Integrand size = 65, antiderivative size = 148 \[ \int \left (a+b x+c x^2\right )^{\frac {-6 c^2 d+3 b c e-2 b^2 f+2 a c f}{4 c^2 d-2 b c e+b^2 f}} \left (d+e x+f x^2\right ) \, dx=-\frac {\left (4 c^2 d-2 b c e+b^2 f\right ) \left (b c d-2 a c e+a b f+\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x\right ) \left (a+b x+c x^2\right )^{-\frac {2 c^2 d-b c e+b^2 f-2 a c f}{4 c^2 d-2 b c e+b^2 f}}}{c \left (b^2-4 a c\right ) \left (2 c^2 d-b c e+b^2 f-2 a c f\right )} \] Output:
-(b^2*f-2*b*c*e+4*c^2*d)*(b*c*d-2*a*c*e+a*b*f+(-2*a*c*f+b^2*f-b*c*e+2*c^2* d)*x)/c/(-4*a*c+b^2)/(-2*a*c*f+b^2*f-b*c*e+2*c^2*d)/((c*x^2+b*x+a)^((-2*a* c*f+b^2*f-b*c*e+2*c^2*d)/(b^2*f-2*b*c*e+4*c^2*d)))
Time = 1.03 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.99 \[ \int \left (a+b x+c x^2\right )^{\frac {-6 c^2 d+3 b c e-2 b^2 f+2 a c f}{4 c^2 d-2 b c e+b^2 f}} \left (d+e x+f x^2\right ) \, dx=\frac {\left (4 c^2 d-2 b c e+b^2 f\right ) (a+x (b+c x))^{\frac {-2 c^2 d+b c e-b^2 f+2 a c f}{4 c^2 d-2 b c e+b^2 f}} \left (a b f+2 c^2 d x+b^2 f x+b c (d-e x)-2 a c (e+f x)\right )}{c \left (-b^2+4 a c\right ) \left (2 c^2 d+b^2 f-c (b e+2 a f)\right )} \] Input:
Integrate[(a + b*x + c*x^2)^((-6*c^2*d + 3*b*c*e - 2*b^2*f + 2*a*c*f)/(4*c ^2*d - 2*b*c*e + b^2*f))*(d + e*x + f*x^2),x]
Output:
((4*c^2*d - 2*b*c*e + b^2*f)*(a + x*(b + c*x))^((-2*c^2*d + b*c*e - b^2*f + 2*a*c*f)/(4*c^2*d - 2*b*c*e + b^2*f))*(a*b*f + 2*c^2*d*x + b^2*f*x + b*c *(d - e*x) - 2*a*c*(e + f*x)))/(c*(-b^2 + 4*a*c)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))
Time = 0.40 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.43, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2192, 25, 27, 1104}
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.
| \(\displaystyle \int \left (d+e x+f x^2\right ) \left (a+b x+c x^2\right )^{\frac {2 a c f-2 b^2 f+3 b c e-6 c^2 d}{b^2 f-2 b c e+4 c^2 d}} \, dx\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle -\frac {\left (b^2 f-2 b c e+4 c^2 d\right ) \int -\frac {f (b c d-2 a c e+a b f) (b+2 c x) \left (c x^2+b x+a\right )^{-\frac {2 f b^2-3 c e b+6 c^2 d-2 a c f}{f b^2-2 c e b+4 c^2 d}}}{f b^2-2 c e b+4 c^2 d}dx}{c f \left (b^2-4 a c\right )}-\frac {x \left (b^2 f-2 b c e+4 c^2 d\right ) \left (a+b x+c x^2\right )^{-\frac {-2 a c f+b^2 f-b c e+2 c^2 d}{b^2 f-2 b c e+4 c^2 d}}}{c \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (b^2 f-2 b c e+4 c^2 d\right ) \int \frac {f (b c d-2 a c e+a b f) (b+2 c x) \left (c x^2+b x+a\right )^{-\frac {2 f b^2-3 c e b+6 c^2 d-2 a c f}{f b^2-2 c e b+4 c^2 d}}}{f b^2-2 c e b+4 c^2 d}dx}{c f \left (b^2-4 a c\right )}-\frac {x \left (b^2 f-2 b c e+4 c^2 d\right ) \left (a+b x+c x^2\right )^{-\frac {-2 a c f+b^2 f-b c e+2 c^2 d}{b^2 f-2 b c e+4 c^2 d}}}{c \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a b f-2 a c e+b c d) \int (b+2 c x) \left (c x^2+b x+a\right )^{-\frac {2 f b^2-3 c e b+6 c^2 d-2 a c f}{f b^2-2 c e b+4 c^2 d}}dx}{c \left (b^2-4 a c\right )}-\frac {x \left (b^2 f-2 b c e+4 c^2 d\right ) \left (a+b x+c x^2\right )^{-\frac {-2 a c f+b^2 f-b c e+2 c^2 d}{b^2 f-2 b c e+4 c^2 d}}}{c \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 1104 |
