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]

1/3*F^(a+b*(d*x+c)^3)*(24-24*b*(d*x+c)^3*ln(F)+12*b^2*(d*x+c)^6*ln(F)^2-4*b^3*(d*x+c)^9*ln(F)^3+b^4*(d*x+c)^12
*ln(F)^4)/b^5/d/ln(F)^5

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Rubi [C]  time = 0.07, 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*}

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Mathematica [C]  time = 0.01, size = 31, normalized size = 0.35 \[ \frac {F^a \Gamma \left (5,-b (c+d x)^3 \log (F)\right )}{3 b^5 d \log ^5(F)} \]

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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fricas [B]  time = 0.45, size = 474, normalized size = 5.39 \[ \frac {{\left ({\left (b^{4} d^{12} x^{12} + 12 \, b^{4} c d^{11} x^{11} + 66 \, b^{4} c^{2} d^{10} x^{10} + 220 \, b^{4} c^{3} d^{9} x^{9} + 495 \, b^{4} c^{4} d^{8} x^{8} + 792 \, b^{4} c^{5} d^{7} x^{7} + 924 \, b^{4} c^{6} d^{6} x^{6} + 792 \, b^{4} c^{7} d^{5} x^{5} + 495 \, b^{4} c^{8} d^{4} x^{4} + 220 \, b^{4} c^{9} d^{3} x^{3} + 66 \, b^{4} c^{10} d^{2} x^{2} + 12 \, b^{4} c^{11} d x + b^{4} c^{12}\right )} \log \relax (F)^{4} - 4 \, {\left (b^{3} d^{9} x^{9} + 9 \, b^{3} c d^{8} x^{8} + 36 \, b^{3} c^{2} d^{7} x^{7} + 84 \, b^{3} c^{3} d^{6} x^{6} + 126 \, b^{3} c^{4} d^{5} x^{5} + 126 \, b^{3} c^{5} d^{4} x^{4} + 84 \, b^{3} c^{6} d^{3} x^{3} + 36 \, b^{3} c^{7} d^{2} x^{2} + 9 \, b^{3} c^{8} d x + b^{3} c^{9}\right )} \log \relax (F)^{3} + 12 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 15 \, b^{2} c^{4} d^{2} x^{2} + 6 \, b^{2} c^{5} d x + b^{2} c^{6}\right )} \log \relax (F)^{2} - 24 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \relax (F) + 24\right )} F^{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}}{3 \, b^{5} d \log \relax (F)^{5}} \]

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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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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 >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Poly
nomial exponent overflow. Error: Bad Argument Value

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maple [B]  time = 0.02, size = 584, normalized size = 6.64 \[ \frac {\left (b^{4} d^{12} x^{12} \ln \relax (F )^{4}+12 b^{4} c \,d^{11} x^{11} \ln \relax (F )^{4}+66 b^{4} c^{2} d^{10} x^{10} \ln \relax (F )^{4}+220 b^{4} c^{3} d^{9} x^{9} \ln \relax (F )^{4}+495 b^{4} c^{4} d^{8} x^{8} \ln \relax (F )^{4}+792 b^{4} c^{5} d^{7} x^{7} \ln \relax (F )^{4}+924 b^{4} c^{6} d^{6} x^{6} \ln \relax (F )^{4}+792 b^{4} c^{7} d^{5} x^{5} \ln \relax (F )^{4}+495 b^{4} c^{8} d^{4} x^{4} \ln \relax (F )^{4}-4 b^{3} d^{9} x^{9} \ln \relax (F )^{3}+220 b^{4} c^{9} d^{3} x^{3} \ln \relax (F )^{4}-36 b^{3} c \,d^{8} x^{8} \ln \relax (F )^{3}+66 b^{4} c^{10} d^{2} x^{2} \ln \relax (F )^{4}-144 b^{3} c^{2} d^{7} x^{7} \ln \relax (F )^{3}+12 b^{4} c^{11} d x \ln \relax (F )^{4}-336 b^{3} c^{3} d^{6} x^{6} \ln \relax (F )^{3}+b^{4} c^{12} \ln \relax (F )^{4}-504 b^{3} c^{4} d^{5} x^{5} \ln \relax (F )^{3}-504 b^{3} c^{5} d^{4} x^{4} \ln \relax (F )^{3}-336 b^{3} c^{6} d^{3} x^{3} \ln \relax (F )^{3}-144 b^{3} c^{7} d^{2} x^{2} \ln \relax (F )^{3}-36 b^{3} c^{8} d x \ln \relax (F )^{3}+12 b^{2} d^{6} x^{6} \ln \relax (F )^{2}-4 b^{3} c^{9} \ln \relax (F )^{3}+72 b^{2} c \,d^{5} x^{5} \ln \relax (F )^{2}+180 b^{2} c^{2} d^{4} x^{4} \ln \relax (F )^{2}+240 b^{2} c^{3} d^{3} x^{3} \ln \relax (F )^{2}+180 b^{2} c^{4} d^{2} x^{2} \ln \relax (F )^{2}+72 b^{2} c^{5} d x \ln \relax (F )^{2}+12 b^{2} c^{6} \ln \relax (F )^{2}-24 b \,d^{3} x^{3} \ln \relax (F )-72 b c \,d^{2} x^{2} \ln \relax (F )-72 b \,c^{2} d x \ln \relax (F )-24 b \,c^{3} \ln \relax (F )+24\right ) F^{b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a}}{3 b^{5} d \ln \relax (F )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+(d*x+c)^3*b)*(d*x+c)^14,x)

