\(\int F^{a+b (c+d x)^3} (c+d x)^{14} \, dx\) [282]

Optimal result

Integrand size = 21, antiderivative size = 88 \[ \int F^{a+b (c+d x)^3} (c+d x)^{14} \, dx=\frac {F^{a+b (c+d x)^3} \left (24-24 b (c+d x)^3 \log (F)+12 b^2 (c+d x)^6 \log ^2(F)-4 b^3 (c+d x)^9 \log ^3(F)+b^4 (c+d x)^{12} \log ^4(F)\right )}{3 b^5 d \log ^5(F)} \]

[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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2249} \[ \int F^{a+b (c+d x)^3} (c+d x)^{14} \, dx=\frac {F^{a+b (c+d x)^3} \left (b^4 \log ^4(F) (c+d x)^{12}-4 b^3 \log ^3(F) (c+d x)^9+12 b^2 \log ^2(F) (c+d x)^6-24 b \log (F) (c+d x)^3+24\right )}{3 b^5 d \log ^5(F)} \]

[In]

Int[F^(a + b*(c + d*x)^3)*(c + d*x)^14,x]

[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)

Rule 2249

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{p = Simplify
[(m + 1)/n]}, Simp[(-F^a)*((f/d)^m/(d*n*((-b)*Log[F])^p))*Simplify[FunctionExpand[Gamma[p, (-b)*(c + d*x)^n*Lo
g[F]]]], x] /; IGtQ[p, 0]] /; FreeQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0] &&  !TrueQ[$UseGamma]

Rubi steps \begin{align*} \text {integral}& = \frac {F^{a+b (c+d x)^3} \left (24-24 b (c+d x)^3 \log (F)+12 b^2 (c+d x)^6 \log ^2(F)-4 b^3 (c+d x)^9 \log ^3(F)+b^4 (c+d x)^{12} \log ^4(F)\right )}{3 b^5 d \log ^5(F)} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.35 \[ \int F^{a+b (c+d x)^3} (c+d x)^{14} \, dx=\frac {F^a \Gamma \left (5,-b (c+d x)^3 \log (F)\right )}{3 b^5 d \log ^5(F)} \]

[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)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(583\) vs. \(2(86)=172\).

Time = 2.61 (sec) , antiderivative size = 584, normalized size of antiderivative = 6.64

