Integrand size = 27, antiderivative size = 123 \[ \int \frac {(d+e x)^2 (f+g x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^2 (b f-2 a g+(2 c f-b g) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {8 (2 c d f-b e f-b d g+2 a e g) (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}} \] Output:
-2/3*(e*x+d)^2*(b*f-2*a*g+(-b*g+2*c*f)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)^(3/2) +8/3*(2*a*e*g-b*d*g-b*e*f+2*c*d*f)*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2) ^2/(c*x^2+b*x+a)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(286\) vs. \(2(123)=246\).
Time = 2.60 (sec) , antiderivative size = 286, normalized size of antiderivative = 2.33 \[ \int \frac {(d+e x)^2 (f+g x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \left (b^3 \left (6 d e x (-f+g x)+e^2 x^2 (3 f+g x)-d^2 (f+3 g x)\right )+4 b \left (3 a c (d-e x)^2 (f-g x)-2 c^2 d x^2 (-3 d f+2 e f x+d g x)+2 a^2 e (e f+2 d g-3 e g x)\right )-8 \left (2 a^3 e^2 g-2 c^3 d^2 f x^3-a c^2 x \left (3 d^2 f+e^2 f x^2+2 d e g x^2\right )+a^2 c \left (2 d e f+d^2 g+3 e^2 g x^2\right )\right )-2 b^2 \left (-c x \left (e^2 f x^2+3 d^2 (f-2 g x)+2 d e x (-6 f+g x)\right )+a \left (d^2 g+2 d e (f-6 g x)+3 e^2 x (-2 f+g x)\right )\right )\right )}{3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \] Input:
Integrate[((d + e*x)^2*(f + g*x))/(a + b*x + c*x^2)^(5/2),x]
Output:
(2*(b^3*(6*d*e*x*(-f + g*x) + e^2*x^2*(3*f + g*x) - d^2*(f + 3*g*x)) + 4*b *(3*a*c*(d - e*x)^2*(f - g*x) - 2*c^2*d*x^2*(-3*d*f + 2*e*f*x + d*g*x) + 2 *a^2*e*(e*f + 2*d*g - 3*e*g*x)) - 8*(2*a^3*e^2*g - 2*c^3*d^2*f*x^3 - a*c^2 *x*(3*d^2*f + e^2*f*x^2 + 2*d*e*g*x^2) + a^2*c*(2*d*e*f + d^2*g + 3*e^2*g* x^2)) - 2*b^2*(-(c*x*(e^2*f*x^2 + 3*d^2*(f - 2*g*x) + 2*d*e*x*(-6*f + g*x) )) + a*(d^2*g + 2*d*e*(f - 6*g*x) + 3*e^2*x*(-2*f + g*x)))))/(3*(b^2 - 4*a *c)^2*(a + x*(b + c*x))^(3/2))
Time = 0.43 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1227, 1158}
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.
| \(\displaystyle \int \frac {(d+e x)^2 (f+g x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1227 |
| \(\displaystyle -\frac {4 (2 a e g-b d g-b e f+2 c d f) \int \frac {d+e x}{\left (c x^2+b x+a\right )^{3/2}}dx}{3 \left (b^2-4 a c\right )}-\frac {2 (d+e x)^2 (-2 a g+x (2 c f-b g)+b f)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1158 |
| \(\displaystyle \frac {8 (-2 a e+x (2 c d-b e)+b d) (2 a e g-b d g-b e f+2 c d f)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 (d+e x)^2 (-2 a g+x (2 c f-b g)+b f)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\) |
Input:
Int[((d + e*x)^2*(f + g*x))/(a + b*x + c*x^2)^(5/2),x]
Output:
(-2*(d + e*x)^2*(b*f - 2*a*g + (2*c*f - b*g)*x))/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(3/2)) + (8*(2*c*d*f - b*e*f - b*d*g + 2*a*e*g)*(b*d - 2*a*e + ( 2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)^2*Sqrt[a + b*x + c*x^2])
Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbo l] :> Simp[-2*((b*d - 2*a*e + (2*c*d - b*e)*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*( (b*f - 2*a*g + (2*c*f - b*g)*x)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[m*((b*( e*f + d*g) - 2*(c*d*f + a*e*g))/((p + 1)*(b^2 - 4*a*c))) Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[Simplify[m + 2*p + 3], 0] && LtQ[p, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(423\) vs. \(2(115)=230\).