[Out]

1/3*(24-24*b*c^3*ln(F)-72*b*c*d^2*x^2*ln(F)-72*b*c^2*d*x*ln(F)-24*b*d^3*x^3*ln(F)-4*b^3*c^9*ln(F)^3+b^4*d^12*x
^12*ln(F)^4-4*b^3*d^9*x^9*ln(F)^3+12*b^2*d^6*x^6*ln(F)^2+12*b^2*c^6*ln(F)^2+b^4*c^12*ln(F)^4+924*b^4*c^6*d^6*x
^6*ln(F)^4+792*b^4*c^7*d^5*x^5*ln(F)^4+495*b^4*c^8*d^4*x^4*ln(F)^4+220*b^4*c^9*d^3*x^3*ln(F)^4-36*b^3*c*d^8*x^
8*ln(F)^3+66*b^4*c^10*d^2*x^2*ln(F)^4-144*b^3*c^2*d^7*x^7*ln(F)^3+12*b^4*c^11*d*x*ln(F)^4-336*b^3*c^3*d^6*x^6*
ln(F)^3-504*b^3*c^4*d^5*x^5*ln(F)^3-504*b^3*c^5*d^4*x^4*ln(F)^3-336*b^3*c^6*d^3*x^3*ln(F)^3-144*b^3*c^7*d^2*x^
2*ln(F)^3-36*b^3*c^8*d*x*ln(F)^3+72*b^2*c*d^5*x^5*ln(F)^2+180*b^2*c^2*d^4*x^4*ln(F)^2+240*b^2*c^3*d^3*x^3*ln(F
)^2+180*b^2*c^4*d^2*x^2*ln(F)^2+72*b^2*c^5*d*x*ln(F)^2+12*b^4*c*d^11*x^11*ln(F)^4+66*b^4*c^2*d^10*x^10*ln(F)^4
+220*b^4*c^3*d^9*x^9*ln(F)^4+495*b^4*c^4*d^8*x^8*ln(F)^4+792*b^4*c^5*d^7*x^7*ln(F)^4)*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 = 2.29, size = 874, normalized size = 9.93 \[ \frac {{\left (F^{b c^{3} + a} b^{4} d^{12} x^{12} \log \relax (F)^{4} + 12 \, F^{b c^{3} + a} b^{4} c d^{11} x^{11} \log \relax (F)^{4} + 66 \, F^{b c^{3} + a} b^{4} c^{2} d^{10} x^{10} \log \relax (F)^{4} + F^{b c^{3} + a} b^{4} c^{12} \log \relax (F)^{4} - 4 \, F^{b c^{3} + a} b^{3} c^{9} \log \relax (F)^{3} + 12 \, F^{b c^{3} + a} b^{2} c^{6} \log \relax (F)^{2} + 4 \, {\left (55 \, F^{b c^{3} + a} b^{4} c^{3} d^{9} \log \relax (F)^{4} - F^{b c^{3} + a} b^{3} d^{9} \log \relax (F)^{3}\right )} x^{9} + 9 \, {\left (55 \, F^{b c^{3} + a} b^{4} c^{4} d^{8} \log \relax (F)^{4} - 4 \, F^{b c^{3} + a} b^{3} c d^{8} \log \relax (F)^{3}\right )} x^{8} + 72 \, {\left (11 \, F^{b c^{3} + a} b^{4} c^{5} d^{7} \log \relax (F)^{4} - 