Optimal. Leaf size=210 \[ -\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {e^2 (50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac {1}{12} d e^3 (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac {25}{8} d^5 e^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {13}{2} d^5 e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )-\frac {1}{8} d^3 e^3 (52 d+25 e x) \sqrt {d^2-e^2 x^2} \]
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Rubi [A] time = 0.31, antiderivative size = 210, 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 {1}{8} d^3 e^3 (52 d+25 e x) \sqrt {d^2-e^2 x^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac {1}{12} d e^3 (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {e^2 (50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x}-\frac {25}{8} d^5 e^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {13}{2} d^5 e^3 \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^4} \, dx &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (-9 d^4 e-5 d^3 e^2 x-3 d^2 e^3 x^2\right )}{x^3} \, dx}{3 d^2}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}+\frac {\int \frac {\left (10 d^5 e^2-39 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^2} \, dx}{6 d^4}\\ &=-\frac {e^2 (50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {\int \frac {\left (78 d^6 e^3+100 d^5 e^4 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x} \, dx}{12 d^4}\\ &=-\frac {1}{12} d e^3 (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac {e^2 (50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}+\frac {\int \frac {\left (-312 d^8 e^5-300 d^7 e^6 x\right ) \sqrt {d^2-e^2 x^2}}{x} \, dx}{48 d^4 e^2}\\ &=-\frac {1}{8} d^3 e^3 (52 d+25 e x) \sqrt {d^2-e^2 x^2}-\frac {1}{12} d e^3 (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac {e^2 (50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {\int \frac {624 d^{10} e^7+300 d^9 e^8 x}{x \sqrt {d^2-e^2 x^2}} \, dx}{96 d^4 e^4}\\ &=-\frac {1}{8} d^3 e^3 (52 d+25 e x) \sqrt {d^2-e^2 x^2}-\frac {1}{12} d e^3 (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac {e^2 (50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {1}{2} \left (13 d^6 e^3\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx-\frac {1}{8} \left (25 d^5 e^4\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=-\frac {1}{8} d^3 e^3 (52 d+25 e x) \sqrt {d^2-e^2 x^2}-\frac {1}{12} d e^3 (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac {e^2 (50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {1}{4} \left (13 d^6 e^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )-\frac {1}{8} \left (25 d^5 e^4\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 {1}{8} d^3 e^3 (52 d+25 e x) \sqrt {d^2-e^2 x^2}-\frac {1}{12} d e^3 (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac {e^2 (50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {25}{8} d^5 e^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {1}{2} \left (13 d^6 e\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 {1}{8} d^3 e^3 (52 d+25 e x) \sqrt {d^2-e^2 x^2}-\frac {1}{12} d e^3 (26 d+25 e x) \left (d^2-e^2 x^2\right )^{3/2}-\frac {e^2 (50 d+39 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{3 x^3}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{2 x^2}-\frac {25}{8} d^5 e^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {13}{2} d^5 e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\\ \end {align*}
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Mathematica [C] time = 0.27, size = 251, normalized size = 1.20 \begin {gather*} -\frac {3 e^3 \left (d^2-e^2 x^2\right )^{7/2} \, _2F_1\left (2,\frac {7}{2};\frac {9}{2};1-\frac {e^2 x^2}{d^2}\right )}{7 d^2}-\frac {d^7 \sqrt {d^2-e^2 x^2} \, _2F_1\left (-\frac {5}{2},-\frac {3}{2};-\frac {1}{2};\frac {e^2 x^2}{d^2}\right )}{3 x^3 \sqrt {1-\frac {e^2 x^2}{d^2}}}-\frac {3 d^5 e^2 \sqrt {d^2-e^2 x^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}}}+\frac {1}{15} e^3 \left (\sqrt {d^2-e^2 x^2} \left (23 d^4-11 d^2 e^2 x^2+3 e^4 x^4\right )-15 d^5 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.62, size = 192, normalized size = 0.91 \begin {gather*} -\frac {25}{8} d^5 e^2 \sqrt {-e^2} \log \left (\sqrt {d^2-e^2 x^2}-\sqrt {-e^2} x\right )-13 d^5 e^3 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x}{d}-\frac {\sqrt {d^2-e^2 x^2}}{d}\right )+\frac {\sqrt {d^2-e^2 x^2} \left (-40 d^7-180 d^6 e x-80 d^5 e^2 x^2-656 d^4 e^3 x^3-345 d^3 e^4 x^4+32 d^2 e^5 x^5+90 d e^6 x^6+24 e^7 x^7\right )}{120 x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 179, normalized size = 0.85 \begin {gather*} \frac {750 \, d^{5} e^{3} x^{3} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - 780 \, d^{5} e^{3} x^{3} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - 656 \, d^{5} e^{3} x^{3} + {\left (24 \, e^{7} x^{7} + 90 \, d e^{6} x^{6} + 32 \, d^{2} e^{5} x^{5} - 345 \, d^{3} e^{4} x^{4} - 656 \, d^{4} e^{3} x^{3} - 80 \, d^{5} e^{2} x^{2} - 180 \, d^{6} e x - 40 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{120 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 318, normalized size = 1.51 \begin {gather*} -\frac {25}{8} \, d^{5} \arcsin \left (\frac {x e}{d}\right ) e^{3} \mathrm {sgn}\relax (d) + \frac {13}{2} \, d^{5} e^{3} \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^{5} e^{8} + \frac {9 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{5} e^{6}}{x} + \frac {9 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{5} e^{4}}{x^{2}}\right )} x^{3} e}{24 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3}} - \frac {1}{24} \, {\left (\frac {9 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{5} e^{16}}{x} + \frac {9 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{5} e^{14}}{x^{2}} + \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{5} e^{12}}{x^{3}}\right )} e^{\left (-15\right )} - \frac {1}{120} \, {\left (656 \, d^{4} e^{3} + {\left (345 \, d^{3} e^{4} - 2 \, {\left (16 \, d^{2} e^{5} + 3 \, {\left (4 \, x e^{7} + 15 \, d e^{6}\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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maple [A] time = 0.02, size = 277, normalized size = 1.32 \begin {gather*} \frac {13 d^{6} e^{3} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 \sqrt {d^{2}}}-\frac {25 d^{5} e^{4} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 \sqrt {e^{2}}}-\frac {25 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{3} e^{4} x}{8}-\frac {13 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{4} e^{3}}{2}-\frac {25 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d \,e^{4} x}{12}-\frac {13 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{2} e^{3}}{6}-\frac {5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{4} x}{3 d}-\frac {13 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{3}}{10}-\frac {5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{2}}{3 d x}-\frac {3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e}{2 x^{2}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 226, normalized size = 1.08 \begin {gather*} -\frac {25}{8} \, d^{5} e^{3} \arcsin \left (\frac {e x}{d}\right ) + \frac {13}{2} \, d^{5} e^{3} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) - \frac {25}{8} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3} e^{4} x - \frac {13}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{4} e^{3} - \frac {25}{12} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d e^{4} x - \frac {13}{6} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} e^{3} - \frac {13}{10} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{3} - \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{2}}{3 \, x} - \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e}{2 \, x^{2}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d}{3 \, x^{3}} \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^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 15.74, size = 911, normalized size = 4.34
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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