Integrand size = 28, antiderivative size = 74 \[ \int \frac {(b+2 c x) (d+e x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^2}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {8 e (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \] Output:
-2/3*(e*x+d)^2/(c*x^2+b*x+a)^(3/2)-8/3*e*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a* c+b^2)/(c*x^2+b*x+a)^(1/2)
Time = 1.82 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.49 \[ \int \frac {(b+2 c x) (d+e x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \left (8 a^2 e^2-8 c^2 d e x^3-b^2 \left (d^2+6 d e x-3 e^2 x^2\right )+4 a c \left (d^2+3 e^2 x^2\right )+4 b e \left (-a (d-3 e x)+c x^2 (-3 d+e x)\right )\right )}{3 \left (b^2-4 a c\right ) (a+x (b+c x))^{3/2}} \] Input:
Integrate[((b + 2*c*x)*(d + e*x)^2)/(a + b*x + c*x^2)^(5/2),x]
Output:
(2*(8*a^2*e^2 - 8*c^2*d*e*x^3 - b^2*(d^2 + 6*d*e*x - 3*e^2*x^2) + 4*a*c*(d ^2 + 3*e^2*x^2) + 4*b*e*(-(a*(d - 3*e*x)) + c*x^2*(-3*d + e*x))))/(3*(b^2 - 4*a*c)*(a + x*(b + c*x))^(3/2))
Time = 0.35 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1222, 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.
\(\displaystyle \int \frac {(b+2 c x) (d+e x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 1222 |
\(\displaystyle \frac {4}{3} e \int \frac {d+e x}{\left (c x^2+b x+a\right )^{3/2}}dx-\frac {2 (d+e x)^2}{3 \left (a+b x+c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 1158 |
\(\displaystyle -\frac {8 e (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 (d+e x)^2}{3 \left (a+b x+c x^2\right )^{3/2}}\) |
Input:
Int[((b + 2*c*x)*(d + e*x)^2)/(a + b*x + c*x^2)^(5/2),x]
Output:
(-2*(d + e*x)^2)/(3*(a + b*x + c*x^2)^(3/2)) - (8*e*(b*d - 2*a*e + (2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*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[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] - Simp[e*g*(m/(2*c*(p + 1))) 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 [2*c*f - b*g, 0] && LtQ[p, -1] && GtQ[m, 0]
Time = 1.68 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.66
method | result | size |
gosper | \(-\frac {2 \left (4 b c \,e^{2} x^{3}-8 c^{2} d e \,x^{3}+12 a c \,e^{2} x^{2}+3 b^{2} e^{2} x^{2}-12 b c d e \,x^{2}+12 x a b \,e^{2}-6 b^{2} d e x +8 e^{2} a^{2}-4 a b d e +4 a \,d^{2} c -b^{2} d^{2}\right )}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (4 a c -b^{2}\right )}\) | \(123\) |
trager | \(-\frac {2 \left (4 b c \,e^{2} x^{3}-8 c^{2} d e \,x^{3}+12 a c \,e^{2} x^{2}+3 b^{2} e^{2} x^{2}-12 b c d e \,x^{2}+12 x a b \,e^{2}-6 b^{2} d e x +8 e^{2} a^{2}-4 a b d e +4 a \,d^{2} c -b^{2} d^{2}\right )}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (4 a c -b^{2}\right )}\) | \(123\) |
orering | \(-\frac {2 \left (4 b c \,e^{2} x^{3}-8 c^{2} d e \,x^{3}+12 a c \,e^{2} x^{2}+3 b^{2} e^{2} x^{2}-12 b c d e \,x^{2}+12 x a b \,e^{2}-6 b^{2} d e x +8 e^{2} a^{2}-4 a b d e +4 a \,d^{2} c -b^{2} d^{2}\right )}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (4 a c -b^{2}\right )}\) | \(123\) |
default | \(b \,d^{2} \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (4 a c -b^{2}\right )}+\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 (b e +4 c d \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 (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (4 a c -b^{2}\right )}+\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 (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (4 a c -b^{2}\right )}+\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 d \left (b e +c d \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 (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (4 a c -b^{2}\right )}+\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 \,e^{2} \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 (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (4 a c -b^{2}\right )}+\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 (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (4 a c -b^{2}\right )}+\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 (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (4 a c -b^{2}\right )}+\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 )\) | \(686\) |
Input:
int((2*c*x+b)*(e*x+d)^2/(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
-2/3/(c*x^2+b*x+a)^(3/2)*(4*b*c*e^2*x^3-8*c^2*d*e*x^3+12*a*c*e^2*x^2+3*b^2 *e^2*x^2-12*b*c*d*e*x^2+12*a*b*e^2*x-6*b^2*d*e*x+8*a^2*e^2-4*a*b*d*e+4*a*c *d^2-b^2*d^2)/(4*a*c-b^2)
Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (66) = 132\).