2 \, F^{b c^{3} + a} b^{3} c^{2} d^{7} \log \relax (F)^{3}\right )} x^{7} + 12 \, {\left (77 \, F^{b c^{3} + a} b^{4} c^{6} d^{6} \log \relax (F)^{4} - 28 \, F^{b c^{3} + a} b^{3} c^{3} d^{6} \log \relax (F)^{3} + F^{b c^{3} + a} b^{2} d^{6} \log \relax (F)^{2}\right )} x^{6} + 72 \, {\left (11 \, F^{b c^{3} + a} b^{4} c^{7} d^{5} \log \relax (F)^{4} - 7 \, F^{b c^{3} + a} b^{3} c^{4} d^{5} \log \relax (F)^{3} + F^{b c^{3} + a} b^{2} c d^{5} \log \relax (F)^{2}\right )} x^{5} - 24 \, F^{b c^{3} + a} b c^{3} \log \relax (F) + 9 \, {\left (55 \, F^{b c^{3} + a} b^{4} c^{8} d^{4} \log \relax (F)^{4} - 56 \, F^{b c^{3} + a} b^{3} c^{5} d^{4} \log \relax (F)^{3} + 20 \, F^{b c^{3} + a} b^{2} c^{2} d^{4} \log \relax (F)^{2}\right )} x^{4} + 4 \, {\left (55 \, F^{b c^{3} + a} b^{4} c^{9} d^{3} \log \relax (F)^{4} - 84 \, F^{b c^{3} + a} b^{3} c^{6} d^{3} \log \relax (F)^{3} + 60 \, F^{b c^{3} + a} b^{2} c^{3} d^{3} \log \relax (F)^{2} - 6 \, F^{b c^{3} + a} b d^{3} \log \relax (F)\right )} x^{3} + 6 \, {\left (11 \, F^{b c^{3} + a} b^{4} c^{10} d^{2} \log \relax (F)^{4} - 24 \, F^{b c^{3} + a} b^{3} c^{7} d^{2} \log \relax (F)^{3} + 30 \, F^{b c^{3} + a} b^{2} c^{4} d^{2} \log \relax (F)^{2} - 12 \, F^{b c^{3} + a} b c d^{2} \log \relax (F)\right )} x^{2} + 12 \, {\left (F^{b c^{3} + a} b^{4} c^{11} d \log \relax (F)^{4} - 3 \, F^{b c^{3} + a} b^{3} c^{8} d \log \relax (F)^{3} + 6 \, F^{b c^{3} + a} b^{2} c^{5} d \log \relax (F)^{2} - 6 \, F^{b c^{3} + a} b c^{2} d \log \relax (F)\right )} x + 24 \, F^{b c^{3} + a}\right )} e^{\left (b d^{3} x^{3} \log \relax (F) + 3 \, b c d^{2} x^{2} \log \relax (F) + 3 \, b c^{2} d x \log \relax (F)\right )}}{3 \, b^{5} d \log \relax (F)^{5}} \]