Time = 2.38 (sec) , antiderivative size = 424, normalized size of antiderivative = 3.45
| method | result | size |
| trager | \(-\frac {2 \left (12 a b c \,e^{2} g \,x^{3}-16 a \,c^{2} d e g \,x^{3}-8 a \,c^{2} e^{2} f \,x^{3}-b^{3} e^{2} g \,x^{3}-4 b^{2} c d e g \,x^{3}-2 b^{2} c \,e^{2} f \,x^{3}+8 b \,c^{2} d^{2} g \,x^{3}+16 b \,c^{2} d e f \,x^{3}-16 c^{3} d^{2} f \,x^{3}+24 a^{2} c \,e^{2} g \,x^{2}+6 a \,b^{2} e^{2} g \,x^{2}-24 a b c d e g \,x^{2}-12 a b c \,e^{2} f \,x^{2}-6 b^{3} d e g \,x^{2}-3 b^{3} e^{2} f \,x^{2}+12 b^{2} c \,d^{2} g \,x^{2}+24 b^{2} c d e f \,x^{2}-24 b \,c^{2} d^{2} f \,x^{2}+24 a^{2} b \,e^{2} g x -24 a \,b^{2} d e g x -12 a \,b^{2} e^{2} f x +12 a b c \,d^{2} g x +24 a b c d e f x -24 a \,c^{2} d^{2} f x +3 b^{3} d^{2} g x +6 b^{3} d e f x -6 b^{2} c \,d^{2} f x +16 a^{3} e^{2} g -16 a^{2} b d e g -8 a^{2} b \,e^{2} f +8 a^{2} c \,d^{2} g +16 a^{2} c d e f +2 a \,b^{2} d^{2} g +4 a \,b^{2} d e f -12 a b c \,d^{2} f +b^{3} d^{2} f \right )}{3 \left (4 a c -b^{2}\right )^{2} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\) | \(424\) |
| gosper | \(-\frac {2 \left (12 a b c \,e^{2} g \,x^{3}-16 a \,c^{2} d e g \,x^{3}-8 a \,c^{2} e^{2} f \,x^{3}-b^{3} e^{2} g \,x^{3}-4 b^{2} c d e g \,x^{3}-2 b^{2} c \,e^{2} f \,x^{3}+8 b \,c^{2} d^{2} g \,x^{3}+16 b \,c^{2} d e f \,x^{3}-16 c^{3} d^{2} f \,x^{3}+24 a^{2} c \,e^{2} g \,x^{2}+6 a \,b^{2} e^{2} g \,x^{2}-24 a b c d e g \,x^{2}-12 a b c \,e^{2} f \,x^{2}-6 b^{3} d e g \,x^{2}-3 b^{3} e^{2} f \,x^{2}+12 b^{2} c \,d^{2} g \,x^{2}+24 b^{2} c d e f \,x^{2}-24 b \,c^{2} d^{2} f \,x^{2}+24 a^{2} b \,e^{2} g x -24 a \,b^{2} d e g x -12 a \,b^{2} e^{2} f x +12 a b c \,d^{2} g x +24 a b c d e f x -24 a \,c^{2} d^{2} f x +3 b^{3} d^{2} g x +6 b^{3} d e f x -6 b^{2} c \,d^{2} f x +16 a^{3} e^{2} g -16 a^{2} b d e g -8 a^{2} b \,e^{2} f +8 a^{2} c \,d^{2} g +16 a^{2} c d e f +2 a \,b^{2} d^{2} g +4 a \,b^{2} d e f -12 a b c \,d^{2} f +b^{3} d^{2} f \right )}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}\) | \(433\) |
