3.281 \(\int F^{a+b (c+d x)^3} (c+d x)^{17} \, dx\)

Optimal. Leaf size=105 \[ -\frac{F^{a+b (c+d x)^3} \left (-b^5 \log ^5(F) (c+d x)^{15}+5 b^4 \log ^4(F) (c+d x)^{12}-20 b^3 \log ^3(F) (c+d x)^9+60 b^2 \log ^2(F) (c+d x)^6-120 b \log (F) (c+d x)^3+120\right )}{3 b^6 d \log ^6(F)} \]

[Out]

-(F^(a + b*(c + d*x)^3)*(120 - 120*b*(c + d*x)^3*Log[F] + 60*b^2*(c + d*x)^6*Log[F]^2 - 20*b^3*(c + d*x)^9*Log
[F]^3 + 5*b^4*(c + d*x)^12*Log[F]^4 - b^5*(c + d*x)^15*Log[F]^5))/(3*b^6*d*Log[F]^6)

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Rubi [C]  time = 0.0669175, antiderivative size = 31, normalized size of antiderivative = 0.3, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {2218} \[ -\frac{F^a \text{Gamma}\left (6,-b \log (F) (c+d x)^3\right )}{3 b^6 d \log ^6(F)} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b*(c + d*x)^3)*(c + d*x)^17,x]

[Out]

-(F^a*Gamma[6, -(b*(c + d*x)^3*Log[F])])/(3*b^6*d*Log[F]^6)

Rule 2218

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Simp[(F^a*(e + f*
x)^(m + 1)*Gamma[(m + 1)/n, -(b*(c + d*x)^n*Log[F])])/(f*n*(-(b*(c + d*x)^n*Log[F]))^((m + 1)/n)), x] /; FreeQ
[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int F^{a+b (c+d x)^3} (c+d x)^{17} \, dx &=-\frac{F^a \Gamma \left (6,-b (c+d x)^3 \log (F)\right )}{3 b^6 d \log ^6(F)}\\ \end{align*}

Mathematica [C]  time = 0.0101806, size = 31, normalized size = 0.3 \[ -\frac{F^a \text{Gamma}\left (6,-b \log (F) (c+d x)^3\right )}{3 b^6 d \log ^6(F)} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*(c + d*x)^3)*(c + d*x)^17,x]

[Out]

-(F^a*Gamma[6, -(b*(c + d*x)^3*Log[F])])/(3*b^6*d*Log[F]^6)

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Maple [B]  time = 0.023, size = 857, normalized size = 8.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b*(d*x+c)^3)*(d*x+c)^17,x)

[Out]

