Integrand size = 25, antiderivative size = 322 \[ \int \left (d+e \left (F^{c (a+b x)}\right )^n\right )^2 (f+g x)^3 \, dx=\frac {d^2 (f+g x)^4}{4 g}-\frac {12 d e \left (F^{a c+b c x}\right )^n g^3}{b^4 c^4 n^4 \log ^4(F)}-\frac {3 e^2 \left (F^{a c+b c x}\right )^{2 n} g^3}{8 b^4 c^4 n^4 \log ^4(F)}+\frac {12 d e \left (F^{a c+b c x}\right )^n g^2 (f+g x)}{b^3 c^3 n^3 \log ^3(F)}+\frac {3 e^2 \left (F^{a c+b c x}\right )^{2 n} g^2 (f+g x)}{4 b^3 c^3 n^3 \log ^3(F)}-\frac {6 d e \left (F^{a c+b c x}\right )^n g (f+g x)^2}{b^2 c^2 n^2 \log ^2(F)}-\frac {3 e^2 \left (F^{a c+b c x}\right )^{2 n} g (f+g x)^2}{4 b^2 c^2 n^2 \log ^2(F)}+\frac {2 d e \left (F^{a c+b c x}\right )^n (f+g x)^3}{b c n \log (F)}+\frac {e^2 \left (F^{a c+b c x}\right )^{2 n} (f+g x)^3}{2 b c n \log (F)} \] Output:
1/4*d^2*(g*x+f)^4/g-12*d*e*(F^(b*c*x+a*c))^n*g^3/b^4/c^4/n^4/ln(F)^4-3/8*e ^2*(F^(b*c*x+a*c))^(2*n)*g^3/b^4/c^4/n^4/ln(F)^4+12*d*e*(F^(b*c*x+a*c))^n* g^2*(g*x+f)/b^3/c^3/n^3/ln(F)^3+3/4*e^2*(F^(b*c*x+a*c))^(2*n)*g^2*(g*x+f)/ b^3/c^3/n^3/ln(F)^3-6*d*e*(F^(b*c*x+a*c))^n*g*(g*x+f)^2/b^2/c^2/n^2/ln(F)^ 2-3/4*e^2*(F^(b*c*x+a*c))^(2*n)*g*(g*x+f)^2/b^2/c^2/n^2/ln(F)^2+2*d*e*(F^( b*c*x+a*c))^n*(g*x+f)^3/b/c/n/ln(F)+1/2*e^2*(F^(b*c*x+a*c))^(2*n)*(g*x+f)^ 3/b/c/n/ln(F)
Time = 1.29 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.74 \[ \int \left (d+e \left (F^{c (a+b x)}\right )^n\right )^2 (f+g x)^3 \, dx=d^2 f^3 x+\frac {3}{2} d^2 f^2 g x^2+d^2 f g^2 x^3+\frac {1}{4} d^2 g^3 x^4+\frac {2 d e \left (F^{c (a+b x)}\right )^n \left (-6 g^3+6 b c g^2 n (f+g x) \log (F)-3 b^2 c^2 g n^2 (f+g x)^2 \log ^2(F)+b^3 c^3 n^3 (f+g x)^3 \log ^3(F)\right )}{b^4 c^4 n^4 \log ^4(F)}+\frac {e^2 \left (F^{c (a+b x)}\right )^{2 n} \left (-3 g^3+6 b c g^2 n (f+g x) \log (F)-6 b^2 c^2 g n^2 (f+g x)^2 \log ^2(F)+4 b^3 c^3 n^3 (f+g x)^3 \log ^3(F)\right )}{8 b^4 c^4 n^4 \log ^4(F)} \] Input:
Integrate[(d + e*(F^(c*(a + b*x)))^n)^2*(f + g*x)^3,x]
Output:
d^2*f^3*x + (3*d^2*f^2*g*x^2)/2 + d^2*f*g^2*x^3 + (d^2*g^3*x^4)/4 + (2*d*e *(F^(c*(a + b*x)))^n*(-6*g^3 + 6*b*c*g^2*n*(f + g*x)*Log[F] - 3*b^2*c^2*g* n^2*(f + g*x)^2*Log[F]^2 + b^3*c^3*n^3*(f + g*x)^3*Log[F]^3))/(b^4*c^4*n^4 *Log[F]^4) + (e^2*(F^(c*(a + b*x)))^(2*n)*(-3*g^3 + 6*b*c*g^2*n*(f + g*x)* Log[F] - 6*b^2*c^2*g*n^2*(f + g*x)^2*Log[F]^2 + 4*b^3*c^3*n^3*(f + g*x)^3* Log[F]^3))/(8*b^4*c^4*n^4*Log[F]^4)
Time = 1.17 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2614, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (f+g x)^3 \left (e \left (F^{c (a+b x)}\right )^n+d\right )^2 \, dx\) |
| \(\Big \downarrow \) 2614 |
| \(\displaystyle \int \left (2 d e (f+g x)^3 \left (F^{a c+b c x}\right )^n+e^2 (f+g x)^3 \left (F^{a c+b c x}\right )^{2 n}+d^2 (f+g x)^3\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {12 d e g^3 \left (F^{a c+b c x}\right )^n}{b^4 c^4 n^4 \log ^4(F)}-\frac {3 e^2 g^3 \left (F^{a c+b c x}\right )^{2 n}}{8 b^4 c^4 n^4 \log ^4(F)}+\frac {12 d e g^2 (f+g x) \left (F^{a c+b c x}\right )^n}{b^3 c^3 n^3 \log ^3(F)}+\frac {3 e^2 g^2 (f+g x) \left (F^{a c+b c x}\right )^{2 n}}{4 b^3 c^3 n^3 \log ^3(F)}-\frac {6 d e g (f+g x)^2 \left (F^{a c+b c x}\right )^n}{b^2 c^2 n^2 \log ^2(F)}-\frac {3 e^2 g (f+g x)^2 \left (F^{a c+b c x}\right )^{2 n}}{4 b^2 c^2 n^2 \log ^2(F)}+\frac {2 d e (f+g x)^3 \left (F^{a c+b c x}\right )^n}{b c n \log (F)}+\frac {e^2 (f+g x)^3 \left (F^{a c+b c x}\right )^{2 n}}{2 b c n \log (F)}+\frac {d^2 (f+g x)^4}{4 g}\) |