method	result	size
gosper	\(\frac {\left (24-24 \ln \left (F \right ) b \,c^{3}-24 \ln \left (F \right ) b \,d^{3} x^{3}+12 c \,d^{11} x^{11} \ln \left (F \right )^{4} b^{4}+66 c^{2} d^{10} x^{10} \ln \left (F \right )^{4} b^{4}+220 \ln \left (F \right )^{4} b^{4} c^{3} d^{9} x^{9}+495 \ln \left (F \right )^{4} b^{4} c^{4} d^{8} x^{8}+792 \ln \left (F \right )^{4} b^{4} c^{5} d^{7} x^{7}+924 \ln \left (F \right )^{4} b^{4} c^{6} d^{6} x^{6}+792 \ln \left (F \right )^{4} b^{4} c^{7} d^{5} x^{5}+495 \ln \left (F \right )^{4} b^{4} c^{8} d^{4} x^{4}+\ln \left (F \right )^{4} b^{4} c^{12}-4 \ln \left (F \right )^{3} b^{3} c^{9}+12 \ln \left (F \right )^{2} b^{2} c^{6}+d^{12} x^{12} \ln \left (F \right )^{4} b^{4}-4 d^{9} x^{9} \ln \left (F \right )^{3} b^{3}+12 d^{6} x^{6} \ln \left (F \right )^{2} b^{2}-72 \ln \left (F \right ) b c \,d^{2} x^{2}-72 \ln \left (F \right ) b \,c^{2} d x +220 \ln \left (F \right )^{4} b^{4} c^{9} d^{3} x^{3}-36 c \,d^{8} x^{8} \ln \left (F \right )^{3} b^{3}+66 \ln \left (F \right )^{4} b^{4} c^{10} d^{2} x^{2}-144 c^{2} d^{7} x^{7} \ln \left (F \right )^{3} b^{3}+12 \ln \left (F \right )^{4} b^{4} c^{11} d x -336 \ln \left (F \right )^{3} b^{3} c^{3} d^{6} x^{6}-504 \ln \left (F \right )^{3} b^{3} c^{4} d^{5} x^{5}-504 \ln \left (F \right )^{3} b^{3} c^{5} d^{4} x^{4}-336 \ln \left (F \right )^{3} b^{3} c^{6} d^{3} x^{3}-144 \ln \left (F \right )^{3} b^{3} c^{7} d^{2} x^{2}-36 \ln \left (F \right )^{3} b^{3} c^{8} d x +72 c \,d^{5} x^{5} \ln \left (F \right )^{2} b^{2}+180 c^{2} d^{4} x^{4} \ln \left (F \right )^{2} b^{2}+240 \ln \left (F \right )^{2} b^{2} c^{3} d^{3} x^{3}+180 \ln \left (F \right )^{2} b^{2} c^{4} d^{2} x^{2}+72 \ln \left (F \right )^{2} b^{2} c^{5} d x \right ) F^{b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a}}{3 d \ln \left (F \right )^{5} b^{5}}\)	\(584\)
risch	\(\frac {\left (24-24 \ln \left (F \right ) b \,c^{3}-24 \ln \left (F \right ) b \,d^{3} x^{3}+12 c \,d^{11} x^{11} \ln \left (F \right )^{4} b^{4}+66 c^{2} d^{10} x^{10} \ln \left (F \right )^{4} b^{4}+220 \ln \left (F \right )^{4} b^{4} c^{3} d^{9} x^{9}+495 \ln \left (F \right )^{4} b^{4} c^{4} d^{8} x^{8}+792 \ln \left (F \right )^{4} b^{4} c^{5} d^{7} x^{7}+924 \ln \left (F \right )^{4} b^{4} c^{6} d^{6} x^{6}+792 \ln \left (F \right )^{4} b^{4} c^{7} d^{5} x^{5}+495 \ln \left (F \right )^{4} b^{4} c^{8} d^{4} x^{4}+\ln \left (F \right )^{4} b^{4} c^{12}-4 \ln \left (F \right )^{3} b^{3} c^{9}+12 \ln \left (F \right )^{2} b^{2} c^{6}+d^{12} x^{12} \ln \left (F \right )^{4} b^{4}-4 d^{9} x^{9} \ln \left (F \right )^{3} b^{3}+12 d^{6} x^{6} \ln \left (F \right )^{2} b^{2}-72 \ln \left (F \right ) b c \,d^{2} x^{2}-72 \ln \left (F \right ) b \,c^{2} d x +220 \ln \left (F \right )^{4} b^{4} c^{9} d^{3} x^{3}-36 c \,d^{8} x^{8} \ln \left (F \right )^{3} b^{3}+66 \ln \left (F \right )^{4} b^{4} c^{10} d^{2} x^{2}-144 c^{2} d^{7} x^{7} \ln \left (F \right )^{3} b^{3}+12 \ln \left (F \right )^{4} b^{4} c^{11} d x -336 \ln \left (F \right )^{3} b^{3} c^{3} d^{6} x^{6}-504 \ln \left (F \right )^{3} b^{3} c^{4} d^{5} x^{5}-504 \ln \left (F \right )^{3} b^{3} c^{5} d^{4} x^{4}-336 \ln \left (F \right )^{3} b^{3} c^{6} d^{3} x^{3}-144 \ln \left (F \right )^{3} b^{3} c^{7} d^{2} x^{2}-36 \ln \left (F \right )^{3} b^{3} c^{8} d x +72 c \,d^{5} x^{5} \ln \left (F \right )^{2} b^{2}+180 c^{2} d^{4} x^{4} \ln \left (F \right )^{2} b^{2}+240 \ln \left (F \right )^{2} b^{2} c^{3} d^{3} x^{3}+180 \ln \left (F \right )^{2} b^{2} c^{4} d^{2} x^{2}+72 \ln \left (F \right )^{2} b^{2} c^{5} d x \right ) F^{b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a}}{3 d \ln \left (F \right )^{5} b^{5}}\)	\(584\)
norman	\(\frac {\left (\ln \left (F \right )^{4} b^{4} c^{12}-4 \ln \left (F \right )^{3} b^{3} c^{9}+12 \ln \left (F \right )^{2} b^{2} c^{6}-24 \ln \left (F \right ) b \,c^{3}+24\right ) {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{3 d \ln \left (F \right )^{5} b^{5}}+\frac {d^{11} x^{12} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{3 \ln \left (F \right ) b}+\frac {4 c^{2} \left (\ln \left (F \right )^{3} b^{3} c^{9}-3 \ln \left (F \right )^{2} b^{2} c^{6}+6 \ln \left (F \right ) b \,c^{3}-6\right ) x \,{\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}+\frac {4 d^{2} \left (55 \ln \left (F \right )^{3} b^{3} c^{9}-84 \ln \left (F \right )^{2} b^{2} c^{6}+60 \ln \left (F \right ) b \,c^{3}-6\right ) x^{3} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{3 \ln \left (F \right )^{4} b^{4}}+\frac {4 d^{5} \left (77 \ln \left (F \right )^{2} b^{2} c^{6}-28 \ln \left (F \right ) b \,c^{3}+1\right ) x^{6} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{3} b^{3}}+\frac {4 d^{8} \left (55 \ln \left (F \right ) b \,c^{3}-1\right ) x^{9} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{3 \ln \left (F \right )^{2} b^{2}}+\frac {22 d^{9} c^{2} x^{10} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right ) b}+\frac {4 d^{10} c \,x^{11} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right ) b}+\frac {2 c d \left (11 \ln \left (F \right )^{3} b^{3} c^{9}-24 \ln \left (F \right )^{2} b^{2} c^{6}+30 \ln \left (F \right ) b \,c^{3}-12\right ) x^{2} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}+\frac {24 c \,d^{4} \left (11 \ln \left (F \right )^{2} b^{2} c^{6}-7 \ln \left (F \right ) b \,c^{3}+1\right ) x^{5} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{3} b^{3}}+\frac {3 c \,d^{7} \left (55 \ln \left (F \right ) b \,c^{3}-4\right ) x^{8} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2}}+\frac {3 c^{2} d^{3} \left (55 \ln \left (F \right )^{2} b^{2} c^{6}-56 \ln \left (F \right ) b \,c^{3}+20\right ) x^{4} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{3} b^{3}}+\frac {24 c^{2} d^{6} \left (11 \ln \left (F \right ) b \,c^{3}-2\right ) x^{7} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2}}\)	\(640\)
parallelrisch	\(\text {Expression too large to display}\)	\(1005\)

