Optimal. Leaf size=229 \[ -\frac {6 F^{c (a+b x)} g}{b^4 c^4 \log ^4(F)}+\frac {2 f F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}+\frac {6 F^{c (a+b x)} g x}{b^3 c^3 \log ^3(F)}-\frac {e F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}-\frac {2 f F^{c (a+b x)} x}{b^2 c^2 \log ^2(F)}-\frac {3 F^{c (a+b x)} g x^2}{b^2 c^2 \log ^2(F)}+\frac {d F^{c (a+b x)}}{b c \log (F)}+\frac {e F^{c (a+b x)} x}{b c \log (F)}+\frac {f F^{c (a+b x)} x^2}{b c \log (F)}+\frac {F^{c (a+b x)} g x^3}{b c \log (F)} \]
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Rubi [A]
time = 0.14, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2227, 2225,
2207} \begin {gather*} -\frac {6 g F^{c (a+b x)}}{b^4 c^4 \log ^4(F)}+\frac {2 f F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}+\frac {6 g x F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}-\frac {e F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}-\frac {2 f x F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}-\frac {3 g x^2 F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}+\frac {d F^{c (a+b x)}}{b c \log (F)}+\frac {e x F^{c (a+b x)}}{b c \log (F)}+\frac {f x^2 F^{c (a+b x)}}{b c \log (F)}+\frac {g x^3 F^{c (a+b x)}}{b c \log (F)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2207
Rule 2225
Rule 2227
Rubi steps
\begin {align*} \int F^{c (a+b x)} \left (d+e x+f x^2+g x^3\right ) \, dx &=\int \left (d F^{c (a+b x)}+e F^{c (a+b x)} x+f F^{c (a+b x)} x^2+F^{c (a+b x)} g x^3\right ) \, dx\\ &=d \int F^{c (a+b x)} \, dx+e \int F^{c (a+b x)} x \, dx+f \int F^{c (a+b x)} x^2 \, dx+g \int F^{c (a+b x)} x^3 \, dx\\ &=\frac {d F^{c (a+b x)}}{b c \log (F)}+\frac {e F^{c (a+b x)} x}{b c \log (F)}+\frac {f F^{c (a+b x)} x^2}{b c \log (F)}+\frac {F^{c (a+b x)} g x^3}{b c \log (F)}-\frac {e \int F^{c (a+b x)} \, dx}{b c \log (F)}-\frac {(2 f) \int F^{c (a+b x)} x \, dx}{b c \log (F)}-\frac {(3 g) \int F^{c (a+b x)} x^2 \, dx}{b c \log (F)}\\ &=-\frac {e F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}-\frac {2 f F^{c (a+b x)} x}{b^2 c^2 \log ^2(F)}-\frac {3 F^{c (a+b x)} g x^2}{b^2 c^2 \log ^2(F)}+\frac {d F^{c (a+b x)}}{b c \log (F)}+\frac {e F^{c (a+b x)} x}{b c \log (F)}+\frac {f F^{c (a+b x)} x^2}{b c \log (F)}+\frac {F^{c (a+b x)} g x^3}{b c \log (F)}+\frac {(2 f) \int F^{c (a+b x)} \, dx}{b^2 c^2 \log ^2(F)}+\frac {(6 g) \int F^{c (a+b x)} x \, dx}{b^2 c^2 \log ^2(F)}\\ &=\frac {2 f F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}+\frac {6 F^{c (a+b x)} g x}{b^3 c^3 \log ^3(F)}-\frac {e F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}-\frac {2 f F^{c (a+b x)} x}{b^2 c^2 \log ^2(F)}-\frac {3 F^{c (a+b x)} g x^2}{b^2 c^2 \log ^2(F)}+\frac {d F^{c (a+b x)}}{b c \log (F)}+\frac {e F^{c (a+b x)} x}{b c \log (F)}+\frac {f F^{c (a+b x)} x^2}{b c \log (F)}+\frac {F^{c (a+b x)} g x^3}{b c \log (F)}-\frac {(6 g) \int F^{c (a+b x)} \, dx}{b^3 c^3 \log ^3(F)}\\ &=-\frac {6 F^{c (a+b x)} g}{b^4 c^4 \log ^4(F)}+\frac {2 f