| \(\displaystyle -\frac {x \left (b^2 f-2 b c e+4 c^2 d\right ) \left (a+b x+c x^2\right )^{-\frac {-2 a c f+b^2 f-b c e+2 c^2 d}{b^2 f-2 b c e+4 c^2 d}}}{c \left (b^2-4 a c\right )}-\frac {\left (b^2 f-2 b c e+4 c^2 d\right ) (a b f-2 a c e+b c d) \left (a+b x+c x^2\right )^{-\frac {-2 a c f+b^2 f-b c e+2 c^2 d}{b^2 f-2 b c e+4 c^2 d}}}{c \left (b^2-4 a c\right ) \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )}\) |
Input:
Int[(a + b*x + c*x^2)^((-6*c^2*d + 3*b*c*e - 2*b^2*f + 2*a*c*f)/(4*c^2*d - 2*b*c*e + b^2*f))*(d + e*x + f*x^2),x]
Output:
-(((b*c*d - 2*a*c*e + a*b*f)*(4*c^2*d - 2*b*c*e + b^2*f))/(c*(b^2 - 4*a*c) *(2*c^2*d - b*c*e + b^2*f - 2*a*c*f)*(a + b*x + c*x^2)^((2*c^2*d - b*c*e + b^2*f - 2*a*c*f)/(4*c^2*d - 2*b*c*e + b^2*f)))) - ((4*c^2*d - 2*b*c*e + b ^2*f)*x)/(c*(b^2 - 4*a*c)*(a + b*x + c*x^2)^((2*c^2*d - b*c*e + b^2*f - 2* a*c*f)/(4*c^2*d - 2*b*c*e + b^2*f)))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol ] :> Simp[d*((a + b*x + c*x^2)^(p + 1)/(b*(p + 1))), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0]
Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1)) Int[(a + b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b *e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c , p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && !LeQ[p, -1]
Time = 1.78 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.18
| method | result | size |
| gosper | \(-\frac {\left (c \,x^{2}+b x +a \right )^{1+\frac {2 a c f -2 b^{2} f +3 b c e -6 c^{2} d}{b^{2} f -2 b c e +4 c^{2} d}} \left (b^{2} f -2 b c e +4 c^{2} d \right ) \left (-2 a c f x +b^{2} f x -b c e x +2 c^{2} d x +a b f -2 a c e +d b c \right )}{c \left (8 a^{2} c^{2} f -6 a \,b^{2} c f +4 a b \,c^{2} e -8 a \,c^{3} d +b^{4} f -b^{3} c e +2 b^{2} c^{2} d \right )}\) | \(174\) |
| risch | \(-\frac {\left (b^{2} f -2 b c e +4 c^{2} d \right ) \left (-2 a \,c^{2} f \,x^{3}+b^{2} c f \,x^{3}-b \,c^{2} x^{3} e +2 c^{3} d \,x^{3}-a b c f \,x^{2}-2 a \,c^{2} e \,x^{2}+b^{3} f \,x^{2}-b^{2} c e \,x^{2}+3 b \,c^{2} d \,x^{2}-2 a^{2} c f x +2 a \,b^{2} f x -3 a b c e x +2 a d x \,c^{2}+b^{2} c x d +f \,a^{2} b -2 a^{2} c e +a b c d \right ) \left (c \,x^{2}+b x +a \right )^{\frac {2 a c f -2 b^{2} f +3 b c e -6 c^{2} d}{b^{2} f -2 b c e +4 c^{2} d}}}{c \left (4 a c -b^{2}\right ) \left (2 a c f -b^{2} f +b c e -2 c^{2} d \right )}\) | \(255\) |