[In]

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

[Out]

1/3*(24-24*ln(F)*b*c^3-24*ln(F)*b*d^3*x^3+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+792*ln(F)^
4*b^4*c^7*d^5*x^5+495*ln(F)^4*b^4*c^8*d^4*x^4+ln(F)^4*b^4*c^12-4*ln(F)^3*b^3*c^9+12*ln(F)^2*b^2*c^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^2-72*ln(F)*b*c*d^2*x^2-72*ln(F)*b*c^2*d*x+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^2-144*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)*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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 474 vs. \(2 (86) = 172\).

Time = 0.27 (sec) , antiderivative size = 474, normalized size of antiderivative = 5.39 \[ \int F^{a+b (c+d x)^3} (c+d x)^{14} \, dx=\frac {{\left ({\left (b^{4} d^{12} x^{12} + 12 \, b^{4} c d^{11} x^{11} + 66 \, b^{4} c^{2} d^{10} x^{10} + 220 \, b^{4} c^{3} d^{9} x^{9} + 495 \, b^{4} c^{4} d^{8} x^{8} + 792 \, b^{4} c^{5} d^{7} x^{7} + 924 \, b^{4} c^{6} d^{6} x^{6} + 792 \, b^{4} c^{7} d^{5} x^{5} + 495 \, b^{4} c^{8} d^{4} x^{4} + 220 \, b^{4} c^{9} d^{3} x^{3} + 66 \, b^{4} c^{10} d^{2} x^{2} + 12 \, b^{4} c^{11} d x + b^{4} c^{12}\right )} \log \left (F\right )^{4} - 4 \, {\left (b^{3} d^{9} x^{9} + 9 \, b^{3} c d^{8} x^{8} + 36 \, b^{3} c^{2} d^{7} x^{7} + 84 \, b^{3} c^{3} d^{6} x^{6} + 126 \, b^{3} c^{4} d^{5} x^{5} + 126 \, b^{3} c^{5} d^{4} x^{4} + 84 \, b^{3} c^{6} d^{3} x^{3} + 36 \, b^{3} c^{7} d^{2} x^{2} + 9 \, b^{3} c^{8} d x + b^{3} c^{9}\right )} \log \left (F\right )^{3} + 12 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 15 \, b^{2} c^{4} d^{2} x^{2} + 6 \, b^{2} c^{5} d x + b^{2} c^{6}\right )} \log \left (F\right )^{2} - 24 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right ) + 24\right )} F^{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}}{3 \, b^{5} d \log \left (F\right )^{5}} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 821 vs. \(2 (87) = 174\).