Time = 0.59 (sec) , antiderivative size = 194, normalized size of antiderivative = 2.62 \[ \int \frac {(b+2 c x) (d+e x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (4 \, a b d e - 8 \, a^{2} e^{2} + 4 \, {\left (2 \, c^{2} d e - b c e^{2}\right )} x^{3} + {\left (b^{2} - 4 \, a c\right )} d^{2} + 3 \, {\left (4 \, b c d e - {\left (b^{2} + 4 \, a c\right )} e^{2}\right )} x^{2} + 6 \, {\left (b^{2} d e - 2 \, a b e^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + a^{2} b^{2} - 4 \, a^{3} c + 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x^{3} + {\left (b^{4} - 2 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} x^{2} + 2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} x\right )}} \] Input:
integrate((2*c*x+b)*(e*x+d)^2/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
Output:
-2/3*(4*a*b*d*e - 8*a^2*e^2 + 4*(2*c^2*d*e - b*c*e^2)*x^3 + (b^2 - 4*a*c)* d^2 + 3*(4*b*c*d*e - (b^2 + 4*a*c)*e^2)*x^2 + 6*(b^2*d*e - 2*a*b*e^2)*x)*s qrt(c*x^2 + b*x + a)/((b^2*c^2 - 4*a*c^3)*x^4 + a^2*b^2 - 4*a^3*c + 2*(b^3 *c - 4*a*b*c^2)*x^3 + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*x^2 + 2*(a*b^3 - 4*a^2 *b*c)*x)
Timed out. \[ \int \frac {(b+2 c x) (d+e x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((2*c*x+b)*(e*x+d)**2/(c*x**2+b*x+a)**(5/2),x)
Output:
Timed out
Exception generated. \[ \int \frac {(b+2 c x) (d+e x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((2*c*x+b)*(e*x+d)^2/(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. 289 vs. \(2 (66) = 132\).
Time = 0.22 (sec) , antiderivative size = 289, normalized size of antiderivative = 3.91 \[ \int \frac {(b+2 c x) (d+e x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left ({\left ({\left (\frac {4 \, {\left (2 \, b^{2} c^{2} d e - 8 \, a c^{3} d e - b^{3} c e^{2} + 4 \, a b c^{2} e^{2}\right )} x}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} + \frac {3 \, {\left (4 \, b^{3} c d e - 16 \, a b c^{2} d e - b^{4} e^{2} + 16 \, a^{2} c^{2} e^{2}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {6 \, {\left (b^{4} d e - 4 \, a b^{2} c d e - 2 \, a b^{3} e^{2} + 8 \, a^{2} b c e^{2}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {b^{4} d^{2} - 8 \, a b^{2} c d^{2} + 16 \, a^{2} c^{2} d^{2} + 4 \, a b^{3} d e - 16 \, a^{2} b c d e - 8 \, a^{2} b^{2} e^{2} + 32 \, a^{3} c e^{2}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \] Input:
integrate((2*c*x+b)*(e*x+d)^2/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
Output:
-2/3*(((4*(2*b^2*c^2*d*e - 8*a*c^3*d*e - b^3*c*e^2 + 4*a*b*c^2*e^2)*x/(b^4 - 8*a*b^2*c + 16*a^2*c^2) + 3*(4*b^3*c*d*e - 16*a*b*c^2*d*e - b^4*e^2 + 1 6*a^2*c^2*e^2)/(b^4 - 8*a*b^2*c + 16*a^2*c^2))*x + 6*(b^4*d*e - 4*a*b^2*c* d*e - 2*a*b^3*e^2 + 8*a^2*b*c*e^2)/(b^4 - 8*a*b^2*c + 16*a^2*c^2))*x + (b^ 4*d^2 - 8*a*b^2*c*d^2 + 16*a^2*c^2*d^2 + 4*a*b^3*d*e - 16*a^2*b*c*d*e - 8* a^2*b^2*e^2 + 32*a^3*c*e^2)/(b^4 - 8*a*b^2*c + 16*a^2*c^2))/(c*x^2 + b*x + a)^(3/2)