| orering | \(-\frac {\left (-2 a \,b^{2} c \,f^{2} x +4 a b \,c^{2} e f x -8 a \,c^{3} d f x +b^{4} f^{2} x -3 b^{3} c e f x +6 b^{2} c^{2} d f x +2 b^{2} c^{2} e^{2} x -8 b \,c^{3} d e x +8 c^{4} d^{2} x +a \,b^{3} f^{2}-4 a \,b^{2} c e f +4 a b \,c^{2} d f +4 a b \,c^{2} e^{2}-8 a \,c^{3} d e +b^{3} c d f -2 b^{2} c^{2} d e +4 b \,c^{3} d^{2}\right ) \left (c \,x^{2}+b x +a \right ) \left (c \,x^{2}+b x +a \right )^{\frac {2 a c f -2 b^{2} f +3 b c e -6 c^{2} d}{b^{2} f -2 b c e +4 c^{2} d}}}{c \left (8 a^{2} c^{2} f -6 a \,b^{2} c f +4 a b \,c^{2} e -8 a \,c^{3} d +b^{4} f -b^{3} c e +2 b^{2} c^{2} d \right )}\) | \(285\) |
| norman | \(\frac {\left (b^{2} f -2 b c e +4 c^{2} d \right ) x^{3} {\mathrm e}^{\frac {\left (2 a c f -2 b^{2} f +3 b c e -6 c^{2} d \right ) \ln \left (c \,x^{2}+b x +a \right )}{b^{2} f -2 b c e +4 c^{2} d}}}{4 a c -b^{2}}+\frac {\left (a \,b^{3} c \,f^{2}+4 a b \,c^{3} d f -4 a b \,c^{3} e^{2}+8 a \,c^{4} d e -b^{5} f^{2}+3 b^{4} c e f -7 b^{3} c^{2} d f -2 b^{3} c^{2} e^{2}+10 b^{2} c^{3} d e -12 b \,c^{4} d^{2}\right ) x^{2} {\mathrm e}^{\frac {\left (2 a c f -2 b^{2} f +3 b c e -6 c^{2} d \right ) \ln \left (c \,x^{2}+b x +a \right )}{b^{2} f -2 b c e +4 c^{2} d}}}{c \left (8 a^{2} c^{2} f -6 a \,b^{2} c f +4 a b \,c^{2} e -8 a \,c^{3} d +b^{4} f -b^{3} c e +2 b^{2} c^{2} d \right )}+\frac {\left (2 a^{2} b^{2} c \,f^{2}-4 a^{2} b \,c^{2} e f +8 a^{2} c^{3} d f -2 a \,b^{4} f^{2}+7 a \,b^{3} c e f -10 a \,b^{2} c^{2} d f -6 a \,b^{2} c^{2} e^{2}+16 c^{3} a d e b -8 a \,c^{4} d^{2}-b^{4} c d f +2 b^{3} c^{2} d e -4 c^{3} b^{2} d^{2}\right ) x \,{\mathrm e}^{\frac {\left (2 a c f -2 b^{2} f +3 b c e -6 c^{2} d \right ) \ln \left (c \,x^{2}+b x +a \right )}{b^{2} f -2 b c e +4 c^{2} d}}}{c \left (8 a^{2} c^{2} f -6 a \,b^{2} c f +4 a b \,c^{2} e -8 a \,c^{3} d +b^{4} f -b^{3} c e +2 b^{2} c^{2} d \right )}-\frac {a \left (a \,b^{3} f^{2}-4 a \,b^{2} c e f +4 a b \,c^{2} d f +4 a b \,c^{2} e^{2}-8 a \,c^{3} d e +b^{3} c d f -2 b^{2} c^{2} d e +4 b \,c^{3} d^{2}\right ) {\mathrm e}^{\frac {\left (2 a c f -2 b^{2} f +3 b c e -6 c^{2} d \right ) \ln \left (c \,x^{2}+b x +a \right )}{b^{2} f -2 b c e +4 c^{2} d}}}{c \left (8 a^{2} c^{2} f -6 a \,b^{2} c f +4 a b \,c^{2} e -8 a \,c^{3} d +b^{4} f -b^{3} c e +2 b^{2} c^{2} d \right )}\) | \(727\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(2620\) |
Input:
int((c*x^2+b*x+a)^((2*a*c*f-2*b^2*f+3*b*c*e-6*c^2*d)/(b^2*f-2*b*c*e+4*c^2* d))*(f*x^2+e*x+d),x,method=_RETURNVERBOSE)
Output:
-1/c*(c*x^2+b*x+a)^(1+(2*a*c*f-2*b^2*f+3*b*c*e-6*c^2*d)/(b^2*f-2*b*c*e+4*c ^2*d))*(b^2*f-2*b*c*e+4*c^2*d)/(8*a^2*c^2*f-6*a*b^2*c*f+4*a*b*c^2*e-8*a*c^ 3*d+b^4*f-b^3*c*e+2*b^2*c^2*d)*(-2*a*c*f*x+b^2*f*x-b*c*e*x+2*c^2*d*x+a*b*f -2*a*c*e+b*c*d)
Leaf count of result is larger than twice the leaf count of optimal. 518 vs. \(2 (149) = 298\).
