Optimal. Leaf size=52 \[ -\frac {F^a \left (-b \log (F) (c+d x)^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},-b (c+d x)^n \log (F)\right )}{d n (c+d x)} \]
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Rubi [A] time = 0.03, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2218} \[ -\frac {F^a \left (-b \log (F) (c+d x)^n\right )^{\frac {1}{n}} \text {Gamma}\left (-\frac {1}{n},-b \log (F) (c+d x)^n\right )}{d n (c+d x)} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin {align*} \int \frac {F^{a+b (c+d x)^n}}{(c+d x)^2} \, dx &=-\frac {F^a \Gamma \left (-\frac {1}{n},-b (c+d x)^n \log (F)\right ) \left (-b (c+d x)^n \log (F)\right )^{\frac {1}{n}}}{d n (c+d x)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 52, normalized size = 1.00 \[ -\frac {F^a \left (-b \log (F) (c+d x)^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},-b (c+d x)^n \log (F)\right )}{d n (c+d x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {F^{{\left (d x + c\right )}^{n} b + a}}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \]
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 \[ \int \frac {F^{{\left (d x + c\right )}^{n} b + a}}{{\left (d x + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {F^{b \left (d x +c \right )^{n}+a}}{\left (d x +c \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{{\left (d x + c\right )}^{n} b + a}}{{\left (d x + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.70, size = 71, normalized size = 1.37 \[ -\frac {F^a\,{\mathrm {e}}^{\frac {b\,\ln \relax (F)\,{\left (c+d\,x\right )}^n}{2}}\,{\left (b\,\ln \relax (F)\,{\left (c+d\,x\right )}^n\right )}^{\frac {1}{2\,n}-\frac {1}{2}}\,{\mathrm {M}}_{\frac {1}{2\,n}+\frac {1}{2},-\frac {1}{2\,n}}\left (b\,\ln \relax (F)\,{\left (c+d\,x\right )}^n\right )}{d\,\left (c+d\,x\right )} \]
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 \[ \begin {cases} \frac {\tilde {\infty } F^{a}}{x} & \text {for}\: c = 0 \wedge d = 0 \wedge n = 1 \\\tilde {\infty } F^{0^{n} b + a} x & \text {for}\: c = - d x \\\frac {F^{a + b c^{n}} x}{c^{2}} & \text {for}\: d = 0 \\\int \frac {F^{a + b \left (c + d x\right )}}{\left (c + d x\right )^{2}}\, dx & \text {for}\: n = 1 \\\frac {F^{a} F^{b \left (c + d x\right )^{n}} b n \left (c + d x\right )^{n} \log {\relax (F )}}{c d n - c d + d^{2} n x - d^{2} x} - \frac {F^{a} F^{b \left (c + d x\right )^{n}} n}{c d n - c d + d^{2} n x - d^{2} x} + \frac {F^{a} F^{b \left (c + d x\right )^{n}}}{c d n - c d + d^{2} n x - d^{2} x} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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