Optimal. Leaf size=242 \[ -\frac {6 f^2 F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}+\frac {4 e f F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac {6 f^2 F^{a+b c+b d x} x}{b^3 d^3 \log ^3(F)}-\frac {e^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac {4 e f F^{a+b c+b d x} x}{b^2 d^2 \log ^2(F)}-\frac {3 f^2 F^{a+b c+b d x} x^2}{b^2 d^2 \log ^2(F)}+\frac {e^2 F^{a+b c+b d x} x}{b d \log (F)}+\frac {2 e f F^{a+b c+b d x} x^2}{b d \log (F)}+\frac {f^2 F^{a+b c+b d x} x^3}{b d \log (F)} \]
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Rubi [A]
time = 0.25, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2227, 2207,
2225} \begin {gather*} -\frac {6 f^2 F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}+\frac {4 e f F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac {6 f^2 x F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}-\frac {e^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac {4 e f x F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac {3 f^2 x^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}+\frac {e^2 x F^{a+b c+b d x}}{b d \log (F)}+\frac {2 e f x^2 F^{a+b c+b d x}}{b d \log (F)}+\frac {f^2 x^3 F^{a+b c+b d x}}{b d \log (F)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2207
Rule 2225
Rule 2227
Rubi steps
\begin {align*} \int F^{a+b (c+d x)} x (e+f x)^2 \, dx &=\int \left (e^2 F^{a+b c+b d x} x+2 e f F^{a+b c+b d x} x^2+f^2 F^{a+b c+b d x} x^3\right ) \, dx\\ &=e^2 \int F^{a+b c+b d x} x \, dx+(2 e f) \int F^{a+b c+b d x} x^2 \, dx+f^2 \int F^{a+b c+b d x} x^3 \, dx\\ &=\frac {e^2 F^{a+b c+b d x} x}{b d \log (F)}+\frac {2 e f F^{a+b c+b d x} x^2}{b d \log (F)}+\frac {f^2 F^{a+b c+b d x} x^3}{b d \log (F)}-\frac {e^2 \int F^{a+b c+b d x} \, dx}{b d \log (F)}-\frac {(4 e f) \int F^{a+b c+b d x} x \, dx}{b d \log (F)}-\frac {\left (3 f^2\right ) \int F^{a+b c+b d x} x^2 \, dx}{b d \log (F)}\\ &=-\frac {e^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac {4 e f F^{a+b c+b d x} x}{b^2 d^2 \log ^2(F)}-\frac {3 f^2 F^{a+b c+b d x} x^2}{b^2 d^2 \log ^2(F)}+\frac {e^2 F^{a+b c+b d x} x}{b d \log (F)}+\frac {2 e f F^{a+b c+b d x} x^2}{b d \log (F)}+\frac {f^2 F^{a+b c+b d x} x^3}{b d \log (F)}+\frac {(4 e f) \int F^{a+b c+b d x} \, dx}{b^2 d^2 \log ^2(F)}+\frac {\left (6 f^2\right ) \int F^{a+b c+b d x} x \, dx}{b^2 d^2 \log ^2(F)}\\ &=\frac {4 e f F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac {6 f^2 F^{a+b c+b d x} x}{b^3 d^3 \log ^3(F)}-\frac {e^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac {4 e f F^{a+b c+b d x} x}{b^2 d^2 \log ^2(F)}-\frac {3 f^2 F^{a+b c+b d x} x^2}{b^2 d^2 \log ^2(F)}+\frac {e^2 F^{a+b c+b d x} x}{b d \log (F)}+\frac {2 e f F^{a+b c+b d x} x^2}{b d \log (F)}+\frac {f^2 F^{a+b c+b d x} x^3}{b d \log (F)}-\frac {\left (6 f^2\right ) \int F^{a+b c+b d x} \, dx}{b^3 d^3 \log ^3(F)}\\ &=-\frac {6 f^2 F^{a+b c+b d x}}{b^4 d^4 \log ^4(F)}+\frac {4 e f F^{a+b c+b d x}}{b^3 d^3 \log ^3(F)}+\frac {6 f^2 