Time = 0.09 (sec) , antiderivative size = 518, normalized size of antiderivative = 3.50 \[ \int \left (a+b x+c x^2\right )^{\frac {-6 c^2 d+3 b c e-2 b^2 f+2 a c f}{4 c^2 d-2 b c e+b^2 f}} \left (d+e x+f x^2\right ) \, dx=-\frac {4 \, a b c^{3} d^{2} + 4 \, a^{2} b c^{2} e^{2} + a^{2} b^{3} f^{2} + {\left (8 \, c^{5} d^{2} - 8 \, b c^{4} d e + 2 \, b^{2} c^{3} e^{2} + {\left (b^{4} c - 2 \, a b^{2} c^{2}\right )} f^{2} + {\left (2 \, {\left (3 \, b^{2} c^{3} - 4 \, a c^{4}\right )} d - {\left (3 \, b^{3} c^{2} - 4 \, a b c^{3}\right )} e\right )} f\right )} x^{3} - 2 \, {\left (a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} d e + {\left (12 \, b c^{4} d^{2} - 2 \, {\left (5 \, b^{2} c^{3} + 4 \, a c^{4}\right )} d e + 2 \, {\left (b^{3} c^{2} + 2 \, a b c^{3}\right )} e^{2} + {\left (b^{5} - a b^{3} c\right )} f^{2} - {\left (3 \, b^{4} c e - {\left (7 \, b^{3} c^{2} - 4 \, a b c^{3}\right )} d\right )} f\right )} x^{2} - {\left (4 \, a^{2} b^{2} c e - {\left (a b^{3} c + 4 \, a^{2} b c^{2}\right )} d\right )} f + {\left (6 \, a b^{2} c^{2} e^{2} + 4 \, {\left (b^{2} c^{3} + 2 \, a c^{4}\right )} d^{2} - 2 \, {\left (b^{3} c^{2} + 8 \, a b c^{3}\right )} d e + 2 \, {\left (a b^{4} - a^{2} b^{2} c\right )} f^{2} + {\left ({\left (b^{4} c + 10 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} d - {\left (7 \, a b^{3} c - 4 \, a^{2} b c^{2}\right )} e\right )} f\right )} x}{{\left (2 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d - {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} e + {\left (b^{4} c - 6 \, a b^{2} c^{2} + 8 \, a^{2} c^{3}\right )} f\right )} {\left (c x^{2} + b x + a\right )}^{\frac {6 \, c^{2} d - 3 \, b c e + 2 \, {\left (b^{2} - a c\right )} f}{4 \, c^{2} d - 2 \, b c e + b^{2} f}}} \] Input:
integrate((c*x^2+b*x+a)^((2*a*c*f-2*b^2*f+3*b*c*e-6*c^2*d)/(b^2*f-2*b*c*e+ 4*c^2*d))*(f*x^2+e*x+d),x, algorithm="fricas")
Output:
-(4*a*b*c^3*d^2 + 4*a^2*b*c^2*e^2 + a^2*b^3*f^2 + (8*c^5*d^2 - 8*b*c^4*d*e + 2*b^2*c^3*e^2 + (b^4*c - 2*a*b^2*c^2)*f^2 + (2*(3*b^2*c^3 - 4*a*c^4)*d - (3*b^3*c^2 - 4*a*b*c^3)*e)*f)*x^3 - 2*(a*b^2*c^2 + 4*a^2*c^3)*d*e + (12* b*c^4*d^2 - 2*(5*b^2*c^3 + 4*a*c^4)*d*e + 2*(b^3*c^2 + 2*a*b*c^3)*e^2 + (b ^5 - a*b^3*c)*f^2 - (3*b^4*c*e - (7*b^3*c^2 - 4*a*b*c^3)*d)*f)*x^2 - (4*a^ 2*b^2*c*e - (a*b^3*c + 4*a^2*b*c^2)*d)*f + (6*a*b^2*c^2*e^2 + 4*(b^2*c^3 + 2*a*c^4)*d^2 - 2*(b^3*c^2 + 8*a*b*c^3)*d*e + 2*(a*b^4 - a^2*b^2*c)*f^2 + ((b^4*c + 10*a*b^2*c^2 - 8*a^2*c^3)*d - (7*a*b^3*c - 4*a^2*b*c^2)*e)*f)*x) /((2*(b^2*c^3 - 4*a*c^4)*d - (b^3*c^2 - 4*a*b*c^3)*e + (b^4*c - 6*a*b^2*c^ 2 + 8*a^2*c^3)*f)*(c*x^2 + b*x + a)^((6*c^2*d - 3*b*c*e + 2*(b^2 - a*c)*f) /(4*c^2*d - 2*b*c*e + b^2*f)))
Timed out. \[ \int \left (a+b x+c x^2\right )^{\frac {-6 c^2 d+3 b c e-2 b^2 f+2 a c f}{4 c^2 d-2 b c e+b^2 f}} \left (d+e x+f x^2\right ) \, dx=\text {Timed out} \] Input:
integrate((c*x**2+b*x+a)**((2*a*c*f-2*b**2*f+3*b*c*e-6*c**2*d)/(b**2*f-2*b *c*e+4*c**2*d))*(f*x**2+e*x+d),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 566 vs. \(2 (149) = 298\).
