Integrand size = 25, antiderivative size = 132 \[ \int \frac {d+e x+f x^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \left (2 a c e-b (c d+a f)-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (8 c^2 d-4 b c e+b^2 f+4 a c f\right ) (b+2 c x)}{3 c \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}} \] Output:
2/3*(2*a*c*e-b*(a*f+c*d)-(-2*a*c*f+b^2*f-b*c*e+2*c^2*d)*x)/c/(-4*a*c+b^2)/ (c*x^2+b*x+a)^(3/2)+2/3*(4*a*c*f+b^2*f-4*b*c*e+8*c^2*d)*(2*c*x+b)/c/(-4*a* c+b^2)^2/(c*x^2+b*x+a)^(1/2)
Time = 1.48 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.11 \[ \int \frac {d+e x+f x^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {-2 b^3 (d+3 x (e-f x))+16 c \left (-a^2 e+2 c^2 d x^3+a c x \left (3 d+f x^2\right )\right )-4 b^2 \left (a (e-6 f x)-c x \left (3 d-6 e x+f x^2\right )\right )+8 b \left (2 a^2 f-2 c^2 x^2 (-3 d+e x)+3 a c \left (d-e x+f x^2\right )\right )}{3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \] Input:
Integrate[(d + e*x + f*x^2)/(a + b*x + c*x^2)^(5/2),x]
Output:
(-2*b^3*(d + 3*x*(e - f*x)) + 16*c*(-(a^2*e) + 2*c^2*d*x^3 + a*c*x*(3*d + f*x^2)) - 4*b^2*(a*(e - 6*f*x) - c*x*(3*d - 6*e*x + f*x^2)) + 8*b*(2*a^2*f - 2*c^2*x^2*(-3*d + e*x) + 3*a*c*(d - e*x + f*x^2)))/(3*(b^2 - 4*a*c)^2*( a + x*(b + c*x))^(3/2))
Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2191, 27, 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.
| \(\displaystyle \int \frac {d+e x+f x^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2191 |
| \(\displaystyle \frac {2 \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {\frac {f b^2}{c}-4 e b+8 c d+4 a f}{2 \left (c x^2+b x+a\right )^{3/2}}dx}{3 \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {\left (4 a f+\frac {b^2 f}{c}-4 b e+8 c d\right ) \int \frac {1}{\left (c x^2+b x+a\right )^{3/2}}dx}{3 \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 1088 |
| \(\displaystyle \frac {2 \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 (b+2 c x) \left (4 a f+\frac {b^2 f}{c}-4 b e+8 c d\right )}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}\) |
Input:
Int[(d + e*x + f*x^2)/(a + b*x + c*x^2)^(5/2),x]
Output:
(2*(c*(2*a*e - b*(d + (a*f)/c)) - (2*c^2*d - b*c*e + b^2*f - 2*a*c*f)*x))/ (3*c*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(3/2)) + (2*(8*c*d - 4*b*e + 4*a*f + (b^2*f)/c)*(b + 2*c*x))/(3*(b^2 - 4*a*c)^2*Sqrt[a + b*x + c*x^2])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[P q, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*c*f - b*g)*x)*((a + b*x + c*x^2)^ (p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int [(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (2*p + 3)* (2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^ 2 - 4*a*c, 0] && LtQ[p, -1]
Time = 1.61 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.33
| method | result | size |
| trager | \(\frac {\frac {16}{3} a \,c^{2} f \,x^{3}+\frac {4}{3} b^{2} c f \,x^{3}-\frac {16}{3} b \,c^{2} x^{3} e +\frac {32}{3} c^{3} d \,x^{3}+8 a b c f \,x^{2}+2 b^{3} f \,x^{2}-8 b^{2} c e \,x^{2}+16 b \,c^{2} d \,x^{2}+8 a \,b^{2} f x -8 a b c e x +16 a d x \,c^{2}-2 b^{3} e x +4 b^{2} c x d +\frac {16}{3} f \,a^{2} b -\frac {16}{3} a^{2} c e -\frac {4}{3} e a \,b^{2}+8 a b c d -\frac {2}{3} b^{3} d}{\left (4 a c -b^{2}\right )^{2} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\) | \(176\) |
