3.3 \(\int F^{c (a+b x)} (d+e x)^3 \, dx\)

Optimal. Leaf size=110 \[ -\frac{6 e^3 F^{c (a+b x)}}{b^4 c^4 \log ^4(F)}+\frac{6 e^2 (d+e x) F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}-\frac{3 e (d+e x)^2 F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}+\frac{(d+e x)^3 F^{c (a+b x)}}{b c \log (F)} \]

[Out]

(-6*e^3*F^(c*(a + b*x)))/(b^4*c^4*Log[F]^4) + (6*e^2*F^(c*(a + b*x))*(d + e*x))/
(b^3*c^3*Log[F]^3) - (3*e*F^(c*(a + b*x))*(d + e*x)^2)/(b^2*c^2*Log[F]^2) + (F^(
c*(a + b*x))*(d + e*x)^3)/(b*c*Log[F])

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Rubi [A]  time = 0.134345, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{6 e^3 F^{c (a+b x)}}{b^4 c^4 \log ^4(F)}+\frac{6 e^2 (d+e x) F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}-\frac{3 e (d+e x)^2 F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}+\frac{(d+e x)^3 F^{c (a+b x)}}{b c \log (F)} \]

Antiderivative was successfully verified.

[In]  Int[F^(c*(a + b*x))*(d + e*x)^3,x]

[Out]

(-6*e^3*F^(c*(a + b*x)))/(b^4*c^4*Log[F]^4) + (6*e^2*F^(c*(a + b*x))*(d + e*x))/
(b^3*c^3*Log[F]^3) - (3*e*F^(c*(a + b*x))*(d + e*x)^2)/(b^2*c^2*Log[F]^2) + (F^(
c*(a + b*x))*(d + e*x)^3)/(b*c*Log[F])

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Rubi in Sympy [A]  time = 27.0858, size = 107, normalized size = 0.97 \[ \frac{F^{c \left (a + b x\right )} \left (d + e x\right )^{3}}{b c \log{\left (F \right )}} - \frac{3 F^{c \left (a + b x\right )} e \left (d + e x\right )^{2}}{b^{2} c^{2} \log{\left (F \right )}^{2}} + \frac{6 F^{c \left (a + b x\right )} e^{2} \left (d + e x\right )}{b^{3} c^{3} \log{\left (F \right )}^{3}} - \frac{6 F^{c \left (a + b x\right )} e^{3}}{b^{4} c^{4} \log{\left (F \right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(c*(b*x+a))*(e*x+d)**3,x)

[Out]

F**(c*(a + b*x))*(d + e*x)**3/(b*c*log(F)) - 3*F**(c*(a + b*x))*e*(d + e*x)**2/(
b**2*c**2*log(F)**2) + 6*F**(c*(a + b*x))*e**2*(d + e*x)/(b**3*c**3*log(F)**3) -
6*F**(c*(a + b*x))*e**3/(b**4*c**4*log(F)**4)

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Mathematica [A]  time = 0.0610608, size = 78, normalized size = 0.71 \[ \frac{F^{c (a+b x)} \left (b^3 c^3 \log ^3(F) (d+e x)^3-3 b^2 c^2 e \log ^2(F) (d+e x)^2+6 b c e^2 \log (F) (d+e x)-6 e^3\right )}{b^4 c^4 \log ^4(F)} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(c*(a + b*x))*(d + e*x)^3,x]

[Out]

(F^(c*(a + b*x))*(-6*e^3 + 6*b*c*e^2*(d + e*x)*Log[F] - 3*b^2*c^2*e*(d + e*x)^2*
Log[F]^2 + b^3*c^3*(d + e*x)^3*Log[F]^3))/(b^4*c^4*Log[F]^4)

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Maple [A]  time = 0.012, size = 165, normalized size = 1.5 \[{\frac{ \left ({e}^{3}{x}^{3}{b}^{3}{c}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}+3\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{3}d{e}^{2}{x}^{2}+3\, \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}{c}^{3}{d}^{2}ex+{b}^{3}{c}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}{d}^{3}-3\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{2}{e}^{3}{x}^{2}-6\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{2}d{e}^{2}x-3\,{b}^{2}{c}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}{d}^{2}e+6\,\ln \left ( F \right ) bc{e}^{3}x+6\,d{e}^{2}bc\ln \left ( F \right ) -6\,{e}^{3} \right ){F}^{c \left ( bx+a \right ) }}{{b}^{4}{c}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(c*(b*x+a))*(e*x+d)^3,x)

[Out]

(e^3*x^3*b^3*c^3*ln(F)^3+3*ln(F)^3*b^3*c^3*d*e^2*x^2+3*ln(F)^3*b^3*c^3*d^2*e*x+b
^3*c^3*ln(F)^3*d^3-3*ln(F)^2*b^2*c^2*e^3*x^2-6*ln(F)^2*b^2*c^2*d*e^2*x-3*b^2*c^2
*ln(F)^2*d^2*e+6*ln(F)*b*c*e^3*x+6*d*e^2*b*c*ln(F)-6*e^3)*F^(c*(b*x+a))/b^4/c^4/
ln(F)^4