1/3*(-120+120*ln(F)*b*c^3+360*ln(F)*b*c*d^2*x^2+360*ln(F)*b*c^2*d*x+120*ln(F)*b*d^3*x^3-3960*ln(F)^4*b^4*c^7*d
^5*x^5-2475*ln(F)^4*b^4*c^8*d^4*x^4-1100*ln(F)^4*b^4*c^9*d^3*x^3+180*c*d^8*x^8*ln(F)^3*b^3-330*ln(F)^4*b^4*c^1
0*d^2*x^2+720*c^2*d^7*x^7*ln(F)^3*b^3-60*ln(F)^4*b^4*c^11*d*x+1680*ln(F)^3*b^3*c^3*d^6*x^6+2520*ln(F)^3*b^3*c^
4*d^5*x^5+2520*ln(F)^3*b^3*c^5*d^4*x^4+1680*ln(F)^3*b^3*c^6*d^3*x^3+720*ln(F)^3*b^3*c^7*d^2*x^2+180*ln(F)^3*b^
3*c^8*d*x-360*c*d^5*x^5*ln(F)^2*b^2-900*c^2*d^4*x^4*ln(F)^2*b^2-1200*ln(F)^2*b^2*c^3*d^3*x^3-900*ln(F)^2*b^2*c
^4*d^2*x^2-360*ln(F)^2*b^2*c^5*d*x+15*d^14*c*x^14*ln(F)^5*b^5+105*d^13*c^2*x^13*ln(F)^5*b^5+455*ln(F)^5*b^5*c^
3*d^12*x^12+1365*ln(F)^5*b^5*c^4*d^11*x^11+3003*ln(F)^5*b^5*c^5*d^10*x^10+5005*ln(F)^5*b^5*c^6*d^9*x^9+6435*ln
(F)^5*b^5*c^7*d^8*x^8+6435*ln(F)^5*b^5*c^8*d^7*x^7+5005*ln(F)^5*b^5*c^9*d^6*x^6-60*c*d^11*x^11*ln(F)^4*b^4+300
3*ln(F)^5*b^5*c^10*d^5*x^5-330*c^2*d^10*x^10*ln(F)^4*b^4+1365*ln(F)^5*b^5*c^11*d^4*x^4-1100*ln(F)^4*b^4*c^3*d^
9*x^9+455*ln(F)^5*b^5*c^12*d^3*x^3-2475*ln(F)^4*b^4*c^4*d^8*x^8+105*ln(F)^5*b^5*c^13*d^2*x^2-3960*ln(F)^4*b^4*
c^5*d^7*x^7+15*ln(F)^5*b^5*c^14*d*x-4620*ln(F)^4*b^4*c^6*d^6*x^6+d^15*x^15*ln(F)^5*b^5-5*d^12*x^12*ln(F)^4*b^4
+20*d^9*x^9*ln(F)^3*b^3-60*d^6*x^6*ln(F)^2*b^2+ln(F)^5*b^5*c^15-5*ln(F)^4*b^4*c^12+20*ln(F)^3*b^3*c^9-60*ln(F)
^2*b^2*c^6)*F^(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)/d/ln(F)^6/b^6

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Maxima [B]  time = 1.67115, size = 1712, normalized size = 16.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^3)*(d*x+c)^17,x, algorithm="maxima")

[Out]

