Optimal. Leaf size=145 \[ \frac {2 e (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^6 x}+\frac {e (30 d+41 e x)}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {2 e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^6}+\frac {e (10 d+13 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.28, antiderivative size = 145, 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 = {1805, 807, 266, 63, 208} \[ \frac {2 e (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^6 x}+\frac {e (30 d+41 e x)}{15 d^6 \sqrt {d^2-e^2 x^2}}+\frac {e (10 d+13 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^6} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 1805
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{x^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {2 e (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {-5 d^2-10 d e x-8 e^2 x^2}{x^2 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2}\\ &=\frac {2 e (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e (10 d+13 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {15 d^2+30 d e x+26 e^2 x^2}{x^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4}\\ &=\frac {2 e (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e (10 d+13 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e (30 d+41 e x)}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-15 d^2-30 d e x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{15 d^6}\\ &=\frac {2 e (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e (10 d+13 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e (30 d+41 e x)}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^6 x}+\frac {(2 e) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^5}\\ &=\frac {2 e (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e (10 d+13 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e (30 d+41 e x)}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^6 x}+\frac {e \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{d^5}\\ &=\frac {2 e (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e (10 d+13 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e (30 d+41 e x)}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^6 x}-\frac {2 \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 )}{d^5 e}\\ &=\frac {2 e (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e (10 d+13 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e (30 d+41 e x)}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^6 x}-\frac {2 e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^6}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 90, normalized size = 0.62 \[ \frac {-15 d^6+105 d^4 e^2 x^2-140 d^2 e^4 x^4+6 d^5 e x \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};1-\frac {e^2 x^2}{d^2}\right )+56 e^6 x^6}{15 d^6 x \left (d^2-e^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 195, normalized size = 1.34 \[ \frac {46 \, e^{5} x^{5} - 92 \, d e^{4} x^{4} + 92 \, d^{3} e^{2} x^{2} - 46 \, d^{4} e x + 30 \, {\left (e^{5} x^{5} - 2 \, d e^{4} x^{4} + 2 \, d^{3} e^{2} x^{2} - d^{4} e x\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (56 \, e^{4} x^{4} - 82 \, d e^{3} x^{3} - 32 \, d^{2} e^{2} x^{2} + 76 \, d^{3} e x - 15 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{6} e^{4} x^{5} - 2 \, d^{7} e^{3} x^{4} + 2 \, d^{9} e x^{2} - d^{10} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 188, normalized size = 1.30 \[ -\frac {\sqrt {-x^{2} e^{2} + d^{2}} {\left ({\left ({\left ({\left (x {\left (\frac {41 \, x e^{6}}{d^{6}} + \frac {30 \, e^{5}}{d^{5}}\right )} - \frac {95 \, e^{4}}{d^{4}}\right )} x - \frac {70 \, e^{3}}{d^{3}}\right )} x + \frac {60 \, e^{2}}{d^{2}}\right )} x + \frac {46 \, e}{d}\right )}}{15 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} - \frac {2 \, e \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right )}{d^{6}} + \frac {x e^{3}}{2 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{6}} - \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-1\right )}}{2 \, d^{6} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 193, normalized size = 1.33 \[ \frac {7 e^{2} x}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d^{2}}+\frac {2 e}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d}+\frac {28 e^{2} x}{15 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{4}}-\frac {1}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} x}+\frac {2 e}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{3}}-\frac {2 e \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}\, d^{5}}+\frac {56 e^{2} x}{15 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{6}}+\frac {2 e}{\sqrt {-e^{2} x^{2}+d^{2}}\, d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 187, normalized size = 1.29 \[ \frac {7 \, e^{2} x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2}} + \frac {2 \, e}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d} + \frac {28 \, e^{2} x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4}} + \frac {2 \, e}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3}} - \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} x} + \frac {56 \, e^{2} x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{6}} - \frac {2 \, e \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{d^{6}} + \frac {2 \, e}{\sqrt {-e^{2} x^{2} + d^{2}} d^{5}} \]
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 \[ \int \frac {{\left (d+e\,x\right )}^2}{x^2\,{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]
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 \[ \int \frac {\left (d + e x\right )^{2}}{x^{2} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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