Input:
Int[(d + e*(F^(c*(a + b*x)))^n)^2*(f + g*x)^3,x]
Output:
(d^2*(f + g*x)^4)/(4*g) - (12*d*e*(F^(a*c + b*c*x))^n*g^3)/(b^4*c^4*n^4*Lo g[F]^4) - (3*e^2*(F^(a*c + b*c*x))^(2*n)*g^3)/(8*b^4*c^4*n^4*Log[F]^4) + ( 12*d*e*(F^(a*c + b*c*x))^n*g^2*(f + g*x))/(b^3*c^3*n^3*Log[F]^3) + (3*e^2* (F^(a*c + b*c*x))^(2*n)*g^2*(f + g*x))/(4*b^3*c^3*n^3*Log[F]^3) - (6*d*e*( F^(a*c + b*c*x))^n*g*(f + g*x)^2)/(b^2*c^2*n^2*Log[F]^2) - (3*e^2*(F^(a*c + b*c*x))^(2*n)*g*(f + g*x)^2)/(4*b^2*c^2*n^2*Log[F]^2) + (2*d*e*(F^(a*c + b*c*x))^n*(f + g*x)^3)/(b*c*n*Log[F]) + (e^2*(F^(a*c + b*c*x))^(2*n)*(f + g*x)^3)/(2*b*c*n*Log[F])
Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (a + b*(F ^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, m, n}, x] && IGtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(738\) vs. \(2(312)=624\).
Time = 1.16 (sec) , antiderivative size = 739, normalized size of antiderivative = 2.30
| method | result | size |
| parallelrisch | \(\frac {16 d e \,g^{3} \left (F^{c \left (b x +a \right )}\right )^{n} x^{3} \ln \left (F \right )^{3} b^{3} c^{3} n^{3}+12 \ln \left (F \right )^{3} x^{2} \left (F^{c \left (b x +a \right )}\right )^{2 n} b^{3} c^{3} e^{2} f \,g^{2} n^{3}+12 \ln \left (F \right )^{3} x \left (F^{c \left (b x +a \right )}\right )^{2 n} b^{3} c^{3} e^{2} f^{2} g \,n^{3}-48 \ln \left (F \right )^{2} x^{2} \left (F^{c \left (b x +a \right )}\right )^{n} b^{2} c^{2} d e \,g^{3} n^{2}-12 \ln \left (F \right )^{2} x \left (F^{c \left (b x +a \right )}\right )^{2 n} b^{2} c^{2} e^{2} f \,g^{2} n^{2}-48 \ln \left (F \right )^{2} \left (F^{c \left (b x +a \right )}\right )^{n} b^{2} c^{2} d e \,f^{2} g \,n^{2}+96 \ln \left (F \right ) x \left (F^{c \left (b x +a \right )}\right )^{n} b c d e \,g^{3} n +96 \ln \left (F \right ) \left (F^{c \left (b x +a \right )}\right )^{n} b c d e f \,g^{2} n +48 \ln \left (F \right )^{3} x^{2} \left (F^{c \left (b x +a \right )}\right )^{n} b^{3} c^{3} d e f \,g^{2} n^{3}+48 \ln \left (F \right )^{3} x \left (F^{c \left (b x +a \right )}\right )^{n} b^{3} c^{3} d e \,f^{2} g \,n^{3}-96 \ln \left (F \right )^{2} x \left (F^{c \left (b x +a \right )}\right )^{n} b^{2} c^{2} d e f \,g^{2} n^{2}+2 d^{2} g^{3} x^{4} \ln \left (F \right )^{4} b^{4} c^{4} n^{4}+8 d^{2} f^{3} x \ln \left (F \right )^{4} b^{4} c^{4} n^{4}+4 \ln \left (F \right )^{3} \left (F^{c \left (b x +a \right )}\right )^{2 n} b^{3} c^{3} e^{2} f^{3} n^{3}+12 d^{2} g \,f^{2} x^{2} \ln \left (F \right )^{4} b^{4} c^{4} n^{4}+4 e^{2} g^{3} \left (F^{c \left (b x +a \right )}\right )^{2 n} x^{3} \ln \left (F \right )^{3} b^{3} c^{3} n^{3}+16 \ln \left (F \right )^{3} \left (F^{c \left (b x +a \right )}\right )^{n} b^{3} c^{3} d e \,f^{3} n^{3}-96 \left (F^{c \left (b x +a \right )}\right )^{n} d e \,g^{3}+8 d^{2} g^{2} f \,x^{3} \ln \left (F \right )^{4} b^{4} c^{4} n^{4}-6 \ln \left (F \right )^{2} x^{2} \left (F^{c \left (b x +a \right )}\right )^{2 n} b^{2} c^{2} e^{2} g^{3} n^{2}-6 \ln \left (F \right )^{2} \left (F^{c \left (b x +a \right )}\right )^{2 n} b^{2} c^{2} e^{2} f^{2} g \,n^{2}+6 \ln \left (F \right ) x \left (F^{c \left (b x +a \right )}\right )^{2 n} b c \,e^{2} g^{3} n +6 \ln \left (F \right ) \left (F^{c \left (b x +a \right )}\right )^{2 n} b c \,e^{2} f \,g^{2} n -3 \left (F^{c \left (b x +a \right )}\right )^{2 n} e^{2} g^{3}}{8 \ln \left (F \right )^{4} b^{4} c^{4} n^{4}}\) | \(739\) |
| orering | \(\text {Expression too large to display}\) | \(1334\) |
Input:
int((d+e*(F^(c*(b*x+a)))^n)^2*(g*x+f)^3,x,method=_RETURNVERBOSE)
Output:
1/8*(16*d*e*g^3*(F^(c*(b*x+a)))^n*x^3*ln(F)^3*b^3*c^3*n^3+12*ln(F)^3*x^2*( (F^(c*(b*x+a)))^n)^2*b^3*c^3*e^2*f*g^2*n^3+12*ln(F)^3*x*((F^(c*(b*x+a)))^n )^2*b^3*c^3*e^2*f^2*g*n^3-48*ln(F)^2*x^2*(F^(c*(b*x+a)))^n*b^2*c^2*d*e*g^3 *n^2-12*ln(F)^2*x*((F^(c*(b*x+a)))^n)^2*b^2*c^2*e^2*f*g^2*n^2-48*ln(F)^2*( F^(c*(b*x+a)))^n*b^2*c^2*d*e*f^2*g*n^2+96*ln(F)*x*(F^(c*(b*x+a)))^n*b*c*d* e*g^3*n+96*ln(F)*(F^(c*(b*x+a)))^n*b*c*d*e*f*g^2*n+48*ln(F)^3*x^2*(F^(c*(b *x+a)))^n*b^3*c^3*d*e*f*g^2*n^3+48*ln(F)^3*x*(F^(c*(b*x+a)))^n*b^3*c^3*d*e *f^2*g*n^3-96*ln(F)^2*x*(F^(c*(b*x+a)))^n*b^2*c^2*d*e*f*g^2*n^2+2*d^2*g^3* x^4*ln(F)^4*b^4*c^4*n^4+8*d^2*f^3*x*ln(F)^4*b^4*c^4*n^4+4*ln(F)^3*((F^(c*( b*x+a)))^n)^2*b^3*c^3*e^2*f^3*n^3+12*d^2*g*f^2*x^2*ln(F)^4*b^4*c^4*n^4+4*e ^2*g^3*((F^(c*(b*x+a)))^n)^2*x^3*ln(F)^3*b^3*c^3*n^3+16*ln(F)^3*(F^(c*(b*x +a)))^n*b^3*c^3*d*e*f^3*n^3-96*(F^(c*(b*x+a)))^n*d*e*g^3+8*d^2*g^2*f*x^3*l n(F)^4*b^4*c^4*n^4-6*ln(F)^2*x^2*((F^(c*(b*x+a)))^n)^2*b^2*c^2*e^2*g^3*n^2 -6*ln(F)^2*((F^(c*(b*x+a)))^n)^2*b^2*c^2*e^2*f^2*g*n^2+6*ln(F)*x*((F^(c*(b *x+a)))^n)^2*b*c*e^2*g^3*n+6*ln(F)*((F^(c*(b*x+a)))^n)^2*b*c*e^2*f*g^2*n-3 *((F^(c*(b*x+a)))^n)^2*e^2*g^3)/ln(F)^4/b^4/c^4/n^4
Time = 0.08 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.50 \[ \int \left (d+e \left (F^{c (a+b x)}\right )^n\right )^2 (f+g x)^3 \, dx=\frac {2 \, {\left (b^{4} c^{4} d^{2} g^{3} n^{4} x^{4} + 4 \, b^{4} c^{4} d^{2} f g^{2} n^{4} x^{3} + 6 \, b^{4} c^{4} d^{2} f^{2} g n^{4} x^{2} + 4 \, b^{4} c^{4} d^{2} f^{3} n^{4} x\right )} \log \left (F\right )^{4} - {\left (3 \, e^{2} g^{3} - 4 \, {\left (b^{3} c^{3} e^{2} g^{3} n^{3} x^{3} + 3 \, b^{3} c^{3} e^{2} f g^{2} n^{3} x^{2} + 3 \, b^{3} c^{3} e^{2} f^{2} g n^{3} x + b^{3} c^{3} e^{2} f^{3} n^{3}\right )} \log \left (F\right )^{3} + 6 \, {\left (b^{2} c^{2} e^{2} g^{3} n^{2} x^{2} + 2 \, b^{2} c^{2} e^{2} f g^{2} n^{2} x + b^{2} c^{2} e^{2} f^{2} g n^{2}\right )} \log \left (F\right )^{2} - 6 \, {\left (b c e^{2} g^{3} n x + b c e^{2} f g^{2} n\right )} \log \left (F\right )\right )} F^{2 \, b c n x + 2 \, a c n} - 16 \, {\left (6 \, d e g^{3} - {\left (b^{3} c^{3} d e g^{3} n^{3} x^{3} + 3 \, b^{3} c^{3} d e f g^{2} n^{3} x^{2} + 3 \, b^{3} c^{3} d e f^{2} g n^{3} x + b^{3} c^{3} d e f^{3} n^{3}\right )} \log \left (F\right )^{3} + 3 \, {\left (b^{2} c^{2} d e g^{3} n^{2} x^{2} + 2 \, b^{2} c^{2} d e f g^{2} n^{2} x + b^{2} c^{2} d e f^{2} g n^{2}\right )} \log \left (F\right )^{2} - 6 \, {\left (b c d e g^{3} n x + b c d e f g^{2} n\right )} \log \left (F\right )\right )} F^{b c n x + a c n}}{8 \, b^{4} c^{4} n^{4} \log \left (F\right )^{4}} \] Input:
