Optimal. Leaf size=137 \[ -\frac {6 F^{a+b (c+d x)^n}}{b^4 d n \log ^4(F)}+\frac {6 F^{a+b (c+d x)^n} (c+d x)^n}{b^3 d n \log ^3(F)}-\frac {3 F^{a+b (c+d x)^n} (c+d x)^{2 n}}{b^2 d n \log ^2(F)}+\frac {F^{a+b (c+d x)^n} (c+d x)^{3 n}}{b d n \log (F)} \]
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Rubi [A]
time = 0.11, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2244, 2240}
\begin {gather*} -\frac {6 F^{a+b (c+d x)^n}}{b^4 d n \log ^4(F)}+\frac {6 (c+d x)^n F^{a+b (c+d x)^n}}{b^3 d n \log ^3(F)}-\frac {3 (c+d x)^{2 n} F^{a+b (c+d x)^n}}{b^2 d n \log ^2(F)}+\frac {(c+d x)^{3 n} F^{a+b (c+d x)^n}}{b d n \log (F)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2240
Rule 2244
Rubi steps
\begin {align*} \int F^{a+b (c+d x)^n} (c+d x)^{-1+4 n} \, dx &=\frac {F^{a+b (c+d x)^n} (c+d x)^{3 n}}{b d n \log (F)}-\frac {3 \int F^{a+b (c+d x)^n} (c+d x)^{-1+3 n} \, dx}{b \log (F)}\\ &=-\frac {3 F^{a+b (c+d x)^n} (c+d x)^{2 n}}{b^2 d n \log ^2(F)}+\frac {F^{a+b (c+d x)^n} (c+d x)^{3 n}}{b d n \log (F)}+\frac {6 \int F^{a+b (c+d x)^n} (c+d x)^{-1+2 n} \, dx}{b^2 \log ^2(F)}\\ &=\frac {6 F^{a+b (c+d x)^n} (c+d x)^n}{b^3 d n \log ^3(F)}-\frac {3 F^{a+b (c+d x)^n} (c+d x)^{2 n}}{b^2 d n \log ^2(F)}+\frac {F^{a+b (c+d x)^n} (c+d x)^{3 n}}{b d n \log (F)}-\frac {6 \int F^{a+b (c+d x)^n} (c+d x)^{-1+n} \, dx}{b^3 \log ^3(F)}\\ &=-\frac {6 F^{a+b (c+d x)^n}}{b^4 d n \log ^4(F)}+\frac {6 F^{a+b (c+d x)^n} (c+d x)^n}{b^3 d n \log ^3(F)}-\frac {3 F^{a+b (c+d x)^n} (c+d x)^{2 n}}{b^2 d n \log ^2(F)}+\frac {F^{a+b (c+d x)^n} (c+d x)^{3 n}}{b d n \log (F)}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.01, size = 32, normalized size = 0.23 \begin {gather*} -\frac {F^a \Gamma \left (4,-b (c+d x)^n \log (F)\right )}{b^4 d n \log ^4(F)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 77, normalized size = 0.56
method | result | size |
risch | \(\frac {\left (b^{3} \left (d x +c \right )^{3 n} \ln \left (F \right )^{3}-3 b^{2} \left (d x +c \right )^{2 n} \ln \left (F \right )^{2}+6 b \left (d x +c \right )^{n} \ln \left (F \right )-6\right ) F^{a +b \left (d x +c \right )^{n}}}{b^{4} \ln \left (F \right )^{4} n d}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 87, normalized size = 0.64 \begin {gather*} \frac {{\left ({\left (d x + c\right )}^{3 \, n} F^{a} b^{3} \log \left (F\right )^{3} - 3 \, {\left (d x + c\right )}^{2 \, n} F^{a} b^{2} \log \left (F\right )^{2} + 6 \, {\left (d x + c\right )}^{n} F^{a} b \log \left (F\right ) - 6 \, F^{a}\right )} F^{{\left (d x + c\right )}^{n} b}}{b^{4} d n \log \left (F\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 80, normalized size = 0.58 \begin {gather*} \frac {{\left ({\left (d x + c\right )}^{3 \, n} b^{3} \log \left (F\right )^{3} - 3 \, {\left (d x + c\right )}^{2 \, n} b^{2} \log \left (F\right )^{2} + 6 \, {\left (d x + c\right )}^{n} b \log \left (F\right ) - 6\right )} e^{\left ({\left (d x + c\right )}^{n} b \log \left (F\right ) + a \log \left (F\right )\right )}}{b^{4} d n \log \left (F\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 F^{a+b\,{\left (c+d\,x\right )}^n}\,{\left (c+d\,x\right )}^{4\,n-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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