Integrand size = 39, antiderivative size = 376 \[ \int \frac {A+B x+C x^2}{(d+e x)^{5/2} (f+g x) \left (2+5 x+3 x^2\right )} \, dx=-\frac {2 \left (C d^2-B d e+A e^2\right )}{3 \left (3 d^2-5 d e+2 e^2\right ) (e f-d g) (d+e x)^{3/2}}+\frac {2 \left (C d \left (4 e^3 f+3 d^3 g-d e^2 (5 f+2 g)\right )+A e^2 \left (9 d^2 g+e^2 (5 f+2 g)-2 d e (3 f+5 g)\right )-B e \left (2 e^3 f+6 d^3 g-d^2 e (3 f+5 g)\right )\right )}{\left (3 d^2-5 d e+2 e^2\right )^2 (e f-d g)^2 \sqrt {d+e x}}+\frac {2 g^{3/2} \left (C f^2-B f g+A g^2\right ) \arctan \left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e f-d g}}\right )}{(3 f-2 g) (f-g) (e f-d g)^{5/2}}-\frac {6 \sqrt {3} (9 A-6 B+4 C) \text {arctanh}\left (\frac {\sqrt {3} \sqrt {d+e x}}{\sqrt {3 d-2 e}}\right )}{(3 d-2 e)^{5/2} (3 f-2 g)}+\frac {2 (A-B+C) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d-e}}\right )}{(d-e)^{5/2} (f-g)} \] Output:
1/3*(-2*A*e^2+2*B*d*e-2*C*d^2)/(3*d^2-5*d*e+2*e^2)/(-d*g+e*f)/(e*x+d)^(3/2 )+2*(C*d*(4*e^3*f+3*d^3*g-d*e^2*(5*f+2*g))+A*e^2*(9*d^2*g+e^2*(5*f+2*g)-2* d*e*(3*f+5*g))-B*e*(2*e^3*f+6*d^3*g-d^2*e*(3*f+5*g)))/(3*d^2-5*d*e+2*e^2)^ 2/(-d*g+e*f)^2/(e*x+d)^(1/2)+2*g^(3/2)*(A*g^2-B*f*g+C*f^2)*arctan(g^(1/2)* (e*x+d)^(1/2)/(-d*g+e*f)^(1/2))/(3*f-2*g)/(f-g)/(-d*g+e*f)^(5/2)-6*3^(1/2) *(9*A-6*B+4*C)*arctanh(3^(1/2)*(e*x+d)^(1/2)/(3*d-2*e)^(1/2))/(3*d-2*e)^(5 /2)/(3*f-2*g)+2*(A-B+C)*arctanh((e*x+d)^(1/2)/(d-e)^(1/2))/(d-e)^(5/2)/(f- g)
Time = 7.93 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.13 \[ \int \frac {A+B x+C x^2}{(d+e x)^{5/2} (f+g x) \left (2+5 x+3 x^2\right )} \, dx=\frac {2 \left (C d \left (12 d^4 g-2 d^2 e^2 (5 f+2 g)+12 e^4 f x+d e^3 (10 f-15 f x-6 g x)+d^3 e (-3 f-5 g+9 g x)\right )+e \left (-B \left (4 d e^3 f+21 d^4 g+6 e^4 f x+d^2 e^2 (5 f+2 g-9 f x-15 g x)-2 d^3 e (6 f+10 g-9 g x)\right )+A e \left (30 d^3 g+2 d e^2 (10 f+4 g-9 f x-15 g x)+e^3 (-2 f+15 f x+6 g x)+d^2 e (-21 f-35 g+27 g x)\right )\right )\right )}{3 (3 d-2 e)^2 (d-e)^2 (e f-d g)^2 (d+e x)^{3/2}}-\frac {2 (A-B+C) \arctan \left (\frac {\sqrt {d+e x}}{\sqrt {-d+e}}\right )}{(-d+e)^{5/2} (f-g)}-\frac {6 \sqrt {3} (9 A-6 B+4 C) \arctan \left (\frac {\sqrt {-9 d+6 e} \sqrt {d+e x}}{3 d-2 e}\right )}{(-3 d+2 e)^{5/2} (3 f-2 g)}+\frac {2 g^{3/2} \left (C f^2+g (-B f+A g)\right ) \arctan \left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e f-d g}}\right )}{(3 f-2 g) (f-g) (e f-d g)^{5/2}} \] Input:
Integrate[(A + B*x + C*x^2)/((d + e*x)^(5/2)*(f + g*x)*(2 + 5*x + 3*x^2)), x]
Output:
(2*(C*d*(12*d^4*g - 2*d^2*e^2*(5*f + 2*g) + 12*e^4*f*x + d*e^3*(10*f - 15* f*x - 6*g*x) + d^3*e*(-3*f - 5*g + 9*g*x)) + e*(-(B*(4*d*e^3*f + 21*d^4*g + 6*e^4*f*x + d^2*e^2*(5*f + 2*g - 9*f*x - 15*g*x) - 2*d^3*e*(6*f + 10*g - 9*g*x))) + A*e*(30*d^3*g + 2*d*e^2*(10*f + 4*g - 9*f*x - 15*g*x) + e^3*(- 2*f + 15*f*x + 6*g*x) + d^2*e*(-21*f - 35*g + 27*g*x)))))/(3*(3*d - 2*e)^2 *(d - e)^2*(e*f - d*g)^2*(d + e*x)^(3/2)) - (2*(A - B + C)*ArcTan[Sqrt[d + e*x]/Sqrt[-d + e]])/((-d + e)^(5/2)*(f - g)) - (6*Sqrt[3]*(9*A - 6*B + 4* C)*ArcTan[(Sqrt[-9*d + 6*e]*Sqrt[d + e*x])/(3*d - 2*e)])/((-3*d + 2*e)^(5/ 2)*(3*f - 2*g)) + (2*g^(3/2)*(C*f^2 + g*(-(B*f) + A*g))*ArcTan[(Sqrt[g]*Sq rt[d + e*x])/Sqrt[e*f - d*g]])/((3*f - 2*g)*(f - g)*(e*f - d*g)^(5/2))
Time = 1.36 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.17, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {2153, 2009}
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.
