Integrand size = 30, antiderivative size = 814 \[ \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) d \sqrt {a+b x+c x^2}}+\frac {2 \left (c e (2 a c e-b (c d+a f))+(b e-a f) \left (2 c^2 d+b^2 f-c (b e+2 a f)\right )+c \left (2 c^2 d e+b f (b e-a f)-b c \left (e^2+d f\right )\right ) x\right )}{\left (b^2-4 a c\right ) d \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {a+b x+c x^2}}-\frac {\text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{a^{3/2} d}-\frac {f \left (2 f \left (b e^2-b d f-a e f\right )-2 c \left (e^3-2 d e f\right )-\left (e-\sqrt {e^2-4 d f}\right ) \left (f (b e-a f)-c \left (e^2-d f\right )\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} d \sqrt {e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}}}+\frac {f \left (2 f \left (b e^2-b d f-a e f\right )-2 c \left (e^3-2 d e f\right )-\left (e+\sqrt {e^2-4 d f}\right ) \left (f (b e-a f)-c \left (e^2-d f\right )\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} d \sqrt {e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}}} \] Output:
2*(b*c*x-2*a*c+b^2)/a/(-4*a*c+b^2)/d/(c*x^2+b*x+a)^(1/2)+2*(c*e*(2*a*c*e-b *(a*f+c*d))+(-a*f+b*e)*(2*c^2*d+b^2*f-c*(2*a*f+b*e))+c*(2*c^2*d*e+b*f*(-a* f+b*e)-b*c*(d*f+e^2))*x)/(-4*a*c+b^2)/d/((-a*f+c*d)^2-(-a*e+b*d)*(-b*f+c*e ))/(c*x^2+b*x+a)^(1/2)-arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))/ a^(3/2)/d-1/2*f*(2*f*(-a*e*f-b*d*f+b*e^2)-2*c*(-2*d*e*f+e^3)-(e-(-4*d*f+e^ 2)^(1/2))*(f*(-a*f+b*e)-c*(-d*f+e^2)))*arctanh(1/4*(4*a*f-b*(e-(-4*d*f+e^2 )^(1/2))+2*(b*f-c*(e-(-4*d*f+e^2)^(1/2)))*x)*2^(1/2)/(c*e^2-2*c*d*f-b*e*f+ 2*a*f^2-(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2)/(c*x^2+b*x+a)^(1/2))*2^(1/2)/ d/(-4*d*f+e^2)^(1/2)/((-a*f+c*d)^2-(-a*e+b*d)*(-b*f+c*e))/(c*e^2-2*c*d*f-b *e*f+2*a*f^2-(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2)+1/2*f*(2*f*(-a*e*f-b*d*f +b*e^2)-2*c*(-2*d*e*f+e^3)-(e+(-4*d*f+e^2)^(1/2))*(f*(-a*f+b*e)-c*(-d*f+e^ 2)))*arctanh(1/4*(4*a*f-b*(e+(-4*d*f+e^2)^(1/2))+2*(b*f-c*(e+(-4*d*f+e^2)^ (1/2)))*x)*2^(1/2)/(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(-b*f+c*e)*(-4*d*f+e^2)^(1 /2))^(1/2)/(c*x^2+b*x+a)^(1/2))*2^(1/2)/d/(-4*d*f+e^2)^(1/2)/((-a*f+c*d)^2 -(-a*e+b*d)*(-b*f+c*e))/(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(-b*f+c*e)*(-4*d*f+e^ 2)^(1/2))^(1/2)
Time = 17.14 (sec) , antiderivative size = 1121, normalized size of antiderivative = 1.38 \[ \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx =\text {Too large to display} \] Input:
Integrate[1/(x*(a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]
Output:
(2*(b^2 - 2*a*c + b*c*x)*(a + b*x + c*x^2))/(a*(b^2 - 4*a*c)*d*(a + x*(b + c*x))^(3/2)) - (2*f*(1 + e/Sqrt[e^2 - 4*d*f])*(2*b^2*f - 4*a*c*f - b*c*(e - Sqrt[e^2 - 4*d*f]) + 2*c*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)*(a + b*x + c*x^2))/((b^2 - 4*a*c)*d*(4*a*f^2 - 2*b*f*(e - Sqrt[e^2 - 4*d*f]) + c*(e - Sqrt[e^2 - 4*d*f])^2)*(a + x*(b + c*x))^(3/2)) - (2*f*(1 - e/Sqrt[e^2 - 4*d*f])*(2*b^2*f - 4*a*c*f - b*c*(e + Sqrt[e^2 - 4*d*f]) + 2*c*(b*f - c*( e + Sqrt[e^2 - 4*d*f]))*x)*(a + b*x + c*x^2))/((b^2 - 4*a*c)*d*(4*a*f^2 - 2*b*f*(e + Sqrt[e^2 - 4*d*f]) + c*(e + Sqrt[e^2 - 4*d*f])^2)*(a + x*(b + c *x))^(3/2)) - ((a + b*x + c*x^2)^(3/2)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt [a + b*x + c*x^2])])/(a^(3/2)*d*(a + x*(b + c*x))^(3/2)) + (16*Sqrt[2]*f^2 *(f + (e*f)/Sqrt[e^2 - 