Time = 0.21 (sec) , antiderivative size = 566, normalized size of antiderivative = 3.82 \[ \int \left (a+b x+c x^2\right )^{\frac {-6 c^2 d+3 b c e-2 b^2 f+2 a c f}{4 c^2 d-2 b c e+b^2 f}} \left (d+e x+f x^2\right ) \, dx=-\frac {{\left ({\left (8 \, c^{5} d^{2} - 8 \, b c^{4} d e - 3 \, b^{3} c^{2} e f + b^{4} c f^{2} + 2 \, {\left (e^{2} + 3 \, d f\right )} b^{2} c^{3} - 2 \, {\left (4 \, c^{4} d f - 2 \, b c^{3} e f + b^{2} c^{2} f^{2}\right )} a\right )} x^{3} - {\left (8 \, c^{3} d e + 4 \, b^{2} c e f - b^{3} f^{2} - 4 \, {\left (e^{2} + d f\right )} b c^{2}\right )} a^{2} + {\left (12 \, b c^{4} d^{2} - 10 \, b^{2} c^{3} d e - 3 \, b^{4} c e f + b^{5} f^{2} + {\left (2 \, e^{2} + 7 \, d f\right )} b^{3} c^{2} - {\left (8 \, c^{4} d e + b^{3} c f^{2} - 4 \, {\left (e^{2} - d f\right )} b c^{3}\right )} a\right )} x^{2} + {\left (4 \, b c^{3} d^{2} - 2 \, b^{2} c^{2} d e + b^{3} c d f\right )} a + {\left (4 \, b^{2} c^{3} d^{2} - 2 \, b^{3} c^{2} d e + b^{4} c d f - 2 \, {\left (4 \, c^{3} d f - 2 \, b c^{2} e f + b^{2} c f^{2}\right )} a^{2} + {\left (8 \, c^{4} d^{2} - 16 \, b c^{3} d e - 7 \, b^{3} c e f + 2 \, b^{4} f^{2} + 2 \, {\left (3 \, e^{2} + 5 \, d f\right )} b^{2} c^{2}\right )} a\right )} x\right )} e^{\left (-\frac {6 \, c^{2} d \log \left (c x^{2} + b x + a\right )}{4 \, c^{2} d - 2 \, b c e + b^{2} f} + \frac {3 \, b c e \log \left (c x^{2} + b x + a\right )}{4 \, c^{2} d - 2 \, b c e + b^{2} f} - \frac {2 \, b^{2} f \log \left (c x^{2} + b x + a\right )}{4 \, c^{2} d - 2 \, b c e + b^{2} f} + \frac {2 \, a c f \log \left (c x^{2} + b x + a\right )}{4 \, c^{2} d - 2 \, b c e + b^{2} f}\right )}}{2 \, b^{2} c^{3} d - b^{3} c^{2} e + b^{4} c f + 8 \, a^{2} c^{3} f - 2 \, {\left (4 \, c^{4} d - 2 \, b c^{3} e + 3 \, b^{2} c^{2} f\right )} a} \] Input:
integrate((c*x^2+b*x+a)^((2*a*c*f-2*b^2*f+3*b*c*e-6*c^2*d)/(b^2*f-2*b*c*e+ 4*c^2*d))*(f*x^2+e*x+d),x, algorithm="maxima")
Output:
-((8*c^5*d^2 - 8*b*c^4*d*e - 3*b^3*c^2*e*f + b^4*c*f^2 + 2*(e^2 + 3*d*f)*b ^2*c^3 - 2*(4*c^4*d*f - 2*b*c^3*e*f + b^2*c^2*f^2)*a)*x^3 - (8*c^3*d*e + 4 *b^2*c*e*f - b^3*f^2 - 4*(e^2 + d*f)*b*c^2)*a^2 + (12*b*c^4*d^2 - 10*b^2*c ^3*d*e - 3*b^4*c*e*f + b^5*f^2 + (2*e^2 + 7*d*f)*b^3*c^2 - (8*c^4*d*e + b^ 3*c*f^2 - 4*(e^2 - d*f)*b*c^3)*a)*x^2 + (4*b*c^3*d^2 - 2*b^2*c^2*d*e + b^3 *c*d*f)*a + (4*b^2*c^3*d^2 - 2*b^3*c^2*d*e + b^4*c*d*f - 2*(4*c^3*d*f - 2* b*c^2*e*f + b^2*c*f^2)*a^2 + (8*c^4*d^2 - 16*b*c^3*d*e - 7*b^3*c*e*f + 2*b ^4*f^2 + 2*(3*e^2 + 5*d*f)*b^2*c^2)*a)*x)*e^(-6*c^2*d*log(c*x^2 + b*x + a) /(4*c^2*d - 2*b*c*e + b^2*f) + 3*b*c*e*log(c*x^2 + b*x + a)/(4*c^2*d - 2*b *c*e + b^2*f) - 2*b^2*f*log(c*x^2 + b*x + a)/(4*c^2*d - 2*b*c*e + b^2*f) + 2*a*c*f*log(c*x^2 + b*x + a)/(4*c^2*d - 2*b*c*e + b^2*f))/(2*b^2*c^3*d - b^3*c^2*e + b^4*c*f + 8*a^2*c^3*f - 2*(4*c^4*d - 2*b*c^3*e + 3*b^2*c^2*f)* a)
Leaf count of result is larger than twice the leaf count of optimal. 3983 vs. \(2 (149) = 298\).