| orering | \(-\frac {2 \left (12 a b c \,e^{2} g \,x^{3}-16 a \,c^{2} d e g \,x^{3}-8 a \,c^{2} e^{2} f \,x^{3}-b^{3} e^{2} g \,x^{3}-4 b^{2} c d e g \,x^{3}-2 b^{2} c \,e^{2} f \,x^{3}+8 b \,c^{2} d^{2} g \,x^{3}+16 b \,c^{2} d e f \,x^{3}-16 c^{3} d^{2} f \,x^{3}+24 a^{2} c \,e^{2} g \,x^{2}+6 a \,b^{2} e^{2} g \,x^{2}-24 a b c d e g \,x^{2}-12 a b c \,e^{2} f \,x^{2}-6 b^{3} d e g \,x^{2}-3 b^{3} e^{2} f \,x^{2}+12 b^{2} c \,d^{2} g \,x^{2}+24 b^{2} c d e f \,x^{2}-24 b \,c^{2} d^{2} f \,x^{2}+24 a^{2} b \,e^{2} g x -24 a \,b^{2} d e g x -12 a \,b^{2} e^{2} f x +12 a b c \,d^{2} g x +24 a b c d e f x -24 a \,c^{2} d^{2} f x +3 b^{3} d^{2} g x +6 b^{3} d e f x -6 b^{2} c \,d^{2} f x +16 a^{3} e^{2} g -16 a^{2} b d e g -8 a^{2} b \,e^{2} f +8 a^{2} c \,d^{2} g +16 a^{2} c d e f +2 a \,b^{2} d^{2} g +4 a \,b^{2} d e f -12 a b c \,d^{2} f +b^{3} d^{2} f \right )}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}\) | \(433\) |
| default | \(d^{2} f \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )+e \left (2 d g +e f \right ) \left (-\frac {x}{2 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )}{4 c}+\frac {a \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )+d \left (d g +2 e f \right ) \left (-\frac {1}{3 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )+e^{2} g \left (-\frac {x^{2}}{c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {b \left (-\frac {x}{2 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )}{4 c}+\frac {a \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )}{2 c}+\frac {2 a \left (-\frac {1}{3 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )}{c}\right )\) | \(685\) |
Input:
int((e*x+d)^2*(g*x+f)/(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
-2/3*(12*a*b*c*e^2*g*x^3-16*a*c^2*d*e*g*x^3-8*a*c^2*e^2*f*x^3-b^3*e^2*g*x^ 3-4*b^2*c*d*e*g*x^3-2*b^2*c*e^2*f*x^3+8*b*c^2*d^2*g*x^3+16*b*c^2*d*e*f*x^3 -16*c^3*d^2*f*x^3+24*a^2*c*e^2*g*x^2+6*a*b^2*e^2*g*x^2-24*a*b*c*d*e*g*x^2- 12*a*b*c*e^2*f*x^2-6*b^3*d*e*g*x^2-3*b^3*e^2*f*x^2+12*b^2*c*d^2*g*x^2+24*b ^2*c*d*e*f*x^2-24*b*c^2*d^2*f*x^2+24*a^2*b*e^2*g*x-24*a*b^2*d*e*g*x-12*a*b ^2*e^2*f*x+12*a*b*c*d^2*g*x+24*a*b*c*d*e*f*x-24*a*c^2*d^2*f*x+3*b^3*d^2*g* x+6*b^3*d*e*f*x-6*b^2*c*d^2*f*x+16*a^3*e^2*g-16*a^2*b*d*e*g-8*a^2*b*e^2*f+ 8*a^2*c*d^2*g+16*a^2*c*d*e*f+2*a*b^2*d^2*g+4*a*b^2*d*e*f-12*a*b*c*d^2*f+b^ 3*d^2*f)/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 474 vs. \(2 (115) = 230\).
