3.282 \(\int F^{a+b (c+d x)^3} (c+d x)^{14} \, dx\)
Optimal. Leaf size=88 \[ \frac{F^{a+b (c+d x)^3} \left (b^4 \log ^4(F) (c+d x)^{12}-4 b^3 \log ^3(F) (c+d x)^9+12 b^2 \log ^2(F) (c+d x)^6-24 b \log (F) (c+d x)^3+24\right )}{3 b^5 d \log ^5(F)} \]
[Out]
(F^(a + b*(c + d*x)^3)*(24 - 24*b*(c + d*x)^3*Log[F] + 12*b^2*(c + d*x)^6*Log[F]^2 - 4*b^3*(c + d*x)^9*Log[F]^
3 + b^4*(c + d*x)^12*Log[F]^4))/(3*b^5*d*Log[F]^5)
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Rubi [C] time = 0.0670609, antiderivative size = 31, normalized size of antiderivative =
0.35, 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} \[ \frac{F^a \text{Gamma}\left (5,-b \log (F) (c+d x)^3\right )}{3 b^5 d \log ^5(F)} \]
Antiderivative was successfully verified.
[In]
Int[F^(a + b*(c + d*x)^3)*(c + d*x)^14,x]
[Out]
(F^a*Gamma[5, -(b*(c + d*x)^3*Log[F])])/(3*b^5*d*Log[F]^5)
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)^{14} \, dx &=\frac{F^a \Gamma \left (5,-b (c+d x)^3 \log (F)\right )}{3 b^5 d \log ^5(F)}\\ \end{align*}
Mathematica [C] time = 0.0089296, size = 31, normalized size = 0.35 \[ \frac{F^a \text{Gamma}\left (5,-b \log (F) (c+d x)^3\right )}{3 b^5 d \log ^5(F)} \]
Antiderivative was successfully verified.
[In]
Integrate[F^(a + b*(c + d*x)^3)*(c + d*x)^14,x]
[Out]
(F^a*Gamma[5, -(b*(c + d*x)^3*Log[F])])/(3*b^5*d*Log[F]^5)
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Maple [B] time = 0.019, size = 584, normalized size = 6.6 \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)^14,x)
[Out]
1/3*(24-24*ln(F)*b*c^3-72*ln(F)*b*c*d^2*x^2-72*ln(F)*b*c^2*d*x-24*ln(F)*b*d^3*x^3+792*ln(F)^4*b^4*c^7*d^5*x^5+
495*ln(F)^4*b^4*c^8*d^4*x^4+220*ln(F)^4*b^4*c^9*d^3*x^3-36*c*d^8*x^8*ln(F)^3*b^3+66*ln(F)^4*b^4*c^10*d^2*x^2-1
44*c^2*d^7*x^7*ln(F)^3*b^3+12*ln(F)^4*b^4*c^11*d*x-336*ln(F)^3*b^3*c^3*d^6*x^6-504*ln(F)^3*b^3*c^4*d^5*x^5-504
*ln(F)^3*b^3*c^5*d^4*x^4-336*ln(F)^3*b^3*c^6*d^3*x^3-144*ln(F)^3*b^3*c^7*d^2*x^2-36*ln(F)^3*b^3*c^8*d*x+72*c*d
^5*x^5*ln(F)^2*b^2+180*c^2*d^4*x^4*ln(F)^2*b^2+240*ln(F)^2*b^2*c^3*d^3*x^3+180*ln(F)^2*b^2*c^4*d^2*x^2+72*ln(F
)^2*b^2*c^5*d*x+12*c*d^11*x^11*ln(F)^4*b^4+66*c^2*d^10*x^10*ln(F)^4*b^4+220*ln(F)^4*b^4*c^3*d^9*x^9+495*ln(F)^
4*b^4*c^4*d^8*x^8+792*ln(F)^4*b^4*c^5*d^7*x^7+924*ln(F)^4*b^4*c^6*d^6*x^6+d^12*x^12*ln(F)^4*b^4-4*d^9*x^9*ln(F
)^3*b^3+12*d^6*x^6*ln(F)^2*b^2+ln(F)^4*b^4*c^12-4*ln(F)^3*b^3*c^9+12*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)^5/b^5
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Maxima [B] time = 1.71363, size = 1180, normalized size = 13.41 \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)^14,x, algorithm="maxima")
[Out]
1/3*(F^(b*c^3 + a)*b^4*d^12*x^12*log(F)^4 + 12*F^(b*c^3 + a)*b^4*c*d^11*x^11*log(F)^4 + 66*F^(b*c^3 + a)*b^4*c
^2*d^10*x^10*log(F)^4 + F^(b*c^3 + a)*b^4*c^12*log(F)^4 - 4*F^(b*c^3 + a)*b^3*c^9*log(F)^3 + 12*F^(b*c^3 + a)*
b^2*c^6*log(F)^2 + 4*(55*F^(b*c^3 + a)*b^4*c^3*d^9*log(F)^4 - F^(b*c^3 + a)*b^3*d^9*log(F)^3)*x^9 + 9*(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 + 72*(11*F^(b*c^3 + a)*b^4*c^5*d^7*lo
g(F)^4 - 2*F^(b*c^3 + a)*b^3*c^2*d^7*log(F)^3)*x^7 + 12*(77*F^(b*c^3 + a)*b^4*c^6*d^6*log(F)^4 - 28*F^(b*c^3 +
a)*b^3*c^3*d^6*log(F)^3 + F^(b*c^3 + a)*b^2*d^6*log(F)^2)*x^6 + 72*(11*F^(b*c^3 + a)*b^4*c^7*d^5*log(F)^4 - 7
*F^(b*c^3 + a)*b^3*c^4*d^5*log(F)^3 + F^(b*c^3 + a)*b^2*c*d^5*log(F)^2)*x^5 - 24*F^(b*c^3 + a)*b*c^3*log(F) +