Time = 1.41 (sec) , antiderivative size = 3983, normalized size of antiderivative = 26.91 \[ \int \left (a+b x+c x^2\right )^{\frac {-6 c^2 d+3 b c e-2 b^2 f+2 a c f}{4 c^2 d-2 b c e+b^2 f}} \left (d+e x+f x^2\right ) \, dx=\text {Too large to display} \] Input:
integrate((c*x^2+b*x+a)^((2*a*c*f-2*b^2*f+3*b*c*e-6*c^2*d)/(b^2*f-2*b*c*e+ 4*c^2*d))*(f*x^2+e*x+d),x, algorithm="giac")
Output:
-(8*c^5*d^2*x^3*e^(-(6*c^2*d*log(c*x^2 + b*x + a) - 3*b*c*e*log(c*x^2 + b* x + a) + 2*b^2*f*log(c*x^2 + b*x + a) - 2*a*c*f*log(c*x^2 + b*x + a))/(4*c ^2*d - 2*b*c*e + b^2*f)) - 8*b*c^4*d*e*x^3*e^(-(6*c^2*d*log(c*x^2 + b*x + a) - 3*b*c*e*log(c*x^2 + b*x + a) + 2*b^2*f*log(c*x^2 + b*x + a) - 2*a*c*f *log(c*x^2 + b*x + a))/(4*c^2*d - 2*b*c*e + b^2*f)) + 2*b^2*c^3*e^2*x^3*e^ (-(6*c^2*d*log(c*x^2 + b*x + a) - 3*b*c*e*log(c*x^2 + b*x + a) + 2*b^2*f*l og(c*x^2 + b*x + a) - 2*a*c*f*log(c*x^2 + b*x + a))/(4*c^2*d - 2*b*c*e + b ^2*f)) + 6*b^2*c^3*d*f*x^3*e^(-(6*c^2*d*log(c*x^2 + b*x + a) - 3*b*c*e*log (c*x^2 + b*x + a) + 2*b^2*f*log(c*x^2 + b*x + a) - 2*a*c*f*log(c*x^2 + b*x + a))/(4*c^2*d - 2*b*c*e + b^2*f)) - 8*a*c^4*d*f*x^3*e^(-(6*c^2*d*log(c*x ^2 + b*x + a) - 3*b*c*e*log(c*x^2 + b*x + a) + 2*b^2*f*log(c*x^2 + b*x + a ) - 2*a*c*f*log(c*x^2 + b*x + a))/(4*c^2*d - 2*b*c*e + b^2*f)) - 3*b^3*c^2 *e*f*x^3*e^(-(6*c^2*d*log(c*x^2 + b*x + a) - 3*b*c*e*log(c*x^2 + b*x + a) + 2*b^2*f*log(c*x^2 + b*x + a) - 2*a*c*f*log(c*x^2 + b*x + a))/(4*c^2*d - 2*b*c*e + b^2*f)) + 4*a*b*c^3*e*f*x^3*e^(-(6*c^2*d*log(c*x^2 + b*x + a) - 3*b*c*e*log(c*x^2 + b*x + a) + 2*b^2*f*log(c*x^2 + b*x + a) - 2*a*c*f*log( c*x^2 + b*x + a))/(4*c^2*d - 2*b*c*e + b^2*f)) + b^4*c*f^2*x^3*e^(-(6*c^2* d*log(c*x^2 + b*x + a) - 3*b*c*e*log(c*x^2 + b*x + a) + 2*b^2*f*log(c*x^2 + b*x + a) - 2*a*c*f*log(c*x^2 + b*x + a))/(4*c^2*d - 2*b*c*e + b^2*f)) - 2*a*b^2*c^2*f^2*x^3*e^(-(6*c^2*d*log(c*x^2 + b*x + a) - 3*b*c*e*log(c*x...
