Integrand size = 20, antiderivative size = 91 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {8 (2 c d-b e) (b+2 c x)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}} \] Output:
1/3*(-2*b*d+4*a*e-2*(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)^(3/2)+8/3*( -b*e+2*c*d)*(2*c*x+b)/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^(1/2)
Time = 0.87 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.21 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \left (b^3 (d+3 e x)+8 c \left (a^2 e-3 a c d x-2 c^2 d x^3\right )+4 b c \left (-3 a (d-e x)+2 c x^2 (-3 d+e x)\right )+2 b^2 (a e+3 c x (-d+2 e x))\right )}{3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \] Input:
Integrate[(d + e*x)/(a + b*x + c*x^2)^(5/2),x]
Output:
(-2*(b^3*(d + 3*e*x) + 8*c*(a^2*e - 3*a*c*d*x - 2*c^2*d*x^3) + 4*b*c*(-3*a *(d - e*x) + 2*c*x^2*(-3*d + e*x)) + 2*b^2*(a*e + 3*c*x*(-d + 2*e*x))))/(3 *(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(3/2))
Time = 0.20 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1159, 1088}
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.
\(\displaystyle \int \frac {d+e x}{\left (a+b x+c x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 1159 |
\(\displaystyle -\frac {4 (2 c d-b e) \int \frac {1}{\left (c x^2+b x+a\right )^{3/2}}dx}{3 \left (b^2-4 a c\right )}-\frac {2 (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 1088 |
\(\displaystyle \frac {8 (b+2 c x) (2 c d-b e)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\) |
Input:
Int[(d + e*x)/(a + b*x + c*x^2)^(5/2),x]
Output:
(-2*(b*d - 2*a*e + (2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(3 /2)) + (8*(2*c*d - b*e)*(b + 2*c*x))/(3*(b^2 - 4*a*c)^2*Sqrt[a + b*x + c*x ^2])
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[((b*d - 2*a*e + (2*c*d - b*e)*x)/((p + 1)*(b^2 - 4*a*c)))*(a + b* x + c*x^2)^(p + 1), x] - Simp[(2*p + 3)*((2*c*d - b*e)/((p + 1)*(b^2 - 4*a* c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] & & LtQ[p, -1] && NeQ[p, -3/2]
Time = 0.95 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.34
method | result | size |
trager | \(-\frac {2 \left (8 b \,c^{2} x^{3} e -16 c^{3} d \,x^{3}+12 b^{2} c e \,x^{2}-24 b \,c^{2} d \,x^{2}+12 a b c e x -24 a d x \,c^{2}+3 b^{3} e x -6 b^{2} c x d +8 a^{2} c e +2 e a \,b^{2}-12 a b c d +b^{3} d \right )}{3 \left (4 a c -b^{2}\right )^{2} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\) | \(122\) |
gosper | \(-\frac {2 \left (8 b \,c^{2} x^{3} e -16 c^{3} d \,x^{3}+12 b^{2} c e \,x^{2}-24 b \,c^{2} d \,x^{2}+12 a b c e x -24 a d x \,c^{2}+3 b^{3} e x -6 b^{2} c x d +8 a^{2} c e +2 e a \,b^{2}-12 a b c d +b^{3} d \right )}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}\) | \(131\) |
orering | \(-\frac {2 \left (8 b \,c^{2} x^{3} e -16 c^{3} d \,x^{3}+12 b^{2} c e \,x^{2}-24 b \,c^{2} d \,x^{2}+12 a b c e x -24 a d x \,c^{2}+3 b^{3} e x -6 b^{2} c x d +8 a^{2} c e +2 e a \,b^{2}-12 a b c d +b^{3} d \right )}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}\) | \(131\) |
default | \(d \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )+e \left (-\frac {1}{3 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )\) | \(162\) |
Input:
int((e*x+d)/(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
-2/3*(8*b*c^2*e*x^3-16*c^3*d*x^3+12*b^2*c*e*x^2-24*b*c^2*d*x^2+12*a*b*c*e* x-24*a*c^2*d*x+3*b^3*e*x-6*b^2*c*d*x+8*a^2*c*e+2*a*b^2*e-12*a*b*c*d+b^3*d) /(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. 247 vs. \(2 (83) = 166\).