F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}+\frac {6 F^{c (a+b x)} g x}{b^3 c^3 \log ^3(F)}-\frac {e F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}-\frac {2 f F^{c (a+b x)} x}{b^2 c^2 \log ^2(F)}-\frac {3 F^{c (a+b x)} g x^2}{b^2 c^2 \log ^2(F)}+\frac {d F^{c (a+b x)}}{b c \log (F)}+\frac {e F^{c (a+b x)} x}{b c \log (F)}+\frac {f F^{c (a+b x)} x^2}{b c \log (F)}+\frac {F^{c (a+b x)} g x^3}{b c \log (F)}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 84, normalized size = 0.37 \begin {gather*} \frac {F^{c (a+b x)} \left (-6 g+2 b c (f+3 g x) \log (F)-b^2 c^2 (e+x (2 f+3 g x)) \log ^2(F)+b^3 c^3 (d+x (e+x (f+g x))) \log ^3(F)\right )}{b^4 c^4 \log ^4(F)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 138, normalized size = 0.60
method | result | size |
gosper | \(\frac {\left (g \,x^{3} c^{3} b^{3} \ln \left (F \right )^{3}+\ln \left (F \right )^{3} b^{3} c^{3} f \,x^{2}+\ln \left (F \right )^{3} b^{3} c^{3} e x +c^{3} b^{3} \ln \left (F \right )^{3} d -3 \ln \left (F \right )^{2} b^{2} c^{2} g \,x^{2}-2 \ln \left (F \right )^{2} b^{2} c^{2} f x -c^{2} b^{2} \ln \left (F \right )^{2} e +6 \ln \left (F \right ) b c g x +2 f c b \ln \left (F \right )-6 g \right ) F^{c \left (b x +a \right )}}{c^{4} b^{4} \ln \left (F \right )^{4}}\) | \(138\) |
risch | \(\frac {\left (g \,x^{3} c^{3} b^{3} \ln \left (F \right )^{3}+\ln \left (F \right )^{3} b^{3} c^{3} f \,x^{2}+\ln \left (F \right )^{3} b^{3} c^{3} e x +c^{3} b^{3} \ln \left (F \right )^{3} d -3 \ln \left (F \right )^{2} b^{2} c^{2} g \,x^{2}-2 \ln \left (F \right )^{2} b^{2} c^{2} f x -c^{2} b^{2} \ln \left (F \right )^{2} e +6 \ln \left (F \right ) b c g x +2 f c b \ln \left (F \right )-6 g \right ) F^{c \left (b x +a \right )}}{c^{4} b^{4} \ln \left (F \right )^{4}}\) | \(138\) |
norman | \(\frac {\left (c^{3} b^{3} \ln \left (F \right )^{3} d -c^{2} b^{2} \ln \left (F \right )^{2} e +2 f c b \ln \left (F \right )-6 g \right ) {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{c^{4} b^{4} \ln \left (F \right )^{4}}+\frac {g \,x^{3} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{c b \ln \left (F \right )}+\frac {\left (f c b \ln \left (F \right )-3 g \right ) x^{2} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{c^{2} b^{2} \ln \left (F \right )^{2}}+\frac {\left (c^{2} b^{2} \ln \left (F \right )^{2} e -2 f c b \ln \left (F \right )+6 g \right ) x \,{\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{c^{3} b^{3} \ln \left (F \right )^{3}}\) | \(163\) |
meijerg | \(\frac {F^{c a} g \left (6-\frac {\left (-4 b^{3} c^{3} x^{3} \ln \left (F \right )^{3}+12 b^{2} c^{2} x^{2} \ln \left (F \right )^{2}-24 b c x \ln \left (F \right )+24\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{4}\right )}{c^{4} b^{4} \ln \left (F \right )^{4}}-\frac {F^{c a} f \left (2-\frac {\left (3 b^{2} c^{2} x^{2} \ln \left (F \right )^{2}-6 b c x \ln \left (F \right )+6\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{3}\right )}{c^{3} b^{3} \ln \left (F \right )^{3}}+\frac {F^{c a} e \left (1-\frac {\left (-2 b c x \ln \left (F \right )+2\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{2}\right )}{c^{2} b^{2} \ln \left (F \right )^{2}}-\frac {F^{c a} d \left (1-{\mathrm e}^{b c x \ln \left (F \right )}\right )}{c b \ln \left (F \right )}\) | \(188\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 196, normalized size = 0.86 \begin {gather*} \frac {F^{b c x + a c} d}{b c \log \left (F\right )} + \frac {{\left (F^{a c} b c x \log \left (F\right ) - F^{a c}\right )} e^{\left (b c x \log \left (F\right ) + 1\right )}}{b^{2} c^{2} \log \left (F\right )^{2}} + \frac {{\left (F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{a c} b c x \log \left (F\right ) + 2 \, F^{a c}\right )} F^{b c x} f}{b^{3} c^{3} \log \left (F\right )^{3}} + \frac {{\left (F^{a c} b^{3} c^{3} x^{3} \log \left (F\right )^{3} - 3 \, F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{a c} b c x \log \left (F\right ) - 6 \, F^{a c}\right )} F^{b c x} g}{b^{4} c^{4} \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 = 124, normalized size = 0.54 \begin {gather*} \frac {{\left ({\left (b^{3} c^{3} g x^{3} + b^{3} c^{3} f x^{2} + b^{3} c^{3} x e + b^{3} c^{3} d\right )} \log \left (F\right )^{3} - {\left (3 \, b^{2} c^{2} g x^{2} + 2 \, b^{2} c^{2} f x + b^{2} c^{2} e\right )} \log \left (F\right )^{2} + 2 \, {\left (3 \, b c g x + b c f\right )} \log \left (F\right ) - 6 \, g\right )} F^{b c x + a c}}{b^{4} c^{4} \log \left (F\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 190, normalized size = 0.83 \begin {gather*} \begin {cases} \frac {F^{c \left (a + b x\right )} \left (b^{3} c^{3} d \log {\left (F \right )}^{3} + b^{3} c^{3} e x \log {\left (F \right )}^{3} + b^{3} c^{3} f x^{2} \log {\left (F \right )}^{3} + b^{3} c^{3} g x^{3} \log {\left (F \right )}^{3} - b^{2} c^{2} e \log {\left (F \right )}^{2} - 2 b^{2} c^{2} f x \log {\left (F \right )}^{2} - 3 b^{2} c^{2} g x^{2} \log {\left (F \right )}^{2} + 2 b c f \log {\left (F \right )} + 6 b c g x \log {\left (F \right )} - 6 g\right )}{b^{4} c^{4} \log {\left (F \right )}^{4}} & \text {for}\: b^{4} c^{4} \log {\left (F \right )}^{4} \neq 0 \\d x + \frac {e x^{2}}{2} + \frac {f x^{3}}{3} + \frac {g x^{4}}{4} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 3.20, size = 4275, normalized size = 18.67 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.54, size = 138, normalized size = 0.60 \begin {gather*} \frac {F^{a\,c+b\,c\,x}\,\left (g\,b^3\,c^3\,x^3\,{\ln \left (F\right )}^3+f\,b^3\,c^3\,x^2\,{\ln \left (F\right )}^3+e\,b^3\,c^3\,x\,{\ln \left (F\right )}^3+d\,b^3\,c^3\,{\ln \left (F\right )}^3-3\,g\,b^2\,c^2\,x^2\,{\ln \left (F\right )}^2-2\,f\,b^2\,c^2\,x\,{\ln \left (F\right )}^2-e\,b^2\,c^2\,{\ln \left (F\right )}^2+6\,g\,b\,c\,x\,\ln \left (F\right )+2\,f\,b\,c\,\ln \left (F\right )-6\,g\right )}{b^4\,c^4\,{\ln \left (F\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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