4*d*f])*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - c* e*Sqrt[e^2 - 4*d*f] + b*f*Sqrt[e^2 - 4*d*f]]*(a + b*x + c*x^2)^(3/2)*ArcTa nh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) - (-2*b*f + 2*c*(e - Sqrt[e^2 - 4*d* f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - c*e*Sqrt[e^2 - 4*d*f] + b*f*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(d*(4*a*f^2 - 2* b*f*(e - Sqrt[e^2 - 4*d*f]) + c*(e - Sqrt[e^2 - 4*d*f])^2)*(16*a*f^2 - 8*b *f*(e - Sqrt[e^2 - 4*d*f]) + 4*c*(e - Sqrt[e^2 - 4*d*f])^2)*(a + x*(b + c* x))^(3/2)) - (16*Sqrt[2]*f^2*(-f + (e*f)/Sqrt[e^2 - 4*d*f])*Sqrt[c*e^2 - 2 *c*d*f - b*e*f + 2*a*f^2 + c*e*Sqrt[e^2 - 4*d*f] - b*f*Sqrt[e^2 - 4*d*f]]* (a + b*x + c*x^2)^(3/2)*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) - (-...
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.
| \(\displaystyle \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {-e-f x}{d \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}+\frac {1}{d x \left (a+b x+c x^2\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {-e-f x}{d \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}+\frac {1}{d x \left (a+b x+c x^2\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {-e-f x}{d \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}+\frac {1}{d x \left (a+b x+c x^2\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {-e-f x}{d \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}+\frac {1}{d x \left (a+b x+c x^2\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {-e-f x}{d \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}+\frac {1}{d x \left (a+b x+c x^2\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {-e-f x}{d \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}+\frac {1}{d x \left (a+b x+c x^2\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {-e-f x}{d \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}+\frac {1}{d x \left (a+b x+c x^2\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {-e-f x}{d \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}+\frac {1}{d x \left (a+b x+c x^2\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {-e-f x}{d \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}+\frac {1}{d x \left (a+b x+c x^2\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {-e-f x}{d \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}+\frac {1}{d x \left (a+b x+c x^2\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {-e-f x}{d \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}+\frac {1}{d x \left (a+b x+c x^2\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {-e-f x}{d \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}+\frac {1}{d x \left (a+b x+c x^2\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {-e-f x}{d \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}+\frac {1}{d x \left (a+b x+c x^2\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {-e-f x}{d \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}+\frac {1}{d x \left (a+b x+c x^2\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {-e-f x}{d \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}+\frac {1}{d x \left (a+b x+c x^2\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}dx\) |
Input:
Int[1/(x*(a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]
Output:
$Aborted
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(2058\) vs. \(2(755)=1510\).