Time = 0.49 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.71 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (8 \, {\left (2 \, c^{3} d - b c^{2} e\right )} x^{3} + 12 \, {\left (2 \, b c^{2} d - b^{2} c e\right )} x^{2} - {\left (b^{3} - 12 \, a b c\right )} d - 2 \, {\left (a b^{2} + 4 \, a^{2} c\right )} e + 3 \, {\left (2 \, {\left (b^{2} c + 4 \, a c^{2}\right )} d - {\left (b^{3} + 4 \, a b c\right )} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )}} \] Input:
integrate((e*x+d)/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
Output:
2/3*(8*(2*c^3*d - b*c^2*e)*x^3 + 12*(2*b*c^2*d - b^2*c*e)*x^2 - (b^3 - 12* a*b*c)*d - 2*(a*b^2 + 4*a^2*c)*e + 3*(2*(b^2*c + 4*a*c^2)*d - (b^3 + 4*a*b *c)*e)*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)
\[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {d + e x}{\left (a + b x + c x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((e*x+d)/(c*x**2+b*x+a)**(5/2),x)
Output:
Integral((d + e*x)/(a + b*x + c*x**2)**(5/2), x)
Exception generated. \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x+d)/(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. 197 vs. \(2 (83) = 166\).
Time = 0.20 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.16 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left (4 \, {\left (\frac {2 \, {\left (2 \, c^{3} d - b c^{2} e\right )} x}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} + \frac {3 \, {\left (2 \, b c^{2} d - b^{2} c e\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {3 \, {\left (2 \, b^{2} c d + 8 \, a c^{2} d - b^{3} e - 4 \, a b c e\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x - \frac {b^{3} d - 12 \, a b c d + 2 \, a b^{2} e + 8 \, a^{2} c e}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \] Input:
integrate((e*x+d)/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
Output:
2/3*((4*(2*(2*c^3*d - b*c^2*e)*x/(b^4 - 8*a*b^2*c + 16*a^2*c^2) + 3*(2*b*c ^2*d - b^2*c*e)/(b^4 - 8*a*b^2*c + 16*a^2*c^2))*x + 3*(2*b^2*c*d + 8*a*c^2 *d - b^3*e - 4*a*b*c*e)/(b^4 - 8*a*b^2*c + 16*a^2*c^2))*x - (b^3*d - 12*a* b*c*d + 2*a*b^2*e + 8*a^2*c*e)/(b^4 - 8*a*b^2*c + 16*a^2*c^2))/(c*x^2 + b* x + a)^(3/2)
Time = 6.34 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.33 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2\,\left (8\,e\,a^2\,c+2\,e\,a\,b^2+12\,e\,a\,b\,c\,x-12\,d\,a\,b\,c-24\,d\,a\,c^2\,x+3\,e\,b^3\,x+d\,b^3+12\,e\,b^2\,c\,x^2-6\,d\,b^2\,c\,x+8\,e\,b\,c^2\,x^3-24\,d\,b\,c^2\,x^2-16\,d\,c^3\,x^3\right )}{3\,{\left (4\,a\,c-b^2\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \] Input:
int((d + e*x)/(a + b*x + c*x^2)^(5/2),x)
Output:
-(2*(b^3*d - 16*c^3*d*x^3 + 2*a*b^2*e + 8*a^2*c*e + 3*b^3*e*x - 24*a*c^2*d *x - 6*b^2*c*d*x - 24*b*c^2*d*x^2 + 12*b^2*c*e*x^2 + 8*b*c^2*e*x^3 - 12*a* b*c*d + 12*a*b*c*e*x))/(3*(4*a*c - b^2)^2*(a + b*x + c*x^2)^(3/2))
Time = 0.73 (sec) , antiderivative size = 511, normalized size of antiderivative = 5.62 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {-16 \sqrt {c \,x^{2}+b x +a}\, a^{2} c e -4 \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} e +24 \sqrt {c \,x^{2}+b x +a}\, a b c d -24 \sqrt {c \,x^{2}+b x +a}\, a b c e x +48 \sqrt {c \,x^{2}+b x +a}\, a \,c^{2} d x -2 \sqrt {c \,x^{2}+b x +a}\, b^{3} d -6 \sqrt {c \,x^{2}+b x +a}\, b^{3} e x +12 \sqrt {c \,x^{2}+b x +a}\, b^{2} c d x -24 \sqrt {c \,x^{2}+b x +a}\, b^{2} c e \,x^{2}+48 \sqrt {c \,x^{2}+b x +a}\, b \,c^{2} d \,x^{2}-16 \sqrt {c \,x^{2}+b x +a}\, b \,c^{2} e \,x^{3}+32 \sqrt {c \,x^{2}+b x +a}\, c^{3} d \,x^{3}+16 \sqrt {c}\, a^{2} b e -32 \sqrt {c}\, a^{2} c d +32 \sqrt {c}\, a \,b^{2} e x -64 \sqrt {c}\, a b c d x +32 \sqrt {c}\, a b c e \,x^{2}-64 \sqrt {c}\, a \,c^{2} d \,x^{2}+16 \sqrt {c}\, b^{3} e \,x^{2}-32 \sqrt {c}\, b^{2} c d \,x^{2}+32 \sqrt {c}\, b^{2} c e \,x^{3}-64 \sqrt {c}\, b \,c^{2} d \,x^{3}+16 \sqrt {c}\, b \,c^{2} e \,x^{4}-32 \sqrt {c}\, c^{3} d \,x^{4}}{48 a^{2} c^{4} x^{4}-24 a \,b^{2} c^{3} x^{4}+3 b^{4} c^{2} x^{4}+96 a^{2} b \,c^{3} x^{3}-48 a \,b^{3} c^{2} x^{3}+6 b^{5} c \,x^{3}+96 a^{3} c^{3} x^{2}-18 a \,b^{4} c \,x^{2}+3 b^{6} x^{2}+96 a^{3} b \,c^{2} x -48 a^{2} b^{3} c x +6 a \,b^{5} x +48 a^{4} c^{2}-24 a^{3} b^{2} c +3 a^{2} b^{4}} \] Input:
int((e*x+d)/(c*x^2+b*x+a)^(5/2),x)
Output:
(2*( - 8*sqrt(a + b*x + c*x**2)*a**2*c*e - 2*sqrt(a + b*x + c*x**2)*a*b**2 *e + 12*sqrt(a + b*x + c*x**2)*a*b*c*d - 12*sqrt(a + b*x + c*x**2)*a*b*c*e *x + 24*sqrt(a + b*x + c*x**2)*a*c**2*d*x - sqrt(a + b*x + c*x**2)*b**3*d - 3*sqrt(a + b*x + c*x**2)*b**3*e*x + 6*sqrt(a + b*x + c*x**2)*b**2*c*d*x - 12*sqrt(a + b*x + c*x**2)*b**2*c*e*x**2 + 24*sqrt(a + b*x + c*x**2)*b*c* *2*d*x**2 - 8*sqrt(a + b*x + c*x**2)*b*c**2*e*x**3 + 16*sqrt(a + b*x + c*x **2)*c**3*d*x**3 + 8*sqrt(c)*a**2*b*e - 16*sqrt(c)*a**2*c*d + 16*sqrt(c)*a *b**2*e*x - 32*sqrt(c)*a*b*c*d*x + 16*sqrt(c)*a*b*c*e*x**2 - 32*sqrt(c)*a* c**2*d*x**2 + 8*sqrt(c)*b**3*e*x**2 - 16*sqrt(c)*b**2*c*d*x**2 + 16*sqrt(c )*b**2*c*e*x**3 - 32*sqrt(c)*b*c**2*d*x**3 + 8*sqrt(c)*b*c**2*e*x**4 - 16* sqrt(c)*c**3*d*x**4))/(3*(16*a**4*c**2 - 8*a**3*b**2*c + 32*a**3*b*c**2*x + 32*a**3*c**3*x**2 + a**2*b**4 - 16*a**2*b**3*c*x + 32*a**2*b*c**3*x**3 + 16*a**2*c**4*x**4 + 2*a*b**5*x - 6*a*b**4*c*x**2 - 16*a*b**3*c**2*x**3 - 8*a*b**2*c**3*x**4 + b**6*x**2 + 2*b**5*c*x**3 + b**4*c**2*x**4))