Time = 0.29 (sec) , antiderivative size = 821, normalized size of antiderivative = 9.33 \[ \int F^{a+b (c+d x)^3} (c+d x)^{14} \, dx=\begin {cases} \frac {F^{a + b \left (c + d x\right )^{3}} \left (b^{4} c^{12} \log {\left (F \right )}^{4} + 12 b^{4} c^{11} d x \log {\left (F \right )}^{4} + 66 b^{4} c^{10} d^{2} x^{2} \log {\left (F \right )}^{4} + 220 b^{4} c^{9} d^{3} x^{3} \log {\left (F \right )}^{4} + 495 b^{4} c^{8} d^{4} x^{4} \log {\left (F \right )}^{4} + 792 b^{4} c^{7} d^{5} x^{5} \log {\left (F \right )}^{4} + 924 b^{4} c^{6} d^{6} x^{6} \log {\left (F \right )}^{4} + 792 b^{4} c^{5} d^{7} x^{7} \log {\left (F \right )}^{4} + 495 b^{4} c^{4} d^{8} x^{8} \log {\left (F \right )}^{4} + 220 b^{4} c^{3} d^{9} x^{9} \log {\left (F \right )}^{4} + 66 b^{4} c^{2} d^{10} x^{10} \log {\left (F \right )}^{4} + 12 b^{4} c d^{11} x^{11} \log {\left (F \right )}^{4} + b^{4} d^{12} x^{12} \log {\left (F \right )}^{4} - 4 b^{3} c^{9} \log {\left (F \right )}^{3} - 36 b^{3} c^{8} d x \log {\left (F \right )}^{3} - 144 b^{3} c^{7} d^{2} x^{2} \log {\left (F \right )}^{3} - 336 b^{3} c^{6} d^{3} x^{3} \log {\left (F \right )}^{3} - 504 b^{3} c^{5} d^{4} x^{4} \log {\left (F \right )}^{3} - 504 b^{3} c^{4} d^{5} x^{5} \log {\left (F \right )}^{3} - 336 b^{3} c^{3} d^{6} x^{6} \log {\left (F \right )}^{3} - 144 b^{3} c^{2} d^{7} x^{7} \log {\left (F \right )}^{3} - 36 b^{3} c d^{8} x^{8} \log {\left (F \right )}^{3} - 4 b^{3} d^{9} x^{9} \log {\left (F \right )}^{3} + 12 b^{2} c^{6} \log {\left (F \right )}^{2} + 72 b^{2} c^{5} d x \log {\left (F \right )}^{2} + 180 b^{2} c^{4} d^{2} x^{2} \log {\left (F \right )}^{2} + 240 b^{2} c^{3} d^{3} x^{3} \log {\left (F \right )}^{2} + 180 b^{2} c^{2} d^{4} x^{4} \log {\left (F \right )}^{2} + 72 b^{2} c d^{5} x^{5} \log {\left (F \right )}^{2} + 12 b^{2} d^{6} x^{6} \log {\left (F \right )}^{2} - 24 b c^{3} \log {\left (F \right )} - 72 b c^{2} d x \log {\left (F \right )} - 72 b c d^{2} x^{2} \log {\left (F \right )} - 24 b d^{3} x^{3} \log {\left (F \right )} + 24\right )}{3 b^{5} d \log {\left (F \right )}^{5}} & \text {for}\: b^{5} d \log {\left (F \right )}^{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} \]

[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(b**5*d*log(F)**5, 0)), (c**14*x + 7*c**13*d*x**2 + 91*c**1
2*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 + 429
*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))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 874 vs. \(2 (86) = 172\).

