 208} \begin {gather*} -\frac {45}{8} d^2 e^4 (d-e x) \sqrt {d^2-e^2 x^2}+\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}+\frac {45}{8} d^4 e^4 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {45}{8} d^4 e^4 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 208
Rule 217
Rule 266
Rule 813
Rule 815
Rule 844
Rule 1807
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^5} \, dx &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (-12 d^4 e-9 d^3 e^2 x-4 d^2 e^3 x^2\right )}{x^4} \, dx}{4 d^2}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}+\frac {\int \frac {\left (27 d^5 e^2-36 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^3} \, dx}{12 d^4}\\ &=-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}-\frac {5 \int \frac {\left (144 d^6 e^3+216 d^5 e^4 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^2} \, dx}{192 d^4}\\ &=\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}+\frac {5 \int \frac {\left (-432 d^7 e^4+864 d^6 e^5 x\right ) \sqrt {d^2-e^2 x^2}}{x} \, dx}{384 d^4}\\ &=-\frac {45}{8} d^2 e^4 (d-e x) \sqrt {d^2-e^2 x^2}+\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}-\frac {5 \int \frac {864 d^9 e^6-864 d^8 e^7 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{768 d^4 e^2}\\ &=-\frac {45}{8} d^2 e^4 (d-e x) \sqrt {d^2-e^2 x^2}+\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}-\frac {1}{8} \left (45 d^5 e^4\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx+\frac {1}{8} \left (45 d^4 e^5\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=-\frac {45}{8} d^2 e^4 (d-e x) \sqrt {d^2-e^2 x^2}+\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}-\frac {1}{16} \left (45 d^5 e^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )+\frac {1}{8} \left (45 d^4 e^5\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 {45}{8} d^2 e^4 (d-e x) \sqrt {d^2-e^2 x^2}+\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}+\frac {45}{8} d^4 e^4 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {1}{8} \left (45 d^5 e^2\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {d^2}{e^2}-\frac {x^2}{e^2}} \, dx,x,\sqrt {d^2-e^2 x^2}\right )\\ &=-\frac {45}{8} d^2 e^4 (d-e x) \sqrt {d^2-e^2 x^2}+\frac {15 d e^3 (2 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{8 x}-\frac {3 e^2 (3 d+2 e x) \left (d^2-e^2 x^2\right )^{5/2}}{8 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{4 x^4}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{x^3}+\frac {45}{8} d^4 e^4 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {45}{8} d^4 e^4 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\\ \end {align*}
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Mathematica [C] time = 0.10, size = 195, normalized size = 0.93 \begin {gather*} \frac {e \sqrt {d^2-e^2 x^2} \left (3 \left (e^3 x^2-d^2 e\right )^3 \, _2F_1\left (2,\frac {7}{2};\frac {9}{2};1-\frac {e^2 x^2}{d^2}\right )+\left (e^3 x^2-d^2 e\right )^3 \, _2F_1\left (3,\frac {7}{2};\frac {9}{2};1-\frac {e^2 x^2}{d^2}\right )-\frac {7 d^9 \, _2F_1\left (-\frac {5}{2},-\frac {3}{2};-\frac {1}{2};\frac {e^2 x^2}{d^2}\right )}{x^3 \sqrt {1-\frac {e^2 x^2}{d^2}}}-\frac {7 d^7 e^2 \, _2F_1\left (-\frac {5}{2},-\frac {1}{2};\frac {1}{2};\frac {e^2 x^2}{d^2}\right )}{x \sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{7 d^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.68, size = 194, normalized size = 0.93 \begin {gather*} -\frac {45}{4} d^4 e^4 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x}{d}-\frac {\sqrt {d^2-e^2 x^2}}{d}\right )+\frac {45}{8} d^4 e^3 \sqrt {-e^2} \log \left (\sqrt {d^2-e^2 x^2}-\sqrt {-e^2} x\right )+\frac {\sqrt {d^2-e^2 x^2} \left (-2 d^7-8 d^6 e x-3 d^5 e^2 x^2+48 d^4 e^3 x^3-48 d^3 e^4 x^4+3 d^2 e^5 x^5+8 d e^6 x^6+2 e^7 x^7\right )}{8 x^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 180, normalized size = 0.86 \begin {gather*} -\frac {90 \, d^{4} e^{4} x^{4} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + 45 \, d^{4} e^{4} x^{4} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + 48 \, d^{4} e^{4} x^{4} - {\left (2 \, e^{7} x^{7} + 8 \, d e^{6} x^{6} + 3 \, d^{2} e^{5} x^{5} - 48 \, d^{3} e^{4} x^{4} + 48 \, d^{4} e^{3} x^{3} - 3 \, d^{5} e^{2} x^{2} - 8 \, d^{6} e x - 2 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{8 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 374, normalized size = 1.79 \begin {gather*} \frac {45}{8} \, d^{4} \arcsin \left (\frac {x e}{d}\right ) e^{4} \mathrm {sgn}\relax (d) + \frac {45}{8} \, d^{4} e^{4} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right ) + \frac {{\left (d^{4} e^{10} + \frac {8 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{4} e^{8}}{x} + \frac {8 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{4} e^{6}}{x^{2}} - \frac {184 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{4} e^{4}}{x^{3}}\right )} x^{4} e^{2}}{64 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4}} + \frac {1}{64} \, {\left (\frac {184 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{4} e^{26}}{x} - \frac {8 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{4} e^{24}}{x^{2}} - \frac {8 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{4} e^{22}}{x^{3}} - \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{4} e^{20}}{x^{4}}\right )} e^{\left (-24\right )} - \frac {1}{8} \, {\left (48 \, d^{3} e^{4} - {\left (3 \, d^{2} e^{5} + 2 \, {\left (x e^{7} + 4 \, d e^{6}\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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maple [A] time = 0.02, size = 302, normalized size = 1.44 \begin {gather*} \frac {45 d^{5} e^{4} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{8 \sqrt {d^{2}}}+\frac {45 d^{4} e^{5} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 \sqrt {e^{2}}}+\frac {45 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{2} e^{5} x}{8}-\frac {45 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{3} e^{4}}{8}+\frac {15 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{5} x}{4}-\frac {15 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d \,e^{4}}{8}+\frac {3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{5} x}{d^{2}}-\frac {9 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{4}}{8 d}+\frac {3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{3}}{d^{2} x}-\frac {9 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{2}}{8 d \,x^{2}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e}{x^{3}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d}{4 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 250, normalized size = 1.20 \begin {gather*} \frac {45}{8} \, d^{4} e^{4} \arcsin \left (\frac {e x}{d}\right ) + \frac {45}{8} \, d^{4} e^{4} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) + \frac {45}{8} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e^{5} x - \frac {45}{8} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3} e^{4} + \frac {15}{4} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{5} x - \frac {15}{8} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d e^{4} - \frac {9 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}}{8 \, d} + \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{3}}{x} - \frac {9 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{2}}{8 \, d x^{2}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e}{x^{3}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d}{4 \, x^{4}} \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 = -0.00 \begin {gather*} \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 20.10, size = 1028, normalized size = 4.92
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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