Time = 4.79 (sec) , antiderivative size = 474, normalized size of antiderivative = 3.85 \[ \int \frac {(d+e x)^2 (f+g x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left (2 \, {\left (8 \, c^{3} d^{2} - 8 \, b c^{2} d e + {\left (b^{2} c + 4 \, a c^{2}\right )} e^{2}\right )} f - {\left (8 \, b c^{2} d^{2} - 4 \, {\left (b^{2} c + 4 \, a c^{2}\right )} d e - {\left (b^{3} - 12 \, a b c\right )} e^{2}\right )} g\right )} x^{3} + 3 \, {\left ({\left (8 \, b c^{2} d^{2} - 8 \, b^{2} c d e + {\left (b^{3} + 4 \, a b c\right )} e^{2}\right )} f - 2 \, {\left (2 \, b^{2} c d^{2} - {\left (b^{3} + 4 \, a b c\right )} d e + {\left (a b^{2} + 4 \, a^{2} c\right )} e^{2}\right )} g\right )} x^{2} + {\left (8 \, a^{2} b e^{2} - {\left (b^{3} - 12 \, a b c\right )} d^{2} - 4 \, {\left (a b^{2} + 4 \, a^{2} c\right )} d e\right )} f + 2 \, {\left (8 \, a^{2} b d e - 8 \, a^{3} e^{2} - {\left (a b^{2} + 4 \, a^{2} c\right )} d^{2}\right )} g + 3 \, {\left (2 \, {\left (2 \, a b^{2} e^{2} + {\left (b^{2} c + 4 \, a c^{2}\right )} d^{2} - {\left (b^{3} + 4 \, a b c\right )} d e\right )} f + {\left (8 \, a b^{2} d e - 8 \, a^{2} b e^{2} - {\left (b^{3} + 4 \, a b c\right )} d^{2}\right )} g\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )}} \] Input:
integrate((e*x+d)^2*(g*x+f)/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
Output:
2/3*((2*(8*c^3*d^2 - 8*b*c^2*d*e + (b^2*c + 4*a*c^2)*e^2)*f - (8*b*c^2*d^2 - 4*(b^2*c + 4*a*c^2)*d*e - (b^3 - 12*a*b*c)*e^2)*g)*x^3 + 3*((8*b*c^2*d^ 2 - 8*b^2*c*d*e + (b^3 + 4*a*b*c)*e^2)*f - 2*(2*b^2*c*d^2 - (b^3 + 4*a*b*c )*d*e + (a*b^2 + 4*a^2*c)*e^2)*g)*x^2 + (8*a^2*b*e^2 - (b^3 - 12*a*b*c)*d^ 2 - 4*(a*b^2 + 4*a^2*c)*d*e)*f + 2*(8*a^2*b*d*e - 8*a^3*e^2 - (a*b^2 + 4*a ^2*c)*d^2)*g + 3*(2*(2*a*b^2*e^2 + (b^2*c + 4*a*c^2)*d^2 - (b^3 + 4*a*b*c) *d*e)*f + (8*a*b^2*d*e - 8*a^2*b*e^2 - (b^3 + 4*a*b*c)*d^2)*g)*x)*sqrt(c*x ^2 + b*x + a)/(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b^6 - 6 *a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x)
Timed out. \[ \int \frac {(d+e x)^2 (f+g x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((e*x+d)**2*(g*x+f)/(c*x**2+b*x+a)**(5/2),x)
Output:
Timed out
Exception generated. \[ \int \frac {(d+e x)^2 (f+g x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x+d)^2*(g*x+f)/(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Leaf count of result is larger than twice the leaf count of optimal. 448 vs. \(2 (115) = 230\).