Time = 0.43 (sec) , antiderivative size = 874, normalized size of antiderivative = 9.93 \[ \int F^{a+b (c+d x)^3} (c+d x)^{14} \, dx=\frac {{\left (F^{b c^{3} + a} b^{4} d^{12} x^{12} \log \left (F\right )^{4} + 12 \, F^{b c^{3} + a} b^{4} c d^{11} x^{11} \log \left (F\right )^{4} + 66 \, F^{b c^{3} + a} b^{4} c^{2} d^{10} x^{10} \log \left (F\right )^{4} + F^{b c^{3} + a} b^{4} c^{12} \log \left (F\right )^{4} - 4 \, F^{b c^{3} + a} b^{3} c^{9} \log \left (F\right )^{3} + 12 \, F^{b c^{3} + a} b^{2} c^{6} \log \left (F\right )^{2} + 4 \, {\left (55 \, F^{b c^{3} + a} b^{4} c^{3} d^{9} \log \left (F\right )^{4} - F^{b c^{3} + a} b^{3} d^{9} \log \left (F\right )^{3}\right )} x^{9} + 9 \, {\left (55 \, F^{b c^{3} + a} b^{4} c^{4} d^{8} \log \left (F\right )^{4} - 4 \, F^{b c^{3} + a} b^{3} c d^{8} \log \left (F\right )^{3}\right )} x^{8} + 72 \, {\left (11 \, F^{b c^{3} + a} b^{4} c^{5} d^{7} \log \left (F\right )^{4} - 2 \, F^{b c^{3} + a} b^{3} c^{2} d^{7} \log \left (F\right )^{3}\right )} x^{7} + 12 \, {\left (77 \, F^{b c^{3} + a} b^{4} c^{6} d^{6} \log \left (F\right )^{4} - 28 \, F^{b c^{3} + a} b^{3} c^{3} d^{6} \log \left (F\right )^{3} + F^{b c^{3} + a} b^{2} d^{6} \log \left (F\right )^{2}\right )} x^{6} + 72 \, {\left (11 \, F^{b c^{3} + a} b^{4} c^{7} d^{5} \log \left (F\right )^{4} - 7 \, F^{b c^{3} + a} b^{3} c^{4} d^{5} \log \left (F\right )^{3} + F^{b c^{3} + a} b^{2} c d^{5} \log \left (F\right )^{2}\right )} x^{5} - 24 \, F^{b c^{3} + a} b c^{3} \log \left (F\right ) + 9 \, {\left (55 \, F^{b c^{3} + a} b^{4} c^{8} d^{4} \log \left (F\right )^{4} - 56 \, F^{b c^{3} + a} b^{3} c^{5} d^{4} \log \left (F\right )^{3} + 20 \, F^{b c^{3} + a} b^{2} c^{2} d^{4} \log \left (F\right )^{2}\right )} x^{4} + 4 \, {\left (55 \, F^{b c^{3} + a} b^{4} c^{9} d^{3} \log \left (F\right )^{4} - 84 \, F^{b c^{3} + a} b^{3} c^{6} d^{3} \log \left (F\right )^{3} + 60 \, F^{b c^{3} + a} b^{2} c^{3} d^{3} \log \left (F\right )^{2} - 6 \, F^{b c^{3} + a} b d^{3} \log \left (F\right )\right )} x^{3} + 6 \, {\left (11 \, F^{b c^{3} + a} b^{4} c^{10} d^{2} \log \left (F\right )^{4} - 24 \, F^{b c^{3} + a} b^{3} c^{7} d^{2} \log \left (F\right )^{3} + 30 \, F^{b c^{3} + a} b^{2} c^{4} d^{2} \log \left (F\right )^{2} - 12 \, F^{b c^{3} + a} b c d^{2} \log \left (F\right )\right )} x^{2} + 12 \, {\left (F^{b c^{3} + a} b^{4} c^{11} d \log \left (F\right )^{4} - 3 \, F^{b c^{3} + a} b^{3} c^{8} d \log \left (F\right )^{3} + 6 \, F^{b c^{3} + a} b^{2} c^{5} d \log \left (F\right )^{2} - 6 \, F^{b c^{3} + a} b c^{2} d \log \left (F\right )\right )} x + 24 \, F^{b c^{3} + a}\right )} e^{\left (b d^{3} x^{3} \log \left (F\right ) + 3 \, b c d^{2} x^{2} \log \left (F\right ) + 3 \, b c^{2} d x \log \left (F\right )\right )}}{3 \, b^{5} d \log \left (F\right )^{5}} \]

