Optimal. Leaf size=117 \[ \frac {2 (d+e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d+8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d+16 e x}{15 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^5} \]
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Rubi [A]
time = 0.10, antiderivative size = 117, 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 = {1819, 837, 12,
272, 65, 214} \begin {gather*} \frac {2 (d+e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {15 d+16 e x}{15 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^5}+\frac {5 d+8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 214
Rule 272
Rule 837
Rule 1819
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{x \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {2 (d+e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {-5 d^2-8 d e x}{x \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2}\\ &=\frac {2 (d+e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d+8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int \frac {-15 d^4 e^2-16 d^3 e^3 x}{x \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^6 e^2}\\ &=\frac {2 (d+e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d+8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d+16 e x}{15 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\int -\frac {15 d^6 e^4}{x \sqrt {d^2-e^2 x^2}} \, dx}{15 d^{10} e^4}\\ &=\frac {2 (d+e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d+8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d+16 e x}{15 d^5 \sqrt {d^2-e^2 x^2}}+\frac {\int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^4}\\ &=\frac {2 (d+e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d+8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d+16 e x}{15 d^5 \sqrt {d^2-e^2 x^2}}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{2 d^4}\\ &=\frac {2 (d+e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d+8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d+16 e x}{15 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\text {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^2}\\ &=\frac {2 (d+e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d+8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d+16 e x}{15 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^5}\\ \end {align*}
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Mathematica [A]
time = 0.47, size = 107, normalized size = 0.91 \begin {gather*} \frac {\frac {\sqrt {d^2-e^2 x^2} \left (26 d^3-22 d^2 e x-17 d e^2 x^2+16 e^3 x^3\right )}{(d-e x)^3 (d+e x)}+30 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )}{15 d^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 202, normalized size = 1.73
method | result | size |
default | \(\frac {1}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+2 e d \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )+d^{2} \left (\frac {1}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {1}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {\frac {1}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}}{d^{2}}}{d^{2}}\right )\) | \(202\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 150, normalized size = 1.28 \begin {gather*} \frac {2 \, x e}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d} + \frac {2}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {8 \, x e}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3}} + \frac {1}{3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}} + \frac {16 \, x e}{15 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{5}} - \frac {\log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}} d}{{\left | x \right |}}\right )}{d^{5}} + \frac {1}{\sqrt {-x^{2} e^{2} + d^{2}} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.33, size = 163, normalized size = 1.39 \begin {gather*} \frac {26 \, x^{4} e^{4} - 52 \, d x^{3} e^{3} + 52 \, d^{3} x e - 26 \, d^{4} + 15 \, {\left (x^{4} e^{4} - 2 \, d x^{3} e^{3} + 2 \, d^{3} x e - d^{4}\right )} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) - {\left (16 \, x^{3} e^{3} - 17 \, d x^{2} e^{2} - 22 \, d^{2} x e + 26 \, d^{3}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d^{5} x^{4} e^{4} - 2 \, d^{6} x^{3} e^{3} + 2 \, d^{8} x e - d^{9}\right )}} \end {gather*}
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 \begin {gather*} \int \frac {\left (d + e x\right )^{2}}{x \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = -0.01 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^2}{x\,{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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