Optimal. Leaf size=114 \[ \frac {4 (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d+11 e x}{15 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d+22 e x}{15 d^4 \sqrt {d^2-e^2 x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^4} \]
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Rubi [A] time = 0.16, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1805, 823, 12, 266, 63, 208} \[ \frac {4 (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {15 d+22 e x}{15 d^4 \sqrt {d^2-e^2 x^2}}+\frac {5 d+11 e x}{15 d^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^4} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 208
Rule 266
Rule 823
Rule 1805
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{x \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {4 (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {-5 d^3-11 d^2 e x}{x \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2}\\ &=\frac {4 (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d+11 e x}{15 d^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int \frac {-15 d^5 e^2-22 d^4 e^3 x}{x \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^6 e^2}\\ &=\frac {4 (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d+11 e x}{15 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d+22 e x}{15 d^4 \sqrt {d^2-e^2 x^2}}-\frac {\int -\frac {15 d^7 e^4}{x \sqrt {d^2-e^2 x^2}} \, dx}{15 d^{10} e^4}\\ &=\frac {4 (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d+11 e x}{15 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d+22 e x}{15 d^4 \sqrt {d^2-e^2 x^2}}+\frac {\int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^3}\\ &=\frac {4 (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d+11 e x}{15 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d+22 e x}{15 d^4 \sqrt {d^2-e^2 x^2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{2 d^3}\\ &=\frac {4 (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d+11 e x}{15 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d+22 e x}{15 d^4 \sqrt {d^2-e^2 x^2}}-\frac {\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^3 e^2}\\ &=\frac {4 (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d+11 e x}{15 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d+22 e x}{15 d^4 \sqrt {d^2-e^2 x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^4}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 81, normalized size = 0.71 \[ \frac {9 d^5+45 d^4 e x-55 d^2 e^3 x^3+3 d^5 \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};1-\frac {e^2 x^2}{d^2}\right )+22 e^5 x^5}{15 d^4 \left (d^2-e^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 158, normalized size = 1.39 \[ \frac {32 \, e^{3} x^{3} - 96 \, d e^{2} x^{2} + 96 \, d^{2} e x - 32 \, d^{3} + 15 \, {\left (e^{3} x^{3} - 3 \, d e^{2} x^{2} + 3 \, d^{2} e x - d^{3}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (22 \, e^{2} x^{2} - 51 \, d e x + 32 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{4} e^{3} x^{3} - 3 \, d^{5} e^{2} x^{2} + 3 \, d^{6} e x - d^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 117, normalized size = 1.03 \[ -\frac {\sqrt {-x^{2} e^{2} + d^{2}} {\left ({\left ({\left ({\left (x {\left (\frac {22 \, x e^{5}}{d^{4}} + \frac {15 \, e^{4}}{d^{3}}\right )} - \frac {55 \, e^{3}}{d^{2}}\right )} x - \frac {35 \, e^{2}}{d}\right )} x + 45 \, e\right )} x + 32 \, d\right )}}{15 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} - \frac {\log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right )}{d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 158, normalized size = 1.39 \[ \frac {4 e x}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {4 d}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {11 e x}{15 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{2}}+\frac {1}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}\, d^{3}}+\frac {22 e x}{15 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{4}}+\frac {1}{\sqrt {-e^{2} x^{2}+d^{2}}\, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 152, normalized size = 1.33 \[ \frac {4 \, e x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {4 \, d}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {11 \, e x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}} + \frac {1}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d} + \frac {22 \, e x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{4}} - \frac {\log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{d^{4}} + \frac {1}{\sqrt {-e^{2} x^{2} + d^{2}} d^{3}} \]
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 \[ \int \frac {{\left (d+e\,x\right )}^3}{x\,{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]
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 \[ \int \frac {\left (d + e x\right )^{3}}{x \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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