Time = 17.49 (sec) , antiderivative size = 348, normalized size of antiderivative = 2.35 \[ \int \left (a+b x+c x^2\right )^{\frac {-6 c^2 d+3 b c e-2 b^2 f+2 a c f}{4 c^2 d-2 b c e+b^2 f}} \left (d+e x+f x^2\right ) \, dx=\frac {\frac {x^3\,\left (f\,b^2-2\,e\,b\,c+4\,d\,c^2\right )}{4\,a\,c-b^2}+\frac {a\,\left (f\,b^2-2\,e\,b\,c+4\,d\,c^2\right )\,\left (a\,b\,f-2\,a\,c\,e+b\,c\,d\right )}{c\,\left (4\,a\,c-b^2\right )\,\left (f\,b^2-e\,b\,c+2\,d\,c^2-2\,a\,f\,c\right )}+\frac {x\,\left (f\,b^2-2\,e\,b\,c+4\,d\,c^2\right )\,\left (-2\,f\,a^2\,c+2\,f\,a\,b^2-3\,e\,a\,b\,c+2\,d\,a\,c^2+d\,b^2\,c\right )}{c\,\left (4\,a\,c-b^2\right )\,\left (f\,b^2-e\,b\,c+2\,d\,c^2-2\,a\,f\,c\right )}-\frac {x^2\,\left (f\,b^2-2\,e\,b\,c+4\,d\,c^2\right )\,\left (-f\,b^3+e\,b^2\,c-3\,d\,b\,c^2+a\,f\,b\,c+2\,a\,e\,c^2\right )}{c\,\left (4\,a\,c-b^2\right )\,\left (f\,b^2-e\,b\,c+2\,d\,c^2-2\,a\,f\,c\right )}}{{\left (c\,x^2+b\,x+a\right )}^{\frac {2\,f\,b^2-3\,e\,b\,c+6\,d\,c^2-2\,a\,f\,c}{f\,b^2-2\,e\,b\,c+4\,d\,c^2}}} \] Input:
int((d + e*x + f*x^2)/(a + b*x + c*x^2)^((6*c^2*d + 2*b^2*f - 2*a*c*f - 3* b*c*e)/(4*c^2*d + b^2*f - 2*b*c*e)),x)
Output:
((x^3*(4*c^2*d + b^2*f - 2*b*c*e))/(4*a*c - b^2) + (a*(4*c^2*d + b^2*f - 2 *b*c*e)*(a*b*f - 2*a*c*e + b*c*d))/(c*(4*a*c - b^2)*(2*c^2*d + b^2*f - 2*a *c*f - b*c*e)) + (x*(4*c^2*d + b^2*f - 2*b*c*e)*(2*a*c^2*d + 2*a*b^2*f + b ^2*c*d - 2*a^2*c*f - 3*a*b*c*e))/(c*(4*a*c - b^2)*(2*c^2*d + b^2*f - 2*a*c *f - b*c*e)) - (x^2*(4*c^2*d + b^2*f - 2*b*c*e)*(2*a*c^2*e - b^3*f - 3*b*c ^2*d + b^2*c*e + a*b*c*f))/(c*(4*a*c - b^2)*(2*c^2*d + b^2*f - 2*a*c*f - b *c*e)))/(a + b*x + c*x^2)^((6*c^2*d + 2*b^2*f - 2*a*c*f - 3*b*c*e)/(4*c^2* d + b^2*f - 2*b*c*e))
\[ \int \left (a+b x+c x^2\right )^{\frac {-6 c^2 d+3 b c e-2 b^2 f+2 a c f}{4 c^2 d-2 b c e+b^2 f}} \left (d+e x+f x^2\right ) \, dx=\left (\int \frac {\left (c \,x^{2}+b x +a \right )^{\frac {2 a c f +2 c^{2} d}{b^{2} f -2 b c e +4 c^{2} d}}}{\left (c \,x^{2}+b x +a \right )^{\frac {b c e}{b^{2} f -2 b c e +4 c^{2} d}} a^{2}+2 \left (c \,x^{2}+b x +a \right )^{\frac {b c e}{b^{2} f -2 b c e +4 c^{2} d}} a b x +2 \left (c \,x^{2}+b x +a \right )^{\frac {b c e}{b^{2} f -2 b c e +4 c^{2} d}} a c \,x^{2}+\left (c \,x^{2}+b x +a \right )^{\frac {b c e}{b^{2} f -2 b c e +4 c^{2} d}} b^{2} x^{2}+2 \left (c \,x^{2}+b x +a \right )^{\frac {b c e}{b^{2} f -2 b c e +4 c^{2} d}} b c \,x^{3}+\left (c \,x^{2}+b x +a \right )^{\frac {b c e}{b^{2} f -2 b c e +4 c^{2} d}} c^{2} x^{4}}d x \right ) d +\left (\int \frac {\left (c \,x^{2}+b x +a \right )^{\frac {2 a c f +2 c^{2} d}{b^{2} f -2 b c e +4 c^{2} d}} x^{2}}{\left (c \,x^{2}+b x +a \right )^{\frac {b c e}{b^{2} f -2 b c e +4 c^{2} d}} a^{2}+2 \left (c \,x^{2}+b