| gosper | \(\frac {\frac {16}{3} a \,c^{2} f \,x^{3}+\frac {4}{3} b^{2} c f \,x^{3}-\frac {16}{3} b \,c^{2} x^{3} e +\frac {32}{3} c^{3} d \,x^{3}+8 a b c f \,x^{2}+2 b^{3} f \,x^{2}-8 b^{2} c e \,x^{2}+16 b \,c^{2} d \,x^{2}+8 a \,b^{2} f x -8 a b c e x +16 a d x \,c^{2}-2 b^{3} e x +4 b^{2} c x d +\frac {16}{3} f \,a^{2} b -\frac {16}{3} a^{2} c e -\frac {4}{3} e a \,b^{2}+8 a b c d -\frac {2}{3} b^{3} d}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}\) | \(185\) |
| orering | \(\frac {\frac {16}{3} a \,c^{2} f \,x^{3}+\frac {4}{3} b^{2} c f \,x^{3}-\frac {16}{3} b \,c^{2} x^{3} e +\frac {32}{3} c^{3} d \,x^{3}+8 a b c f \,x^{2}+2 b^{3} f \,x^{2}-8 b^{2} c e \,x^{2}+16 b \,c^{2} d \,x^{2}+8 a \,b^{2} f x -8 a b c e x +16 a d x \,c^{2}-2 b^{3} e x +4 b^{2} c x d +\frac {16}{3} f \,a^{2} b -\frac {16}{3} a^{2} c e -\frac {4}{3} e a \,b^{2}+8 a b c d -\frac {2}{3} b^{3} d}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}\) | \(185\) |
| default | \(d \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )+e \left (-\frac {1}{3 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )+f \left (-\frac {x}{2 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )}{4 c}+\frac {a \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )\) | \(351\) |
Input:
int((f*x^2+e*x+d)/(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/3*(8*a*c^2*f*x^3+2*b^2*c*f*x^3-8*b*c^2*e*x^3+16*c^3*d*x^3+12*a*b*c*f*x^2 +3*b^3*f*x^2-12*b^2*c*e*x^2+24*b*c^2*d*x^2+12*a*b^2*f*x-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*b*f-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. 286 vs. \(2 (124) = 248\).
Time = 1.28 (sec) , antiderivative size = 286, normalized size of antiderivative = 2.17 \[ \int \frac {d+e x+f x^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (8 \, a^{2} b f + 2 \, {\left (8 \, c^{3} d - 4 \, b c^{2} e + {\left (b^{2} c + 4 \, a c^{2}\right )} f\right )} x^{3} + 3 \, {\left (8 \, b c^{2} d - 4 \, b^{2} c e + {\left (b^{3} + 4 \, a b c\right )} f\right )} x^{2} - {\left (b^{3} - 12 \, a b c\right )} d - 2 \, {\left (a b^{2} + 4 \, a^{2} c\right )} e + 3 \, {\left (4 \, a b^{2} f + 2 \, {\left (b^{2} c + 4 \, a c^{2}\right )} d - {\left (b^{3} + 4 \, a b c\right )} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )}} \] Input:
integrate((f*x^2+e*x+d)/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
Output:
2/3*(8*a^2*b*f + 2*(8*c^3*d - 4*b*c^2*e + (b^2*c + 4*a*c^2)*f)*x^3 + 3*(8* b*c^2*d - 4*b^2*c*e + (b^3 + 4*a*b*c)*f)*x^2 - (b^3 - 12*a*b*c)*d - 2*(a*b ^2 + 4*a^2*c)*e + 3*(4*a*b^2*f + 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)
\[ \int \frac {d+e x+f x^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {d + e x + f x^{2}}{\left (a + b x + c x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((f*x**2+e*x+d)/(c*x**2+b*x+a)**(5/2),x)
Output:
Integral((d + e*x + f*x**2)/(a + b*x + c*x**2)**(5/2), x)
Exception generated. \[ \int \frac {d+e x+f x^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((f*x^2+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