[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)

Giac [F(-2)]

Exception generated. \[ \int F^{a+b (c+d x)^3} (c+d x)^{14} \, dx=\text {Exception raised: TypeError} \]

[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,sageVARx):;OUTP
UT:Polynomial exponent overflow. Error: Bad Argument Value

Mupad [B] (verification not implemented)

Time = 0.74 (sec) , antiderivative size = 487, normalized size of antiderivative = 5.53 \[ \int F^{a+b (c+d x)^3} (c+d x)^{14} \, dx=F^{b\,d^3\,x^3}\,F^{3\,b\,c^2\,d\,x}\,F^a\,F^{b\,c^3}\,F^{3\,b\,c\,d^2\,x^2}\,\left (\frac {b^4\,c^{12}\,{\ln \left (F\right )}^4-4\,b^3\,c^9\,{\ln \left (F\right )}^3+12\,b^2\,c^6\,{\ln \left (F\right )}^2-24\,b\,c^3\,\ln \left (F\right )+24}{3\,b^5\,d\,{\ln \left (F\right )}^5}+\frac {d^{11}\,x^{12}}{3\,b\,\ln \left (F\right )}+\frac {4\,c\,d^{10}\,x^{11}}{b\,\ln \left (F\right )}+\frac {4\,d^2\,x^3\,\left (55\,b^3\,c^9\,{\ln \left (F\right )}^3-84\,b^2\,c^6\,{\ln \left (F\right )}^2+60\,b\,c^3\,\ln \left (F\right )-6\right )}{3\,b^4\,{\ln \left (F\right )}^4}+\frac {4\,d^5\,x^6\,\left (77\,b^2\,c^6\,{\ln \left (F\right )}^2-28\,b\,c^3\,\ln \left (F\right )+1\right )}{b^3\,{\ln \left (F\right )}^3}+\frac {4\,d^8\,x^9\,\left (55\,b\,c^3\,\ln \left (F\right )-1\right )}{3\,b^2\,{\ln \left (F\right )}^2}+\frac {22\,c^2\,d^9\,x^{10}}{b\,\ln \left (F\right )}+\frac {4\,c^2\,x\,\left (b^3\,c^9\,{\ln \left (F\right )}^3-3\,b^2\,c^6\,{\ln \left (F\right )}^2+6\,b\,c^3\,\ln \left (F\right )-6\right )}{b^4\,{\ln \left (F\right )}^4}+\frac {3\,c^2\,d^3\,x^4\,\left (55\,b^2\,c^6\,{\ln \left (F\right )}^2-56\,b\,c^3\,\ln \left (F\right )+20\right )}{b^3\,{\ln \left (F\right )}^3}+\frac {24\,c^2\,d^6\,x^7\,\left (11\,b\,c^3\,\ln \left (F\right )-2\right )}{b^2\,{\ln \left (F\right )}^2}+\frac {2\,c\,d\,x^2\,\left (11\,b^3\,c^9\,{\ln \left (F\right )}^3-24\,b^2\,c^6\,{\ln \left (F\right )}^2+30\,b\,c^3\,\ln \left (F\right )-12\right )}{b^4\,{\ln \left (F\right )}^4}+\frac {24\,c\,d^4\,x^5\,\left (11\,b^2\,c^6\,{\ln \left (F\right )}^2-7\,b\,c^3\,\ln \left (F\right )+1\right )}{b^3\,{\ln \left (F\right )}^3}+\frac {3\,c\,d^7\,x^8\,\left (55\,b\,c^3\,\ln \left (F\right )-4\right )}{b^2\,{\ln \left (F\right )}^2}\right ) \]

[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))