x +a \right )^{\frac {b c e}{b^{2} f -2 b c e +4 c^{2} d}} a b x +2 \left (c \,x^{2}+b x +a \right )^{\frac {b c e}{b^{2} f -2 b c e +4 c^{2} d}} a c \,x^{2}+\left (c \,x^{2}+b x +a \right )^{\frac {b c e}{b^{2} f -2 b c e +4 c^{2} d}} b^{2} x^{2}+2 \left (c \,x^{2}+b x +a \right )^{\frac {b c e}{b^{2} f -2 b c e +4 c^{2} d}} b c \,x^{3}+\left (c \,x^{2}+b x +a \right )^{\frac {b c e}{b^{2} f -2 b c e +4 c^{2} d}} c^{2} x^{4}}d x \right ) f +\left (\int \frac {\left (c \,x^{2}+b x +a \right )^{\frac {2 a c f +2 c^{2} d}{b^{2} f -2 b c e +4 c^{2} d}} x}{\left (c \,x^{2}+b x +a \right )^{\frac {b c e}{b^{2} f -2 b c e +4 c^{2} d}} a^{2}+2 \left (c \,x^{2}+b x +a \right )^{\frac {b c e}{b^{2} f -2 b c e +4 c^{2} d}} a b x +2 \left (c \,x^{2}+b x +a \right )^{\frac {b c e}{b^{2} f -2 b c e +4 c^{2} d}} a c \,x^{2}+\left (c \,x^{2}+b x +a \right )^{\frac {b c e}{b^{2} f -2 b c e +4 c^{2} d}} b^{2} x^{2}+2 \left (c \,x^{2}+b x +a \right )^{\frac {b c e}{b^{2} f -2 b c e +4 c^{2} d}} b c \,x^{3}+\left (c \,x^{2}+b x +a \right )^{\frac {b c e}{b^{2} f -2 b c e +4 c^{2} d}} c^{2} x^{4}}d x \right ) e \] Input:
int((c*x^2+b*x+a)^((2*a*c*f-2*b^2*f+3*b*c*e-6*c^2*d)/(b^2*f-2*b*c*e+4*c^2* d))*(f*x^2+e*x+d),x)
Output:
int((a + b*x + c*x**2)**((2*a*c*f + 2*c**2*d)/(b**2*f - 2*b*c*e + 4*c**2*d ))/((a + b*x + c*x**2)**((b*c*e)/(b**2*f - 2*b*c*e + 4*c**2*d))*a**2 + 2*( a + b*x + c*x**2)**((b*c*e)/(b**2*f - 2*b*c*e + 4*c**2*d))*a*b*x + 2*(a + b*x + c*x**2)**((b*c*e)/(b**2*f - 2*b*c*e + 4*c**2*d))*a*c*x**2 + (a + b*x + c*x**2)**((b*c*e)/(b**2*f - 2*b*c*e + 4*c**2*d))*b**2*x**2 + 2*(a + b*x + c*x**2)**((b*c*e)/(b**2*f - 2*b*c*e + 4*c**2*d))*b*c*x**3 + (a + b*x + c*x**2)**((b*c*e)/(b**2*f - 2*b*c*e + 4*c**2*d))*c**2*x**4),x)*d + int(((a + b*x + c*x**2)**((2*a*c*f + 2*c**2*d)/(b**2*f - 2*b*c*e + 4*c**2*d))*x** 2)/((a + b*x + c*x**2)**((b*c*e)/(b**2*f - 2*b*c*e + 4*c**2*d))*a**2 + 2*( a + b*x + c*x**2)**((b*c*e)/(b**2*f - 2*b*c*e + 4*c**2*d))*a*b*x + 2*(a + b*x + c*x**2)**((b*c*e)/(b**2*f - 2*b*c*e + 4*c**2*d))*a*c*x**2 + (a + b*x + c*x**2)**((b*c*e)/(b**2*f - 2*b*c*e + 4*c**2*d))*b**2*x**2 + 2*(a + b*x + c*x**2)**((b*c*e)/(b**2*f - 2*b*c*e + 4*c**2*d))*b*c*x**3 + (a + b*x + c*x**2)**((b*c*e)/(b**2*f - 2*b*c*e + 4*c**2*d))*c**2*x**4),x)*f + int(((a + b*x + c*x**2)**((2*a*c*f + 2*c**2*d)/(b**2*f - 2*b*c*e + 4*c**2*d))*x)/ ((a + b*x + c*x**2)**((b*c*e)/(b**2*f - 2*b*c*e + 4*c**2*d))*a**2 + 2*(a + b*x + c*x**2)**((b*c*e)/(b**2*f - 2*b*c*e + 4*c**2*d))*a*b*x + 2*(a + b*x + c*x**2)**((b*c*e)/(b**2*f - 2*b*c*e + 4*c**2*d))*a*c*x**2 + (a + b*x + c*x**2)**((b*c*e)/(b**2*f - 2*b*c*e + 4*c**2*d))*b**2*x**2 + 2*(a + b*x + c*x**2)**((b*c*e)/(b**2*f - 2*b*c*e + 4*c**2*d))*b*c*x**3 + (a + b*x + ...