integrate((d+e*(F^((b*x+a)*c))^n)^2*(g*x+f)^3,x, algorithm="fricas")
Output:
1/8*(2*(b^4*c^4*d^2*g^3*n^4*x^4 + 4*b^4*c^4*d^2*f*g^2*n^4*x^3 + 6*b^4*c^4* d^2*f^2*g*n^4*x^2 + 4*b^4*c^4*d^2*f^3*n^4*x)*log(F)^4 - (3*e^2*g^3 - 4*(b^ 3*c^3*e^2*g^3*n^3*x^3 + 3*b^3*c^3*e^2*f*g^2*n^3*x^2 + 3*b^3*c^3*e^2*f^2*g* n^3*x + b^3*c^3*e^2*f^3*n^3)*log(F)^3 + 6*(b^2*c^2*e^2*g^3*n^2*x^2 + 2*b^2 *c^2*e^2*f*g^2*n^2*x + b^2*c^2*e^2*f^2*g*n^2)*log(F)^2 - 6*(b*c*e^2*g^3*n* x + b*c*e^2*f*g^2*n)*log(F))*F^(2*b*c*n*x + 2*a*c*n) - 16*(6*d*e*g^3 - (b^ 3*c^3*d*e*g^3*n^3*x^3 + 3*b^3*c^3*d*e*f*g^2*n^3*x^2 + 3*b^3*c^3*d*e*f^2*g* n^3*x + b^3*c^3*d*e*f^3*n^3)*log(F)^3 + 3*(b^2*c^2*d*e*g^3*n^2*x^2 + 2*b^2 *c^2*d*e*f*g^2*n^2*x + b^2*c^2*d*e*f^2*g*n^2)*log(F)^2 - 6*(b*c*d*e*g^3*n* x + b*c*d*e*f*g^2*n)*log(F))*F^(b*c*n*x + a*c*n))/(b^4*c^4*n^4*log(F)^4)
Leaf count of result is larger than twice the leaf count of optimal. 913 vs. \(2 (323) = 646\).
Time = 4.36 (sec) , antiderivative size = 913, normalized size of antiderivative = 2.84 \[ \int \left (d+e \left (F^{c (a+b x)}\right )^n\right )^2 (f+g x)^3 \, dx =\text {Too large to display} \] Input:
integrate((d+e*(F**((b*x+a)*c))**n)**2*(g*x+f)**3,x)
Output:
Piecewise(((d + e)**2*(f**3*x + 3*f**2*g*x**2/2 + f*g**2*x**3 + g**3*x**4/ 4), Eq(F, 1) & Eq(b, 0) & Eq(c, 0) & Eq(n, 0)), ((d + e*(F**(a*c))**n)**2* (f**3*x + 3*f**2*g*x**2/2 + f*g**2*x**3 + g**3*x**4/4), Eq(b, 0)), ((d + e )**2*(f**3*x + 3*f**2*g*x**2/2 + f*g**2*x**3 + g**3*x**4/4), Eq(F, 1) | Eq (c, 0) | Eq(n, 0)), (d**2*f**3*x + 3*d**2*f**2*g*x**2/2 + d**2*f*g**2*x**3 + d**2*g**3*x**4/4 + 2*d*e*f**3*(F**(a*c + b*c*x))**n/(b*c*n*log(F)) + 6* d*e*f**2*g*x*(F**(a*c + b*c*x))**n/(b*c*n*log(F)) + 6*d*e*f*g**2*x**2*(F** (a*c + b*c*x))**n/(b*c*n*log(F)) + 2*d*e*g**3*x**3*(F**(a*c + b*c*x))**n/( b*c*n*log(F)) + e**2*f**3*(F**(a*c + b*c*x))**(2*n)/(2*b*c*n*log(F)) + 3*e **2*f**2*g*x*(F**(a*c + b*c*x))**(2*n)/(2*b*c*n*log(F)) + 3*e**2*f*g**2*x* *2*(F**(a*c + b*c*x))**(2*n)/(2*b*c*n*log(F)) + e**2*g**3*x**3*(F**(a*c + b*c*x))**(2*n)/(2*b*c*n*log(F)) - 6*d*e*f**2*g*(F**(a*c + b*c*x))**n/(b**2 *c**2*n**2*log(F)**2) - 12*d*e*f*g**2*x*(F**(a*c + b*c*x))**n/(b**2*c**2*n **2*log(F)**2) - 6*d*e*g**3*x**2*(F**(a*c + b*c*x))**n/(b**2*c**2*n**2*log (F)**2) - 3*e**2*f**2*g*(F**(a*c + b*c*x))**(2*n)/(4*b**2*c**2*n**2*log(F) **2) - 3*e**2*f*g**2*x*(F**(a*c + b*c*x))**(2*n)/(2*b**2*c**2*n**2*log(F)* *2) - 3*e**2*g**3*x**2*(F**(a*c + b*c*x))**(2*n)/(4*b**2*c**2*n**2*log(F)* *2) + 12*d*e*f*g**2*(F**(a*c + b*c*x))**n/(b**3*c**3*n**3*log(F)**3) + 12* d*e*g**3*x*(F**(a*c + b*c*x))**n/(b**3*c**3*n**3*log(F)**3) + 3*e**2*f*g** 2*(F**(a*c + b*c*x))**(2*n)/(4*b**3*c**3*n**3*log(F)**3) + 3*e**2*g**3*...