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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mupad [B]  time = 4.08, size = 487, normalized size = 5.53 \[ F^{b\,d^3\,x^3}\,F^{3\,b\,c^2\,d\,x}\,F^a\,F^{b\,c^3}\,F^{3\,b\,c\,d^2\,x^2}\,\left (\frac {b^4\,c^{12}\,{\ln \relax (F)}^4-4\,b^3\,c^9\,{\ln \relax (F)}^3+12\,b^2\,c^6\,{\ln \relax (F)}^2-24\,b\,c^3\,\ln \relax (F)+24}{3\,b^5\,d\,{\ln \relax (F)}^5}+\frac {d^{11}\,x^{12}}{3\,b\,\ln \relax (F)}+\frac {4\,c\,d^{10}\,x^{11}}{b\,\ln \relax (F)}+\frac {4\,d^2\,x^3\,\left (55\,b^3\,c^9\,{\ln \relax (F)}^3-84\,b^2\,c^6\,{\ln \relax (F)}^2+60\,b\,c^3\,\ln \relax (F)-6\right )}{3\,b^4\,{\ln \relax (F)}^4}+\frac {4\,d^5\,x^6\,\left (77\,b^2\,c^6\,{\ln \relax (F)}^2-28\,b\,c^3\,\ln \relax (F)+1\right )}{b^3\,{\ln \relax (F)}^3}+\frac {4\,d^8\,x^9\,\left (55\,b\,c^3\,\ln \relax (F)-1\right )}{3\,b^2\,{\ln \relax (F)}^2}+\frac {22\,c^2\,d^9\,x^{10}}{b\,\ln \relax (F)}+\frac {4\,c^2\,x\,\left (b^3\,c^9\,{\ln \relax (F)}^3-3\,b^2\,c^6\,{\ln \relax (F)}^2+6\,b\,c^3\,\ln \relax (F)-6\right )}{b^4\,{\ln \relax (F)}^4}+\frac {3\,c^2\,d^3\,x^4\,\left (55\,b^2\,c^6\,{\ln \relax (F)}^2-56\,b\,c^3\,\ln \relax (F)+20\right )}{b^3\,{\ln \relax (F)}^3}+\frac {24\,c^2\,d^6\,x^7\,\left (11\,b\,c^3\,\ln \relax (F)-2\right )}{b^2\,{\ln \relax (F)}^2}+\frac {2\,c\,d\,x^2\,\left (11\,b^3\,c^9\,{\ln \relax (F)}^3-24\,b^2\,c^6\,{\ln \relax (F)}^2+30\,b\,c^3\,\ln \relax (F)-12\right )}{b^4\,{\ln \relax (F)}^4}+\frac {24\,c\,d^4\,x^5\,\left (11\,b^2\,c^6\,{\ln \relax (F)}^2-7\,b\,c^3\,\ln \relax (F)+1\right )}{b^3\,{\ln \relax (F)}^3}+\frac {3\,c\,d^7\,x^8\,\left (55\,b\,c^3\,\ln \relax (F)-4\right )}{b^2\,{\ln \relax (F)}^2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a + b*(c + d*x)^3)*(c + d*x)^14,x)

[Out]

F^(b*d^3*x^3)*F^(3*b*c^2*d*x)*F^a*F^(b*c^3)*F^(3*b*c*d^2*x^2)*((12*b^2*c^6*log(F)^2 - 24*b*c^3*log(F) - 4*b^3*
c^9*log(F)^3 + b^4*c^12*log(F)^4 + 24)/(3*b^5*d*log(F)^5) + (d^11*x^12)/(3*b*log(F)) + (4*c*d^10*x^11)/(b*log(
F)) + (4*d^2*x^3*(60*b*c^3*log(F) - 84*b^2*c^6*log(F)^2 + 55*b^3*c^9*log(F)^3 - 6))/(3*b^4*log(F)^4) + (4*d^5*
x^6*(77*b^2*c^6*log(F)^2 - 28*b*c^3*log(F) + 1))/(b^3*log(F)^3) + (4*d^8*x^9*(55*b*c^3*log(F) - 1))/(3*b^2*log
(F)^2) + (22*c^2*d^9*x^10)/(b*log(F)) + (4*c^2*x*(6*b*c^3*log(F) - 3*b^2*c^6*log(F)^2 + b^3*c^9*log(F)^3 - 6))
/(b^4*log(F)^4) + (3*c^2*d^3*x^4*(55*b^2*c^6*log(F)^2 - 56*b*c^3*log(F) + 20))/(b^3*log(F)^3) + (24*c^2*d^6*x^
7*(11*b*c^3*log(F) - 2))/(b^2*log(F)^2) + (2*c*d*x^2*(30*b*c^3*log(F) - 24*b^2*c^6*log(F)^2 + 11*b^3*c^9*log(F
)^3 - 12))/(b^4*log(F)^4) + (24*c*d^4*x^5*(11*b^2*c^6*log(F)^2 - 7*b*c^3*log(F) + 1))/(b^3*log(F)^3) + (3*c*d^
7*x^8*(55*b*c^3*log(F) - 4))/(b^2*log(F)^2))