1/3*(F^(b*c^3 + a)*b^5*d^15*x^15*log(F)^5 + 15*F^(b*c^3 + a)*b^5*c*d^14*x^14*log(F)^5 + 105*F^(b*c^3 + a)*b^5*
c^2*d^13*x^13*log(F)^5 + F^(b*c^3 + a)*b^5*c^15*log(F)^5 - 5*F^(b*c^3 + a)*b^4*c^12*log(F)^4 + 20*F^(b*c^3 + a
)*b^3*c^9*log(F)^3 + 5*(91*F^(b*c^3 + a)*b^5*c^3*d^12*log(F)^5 - F^(b*c^3 + a)*b^4*d^12*log(F)^4)*x^12 + 15*(9
1*F^(b*c^3 + a)*b^5*c^4*d^11*log(F)^5 - 4*F^(b*c^3 + a)*b^4*c*d^11*log(F)^4)*x^11 + 33*(91*F^(b*c^3 + a)*b^5*c
^5*d^10*log(F)^5 - 10*F^(b*c^3 + a)*b^4*c^2*d^10*log(F)^4)*x^10 - 60*F^(b*c^3 + a)*b^2*c^6*log(F)^2 + 5*(1001*
F^(b*c^3 + a)*b^5*c^6*d^9*log(F)^5 - 220*F^(b*c^3 + a)*b^4*c^3*d^9*log(F)^4 + 4*F^(b*c^3 + a)*b^3*d^9*log(F)^3
)*x^9 + 45*(143*F^(b*c^3 + a)*b^5*c^7*d^8*log(F)^5 - 55*F^(b*c^3 + a)*b^4*c^4*d^8*log(F)^4 + 4*F^(b*c^3 + a)*b
^3*c*d^8*log(F)^3)*x^8 + 45*(143*F^(b*c^3 + a)*b^5*c^8*d^7*log(F)^5 - 88*F^(b*c^3 + a)*b^4*c^5*d^7*log(F)^4 +
16*F^(b*c^3 + a)*b^3*c^2*d^7*log(F)^3)*x^7 + 5*(1001*F^(b*c^3 + a)*b^5*c^9*d^6*log(F)^5 - 924*F^(b*c^3 + a)*b^
4*c^6*d^6*log(F)^4 + 336*F^(b*c^3 + a)*b^3*c^3*d^6*log(F)^3 - 12*F^(b*c^3 + a)*b^2*d^6*log(F)^2)*x^6 + 3*(1001
*F^(b*c^3 + a)*b^5*c^10*d^5*log(F)^5 - 1320*F^(b*c^3 + a)*b^4*c^7*d^5*log(F)^4 + 840*F^(b*c^3 + a)*b^3*c^4*d^5
*log(F)^3 - 120*F^(b*c^3 + a)*b^2*c*d^5*log(F)^2)*x^5 + 120*F^(b*c^3 + a)*b*c^3*log(F) + 15*(91*F^(b*c^3 + a)*
b^5*c^11*d^4*log(F)^5 - 165*F^(b*c^3 + a)*b^4*c^8*d^4*log(F)^4 + 168*F^(b*c^3 + a)*b^3*c^5*d^4*log(F)^3 - 60*F
^(b*c^3 + a)*b^2*c^2*d^4*log(F)^2)*x^4 + 5*(91*F^(b*c^3 + a)*b^5*c^12*d^3*log(F)^5 - 220*F^(b*c^3 + a)*b^4*c^9
*d^3*log(F)^4 + 336*F^(b*c^3 + a)*b^3*c^6*d^3*log(F)^3 - 240*F^(b*c^3 + a)*b^2*c^3*d^3*log(F)^2 + 24*F^(b*c^3
+ a)*b*d^3*log(F))*x^3 + 15*(7*F^(b*c^3 + a)*b^5*c^13*d^2*log(F)^5 - 22*F^(b*c^3 + a)*b^4*c^10*d^2*log(F)^4 +
48*F^(b*c^3 + a)*b^3*c^7*d^2*log(F)^3 - 60*F^(b*c^3 + a)*b^2*c^4*d^2*log(F)^2 + 24*F^(b*c^3 + a)*b*c*d^2*log(F
))*x^2 + 15*(F^(b*c^3 + a)*b^5*c^14*d*log(F)^5 - 4*F^(b*c^3 + a)*b^4*c^11*d*log(F)^4 + 12*F^(b*c^3 + a)*b^3*c^
8*d*log(F)^3 - 24*F^(b*c^3 + a)*b^2*c^5*d*log(F)^2 + 24*F^(b*c^3 + a)*b*c^2*d*log(F))*x - 120*F^(b*c^3 + a))*e
^(b*d^3*x^3*log(F) + 3*b*c*d^2*x^2*log(F) + 3*b*c^2*d*x*log(F))/(b^6*d*log(F)^6)

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Fricas [B]  time = 1.89123, size = 1513, normalized size = 14.41 \begin{align*} \frac{{\left ({\left (b^{5} d^{15} x^{15} + 15 \, b^{5} c d^{14} x^{14} + 105 \, b^{5} c^{2} d^{13} x^{13} + 455 \, b^{5} c^{3} d^{12} x^{12} + 1365 \, b^{5} c^{4} d^{11} x^{11} + 3003 \, b^{5} c^{5} d^{10} x^{10} + 5005 \, b^{5} c^{6} d^{9} x^{9} + 6435 \, b^{5} c^{7} d^{8} x^{8} + 6435 \, b^{5} c^{8} d^{7} x^{7} + 5005 \, b^{5} c^{9} d^{6} x^{6} + 3003 \, b^{5} c^{10} d^{5} x^{5} + 1365 \, b^{5} c^{11} d^{4} x^{4} + 455 \, b^{5} c^{12} d^{3} x^{3} + 105 \, b^{5} c^{13} d^{2} x^{2} + 15 \, b^{5} c^{14} d x + b^{5} c^{15}\right )} \log \left (F\right )^{5} - 5 \,{\left (b^{4} d^{12} x^{12} + 12 \, b^{4} c d^{11} x^{11} + 66 \, b^{4} c^{2} d^{10} x^{10} + 220 \, b^{4} c^{3} d^{9} x^{9} + 495 \, b^{4} c^{4} d^{8} x^{8} + 792 \, b^{4} c^{5} d^{7} x^{7} + 924 \, b^{4} c^{6} d^{6} x^{6} + 792 \, b^{4} c^{7} d^{5} x^{5} + 495 \, b^{4} c^{8} d^{4} x^{4} + 220 \, b^{4} c^{9} d^{3} x^{3} + 66 \, b^{4} c^{10} d^{2} x^{2} + 12 \, b^{4} c^{11} d x + b^{4} c^{12}\right )} \log \left (F\right )^{4} + 20 \,{\left (b^{3} d^{9} x^{9} + 9 \, b^{3} c d^{8} x^{8} + 36 \, b^{3} c^{2} d^{7} x^{7} + 84 \, b^{3} c^{3} d^{6} x^{6} + 126 \, b^{3} c^{4} d^{5} x^{5} + 126 \, b^{3} c^{5} d^{4} x^{4} + 84 \, b^{3} c^{6} d^{3} x^{3} + 36 \, b^{3} c^{7} d^{2} x^{2} + 9 \, b^{3} c^{8} d x + b^{3} c^{9}\right )} \log \left (F\right )^{3} - 60 \,{\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 15 \, b^{2} c^{4} d^{2} x^{2} + 6 \, b^{2} c^{5} d x + b^{2} c^{6}\right )} \log \left (F\right )^{2} + 120 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right ) - 120\right )} F^{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}}{3 \, b^{6} d \log \left (F\right )^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^3)*(d*x+c)^17,x, algorithm="fricas")