Time = 0.07 (sec) , antiderivative size = 568, normalized size of antiderivative = 1.76 \[ \int \left (d+e \left (F^{c (a+b x)}\right )^n\right )^2 (f+g x)^3 \, dx=\frac {1}{4} \, d^{2} g^{3} x^{4} + d^{2} f g^{2} x^{3} + \frac {3}{2} \, d^{2} f^{2} g x^{2} + d^{2} f^{3} x + \frac {2 \, F^{b c n x + a c n} d e f^{3}}{b c n \log \left (F\right )} + \frac {F^{2 \, b c n x + 2 \, a c n} e^{2} f^{3}}{2 \, b c n \log \left (F\right )} + \frac {6 \, {\left (F^{a c n} b c n x \log \left (F\right ) - F^{a c n}\right )} F^{b c n x} d e f^{2} g}{b^{2} c^{2} n^{2} \log \left (F\right )^{2}} + \frac {3 \, {\left (2 \, F^{2 \, a c n} b c n x \log \left (F\right ) - F^{2 \, a c n}\right )} F^{2 \, b c n x} e^{2} f^{2} g}{4 \, b^{2} c^{2} n^{2} \log \left (F\right )^{2}} + \frac {6 \, {\left (F^{a c n} b^{2} c^{2} n^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{a c n} b c n x \log \left (F\right ) + 2 \, F^{a c n}\right )} F^{b c n x} d e f g^{2}}{b^{3} c^{3} n^{3} \log \left (F\right )^{3}} + \frac {3 \, {\left (2 \, F^{2 \, a c n} b^{2} c^{2} n^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{2 \, a c n} b c n x \log \left (F\right ) + F^{2 \, a c n}\right )} F^{2 \, b c n x} e^{2} f g^{2}}{4 \, b^{3} c^{3} n^{3} \log \left (F\right )^{3}} + \frac {2 \, {\left (F^{a c n} b^{3} c^{3} n^{3} x^{3} \log \left (F\right )^{3} - 3 \, F^{a c n} b^{2} c^{2} n^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{a c n} b c n x \log \left (F\right ) - 6 \, F^{a c n}\right )} F^{b c n x} d e g^{3}}{b^{4} c^{4} n^{4} \log \left (F\right )^{4}} + \frac {{\left (4 \, F^{2 \, a c n} b^{3} c^{3} n^{3} x^{3} \log \left (F\right )^{3} - 6 \, F^{2 \, a c n} b^{2} c^{2} n^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{2 \, a c n} b c n x \log \left (F\right ) - 3 \, F^{2 \, a c n}\right )} F^{2 \, b c n x} e^{2} g^{3}}{8 \, b^{4} c^{4} n^{4} \log \left (F\right )^{4}} \] Input:
integrate((d+e*(F^((b*x+a)*c))^n)^2*(g*x+f)^3,x, algorithm="maxima")
Output:
1/4*d^2*g^3*x^4 + d^2*f*g^2*x^3 + 3/2*d^2*f^2*g*x^2 + d^2*f^3*x + 2*F^(b*c *n*x + a*c*n)*d*e*f^3/(b*c*n*log(F)) + 1/2*F^(2*b*c*n*x + 2*a*c*n)*e^2*f^3 /(b*c*n*log(F)) + 6*(F^(a*c*n)*b*c*n*x*log(F) - F^(a*c*n))*F^(b*c*n*x)*d*e *f^2*g/(b^2*c^2*n^2*log(F)^2) + 3/4*(2*F^(2*a*c*n)*b*c*n*x*log(F) - F^(2*a *c*n))*F^(2*b*c*n*x)*e^2*f^2*g/(b^2*c^2*n^2*log(F)^2) + 6*(F^(a*c*n)*b^2*c ^2*n^2*x^2*log(F)^2 - 2*F^(a*c*n)*b*c*n*x*log(F) + 2*F^(a*c*n))*F^(b*c*n*x )*d*e*f*g^2/(b^3*c^3*n^3*log(F)^3) + 3/4*(2*F^(2*a*c*n)*b^2*c^2*n^2*x^2*lo g(F)^2 - 2*F^(2*a*c*n)*b*c*n*x*log(F) + F^(2*a*c*n))*F^(2*b*c*n*x)*e^2*f*g ^2/(b^3*c^3*n^3*log(F)^3) + 2*(F^(a*c*n)*b^3*c^3*n^3*x^3*log(F)^3 - 3*F^(a *c*n)*b^2*c^2*n^2*x^2*log(F)^2 + 6*F^(a*c*n)*b*c*n*x*log(F) - 6*F^(a*c*n)) *F^(b*c*n*x)*d*e*g^3/(b^4*c^4*n^4*log(F)^4) + 1/8*(4*F^(2*a*c*n)*b^3*c^3*n ^3*x^3*log(F)^3 - 6*F^(2*a*c*n)*b^2*c^2*n^2*x^2*log(F)^2 + 6*F^(2*a*c*n)*b *c*n*x*log(F) - 3*F^(2*a*c*n))*F^(2*b*c*n*x)*e^2*g^3/(b^4*c^4*n^4*log(F)^4 )
Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 12013, normalized size of antiderivative = 37.31 \[ \int \left (d+e \left (F^{c (a+b x)}\right )^n\right )^2 (f+g x)^3 \, dx=\text {Too large to display} \] Input:
integrate((d+e*(F^((b*x+a)*c))^n)^2*(g*x+f)^3,x, algorithm="giac")
Output:
1/4*d^2*g^3*x^4 + d^2*f*g^2*x^3 + 3/2*d^2*f^2*g*x^2 + d^2*f^3*x - 1/4*(((6 *pi^2*b^3*c^3*e^2*g^3*n^3*x^3*log(abs(F))*sgn(F) - 6*pi^2*b^3*c^3*e^2*g^3* n^3*x^3*log(abs(F)) + 4*b^3*c^3*e^2*g^3*n^3*x^3*log(abs(F))^3 + 18*pi^2*b^ 3*c^3*e^2*f*g^2*n^3*x^2*log(abs(F))*sgn(F) - 18*pi^2*b^3*c^3*e^2*f*g^2*n^3 *x^2*log(abs(F)) + 12*b^3*c^3*e^2*f*g^2*n^3*x^2*log(abs(F))^3 + 18*pi^2*b^ 3*c^3*e^2*f^2*g*n^3*x*log(abs(F))*sgn(F) - 18*pi^2*b^3*c^3*e^2*f^2*g*n^3*x *log(abs(F)) + 12*b^3*c^3*e^2*f^2*g*n^3*x*log(abs(F))^3 + 6*pi^2*b^3*c^3*e ^2*f^3*n^3*log(abs(F))*sgn(F) - 6*pi^2*b^3*c^3*e^2*f^3*n^3*log(abs(F)) + 4 *b^3*c^3*e^2*f^3*n^3*log(abs(F))^3 - 3*pi^2*b^2*c^2*e^2*g^3*n^2*x^2*sgn(F) + 3*pi^2*b^2*c^2*e^2*g^3*n^2*x^2 - 6*b^2*c^2*e^2*g^3*n^2*x^2*log(abs(F))^ 2 - 6*pi^2*b^2*c^2*e^2*f*g^2*n^2*x*sgn(F) + 6*pi^2*b^2*c^2*e^2*f*g^2*n^2*x - 12*b^2*c^2*e^2*f*g^2*n^2*x*log(abs(F))^2 - 3*pi^2*b^2*c^2*e^2*f^2*g*n^2 *sgn(F) + 3*pi^2*b^2*c^2*e^2*f^2*g*n^2 - 6*b^2*c^2*e^2*f^2*g*n^2*log(abs(F ))^2 + 6*b*c*e^2*g^3*n*x*log(abs(F)) + 6*b*c*e^2*f*g^2*n*log(abs(F)) - 3*e ^2*g^3)*(pi^4*b^4*c^4*n^4*sgn(F) - 6*pi^2*b^4*c^4*n^4*log(abs(F))^2*sgn(F) - pi^4*b^4*c^4*n^4 + 6*pi^2*b^4*c^4*n^4*log(abs(F))^2 - 2*b^4*c^4*n^4*log (abs(F))^4)/((pi^4*b^4*c^4*n^4*sgn(F) - 6*pi^2*b^4*c^4*n^4*log(abs(F))^2*s gn(F) - pi^4*b^4*c^4*n^4 + 6*pi^2*b^4*c^4*n^4*log(abs(F))^2 - 2*b^4*c^4*n^ 4*log(abs(F))^4)^2 + 16*(pi^3*b^4*c^4*n^4*log(abs(F))*sgn(F) - pi*b^4*c^4* n^4*log(abs(F))^3*sgn(F) - pi^3*b^4*c^4*n^4*log(abs(F)) + pi*b^4*c^4*n^...