Time = 12.09 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.65 \[ \int \frac {(b+2 c x) (d+e x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2\,\left (8\,a^2\,e^2-4\,a\,b\,d\,e+12\,a\,b\,e^2\,x+4\,a\,c\,d^2+12\,a\,c\,e^2\,x^2-b^2\,d^2-6\,b^2\,d\,e\,x+3\,b^2\,e^2\,x^2-12\,b\,c\,d\,e\,x^2+4\,b\,c\,e^2\,x^3-8\,c^2\,d\,e\,x^3\right )}{3\,\left (4\,a\,c-b^2\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \] Input:
int(((b + 2*c*x)*(d + e*x)^2)/(a + b*x + c*x^2)^(5/2),x)
Output:
-(2*(8*a^2*e^2 - b^2*d^2 + 3*b^2*e^2*x^2 + 4*a*c*d^2 + 12*a*b*e^2*x - 6*b^ 2*d*e*x + 12*a*c*e^2*x^2 + 4*b*c*e^2*x^3 - 8*c^2*d*e*x^3 - 4*a*b*d*e - 12* b*c*d*e*x^2))/(3*(4*a*c - b^2)*(a + b*x + c*x^2)^(3/2))
Time = 0.34 (sec) , antiderivative size = 483, normalized size of antiderivative = 6.53 \[ \int \frac {(b+2 c x) (d+e x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {-\frac {16 \sqrt {c \,x^{2}+b x +a}\, a^{2} c \,e^{2}}{3}+\frac {8 \sqrt {c \,x^{2}+b x +a}\, a b c d e}{3}-8 \sqrt {c \,x^{2}+b x +a}\, a b c \,e^{2} x -\frac {8 \sqrt {c \,x^{2}+b x +a}\, a \,c^{2} d^{2}}{3}-8 \sqrt {c \,x^{2}+b x +a}\, a \,c^{2} e^{2} x^{2}+\frac {2 \sqrt {c \,x^{2}+b x +a}\, b^{2} c \,d^{2}}{3}+4 \sqrt {c \,x^{2}+b x +a}\, b^{2} c d e x -2 \sqrt {c \,x^{2}+b x +a}\, b^{2} c \,e^{2} x^{2}+8 \sqrt {c \,x^{2}+b x +a}\, b \,c^{2} d e \,x^{2}-\frac {8 \sqrt {c \,x^{2}+b x +a}\, b \,c^{2} e^{2} x^{3}}{3}+\frac {16 \sqrt {c \,x^{2}+b x +a}\, c^{3} d e \,x^{3}}{3}-\frac {8 \sqrt {c}\, a^{2} b \,e^{2}}{3}+\frac {16 \sqrt {c}\, a^{2} c d e}{3}-\frac {16 \sqrt {c}\, a \,b^{2} e^{2} x}{3}+\frac {32 \sqrt {c}\, a b c d e x}{3}-\frac {16 \sqrt {c}\, a b c \,e^{2} x^{2}}{3}+\frac {32 \sqrt {c}\, a \,c^{2} d e \,x^{2}}{3}-\frac {8 \sqrt {c}\, b^{3} e^{2} x^{2}}{3}+\frac {16 \sqrt {c}\, b^{2} c d e \,x^{2}}{3}-\frac {16 \sqrt {c}\, b^{2} c \,e^{2} x^{3}}{3}+\frac {32 \sqrt {c}\, b \,c^{2} d e \,x^{3}}{3}-\frac {8 \sqrt {c}\, b \,c^{2} e^{2} x^{4}}{3}+\frac {16 \sqrt {c}\, c^{3} d e \,x^{4}}{3}}{c \left (4 a \,c^{3} x^{4}-b^{2} c^{2} x^{4}+8 a b \,c^{2} x^{3}-2 b^{3} c \,x^{3}+8 a^{2} c^{2} x^{2}+2 a \,b^{2} c \,x^{2}-b^{4} x^{2}+8 a^{2} b c x -2 a \,b^{3} x +4 a^{3} c -a^{2} b^{2}\right )} \] Input:
int((2*c*x+b)*(e*x+d)^2/(c*x^2+b*x+a)^(5/2),x)
Output:
(2*( - 8*sqrt(a + b*x + c*x**2)*a**2*c*e**2 + 4*sqrt(a + b*x + c*x**2)*a*b *c*d*e - 12*sqrt(a + b*x + c*x**2)*a*b*c*e**2*x - 4*sqrt(a + b*x + c*x**2) *a*c**2*d**2 - 12*sqrt(a + b*x + c*x**2)*a*c**2*e**2*x**2 + sqrt(a + b*x + c*x**2)*b**2*c*d**2 + 6*sqrt(a + b*x + c*x**2)*b**2*c*d*e*x - 3*sqrt(a + b*x + c*x**2)*b**2*c*e**2*x**2 + 12*sqrt(a + b*x + c*x**2)*b*c**2*d*e*x**2 - 4*sqrt(a + b*x + c*x**2)*b*c**2*e**2*x**3 + 8*sqrt(a + b*x + c*x**2)*c* *3*d*e*x**3 - 4*sqrt(c)*a**2*b*e**2 + 8*sqrt(c)*a**2*c*d*e - 8*sqrt(c)*a*b **2*e**2*x + 16*sqrt(c)*a*b*c*d*e*x - 8*sqrt(c)*a*b*c*e**2*x**2 + 16*sqrt( c)*a*c**2*d*e*x**2 - 4*sqrt(c)*b**3*e**2*x**2 + 8*sqrt(c)*b**2*c*d*e*x**2 - 8*sqrt(c)*b**2*c*e**2*x**3 + 16*sqrt(c)*b*c**2*d*e*x**3 - 4*sqrt(c)*b*c* *2*e**2*x**4 + 8*sqrt(c)*c**3*d*e*x**4))/(3*c*(4*a**3*c - a**2*b**2 + 8*a* *2*b*c*x + 8*a**2*c**2*x**2 - 2*a*b**3*x + 2*a*b**2*c*x**2 + 8*a*b*c**2*x* *3 + 4*a*c**3*x**4 - b**4*x**2 - 2*b**3*c*x**3 - b**2*c**2*x**4))