Optimal. Leaf size=100 \[ \frac {2 F^{a+b (c+d x)^n}}{b^3 d n \log ^3(F)}-\frac {2 (c+d x)^n F^{a+b (c+d x)^n}}{b^2 d n \log ^2(F)}+\frac {(c+d x)^{2 n} F^{a+b (c+d x)^n}}{b d n \log (F)} \]
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Rubi [A] time = 0.12, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2213, 2209} \[ -\frac {2 (c+d x)^n F^{a+b (c+d x)^n}}{b^2 d n \log ^2(F)}+\frac {2 F^{a+b (c+d x)^n}}{b^3 d n \log ^3(F)}+\frac {(c+d x)^{2 n} F^{a+b (c+d x)^n}}{b d n \log (F)} \]
Antiderivative was successfully verified.
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Rule 2209
Rule 2213
Rubi steps
\begin {align*} \int F^{a+b (c+d x)^n} (c+d x)^{-1+3 n} \, dx &=\frac {F^{a+b (c+d x)^n} (c+d x)^{2 n}}{b d n \log (F)}-\frac {2 \int F^{a+b (c+d x)^n} (c+d x)^{-1+2 n} \, dx}{b \log (F)}\\ &=-\frac {2 F^{a+b (c+d x)^n} (c+d x)^n}{b^2 d n \log ^2(F)}+\frac {F^{a+b (c+d x)^n} (c+d x)^{2 n}}{b d n \log (F)}+\frac {2 \int F^{a+b (c+d x)^n} (c+d x)^{-1+n} \, dx}{b^2 \log ^2(F)}\\ &=\frac {2 F^{a+b (c+d x)^n}}{b^3 d n \log ^3(F)}-\frac {2 F^{a+b (c+d x)^n} (c+d x)^n}{b^2 d n \log ^2(F)}+\frac {F^{a+b (c+d x)^n} (c+d x)^{2 n}}{b d n \log (F)}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 31, normalized size = 0.31 \[ \frac {F^a \Gamma \left (3,-b (c+d x)^n \log (F)\right )}{b^3 d n \log ^3(F)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 62, normalized size = 0.62 \[ \frac {{\left ({\left (d x + c\right )}^{2 \, n} b^{2} \log \relax (F)^{2} - 2 \, {\left (d x + c\right )}^{n} b \log \relax (F) + 2\right )} e^{\left ({\left (d x + c\right )}^{n} b \log \relax (F) + a \log \relax (F)\right )}}{b^{3} d n \log \relax (F)^{3}} \]
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 \[ \int {\left (d x + c\right )}^{3 \, n - 1} F^{{\left (d x + c\right )}^{n} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 59, normalized size = 0.59 \[ \frac {\left (b^{2} \left (d x +c \right )^{2 n} \ln \relax (F )^{2}-2 b \left (d x +c \right )^{n} \ln \relax (F )+2\right ) F^{b \left (d x +c \right )^{n}+a}}{b^{3} d n \ln \relax (F )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 66, normalized size = 0.66 \[ \frac {{\left ({\left (d x + c\right )}^{2 \, n} F^{a} b^{2} \log \relax (F)^{2} - 2 \, {\left (d x + c\right )}^{n} F^{a} b \log \relax (F) + 2 \, F^{a}\right )} F^{{\left (d x + c\right )}^{n} b}}{b^{3} d n \log \relax (F)^{3}} \]
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 \[ \int F^{a+b\,{\left (c+d\,x\right )}^n}\,{\left (c+d\,x\right )}^{3\,n-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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