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Maxima [A]  time = 0.703305, size = 278, normalized size = 2.53 \[ \frac{F^{b c x + a c} d^{3}}{b c \log \left (F\right )} + \frac{3 \,{\left (F^{a c} b c x \log \left (F\right ) - F^{a c}\right )} F^{b c x} d^{2} e}{b^{2} c^{2} \log \left (F\right )^{2}} + \frac{3 \,{\left (F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{a c} b c x \log \left (F\right ) + 2 \, F^{a c}\right )} F^{b c x} d e^{2}}{b^{3} c^{3} \log \left (F\right )^{3}} + \frac{{\left (F^{a c} b^{3} c^{3} x^{3} \log \left (F\right )^{3} - 3 \, F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{a c} b c x \log \left (F\right ) - 6 \, F^{a c}\right )} F^{b c x} e^{3}}{b^{4} c^{4} \log \left (F\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3*F^((b*x + a)*c),x, algorithm="maxima")

[Out]

F^(b*c*x + a*c)*d^3/(b*c*log(F)) + 3*(F^(a*c)*b*c*x*log(F) - F^(a*c))*F^(b*c*x)*
d^2*e/(b^2*c^2*log(F)^2) + 3*(F^(a*c)*b^2*c^2*x^2*log(F)^2 - 2*F^(a*c)*b*c*x*log
(F) + 2*F^(a*c))*F^(b*c*x)*d*e^2/(b^3*c^3*log(F)^3) + (F^(a*c)*b^3*c^3*x^3*log(F
)^3 - 3*F^(a*c)*b^2*c^2*x^2*log(F)^2 + 6*F^(a*c)*b*c*x*log(F) - 6*F^(a*c))*F^(b*
c*x)*e^3/(b^4*c^4*log(F)^4)

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Fricas [A]  time = 0.237181, size = 198, normalized size = 1.8 \[ \frac{{\left ({\left (b^{3} c^{3} e^{3} x^{3} + 3 \, b^{3} c^{3} d e^{2} x^{2} + 3 \, b^{3} c^{3} d^{2} e x + b^{3} c^{3} d^{3}\right )} \log \left (F\right )^{3} - 6 \, e^{3} - 3 \,{\left (b^{2} c^{2} e^{3} x^{2} + 2 \, b^{2} c^{2} d e^{2} x + b^{2} c^{2} d^{2} e\right )} \log \left (F\right )^{2} + 6 \,{\left (b c e^{3} x + b c d e^{2}\right )} \log \left (F\right )\right )} F^{b c x + a c}}{b^{4} c^{4} \log \left (F\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3*F^((b*x + a)*c),x, algorithm="fricas")

[Out]

((b^3*c^3*e^3*x^3 + 3*b^3*c^3*d*e^2*x^2 + 3*b^3*c^3*d^2*e*x + b^3*c^3*d^3)*log(F
)^3 - 6*e^3 - 3*(b^2*c^2*e^3*x^2 + 2*b^2*c^2*d*e^2*x + b^2*c^2*d^2*e)*log(F)^2 +
6*(b*c*e^3*x + b*c*d*e^2)*log(F))*F^(b*c*x + a*c)/(b^4*c^4*log(F)^4)

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Sympy [A]  time = 0.494307, size = 231, normalized size = 2.1 \[ \begin{cases} \frac{F^{c \left (a + b x\right )} \left (b^{3} c^{3} d^{3} \log{\left (F \right )}^{3} + 3 b^{3} c^{3} d^{2} e x \log{\left (F \right )}^{3} + 3 b^{3} c^{3} d e^{2} x^{2} \log{\left (F \right )}^{3} + b^{3} c^{3} e^{3} x^{3} \log{\left (F \right )}^{3} - 3 b^{2} c^{2} d^{2} e \log{\left (F \right )}^{2} - 6 b^{2} c^{2} d e^{2} x \log{\left (F \right )}^{2} - 3 b^{2} c^{2} e^{3} x^{2} \log{\left (F \right )}^{2} + 6 b c d e^{2} \log{\left (F \right )} + 6 b c e^{3} x \log{\left (F \right )} - 6 e^{3}\right )}{b^{4} c^{4} \log{\left (F \right )}^{4}} & \text{for}\: b^{4} c^{4} \log{\left (F \right )}^{4} \neq 0 \\d^{3} x + \frac{3 d^{2} e x^{2}}{2} + d e^{2} x^{3} + \frac{e^{3} x^{4}}{4} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F**(c*(b*x+a))*(e*x+d)**3,x)

[Out]

Piecewise((F**(c*(a + b*x))*(b**3*c**3*d**3*log(F)**3 + 3*b**3*c**3*d**2*e*x*log
(F)**3 + 3*b**3*c**3*d*e**2*x**2*log(F)**3 + b**3*c**3*e**3*x**3*log(F)**3 - 3*b
**2*c**2*d**2*e*log(F)**2 - 6*b**2*c**2*d*e**2*x*log(F)**2 - 3*b**2*c**2*e**3*x*
*2*log(F)**2 + 6*b*c*d*e**2*log(F) + 6*b*c*e**3*x*log(F) - 6*e**3)/(b**4*c**4*lo
g(F)**4), Ne(b**4*c**4*log(F)**4, 0)), (d**3*x + 3*d**2*e*x**2/2 + d*e**2*x**3 +
e**3*x**4/4, True))

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GIAC/XCAS [A]  time = 0.325203, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3*F^((b*x + a)*c),x, algorithm="giac")

[Out]

Done