[Out]

1/3*((b^5*d^15*x^15 + 15*b^5*c*d^14*x^14 + 105*b^5*c^2*d^13*x^13 + 455*b^5*c^3*d^12*x^12 + 1365*b^5*c^4*d^11*x
^11 + 3003*b^5*c^5*d^10*x^10 + 5005*b^5*c^6*d^9*x^9 + 6435*b^5*c^7*d^8*x^8 + 6435*b^5*c^8*d^7*x^7 + 5005*b^5*c
^9*d^6*x^6 + 3003*b^5*c^10*d^5*x^5 + 1365*b^5*c^11*d^4*x^4 + 455*b^5*c^12*d^3*x^3 + 105*b^5*c^13*d^2*x^2 + 15*
b^5*c^14*d*x + b^5*c^15)*log(F)^5 - 5*(b^4*d^12*x^12 + 12*b^4*c*d^11*x^11 + 66*b^4*c^2*d^10*x^10 + 220*b^4*c^3
*d^9*x^9 + 495*b^4*c^4*d^8*x^8 + 792*b^4*c^5*d^7*x^7 + 924*b^4*c^6*d^6*x^6 + 792*b^4*c^7*d^5*x^5 + 495*b^4*c^8
*d^4*x^4 + 220*b^4*c^9*d^3*x^3 + 66*b^4*c^10*d^2*x^2 + 12*b^4*c^11*d*x + b^4*c^12)*log(F)^4 + 20*(b^3*d^9*x^9
+ 9*b^3*c*d^8*x^8 + 36*b^3*c^2*d^7*x^7 + 84*b^3*c^3*d^6*x^6 + 126*b^3*c^4*d^5*x^5 + 126*b^3*c^5*d^4*x^4 + 84*b
^3*c^6*d^3*x^3 + 36*b^3*c^7*d^2*x^2 + 9*b^3*c^8*d*x + b^3*c^9)*log(F)^3 - 60*(b^2*d^6*x^6 + 6*b^2*c*d^5*x^5 +
15*b^2*c^2*d^4*x^4 + 20*b^2*c^3*d^3*x^3 + 15*b^2*c^4*d^2*x^2 + 6*b^2*c^5*d*x + b^2*c^6)*log(F)^2 + 120*(b*d^3*
x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3)*log(F) - 120)*F^(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 +
a)/(b^6*d*log(F)^6)

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Sympy [A]  time = 0.549061, size = 1171, normalized size = 11.15 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b*(d*x+c)**3)*(d*x+c)**17,x)

[Out]