Time = 23.23 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.36 \[ \int \left (d+e \left (F^{c (a+b x)}\right )^n\right )^2 (f+g x)^3 \, dx=d^2\,f^3\,x-{\left (F^{b\,c\,x}\,F^{a\,c}\right )}^n\,\left (\frac {2\,d\,e\,\left (-b^3\,c^3\,f^3\,n^3\,{\ln \left (F\right )}^3+3\,b^2\,c^2\,f^2\,g\,n^2\,{\ln \left (F\right )}^2-6\,b\,c\,f\,g^2\,n\,\ln \left (F\right )+6\,g^3\right )}{b^4\,c^4\,n^4\,{\ln \left (F\right )}^4}-\frac {2\,d\,e\,g^3\,x^3}{b\,c\,n\,\ln \left (F\right )}-\frac {6\,d\,e\,g\,x\,\left (b^2\,c^2\,f^2\,n^2\,{\ln \left (F\right )}^2-2\,b\,c\,f\,g\,n\,\ln \left (F\right )+2\,g^2\right )}{b^3\,c^3\,n^3\,{\ln \left (F\right )}^3}+\frac {6\,d\,e\,g^2\,x^2\,\left (g-b\,c\,f\,n\,\ln \left (F\right )\right )}{b^2\,c^2\,n^2\,{\ln \left (F\right )}^2}\right )-{\left (F^{b\,c\,x}\,F^{a\,c}\right )}^{2\,n}\,\left (\frac {e^2\,\left (-4\,b^3\,c^3\,f^3\,n^3\,{\ln \left (F\right )}^3+6\,b^2\,c^2\,f^2\,g\,n^2\,{\ln \left (F\right )}^2-6\,b\,c\,f\,g^2\,n\,\ln \left (F\right )+3\,g^3\right )}{8\,b^4\,c^4\,n^4\,{\ln \left (F\right )}^4}-\frac {e^2\,g^3\,x^3}{2\,b\,c\,n\,\ln \left (F\right )}-\frac {3\,e^2\,g\,x\,\left (2\,b^2\,c^2\,f^2\,n^2\,{\ln \left (F\right )}^2-2\,b\,c\,f\,g\,n\,\ln \left (F\right )+g^2\right )}{4\,b^3\,c^3\,n^3\,{\ln \left (F\right )}^3}+\frac {3\,e^2\,g^2\,x^2\,\left (g-2\,b\,c\,f\,n\,\ln \left (F\right )\right )}{4\,b^2\,c^2\,n^2\,{\ln \left (F\right )}^2}\right )+\frac {d^2\,g^3\,x^4}{4}+\frac {3\,d^2\,f^2\,g\,x^2}{2}+d^2\,f\,g^2\,x^3 \] Input:
int((f + g*x)^3*(d + e*(F^(c*(a + b*x)))^n)^2,x)
Output:
d^2*f^3*x - (F^(b*c*x)*F^(a*c))^n*((2*d*e*(6*g^3 - b^3*c^3*f^3*n^3*log(F)^ 3 - 6*b*c*f*g^2*n*log(F) + 3*b^2*c^2*f^2*g*n^2*log(F)^2))/(b^4*c^4*n^4*log (F)^4) - (2*d*e*g^3*x^3)/(b*c*n*log(F)) - (6*d*e*g*x*(2*g^2 + b^2*c^2*f^2* n^2*log(F)^2 - 2*b*c*f*g*n*log(F)))/(b^3*c^3*n^3*log(F)^3) + (6*d*e*g^2*x^ 2*(g - b*c*f*n*log(F)))/(b^2*c^2*n^2*log(F)^2)) - (F^(b*c*x)*F^(a*c))^(2*n )*((e^2*(3*g^3 - 4*b^3*c^3*f^3*n^3*log(F)^3 - 6*b*c*f*g^2*n*log(F) + 6*b^2 *c^2*f^2*g*n^2*log(F)^2))/(8*b^4*c^4*n^4*log(F)^4) - (e^2*g^3*x^3)/(2*b*c* n*log(F)) - (3*e^2*g*x*(g^2 + 2*b^2*c^2*f^2*n^2*log(F)^2 - 2*b*c*f*g*n*log (F)))/(4*b^3*c^3*n^3*log(F)^3) + (3*e^2*g^2*x^2*(g - 2*b*c*f*n*log(F)))/(4 *b^2*c^2*n^2*log(F)^2)) + (d^2*g^3*x^4)/4 + (3*d^2*f^2*g*x^2)/2 + d^2*f*g^ 2*x^3
Time = 0.17 (sec) , antiderivative size = 758, normalized size of antiderivative = 2.35 \[ \int \left (d+e \left (F^{c (a+b x)}\right )^n\right )^2 (f+g x)^3 \, dx=\frac {4 f^{2 b c n x +2 a c n} \mathrm {log}\left (f \right )^{3} b^{3} c^{3} e^{2} f^{3} n^{3}+12 f^{2 b c n x +2 a c n} \mathrm {log}\left (f \right )^{3} b^{3} c^{3} e^{2} f^{2} g \,n^{3} x +12 f^{2 b c n x +2 a c n} \mathrm {log}\left (f \right )^{3} b^{3} c^{3} e^{2} f \,g^{2} n^{3} x^{2}+4 f^{2 b c n x +2 a c n} \mathrm {log}\left (f \right )^{3} b^{3} c^{3} e^{2} g^{3} n^{3} x^{3}-6 f^{2 b c n x +2 a c n} \mathrm {log}\left (f \right )^{2} b^{2} c^{2} e^{2} f^{2} g \,n^{2}-12 f^{2 b c n x +2 a c n} \mathrm {log}\left (f \right )^{2} b^{2} c^{2} e^{2} f \,g^{2} n^{2} x -6 f^{2 b c n x +2 a c n} \mathrm {log}\left (f \right )^{2} b^{2} c^{2} e^{2} g^{3} n^{2} x^{2}+6 f^{2 b c n x +2 a c n} \mathrm {log}\left (f \right ) b c \,e^{2} f \,g^{2} n +6 f^{2 b c n x +2 a c n} \mathrm {log}\left (f \right ) b c \,e^{2} g^{3} n x -3 f^{2 b c n x +2 a c n} e^{2} g^{3}+16 f^{b c n x +a c n} \mathrm {log}\left (f \right )^{3} b^{3} c^{3} d e \,f^{3} n^{3}+48 f^{b c n x +a c n} \mathrm {log}\left (f \right )^{3} b^{3} c^{3} d e \,f^{2} g \,n^{3} x +48 f^{b c n x +a c n} \mathrm {log}\left (f \right )^{3} b^{3} c^{3} d e f \,g^{2} n^{3} x^{2}+16 f^{b c n x +a c n} \mathrm {log}\left (f \right )^{3} b^{3} c^{3} d e \,g^{3} n^{3} x^{3}-48 f^{b c n x +a c n} \mathrm {log}\left (f \right )^{2} b^{2} c^{2} d e \,f^{2} g \,n^{2}-96 f^{b c n x +a c n} \mathrm {log}\left (f \right )^{2} b^{2} c^{2} d e f \,g^{2} n^{2} x -48 f^{b c n x +a c n} \mathrm {log}\left (f \right )^{2} b^{2} c^{2} d e \,g^{3} n^{2} x^{2}+96 f^{b c n x +a c n} \mathrm {log}\left (f \right ) b c d e f \,g^{2} n +96 f^{b c n x +a c n} \mathrm {log}\left (f \right ) b c d e \,g^{3} n x -96 f^{b c n x +a c n} d e \,g^{3}+8 \mathrm {log}\left (f \right )^{4} b^{4} c^{4} d^{2} f^{3} n^{4} x +12 \mathrm {log}\left (f \right )^{4} b^{4} c^{4} d^{2} f^{2} g \,n^{4} x^{2}+8 \mathrm {log}\left (f \right )^{4} b^{4} c^{4} d^{2} f \,g^{2} n^{4} x^{3}+2 \mathrm {log}\left (f \right )^{4} b^{4} c^{4} d^{2} g^{3} n^{4} x^{4}}{8 \mathrm {log}\left (f \right )^{4} b^{4} c^{4} n^{4}} \] Input:
int((d+e*(F^((b*x+a)*c))^n)^2*(g*x+f)^3,x)
Output:
(4*f**(2*a*c*n + 2*b*c*n*x)*log(f)**3*b**3*c**3*e**2*f**3*n**3 + 12*f**(2* a*c*n + 2*b*c*n*x)*log(f)**3*b**3*c**3*e**2*f**2*g*n**3*x + 12*f**(2*a*c*n + 2*b*c*n*x)*log(f)**3*b**3*c**3*e**2*f*g**2*n**3*x**2 + 4*f**(2*a*c*n + 2*b*c*n*x)*log(f)**3*b**3*c**3*e**2*g**3*n**3*x**3 - 6*f**(2*a*c*n + 2*b*c *n*x)*log(f)**2*b**2*c**2*e**2*f**2*g*n**2 - 12*f**(2*a*c*n + 2*b*c*n*x)*l og(f)**2*b**2*c**2*e**2*f*g**2*n**2*x - 6*f**(2*a*c*n + 2*b*c*n*x)*log(f)* *2*b**2*c**2*e**2*g**3*n**2*x**2 + 6*f**(2*a*c*n + 2*b*c*n*x)*log(f)*b*c*e **2*f*g**2*n + 6*f**(2*a*c*n + 2*b*c*n*x)*log(f)*b*c*e**2*g**3*n*x - 3*f** (2*a*c*n + 2*b*c*n*x)*e**2*g**3 + 16*f**(a*c*n + b*c*n*x)*log(f)**3*b**3*c **3*d*e*f**3*n**3 + 48*f**(a*c*n + b*c*n*x)*log(f)**3*b**3*c**3*d*e*f**2*g *n**3*x + 48*f**(a*c*n + b*c*n*x)*log(f)**3*b**3*c**3*d*e*f*g**2*n**3*x**2 + 16*f**(a*c*n + b*c*n*x)*log(f)**3*b**3*c**3*d*e*g**3*n**3*x**3 - 48*f** (a*c*n + b*c*n*x)*log(f)**2*b**2*c**2*d*e*f**2*g*n**2 - 96*f**(a*c*n + b*c *n*x)*log(f)**2*b**2*c**2*d*e*f*g**2*n**2*x - 48*f**(a*c*n + b*c*n*x)*log( f)**2*b**2*c**2*d*e*g**3*n**2*x**2 + 96*f**(a*c*n + b*c*n*x)*log(f)*b*c*d* e*f*g**2*n + 96*f**(a*c*n + b*c*n*x)*log(f)*b*c*d*e*g**3*n*x - 96*f**(a*c* n + b*c*n*x)*d*e*g**3 + 8*log(f)**4*b**4*c**4*d**2*f**3*n**4*x + 12*log(f) **4*b**4*c**4*d**2*f**2*g*n**4*x**2 + 8*log(f)**4*b**4*c**4*d**2*f*g**2*n* *4*x**3 + 2*log(f)**4*b**4*c**4*d**2*g**3*n**4*x**4)/(8*log(f)**4*b**4*c** 4*n**4)