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sympy [A]  time = 0.61, size = 823, normalized size = 9.35 \[ \begin {cases} \frac {F^{a + b \left (c + d x\right )^{3}} \left (b^{4} c^{12} \log {\relax (F )}^{4} + 12 b^{4} c^{11} d x \log {\relax (F )}^{4} + 66 b^{4} c^{10} d^{2} x^{2} \log {\relax (F )}^{4} + 220 b^{4} c^{9} d^{3} x^{3} \log {\relax (F )}^{4} + 495 b^{4} c^{8} d^{4} x^{4} \log {\relax (F )}^{4} + 792 b^{4} c^{7} d^{5} x^{5} \log {\relax (F )}^{4} + 924 b^{4} c^{6} d^{6} x^{6} \log {\relax (F )}^{4} + 792 b^{4} c^{5} d^{7} x^{7} \log {\relax (F )}^{4} + 495 b^{4} c^{4} d^{8} x^{8} \log {\relax (F )}^{4} + 220 b^{4} c^{3} d^{9} x^{9} \log {\relax (F )}^{4} + 66 b^{4} c^{2} d^{10} x^{10} \log {\relax (F )}^{4} + 12 b^{4} c d^{11} x^{11} \log {\relax (F )}^{4} + b^{4} d^{12} x^{12} \log {\relax (F )}^{4} - 4 b^{3} c^{9} \log {\relax (F )}^{3} - 36 b^{3} c^{8} d x \log {\relax (F )}^{3} - 144 b^{3} c^{7} d^{2} x^{2} \log {\relax (F )}^{3} - 336 b^{3} c^{6} d^{3} x^{3} \log {\relax (F )}^{3} - 504 b^{3} c^{5} d^{4} x^{4} \log {\relax (F )}^{3} - 504 b^{3} c^{4} d^{5} x^{5} \log {\relax (F )}^{3} - 336 b^{3} c^{3} d^{6} x^{6} \log {\relax (F )}^{3} - 144 b^{3} c^{2} d^{7} x^{7} \log {\relax (F )}^{3} - 36 b^{3} c d^{8} x^{8} \log {\relax (F )}^{3} - 4 b^{3} d^{9} x^{9} \log {\relax (F )}^{3} + 12 b^{2} c^{6} \log {\relax (F )}^{2} + 72 b^{2} c^{5} d x \log {\relax (F )}^{2} + 180 b^{2} c^{4} d^{2} x^{2} \log {\relax (F )}^{2} + 240 b^{2} c^{3} d^{3} x^{3} \log {\relax (F )}^{2} + 180 b^{2} c^{2} d^{4} x^{4} \log {\relax (F )}^{2} + 72 b^{2} c d^{5} x^{5} \log {\relax (F )}^{2} + 12 b^{2} d^{6} x^{6} \log {\relax (F )}^{2} - 24 b c^{3} \log {\relax (F )} - 72 b c^{2} d x \log {\relax (F )} - 72 b c d^{2} x^{2} \log {\relax (F )} - 24 b d^{3} x^{3} \log {\relax (F )} + 24\right )}{3 b^{5} d \log {\relax (F )}^{5}} & \text {for}\: 3 b^{5} d \log {\relax (F )}^{5} \neq 0 \\c^{14} x + 7 c^{13} d x^{2} + \frac {91 c^{12} d^{2} x^{3}}{3} + 91 c^{11} d^{3} x^{4} + \frac {1001 c^{10} d^{4} x^{5}}{5} + \frac {1001 c^{9} d^{5} x^{6}}{3} + 429 c^{8} d^{6} x^{7} + 429 c^{7} d^{7} x^{8} + \frac {1001 c^{6} d^{8} x^{9}}{3} + \frac {1001 c^{5} d^{9} x^{10}}{5} + 91 c^{4} d^{10} x^{11} + \frac {91 c^{3} d^{11} x^{12}}{3} + 7 c^{2} d^{12} x^{13} + c d^{13} x^{14} + \frac {d^{14} x^{15}}{15} & \text {otherwise} \end {cases} \]

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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