Optimal. Leaf size=145 \[ \frac {4 e (d+e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^5 x}+\frac {e (15 d+19 e x)}{5 d^5 \sqrt {d^2-e^2 x^2}}-\frac {3 e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^5}+\frac {e (5 d+7 e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.29, 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} \[ \frac {4 e (d+e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^5 x}+\frac {e (15 d+19 e x)}{5 d^5 \sqrt {d^2-e^2 x^2}}+\frac {e (5 d+7 e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {3 e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^5} \]
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)^3}{x^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {4 e (d+e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {-5 d^3-15 d^2 e x-16 d e^2 x^2}{x^2 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2}\\ &=\frac {4 e (d+e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e (5 d+7 e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {15 d^3+45 d^2 e x+42 d e^2 x^2}{x^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4}\\ &=\frac {4 e (d+e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e (5 d+7 e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e (15 d+19 e x)}{5 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-15 d^3-45 d^2 e x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{15 d^6}\\ &=\frac {4 e (d+e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e (5 d+7 e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e (15 d+19 e x)}{5 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^5 x}+\frac {(3 e) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^4}\\ &=\frac {4 e (d+e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e (5 d+7 e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e (15 d+19 e x)}{5 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^5 x}+\frac {(3 e) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{2 d^4}\\ &=\frac {4 e (d+e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e (5 d+7 e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e (15 d+19 e x)}{5 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^5 x}-\frac {3 \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^4 e}\\ &=\frac {4 e (d+e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e (5 d+7 e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e (15 d+19 e x)}{5 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{d^5 x}-\frac {3 e \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^5}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 96, normalized size = 0.66 \[ \frac {-5 d^6+d^5 e x+45 d^4 e^2 x^2-60 d^2 e^4 x^4+3 d^5 e x \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};1-\frac {e^2 x^2}{d^2}\right )+24 e^6 x^6}{5 d^5 x \left (d^2-e^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 184, normalized size = 1.27 \[ \frac {24 \, e^{4} x^{4} - 72 \, d e^{3} x^{3} + 72 \, d^{2} e^{2} x^{2} - 24 \, d^{3} e x + 15 \, {\left (e^{4} x^{4} - 3 \, d e^{3} x^{3} + 3 \, d^{2} e^{2} x^{2} - d^{3} e x\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (24 \, e^{3} x^{3} - 57 \, d e^{2} x^{2} + 39 \, d^{2} e x - 5 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{5 \, {\left (d^{5} e^{3} x^{4} - 3 \, d^{6} e^{2} x^{3} + 3 \, d^{7} e x^{2} - d^{8} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 185, normalized size = 1.28 \[ -\frac {\sqrt {-x^{2} e^{2} + d^{2}} {\left ({\left ({\left ({\left (x {\left (\frac {19 \, x e^{6}}{d^{5}} + \frac {15 \, e^{5}}{d^{4}}\right )} - \frac {45 \, e^{4}}{d^{3}}\right )} x - \frac {35 \, e^{3}}{d^{2}}\right )} x + \frac {30 \, e^{2}}{d}\right )} x + 24 \, e\right )}}{5 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} - \frac {3 \, e \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right )}{d^{5}} + \frac {x e^{3}}{2 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{5}} - \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-1\right )}}{2 \, d^{5} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 190, normalized size = 1.31 \[ \frac {9 e^{2} x}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d}+\frac {4 e}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} x}+\frac {12 e^{2} x}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{3}}+\frac {e}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{2}}-\frac {3 e \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}\, d^{4}}+\frac {24 e^{2} x}{5 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{5}}+\frac {3 e}{\sqrt {-e^{2} x^{2}+d^{2}}\, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 184, normalized size = 1.27 \[ \frac {9 \, e^{2} x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d} + \frac {4 \, e}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {12 \, e^{2} x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3}} + \frac {e}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}} - \frac {d}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} x} + \frac {24 \, e^{2} x}{5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{5}} - \frac {3 \, e \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{d^{5}} + \frac {3 \, e}{\sqrt {-e^{2} x^{2} + d^{2}} d^{4}} \]
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 \[ \int \frac {{\left (d+e\,x\right )}^3}{x^2\,{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]
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 \[ \int \frac {\left (d + e x\right )^{3}}{x^{2} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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