Time = 0.15 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.77 \[ \int \frac {d+e x+f x^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left ({\left (\frac {2 \, {\left (8 \, c^{3} d - 4 \, b c^{2} e + b^{2} c f + 4 \, a c^{2} f\right )} x}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} + \frac {3 \, {\left (8 \, b c^{2} d - 4 \, b^{2} c e + b^{3} f + 4 \, a b c f\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {3 \, {\left (2 \, b^{2} c d + 8 \, a c^{2} d - b^{3} e - 4 \, a b c e + 4 \, a b^{2} f\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x - \frac {b^{3} d - 12 \, a b c d + 2 \, a b^{2} e + 8 \, a^{2} c e - 8 \, a^{2} b f}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \] Input:
integrate((f*x^2+e*x+d)/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
Output:
2/3*(((2*(8*c^3*d - 4*b*c^2*e + b^2*c*f + 4*a*c^2*f)*x/(b^4 - 8*a*b^2*c + 16*a^2*c^2) + 3*(8*b*c^2*d - 4*b^2*c*e + b^3*f + 4*a*b*c*f)/(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 + 4*a*b ^2*f)/(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 - 8*a^2*b*f)/(b^4 - 8*a*b^2*c + 16*a^2*c^2))/(c*x^2 + b*x + a) ^(3/2)
Time = 15.42 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.33 \[ \int \frac {d+e x+f x^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2\,\left (8\,f\,a^2\,b-8\,e\,a^2\,c+12\,f\,a\,b^2\,x-2\,e\,a\,b^2+12\,f\,a\,b\,c\,x^2-12\,e\,a\,b\,c\,x+12\,d\,a\,b\,c+8\,f\,a\,c^2\,x^3+24\,d\,a\,c^2\,x+3\,f\,b^3\,x^2-3\,e\,b^3\,x-d\,b^3+2\,f\,b^2\,c\,x^3-12\,e\,b^2\,c\,x^2+6\,d\,b^2\,c\,x-8\,e\,b\,c^2\,x^3+24\,d\,b\,c^2\,x^2+16\,d\,c^3\,x^3\right )}{3\,{\left (4\,a\,c-b^2\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \] Input:
int((d + e*x + f*x^2)/(a + b*x + c*x^2)^(5/2),x)
Output:
(2*(16*c^3*d*x^3 - b^3*d + 3*b^3*f*x^2 - 2*a*b^2*e + 8*a^2*b*f - 8*a^2*c*e - 3*b^3*e*x + 24*a*c^2*d*x + 12*a*b^2*f*x + 6*b^2*c*d*x + 24*b*c^2*d*x^2 - 12*b^2*c*e*x^2 + 8*a*c^2*f*x^3 - 8*b*c^2*e*x^3 + 2*b^2*c*f*x^3 + 12*a*b* c*d - 12*a*b*c*e*x + 12*a*b*c*f*x^2))/(3*(4*a*c - b^2)^2*(a + b*x + c*x^2) ^(3/2))
Time = 0.20 (sec) , antiderivative size = 793, normalized size of antiderivative = 6.01 \[ \int \frac {d+e x+f x^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
int((f*x^2+e*x+d)/(c*x^2+b*x+a)^(5/2),x)
Output:
(2*(8*sqrt(a + b*x + c*x**2)*a**2*b*c*f - 8*sqrt(a + b*x + c*x**2)*a**2*c* *2*e - 2*sqrt(a + b*x + c*x**2)*a*b**2*c*e + 12*sqrt(a + b*x + c*x**2)*a*b **2*c*f*x + 12*sqrt(a + b*x + c*x**2)*a*b*c**2*d - 12*sqrt(a + b*x + c*x** 2)*a*b*c**2*e*x + 12*sqrt(a + b*x + c*x**2)*a*b*c**2*f*x**2 + 24*sqrt(a + b*x + c*x**2)*a*c**3*d*x + 8*sqrt(a + b*x + c*x**2)*a*c**3*f*x**3 - sqrt(a + b*x + c*x**2)*b**3*c*d - 3*sqrt(a + b*x + c*x**2)*b**3*c*e*x + 3*sqrt(a + b*x + c*x**2)*b**3*c*f*x**2 + 6*sqrt(a + b*x + c*x**2)*b**2*c**2*d*x - 12*sqrt(a + b*x + c*x**2)*b**2*c**2*e*x**2 + 2*sqrt(a + b*x + c*x**2)*b**2 *c**2*f*x**3 + 24*sqrt(a + b*x + c*x**2)*b*c**3*d*x**2 - 8*sqrt(a + b*x + c*x**2)*b*c**3*e*x**3 + 16*sqrt(a + b*x + c*x**2)*c**4*d*x**3 + 8*sqrt(c)* a**3*c*f - 6*sqrt(c)*a**2*b**2*f + 8*sqrt(c)*a**2*b*c*e + 16*sqrt(c)*a**2* b*c*f*x - 16*sqrt(c)*a**2*c**2*d + 16*sqrt(c)*a**2*c**2*f*x**2 - 12*sqrt(c )*a*b**3*f*x + 16*sqrt(c)*a*b**2*c*e*x - 4*sqrt(c)*a*b**2*c*f*x**2 - 32*sq rt(c)*a*b*c**2*d*x + 16*sqrt(c)*a*b*c**2*e*x**2 + 16*sqrt(c)*a*b*c**2*f*x* *3 - 32*sqrt(c)*a*c**3*d*x**2 + 8*sqrt(c)*a*c**3*f*x**4 - 6*sqrt(c)*b**4*f *x**2 + 8*sqrt(c)*b**3*c*e*x**2 - 12*sqrt(c)*b**3*c*f*x**3 - 16*sqrt(c)*b* *2*c**2*d*x**2 + 16*sqrt(c)*b**2*c**2*e*x**3 - 6*sqrt(c)*b**2*c**2*f*x**4 - 32*sqrt(c)*b*c**3*d*x**3 + 8*sqrt(c)*b*c**3*e*x**4 - 16*sqrt(c)*c**4*d*x **4))/(3*c*(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*...