Time = 0.35 (sec) , antiderivative size = 448, normalized size of antiderivative = 3.64 \[ \int \frac {(d+e x)^2 (f+g x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left ({\left (\frac {{\left (16 \, c^{3} d^{2} f - 16 \, b c^{2} d e f + 2 \, b^{2} c e^{2} f + 8 \, a c^{2} e^{2} f - 8 \, b c^{2} d^{2} g + 4 \, b^{2} c d e g + 16 \, a c^{2} d e g + b^{3} e^{2} g - 12 \, a b c e^{2} g\right )} x}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} + \frac {3 \, {\left (8 \, b c^{2} d^{2} f - 8 \, b^{2} c d e f + b^{3} e^{2} f + 4 \, a b c e^{2} f - 4 \, b^{2} c d^{2} g + 2 \, b^{3} d e g + 8 \, a b c d e g - 2 \, a b^{2} e^{2} g - 8 \, a^{2} c e^{2} g\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {3 \, {\left (2 \, b^{2} c d^{2} f + 8 \, a c^{2} d^{2} f - 2 \, b^{3} d e f - 8 \, a b c d e f + 4 \, a b^{2} e^{2} f - b^{3} d^{2} g - 4 \, a b c d^{2} g + 8 \, a b^{2} d e g - 8 \, a^{2} b e^{2} g\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x - \frac {b^{3} d^{2} f - 12 \, a b c d^{2} f + 4 \, a b^{2} d e f + 16 \, a^{2} c d e f - 8 \, a^{2} b e^{2} f + 2 \, a b^{2} d^{2} g + 8 \, a^{2} c d^{2} g - 16 \, a^{2} b d e g + 16 \, a^{3} e^{2} g}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \] Input:
integrate((e*x+d)^2*(g*x+f)/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
Output:
2/3*((((16*c^3*d^2*f - 16*b*c^2*d*e*f + 2*b^2*c*e^2*f + 8*a*c^2*e^2*f - 8* b*c^2*d^2*g + 4*b^2*c*d*e*g + 16*a*c^2*d*e*g + b^3*e^2*g - 12*a*b*c*e^2*g) *x/(b^4 - 8*a*b^2*c + 16*a^2*c^2) + 3*(8*b*c^2*d^2*f - 8*b^2*c*d*e*f + b^3 *e^2*f + 4*a*b*c*e^2*f - 4*b^2*c*d^2*g + 2*b^3*d*e*g + 8*a*b*c*d*e*g - 2*a *b^2*e^2*g - 8*a^2*c*e^2*g)/(b^4 - 8*a*b^2*c + 16*a^2*c^2))*x + 3*(2*b^2*c *d^2*f + 8*a*c^2*d^2*f - 2*b^3*d*e*f - 8*a*b*c*d*e*f + 4*a*b^2*e^2*f - b^3 *d^2*g - 4*a*b*c*d^2*g + 8*a*b^2*d*e*g - 8*a^2*b*e^2*g)/(b^4 - 8*a*b^2*c + 16*a^2*c^2))*x - (b^3*d^2*f - 12*a*b*c*d^2*f + 4*a*b^2*d*e*f + 16*a^2*c*d *e*f - 8*a^2*b*e^2*f + 2*a*b^2*d^2*g + 8*a^2*c*d^2*g - 16*a^2*b*d*e*g + 16 *a^3*e^2*g)/(b^4 - 8*a*b^2*c + 16*a^2*c^2))/(c*x^2 + b*x + a)^(3/2)
Time = 12.77 (sec) , antiderivative size = 423, normalized size of antiderivative = 3.44 \[ \int \frac {(d+e x)^2 (f+g x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2\,\left (16\,g\,a^3\,e^2-16\,g\,a^2\,b\,d\,e+24\,g\,a^2\,b\,e^2\,x-8\,f\,a^2\,b\,e^2+8\,g\,a^2\,c\,d^2+16\,f\,a^2\,c\,d\,e+24\,g\,a^2\,c\,e^2\,x^2+2\,g\,a\,b^2\,d^2-24\,g\,a\,b^2\,d\,e\,x+4\,f\,a\,b^2\,d\,e+6\,g\,a\,b^2\,e^2\,x^2-12\,f\,a\,b^2\,e^2\,x+12\,g\,a\,b\,c\,d^2\,x-12\,f\,a\,b\,c\,d^2-24\,g\,a\,b\,c\,d\,e\,x^2+24\,f\,a\,b\,c\,d\,e\,x+12\,g\,a\,b\,c\,e^2\,x^3-12\,f\,a\,b\,c\,e^2\,x^2-24\,f\,a\,c^2\,d^2\,x-16\,g\,a\,c^2\,d\,e\,x^3-8\,f\,a\,c^2\,e^2\,x^3+3\,g\,b^3\,d^2\,x+f\,b^3\,d^2-6\,g\,b^3\,d\,e\,x^2+6\,f\,b^3\,d\,e\,x-g\,b^3\,e^2\,x^3-3\,f\,b^3\,e^2\,x^2+12\,g\,b^2\,c\,d^2\,x^2-6\,f\,b^2\,c\,d^2\,x-4\,g\,b^2\,c\,d\,e\,x^3+24\,f\,b^2\,c\,d\,e\,x^2-2\,f\,b^2\,c\,e^2\,x^3+8\,g\,b\,c^2\,d^2\,x^3-24\,f\,b\,c^2\,d^2\,x^2+16\,f\,b\,c^2\,d\,e\,x^3-16\,f\,c^3\,d^2\,x^3\right )}{3\,{\left (4\,a\,c-b^2\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \] Input:
