Optimal. Leaf size=122 \[ \frac{6 F^{a+\frac{b}{c+d x}}}{b^4 d \log ^4(F)}-\frac{6 F^{a+\frac{b}{c+d x}}}{b^3 d \log ^3(F) (c+d x)}+\frac{3 F^{a+\frac{b}{c+d x}}}{b^2 d \log ^2(F) (c+d x)^2}-\frac{F^{a+\frac{b}{c+d x}}}{b d \log (F) (c+d x)^3} \]
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Rubi [A] time = 0.297368, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{6 F^{a+\frac{b}{c+d x}}}{b^4 d \log ^4(F)}-\frac{6 F^{a+\frac{b}{c+d x}}}{b^3 d \log ^3(F) (c+d x)}+\frac{3 F^{a+\frac{b}{c+d x}}}{b^2 d \log ^2(F) (c+d x)^2}-\frac{F^{a+\frac{b}{c+d x}}}{b d \log (F) (c+d x)^3} \]
Antiderivative was successfully verified.
[In] Int[F^(a + b/(c + d*x))/(c + d*x)^5,x]
[Out]
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Rubi in Sympy [A] time = 27.5714, size = 100, normalized size = 0.82 \[ - \frac{F^{a + \frac{b}{c + d x}}}{b d \left (c + d x\right )^{3} \log{\left (F \right )}} + \frac{3 F^{a + \frac{b}{c + d x}}}{b^{2} d \left (c + d x\right )^{2} \log{\left (F \right )}^{2}} - \frac{6 F^{a + \frac{b}{c + d x}}}{b^{3} d \left (c + d x\right ) \log{\left (F \right )}^{3}} + \frac{6 F^{a + \frac{b}{c + d x}}}{b^{4} d \log{\left (F \right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(F**(a+b/(d*x+c))/(d*x+c)**5,x)
[Out]
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Mathematica [A] time = 0.058682, size = 76, normalized size = 0.62 \[ \frac{F^{a+\frac{b}{c+d x}} \left (-b^3 \log ^3(F)+3 b^2 \log ^2(F) (c+d x)-6 b \log (F) (c+d x)^2+6 (c+d x)^3\right )}{b^4 d \log ^4(F) (c+d x)^3} \]
Antiderivative was successfully verified.
[In] Integrate[F^(a + b/(c + d*x))/(c + d*x)^5,x]
[Out]
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Maple [A] time = 0.053, size = 243, normalized size = 2. \[{\frac{1}{ \left ( dx+c \right ) ^{4}} \left ( -{\frac{ \left ( \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}-6\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}c+18\,\ln \left ( F \right ) b{c}^{2}-24\,{c}^{3} \right ) x}{ \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}}{{\rm e}^{ \left ( a+{\frac{b}{dx+c}} \right ) \ln \left ( F \right ) }}}+6\,{\frac{{d}^{3}{x}^{4}}{ \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}}{{\rm e}^{ \left ( a+{\frac{b}{dx+c}} \right ) \ln \left ( F \right ) }}}+3\,{\frac{d \left ( \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}-6\,cb\ln \left ( F \right ) +12\,{c}^{2} \right ){x}^{2}}{ \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}}{{\rm e}^{ \left ( a+{\frac{b}{dx+c}} \right ) \ln \left ( F \right ) }}}-6\,{\frac{{d}^{2} \left ( b\ln \left ( F \right ) -4\,c \right ){x}^{3}}{ \left ( \ln \left ( F \right ) \right ) ^{4}{b}^{4}}{{\rm e}^{ \left ( a+{\frac{b}{dx+c}} \right ) \ln \left ( F \right ) }}}-{\frac{ \left ( \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}-3\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}c+6\,\ln \left ( F \right ) b{c}^{2}-6\,{c}^{3} \right ) c}{d{b}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}}{{\rm e}^{ \left ( a+{\frac{b}{dx+c}} \right ) \ln \left ( F \right ) }}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(F^(a+b/(d*x+c))/(d*x+c)^5,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{a + \frac{b}{d x + c}}}{{\left (d x + c\right )}^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^(a + b/(d*x + c))/(d*x + c)^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.25486, size = 203, normalized size = 1.66 \[ \frac{{\left (6 \, d^{3} x^{3} - b^{3} \log \left (F\right )^{3} + 18 \, c d^{2} x^{2} + 18 \, c^{2} d x + 6 \, c^{3} + 3 \,{\left (b^{2} d x + b^{2} c\right )} \log \left (F\right )^{2} - 6 \,{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \left (F\right )\right )} F^{\frac{a d x + a c + b}{d x + c}}}{{\left (b^{4} d^{4} x^{3} + 3 \, b^{4} c d^{3} x^{2} + 3 \, b^{4} c^{2} d^{2} x + b^{4} c^{3} d\right )} \log \left (F\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^(a + b/(d*x + c))/(d*x + c)^5,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.412339, size = 177, normalized size = 1.45 \[ \frac{F^{a + \frac{b}{c + d x}} \left (- b^{3} \log{\left (F \right )}^{3} + 3 b^{2} c \log{\left (F \right )}^{2} + 3 b^{2} d x \log{\left (F \right )}^{2} - 6 b c^{2} \log{\left (F \right )} - 12 b c d x \log{\left (F \right )} - 6 b d^{2} x^{2} \log{\left (F \right )} + 6 c^{3} + 18 c^{2} d x + 18 c d^{2} x^{2} + 6 d^{3} x^{3}\right )}{b^{4} c^{3} d \log{\left (F \right )}^{4} + 3 b^{4} c^{2} d^{2} x \log{\left (F \right )}^{4} + 3 b^{4} c d^{3} x^{2} \log{\left (F \right )}^{4} + b^{4} d^{4} x^{3} \log{\left (F \right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F**(a+b/(d*x+c))/(d*x+c)**5,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{a + \frac{b}{d x + c}}}{{\left (d x + c\right )}^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(F^(a + b/(d*x + c))/(d*x + c)^5,x, algorithm="giac")
[Out]