F^{a+b c+b d x} x}{b^3 d^3 \log ^3(F)}-\frac {e^2 F^{a+b c+b d x}}{b^2 d^2 \log ^2(F)}-\frac {4 e f F^{a+b c+b d x} x}{b^2 d^2 \log ^2(F)}-\frac {3 f^2 F^{a+b c+b d x} x^2}{b^2 d^2 \log ^2(F)}+\frac {e^2 F^{a+b c+b d x} x}{b d \log (F)}+\frac {2 e f F^{a+b c+b d x} x^2}{b d \log (F)}+\frac {f^2 F^{a+b c+b d x} x^3}{b d \log (F)}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 91, normalized size = 0.38 \begin {gather*} \frac {F^{a+b (c+d x)} \left (-6 f^2+2 b d f (2 e+3 f x) \log (F)-b^2 d^2 \left (e^2+4 e f x+3 f^2 x^2\right ) \log ^2(F)+b^3 d^3 x (e+f x)^2 \log ^3(F)\right )}{b^4 d^4 \log ^4(F)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 144, normalized size = 0.60
method | result | size |
gosper | \(\frac {\left (\ln \left (F \right )^{3} b^{3} d^{3} f^{2} x^{3}+2 \ln \left (F \right )^{3} b^{3} d^{3} e f \,x^{2}+\ln \left (F \right )^{3} b^{3} d^{3} e^{2} x -3 \ln \left (F \right )^{2} b^{2} d^{2} f^{2} x^{2}-4 \ln \left (F \right )^{2} b^{2} d^{2} e f x -\ln \left (F \right )^{2} b^{2} d^{2} e^{2}+6 \ln \left (F \right ) b d \,f^{2} x +4 e f \ln \left (F \right ) b d -6 f^{2}\right ) F^{b d x +c b +a}}{\ln \left (F \right )^{4} b^{4} d^{4}}\) | \(144\) |
risch | \(\frac {\left (\ln \left (F \right )^{3} b^{3} d^{3} f^{2} x^{3}+2 \ln \left (F \right )^{3} b^{3} d^{3} e f \,x^{2}+\ln \left (F \right )^{3} b^{3} d^{3} e^{2} x -3 \ln \left (F \right )^{2} b^{2} d^{2} f^{2} x^{2}-4 \ln \left (F \right )^{2} b^{2} d^{2} e f x -\ln \left (F \right )^{2} b^{2} d^{2} e^{2}+6 \ln \left (F \right ) b d \,f^{2} x +4 e f \ln \left (F \right ) b d -6 f^{2}\right ) F^{b d x +c b +a}}{\ln \left (F \right )^{4} b^{4} d^{4}}\) | \(144\) |
meijerg | \(\frac {F^{c b +a} f^{2} \left (6-\frac {\left (-4 b^{3} d^{3} x^{3} \ln \left (F \right )^{3}+12 b^{2} d^{2} x^{2} \ln \left (F \right )^{2}-24 b d x \ln \left (F \right )+24\right ) {\mathrm e}^{b d x \ln \left (F \right )}}{4}\right )}{\ln \left (F \right )^{4} b^{4} d^{4}}-\frac {2 F^{c b +a} f e \left (2-\frac {\left (3 b^{2} d^{2} x^{2} \ln \left (F \right )^{2}-6 b d x \ln \left (F \right )+6\right ) {\mathrm e}^{b d x \ln \left (F \right )}}{3}\right )}{b^{3} d^{3} \ln \left (F \right )^{3}}+\frac {F^{c b +a} e^{2} \left (1-\frac {\left (-2 b d x \ln \left (F \right )+2\right ) {\mathrm e}^{b d x \ln \left (F \right )}}{2}\right )}{b^{2} d^{2} \ln \left (F \right )^{2}}\) | \(170\) |
norman | \(\frac {f^{2} x^{3} {\mathrm e}^{\left (a +b \left (d x +c \right )\right ) \ln \left (F \right )}}{b d \ln \left (F \right )}+\frac {\left (\ln \left (F \right )^{2} b^{2} d^{2} e^{2}-4 e f \ln \left (F \right ) b d +6 f^{2}\right ) x \,{\mathrm e}^{\left (a +b \left (d x +c \right )\right ) \ln \left (F \right )}}{\ln \left (F \right )^{3} b^{3} d^{3}}+\frac {f \left (2 \ln \left (F \right ) b d e -3 f \right ) x^{2} {\mathrm e}^{\left (a +b \left (d x +c \right )\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2} d^{2}}-\frac {\left (\ln \left (F \right )^{2} b^{2} d^{2} e^{2}-4 e f \ln \left (F \right ) b d +6 f^{2}\right ) {\mathrm e}^{\left (a +b \left (d x +c \right )\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4} d^{4}}\) | \(177\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.43, size = 198, normalized size = 0.82 \begin {gather*} \frac {{\left (F^{b c + a} b d x \log \left (F\right ) - F^{b c + a}\right )} e^{\left (b d x \log \left (F\right ) + 2\right )}}{b^{2} d^{2} \log \left (F\right )^{2}} + \frac {2 \, {\left (F^{b c + a} b^{2} d^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{b c + a} b d x \log \left (F\right ) + 2 \, F^{b c + a}\right )} f e^{\left (b d x \log \left (F\right ) + 1\right )}}{b^{3} d^{3} \log \left (F\right )^{3}} + \frac {{\left (F^{b c + a} b^{3} d^{3} x^{3} \log \left (F\right )^{3} - 3 \, F^{b c + a} b^{2} d^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{b c + a} b d x \log \left (F\right ) - 6 \, F^{b c + a}\right )} F^{b d x} f^{2}}{b^{4} d^{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 = 0.39, size = 133, normalized size = 0.55 \begin {gather*} \frac {{\left ({\left (b^{3} d^{3} f^{2} x^{3} + 2 \, b^{3} d^{3} f x^{2} e + b^{3} d^{3} x e^{2}\right )} \log \left (F\right )^{3} - {\left (3 \, b^{2} d^{2} f^{2} x^{2} + 4 \, b^{2} d^{2} f x e + b^{2} d^{2} e^{2}\right )} \log \left (F\right )^{2} - 6 \, f^{2} + 2 \, {\left (3 \, b d f^{2} x + 2 \, b d f e\right )} \log \left (F\right )\right )} F^{b d x + b c + a}}{b^{4} d^{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 = 0.08, size = 199, normalized size = 0.82 \begin {gather*} \begin {cases} \frac {F^{a + b \left (c + d x\right )} \left (b^{3} d^{3} e^{2} x \log {\left (F \right )}^{3} + 2 b^{3} d^{3} e f x^{2} \log {\left (F \right )}^{3} + b^{3} d^{3} f^{2} x^{3} \log {\left (F \right )}^{3} - b^{2} d^{2} e^{2} \log {\left (F \right )}^{2} - 4 b^{2} d^{2} e f x \log {\left (F \right )}^{2} - 3 b^{2} d^{2} f^{2} x^{2} \log {\left (F \right )}^{2} + 4 b d e f \log {\left (F \right )} + 6 b d f^{2} x \log {\left (F \right )} - 6 f^{2}\right )}{b^{4} d^{4} \log {\left (F \right )}^{4}} & \text {for}\: b^{4} d^{4} \log {\left (F \right )}^{4} \neq 0 \\\frac {e^{2} x^{2}}{2} + \frac {2 e f x^{3}}{3} + \frac {f^{2} 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 = 2.19, size = 4688, normalized size = 19.37 \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 = 3.52, size = 143, normalized size = 0.59 \begin {gather*} \frac {F^{a+b\,c+b\,d\,x}\,\left (b^3\,d^3\,e^2\,x\,{\ln \left (F\right )}^3+2\,b^3\,d^3\,e\,f\,x^2\,{\ln \left (F\right )}^3+b^3\,d^3\,f^2\,x^3\,{\ln \left (F\right )}^3-b^2\,d^2\,e^2\,{\ln \left (F\right )}^2-4\,b^2\,d^2\,e\,f\,x\,{\ln \left (F\right )}^2-3\,b^2\,d^2\,f^2\,x^2\,{\ln \left (F\right )}^2+4\,b\,d\,e\,f\,\ln \left (F\right )+6\,b\,d\,f^2\,x\,\ln \left (F\right )-6\,f^2\right )}{b^4\,d^4\,{\ln \left (F\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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