Time = 2.84 (sec) , antiderivative size = 2059, normalized size of antiderivative = 2.53
Input:
int(1/x/(c*x^2+b*x+a)^(3/2)/(f*x^2+e*x+d),x,method=_RETURNVERBOSE)
Output:
-4*f/(-e+(-4*d*f+e^2)^(1/2))/(e+(-4*d*f+e^2)^(1/2))*(1/a/(c*x^2+b*x+a)^(1/ 2)-b/a*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-1/a^(3/2)*ln((2*a+b*x+2*a ^(1/2)*(c*x^2+b*x+a)^(1/2))/x))+2*f/(-e+(-4*d*f+e^2)^(1/2))/(-4*d*f+e^2)^( 1/2)*(2/(f*b*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*d*f *c+c*e^2)*f^2/(c*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))^2+(c*(-4*d*f+e^2)^(1/2) +f*b-c*e)/f*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+1/2*(f*b*(-4*d*f+e^2)^(1/2)- (-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*d*f*c+c*e^2)/f^2)^(1/2)-2*(c*(-4*d* f+e^2)^(1/2)+f*b-c*e)*f/(f*b*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c*e+2*a *f^2-b*e*f-2*d*f*c+c*e^2)*(2*c*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+(c*(-4*d* f+e^2)^(1/2)+f*b-c*e)/f)/(2*c*(f*b*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c *e+2*a*f^2-b*e*f-2*d*f*c+c*e^2)/f^2-(c*(-4*d*f+e^2)^(1/2)+f*b-c*e)^2/f^2)/ (c*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))^2+(c*(-4*d*f+e^2)^(1/2)+f*b-c*e)/f*(x -1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+1/2*(f*b*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^( 1/2)*c*e+2*a*f^2-b*e*f-2*d*f*c+c*e^2)/f^2)^(1/2)-2/(f*b*(-4*d*f+e^2)^(1/2) -(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*d*f*c+c*e^2)*f^2*2^(1/2)/((f*b*(-4 *d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*d*f*c+c*e^2)/f^2)^( 1/2)*ln(((f*b*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*d* f*c+c*e^2)/f^2+(c*(-4*d*f+e^2)^(1/2)+f*b-c*e)/f*(x-1/2/f*(-e+(-4*d*f+e^2)^ (1/2)))+1/2*2^(1/2)*((f*b*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^ 2-b*e*f-2*d*f*c+c*e^2)/f^2)^(1/2)*(4*c*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2))...
Timed out. \[ \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate(1/x/(c*x^2+b*x+a)^(3/2)/(f*x^2+e*x+d),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\int \frac {1}{x \left (a + b x + c x^{2}\right )^{\frac {3}{2}} \left (d + e x + f x^{2}\right )}\, dx \] Input:
integrate(1/x/(c*x**2+b*x+a)**(3/2)/(f*x**2+e*x+d),x)
Output:
Integral(1/(x*(a + b*x + c*x**2)**(3/2)*(d + e*x + f*x**2)), x)
\[ \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e x + d\right )} x} \,d x } \] Input:
integrate(1/x/(c*x^2+b*x+a)^(3/2)/(f*x^2+e*x+d),x, algorithm="maxima")
Output:
integrate(1/((c*x^2 + b*x + a)^(3/2)*(f*x^2 + e*x + d)*x), x)
Exception generated. \[ \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/x/(c*x^2+b*x+a)^(3/2)/(f*x^2+e*x+d),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Degree mismatch inside factorisatio n over extensionNot implemented, e.g. for multivariate mod/approx polynomi alsError:
Timed out. \[ \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\int \frac {1}{x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}\,\left (f\,x^2+e\,x+d\right )} \,d x \] Input:
int(1/(x*(a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2)),x)
Output:
int(1/(x*(a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2)), x)
\[ \int \frac {1}{x \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\int \frac {1}{x \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (f \,x^{2}+e x +d \right )}d x \] Input:
int(1/x/(c*x^2+b*x+a)^(3/2)/(f*x^2+e*x+d),x)
Output:
int(1/x/(c*x^2+b*x+a)^(3/2)/(f*x^2+e*x+d),x)