int(((f + g*x)*(d + e*x)^2)/(a + b*x + c*x^2)^(5/2),x)
Output:
-(2*(b^3*d^2*f + 16*a^3*e^2*g - 3*b^3*e^2*f*x^2 - 16*c^3*d^2*f*x^3 - b^3*e ^2*g*x^3 + 2*a*b^2*d^2*g - 8*a^2*b*e^2*f + 8*a^2*c*d^2*g + 3*b^3*d^2*g*x - 12*a*b^2*e^2*f*x - 24*a*c^2*d^2*f*x + 24*a^2*b*e^2*g*x - 6*b^2*c*d^2*f*x - 6*b^3*d*e*g*x^2 + 6*a*b^2*e^2*g*x^2 - 24*b*c^2*d^2*f*x^2 - 8*a*c^2*e^2*f *x^3 + 24*a^2*c*e^2*g*x^2 + 12*b^2*c*d^2*g*x^2 + 8*b*c^2*d^2*g*x^3 - 2*b^2 *c*e^2*f*x^3 - 12*a*b*c*d^2*f + 4*a*b^2*d*e*f - 16*a^2*b*d*e*g + 16*a^2*c* d*e*f + 6*b^3*d*e*f*x + 12*a*b*c*d^2*g*x - 24*a*b^2*d*e*g*x - 12*a*b*c*e^2 *f*x^2 + 12*a*b*c*e^2*g*x^3 + 24*b^2*c*d*e*f*x^2 - 16*a*c^2*d*e*g*x^3 + 16 *b*c^2*d*e*f*x^3 - 4*b^2*c*d*e*g*x^3 + 24*a*b*c*d*e*f*x - 24*a*b*c*d*e*g*x ^2))/(3*(4*a*c - b^2)^2*(a + b*x + c*x^2)^(3/2))
Time = 0.34 (sec) , antiderivative size = 1801, normalized size of antiderivative = 14.64 \[ \int \frac {(d+e x)^2 (f+g x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
int((e*x+d)^2*(g*x+f)/(c*x^2+b*x+a)^(5/2),x)
Output:
(2*( - 16*sqrt(a + b*x + c*x**2)*a**3*c**2*e**2*g + 16*sqrt(a + b*x + c*x* *2)*a**2*b*c**2*d*e*g + 8*sqrt(a + b*x + c*x**2)*a**2*b*c**2*e**2*f - 24*s qrt(a + b*x + c*x**2)*a**2*b*c**2*e**2*g*x - 8*sqrt(a + b*x + c*x**2)*a**2 *c**3*d**2*g - 16*sqrt(a + b*x + c*x**2)*a**2*c**3*d*e*f - 24*sqrt(a + b*x + c*x**2)*a**2*c**3*e**2*g*x**2 - 2*sqrt(a + b*x + c*x**2)*a*b**2*c**2*d* *2*g - 4*sqrt(a + b*x + c*x**2)*a*b**2*c**2*d*e*f + 24*sqrt(a + b*x + c*x* *2)*a*b**2*c**2*d*e*g*x + 12*sqrt(a + b*x + c*x**2)*a*b**2*c**2*e**2*f*x - 6*sqrt(a + b*x + c*x**2)*a*b**2*c**2*e**2*g*x**2 + 12*sqrt(a + b*x + c*x* *2)*a*b*c**3*d**2*f - 12*sqrt(a + b*x + c*x**2)*a*b*c**3*d**2*g*x - 24*sqr t(a + b*x + c*x**2)*a*b*c**3*d*e*f*x + 24*sqrt(a + b*x + c*x**2)*a*b*c**3* d*e*g*x**2 + 12*sqrt(a + b*x + c*x**2)*a*b*c**3*e**2*f*x**2 - 12*sqrt(a + b*x + c*x**2)*a*b*c**3*e**2*g*x**3 + 24*sqrt(a + b*x + c*x**2)*a*c**4*d**2 *f*x + 16*sqrt(a + b*x + c*x**2)*a*c**4*d*e*g*x**3 + 8*sqrt(a + b*x + c*x* *2)*a*c**4*e**2*f*x**3 - sqrt(a + b*x + c*x**2)*b**3*c**2*d**2*f - 3*sqrt( a + b*x + c*x**2)*b**3*c**2*d**2*g*x - 6*sqrt(a + b*x + c*x**2)*b**3*c**2* d*e*f*x + 6*sqrt(a + b*x + c*x**2)*b**3*c**2*d*e*g*x**2 + 3*sqrt(a + b*x + c*x**2)*b**3*c**2*e**2*f*x**2 + sqrt(a + b*x + c*x**2)*b**3*c**2*e**2*g*x **3 + 6*sqrt(a + b*x + c*x**2)*b**2*c**3*d**2*f*x - 12*sqrt(a + b*x + c*x* *2)*b**2*c**3*d**2*g*x**2 - 24*sqrt(a + b*x + c*x**2)*b**2*c**3*d*e*f*x**2 + 4*sqrt(a + b*x + c*x**2)*b**2*c**3*d*e*g*x**3 + 2*sqrt(a + b*x + c*x...