9*(55*F^(b*c^3 + a)*b^4*c^8*d^4*log(F)^4 - 56*F^(b*c^3 + a)*b^3*c^5*d^4*log(F)^3 + 20*F^(b*c^3 + a)*b^2*c^2*d^
4*log(F)^2)*x^4 + 4*(55*F^(b*c^3 + a)*b^4*c^9*d^3*log(F)^4 - 84*F^(b*c^3 + a)*b^3*c^6*d^3*log(F)^3 + 60*F^(b*c
^3 + a)*b^2*c^3*d^3*log(F)^2 - 6*F^(b*c^3 + a)*b*d^3*log(F))*x^3 + 6*(11*F^(b*c^3 + a)*b^4*c^10*d^2*log(F)^4 -
24*F^(b*c^3 + a)*b^3*c^7*d^2*log(F)^3 + 30*F^(b*c^3 + a)*b^2*c^4*d^2*log(F)^2 - 12*F^(b*c^3 + a)*b*c*d^2*log(
F))*x^2 + 12*(F^(b*c^3 + a)*b^4*c^11*d*log(F)^4 - 3*F^(b*c^3 + a)*b^3*c^8*d*log(F)^3 + 6*F^(b*c^3 + a)*b^2*c^5
*d*log(F)^2 - 6*F^(b*c^3 + a)*b*c^2*d*log(F))*x + 24*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^5*d*log(F)^5)
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Fricas [B] time = 1.8485, size = 1019, normalized size = 11.58 \begin{align*} \frac{{\left ({\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} - 4 \,{\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} + 12 \,{\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} - 24 \,{\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 ) + 24\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^{5} d \log \left (F\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F^(a+b*(d*x+c)^3)*(d*x+c)^14,x, algorithm="fricas")
[Out]
1/3*((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 - 4*(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 + 12*(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 - 24*(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x +
b*c^3)*log(F) + 24)*F^(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)/(b^5*d*log(F)^5)
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Sympy [A] time = 0.439723, size = 823, normalized size = 9.35 \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)**14,x)
[Out]
Piecewise((F**(a + b*(c + d*x)**3)*(b**4*c**12*log(F)**4 + 12*b**4*c**11*d*x*log(F)**4 + 66*b**4*c**10*d**2*x*
*2*log(F)**4 + 220*b**4*c**9*d**3*x**3*log(F)**4 + 495*b**4*c**8*d**4*x**4*log(F)**4 + 792*b**4*c**7*d**5*x**5
*log(F)**4 + 924*b**4*c**6*d**6*x**6*log(F)**4 + 792*b**4*c**5*d**7*x**7*log(F)**4 + 495*b**4*c**4*d**8*x**8*l
og(F)**4 + 220*b**4*c**3*d**9*x**9*log(F)**4 + 66*b**4*c**2*d**10*x**10*log(F)**4 + 12*b**4*c*d**11*x**11*log(
F)**4 + b**4*d**12*x**12*log(F)**4 - 4*b**3*c**9*log(F)**3 - 36*b**3*c**8*d*x*log(F)**3 - 144*b**3*c**7*d**2*x
**2*log(F)**3 - 336*b**3*c**6*d**3*x**3*log(F)**3 - 504*b**3*c**5*d**4*x**4*log(F)**3 - 504*b**3*c**4*d**5*x**
5*log(F)**3 - 336*b**3*c**3*d**6*x**6*log(F)**3 - 144*b**3*c**2*d**7*x**7*log(F)**3 - 36*b**3*c*d**8*x**8*log(
F)**3 - 4*b**3*d**9*x**9*log(F)**3 + 12*b**2*c**6*log(F)**2 + 72*b**2*c**5*d*x*log(F)**2 + 180*b**2*c**4*d**2*
x**2*log(F)**2 + 240*b**2*c**3*d**3*x**3*log(F)**2 + 180*b**2*c**2*d**4*x**4*log(F)**2 + 72*b**2*c*d**5*x**5*l
og(F)**2 + 12*b**2*d**6*x**6*log(F)**2 - 24*b*c**3*log(F) - 72*b*c**2*d*x*log(F) - 72*b*c*d**2*x**2*log(F) - 2
4*b*d**3*x**3*log(F) + 24)/(3*b**5*d*log(F)**5), Ne(3*b**5*d*log(F)**5, 0)), (c**14*x + 7*c**13*d*x**2 + 91*c*
*12*d**2*x**3/3 + 91*c**11*d**3*x**4 + 1001*c**10*d**4*x**5/5 + 1001*c**9*d**5*x**6/3 + 429*c**8*d**6*x**7 + 4
29*c**7*d**7*x**8 + 1001*c**6*d**8*x**9/3 + 1001*c**5*d**9*x**10/5 + 91*c**4*d**10*x**11 + 91*c**3*d**11*x**12
/3 + 7*c**2*d**12*x**13 + c*d**13*x**14 + d**14*x**15/15, 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)^14,x, algorithm="giac")
[Out]
Exception raised: TypeError