Piecewise((F**(a + b*(c + d*x)**3)*(b**5*c**15*log(F)**5 + 15*b**5*c**14*d*x*log(F)**5 + 105*b**5*c**13*d**2*x
**2*log(F)**5 + 455*b**5*c**12*d**3*x**3*log(F)**5 + 1365*b**5*c**11*d**4*x**4*log(F)**5 + 3003*b**5*c**10*d**
5*x**5*log(F)**5 + 5005*b**5*c**9*d**6*x**6*log(F)**5 + 6435*b**5*c**8*d**7*x**7*log(F)**5 + 6435*b**5*c**7*d*
*8*x**8*log(F)**5 + 5005*b**5*c**6*d**9*x**9*log(F)**5 + 3003*b**5*c**5*d**10*x**10*log(F)**5 + 1365*b**5*c**4
*d**11*x**11*log(F)**5 + 455*b**5*c**3*d**12*x**12*log(F)**5 + 105*b**5*c**2*d**13*x**13*log(F)**5 + 15*b**5*c
*d**14*x**14*log(F)**5 + b**5*d**15*x**15*log(F)**5 - 5*b**4*c**12*log(F)**4 - 60*b**4*c**11*d*x*log(F)**4 - 3
30*b**4*c**10*d**2*x**2*log(F)**4 - 1100*b**4*c**9*d**3*x**3*log(F)**4 - 2475*b**4*c**8*d**4*x**4*log(F)**4 -
3960*b**4*c**7*d**5*x**5*log(F)**4 - 4620*b**4*c**6*d**6*x**6*log(F)**4 - 3960*b**4*c**5*d**7*x**7*log(F)**4 -
2475*b**4*c**4*d**8*x**8*log(F)**4 - 1100*b**4*c**3*d**9*x**9*log(F)**4 - 330*b**4*c**2*d**10*x**10*log(F)**4
- 60*b**4*c*d**11*x**11*log(F)**4 - 5*b**4*d**12*x**12*log(F)**4 + 20*b**3*c**9*log(F)**3 + 180*b**3*c**8*d*x
*log(F)**3 + 720*b**3*c**7*d**2*x**2*log(F)**3 + 1680*b**3*c**6*d**3*x**3*log(F)**3 + 2520*b**3*c**5*d**4*x**4
*log(F)**3 + 2520*b**3*c**4*d**5*x**5*log(F)**3 + 1680*b**3*c**3*d**6*x**6*log(F)**3 + 720*b**3*c**2*d**7*x**7
*log(F)**3 + 180*b**3*c*d**8*x**8*log(F)**3 + 20*b**3*d**9*x**9*log(F)**3 - 60*b**2*c**6*log(F)**2 - 360*b**2*
c**5*d*x*log(F)**2 - 900*b**2*c**4*d**2*x**2*log(F)**2 - 1200*b**2*c**3*d**3*x**3*log(F)**2 - 900*b**2*c**2*d*
*4*x**4*log(F)**2 - 360*b**2*c*d**5*x**5*log(F)**2 - 60*b**2*d**6*x**6*log(F)**2 + 120*b*c**3*log(F) + 360*b*c
**2*d*x*log(F) + 360*b*c*d**2*x**2*log(F) + 120*b*d**3*x**3*log(F) - 120)/(3*b**6*d*log(F)**6), Ne(3*b**6*d*lo
g(F)**6, 0)), (c**17*x + 17*c**16*d*x**2/2 + 136*c**15*d**2*x**3/3 + 170*c**14*d**3*x**4 + 476*c**13*d**4*x**5
+ 3094*c**12*d**5*x**6/3 + 1768*c**11*d**6*x**7 + 2431*c**10*d**7*x**8 + 24310*c**9*d**8*x**9/9 + 2431*c**8*d
**9*x**10 + 1768*c**7*d**10*x**11 + 3094*c**6*d**11*x**12/3 + 476*c**5*d**12*x**13 + 170*c**4*d**13*x**14 + 13
6*c**3*d**14*x**15/3 + 17*c**2*d**15*x**16/2 + c*d**16*x**17 + d**17*x**18/18, True))

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^3)*(d*x+c)^17,x, algorithm="giac")

[Out]

Exception raised: TypeError