| \(\displaystyle \int \frac {A+B x+C x^2}{\left (3 x^2+5 x+2\right ) (d+e x)^{5/2} (f+g x)} \, dx\) |
| \(\Big \downarrow \) 2153 |
| \(\displaystyle \int \left (\frac {A g^2-B f g+C f^2}{(3 f-2 g) (f-g) (d+e x)^{5/2} (f+g x)}+\frac {-A+B-C}{(x+1) (f-g) (d+e x)^{5/2}}+\frac {9 A-6 B+4 C}{(3 x+2) (3 f-2 g) (d+e x)^{5/2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 g^{3/2} \left (A g^2-B f g+C f^2\right ) \arctan \left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e f-d g}}\right )}{(3 f-2 g) (f-g) (e f-d g)^{5/2}}-\frac {6 \sqrt {3} (9 A-6 B+4 C) \text {arctanh}\left (\frac {\sqrt {3} \sqrt {d+e x}}{\sqrt {3 d-2 e}}\right )}{(3 d-2 e)^{5/2} (3 f-2 g)}+\frac {2 (A-B+C) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d-e}}\right )}{(d-e)^{5/2} (f-g)}+\frac {2 g \left (A g^2-B f g+C f^2\right )}{(3 f-2 g) (f-g) \sqrt {d+e x} (e f-d g)^2}-\frac {2 \left (A g^2-B f g+C f^2\right )}{3 (3 f-2 g) (f-g) (d+e x)^{3/2} (e f-d g)}+\frac {6 (9 A-6 B+4 C)}{(3 d-2 e)^2 (3 f-2 g) \sqrt {d+e x}}-\frac {2 (A-B+C)}{(d-e)^2 (f-g) \sqrt {d+e x}}+\frac {2 (9 A-6 B+4 C)}{3 (3 d-2 e) (3 f-2 g) (d+e x)^{3/2}}-\frac {2 (A-B+C)}{3 (d-e) (f-g) (d+e x)^{3/2}}\) |
Input:
Int[(A + B*x + C*x^2)/((d + e*x)^(5/2)*(f + g*x)*(2 + 5*x + 3*x^2)),x]
Output:
(2*(9*A - 6*B + 4*C))/(3*(3*d - 2*e)*(3*f - 2*g)*(d + e*x)^(3/2)) - (2*(A - B + C))/(3*(d - e)*(f - g)*(d + e*x)^(3/2)) - (2*(C*f^2 - B*f*g + A*g^2) )/(3*(3*f - 2*g)*(f - g)*(e*f - d*g)*(d + e*x)^(3/2)) + (6*(9*A - 6*B + 4* C))/((3*d - 2*e)^2*(3*f - 2*g)*Sqrt[d + e*x]) - (2*(A - B + C))/((d - e)^2 *(f - g)*Sqrt[d + e*x]) + (2*g*(C*f^2 - B*f*g + A*g^2))/((3*f - 2*g)*(f - g)*(e*f - d*g)^2*Sqrt[d + e*x]) + (2*g^(3/2)*(C*f^2 - B*f*g + A*g^2)*ArcTa n[(Sqrt[g]*Sqrt[d + e*x])/Sqrt[e*f - d*g]])/((3*f - 2*g)*(f - g)*(e*f - d* g)^(5/2)) - (6*Sqrt[3]*(9*A - 6*B + 4*C)*ArcTanh[(Sqrt[3]*Sqrt[d + e*x])/S qrt[3*d - 2*e]])/((3*d - 2*e)^(5/2)*(3*f - 2*g)) + (2*(A - B + C)*ArcTanh[ Sqrt[d + e*x]/Sqrt[d - e]])/((d - e)^(5/2)*(f - g))
Int[(Px_)*((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b _.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(d + e* x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && PolyQ[Px, x] && (IntegerQ[p] || (IntegerQ[2*p] && IntegerQ [m] && ILtQ[n, 0])) && !(IGtQ[m, 0] && IGtQ[n, 0])
Time = 3.42 (sec) , antiderivative size = 338, normalized size of antiderivative = 0.90
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (A -B +C \right ) \arctan \left (\frac {\sqrt {e x +d}}{\sqrt {-d +e}}\right )}{\left (f -g \right ) \left (-d +e \right )^{\frac {5}{2}}}-\frac {2 g^{2} \left (A \,g^{2}-B f g +C \,f^{2}\right ) \operatorname {arctanh}\left (\frac {g \sqrt {e x +d}}{\sqrt {\left (d g -e f \right ) g}}\right )}{\left (f -g \right ) \left (3 f -2 g \right ) \left (d g -e f \right )^{2} \sqrt {\left (d g -e f \right ) g}}+\frac {\frac {2}{9} A \,e^{2}-\frac {2}{9} B d e +\frac {2}{9} C \,d^{2}}{\left (e x +d \right )^{\frac {3}{2}} \left (d -e \right ) \left (d g -e f \right ) \left (d -\frac {2 e}{3}\right )}+\frac {\frac {2 C \,d^{4} g}{3}-\frac {4 B \,d^{3} e g}{3}+2 d^{2} \left (\left (A +\frac {5 B}{9}-\frac {2 C}{9}\right ) g +\frac {\left (B -\frac {5 C}{3}\right ) f}{3}\right ) e^{2}-\frac {4 d \left (\frac {5 A g}{3}+f \left (A -\frac {2 C}{3}\right )\right ) e^{3}}{3}+\frac {10 e^{4} \left (\frac {2 A g}{5}+f \left (A -\frac {2 B}{5}\right )\right )}{9}}{\sqrt {e x +d}\, \left (d -e \right )^{2} \left (d g -e f \right )^{2} \left (d -\frac {2 e}{3}\right )^{2}}+\frac {\left (162 A -108 B +72 C \right ) \arctan \left (\frac {3 \sqrt {e x +d}}{\sqrt {-9 d +6 e}}\right )}{\sqrt {-9 d +6 e}\, \left (3 f -2 g \right ) \left (3 d -2 e \right )^{2}}\) | \(338\) |
| derivativedivides | \(\frac {2 \left (-A +B -C \right ) \arctan \left (\frac {\sqrt {e x +d}}{\sqrt {-d +e}}\right )}{\left (f -g \right ) \left (d -e \right )^{2} \sqrt {-d +e}}-\frac {2 g^{2} \left (A \,g^{2}-B f g +C \,f^{2}\right ) \operatorname {arctanh}\left (\frac {g \sqrt {e x +d}}{\sqrt {\left (d g -e f \right ) g}}\right )}{\left (f -g \right ) \left (3 f -2 g \right ) \left (d g -e f \right )^{2} \sqrt {\left (d g -e f \right ) g}}-\frac {2 \left (-A \,e^{2}+B d e -C \,d^{2}\right )}{3 \left (d g -e f \right ) \left (3 d -2 e \right ) \left (d -e \right ) \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (-9 A \,d^{2} e^{2} g +6 A d \,e^{3} f +10 A d \,e^{3} g -5 A \,e^{4} f -2 A \,e^{4} g +6 B \,d^{3} e g -3 B \,d^{2} e^{2} f -5 B \,d^{2} e^{2} g +2 B \,e^{4} f -3 C \,d^{4} g +5 C \,d^{2} e^{2} f +2 C \,d^{2} e^{2} g -4 C d \,e^{3} f \right )}{\left (d g -e f \right )^{2} \left (3 d -2 e \right )^{2} \left (d -e \right )^{2} \sqrt {e x +d}}+\frac {2 \left (81 A -54 B +36 C \right ) \arctan \left (\frac {3 \sqrt {e x +d}}{\sqrt {-9 d +6 e}}\right )}{\left (3 f -2 g \right ) \left (3 d -2 e \right )^{2} \sqrt {-9 d +6 e}}\) | \(386\) |
| default | \(\frac {2 \left (-A +B -C \right ) \arctan \left (\frac {\sqrt {e x +d}}{\sqrt {-d +e}}\right )}{\left (f -g \right ) \left (d -e \right )^{2} \sqrt {-d +e}}-\frac {2 g^{2} \left (A \,g^{2}-B f g +C \,f^{2}\right ) \operatorname {arctanh}\left (\frac {g \sqrt {e x +d}}{\sqrt {\left (d g -e f \right ) g}}\right )}{\left (f -g \right ) \left (3 f -2 g \right ) \left (d g -e f \right )^{2} \sqrt {\left (d g -e f \right ) g}}-\frac {2 \left (-A \,e^{2}+B d e -C \,d^{2}\right )}{3 \left (d g -e f \right ) \left (3 d -2 e \right ) \left (d -e \right ) \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (-9 A \,d^{2} e^{2} g +6 A d \,e^{3} f +10 A d \,e^{3} g -5 A \,e^{4} f -2 A \,e^{4} g +6 B \,d^{3} e g -3 B \,d^{2} e^{2} f -5 B \,d^{2} e^{2} g +2 B \,e^{4} f -3 C \,d^{4} g +5 C \,d^{2} e^{2} f +2 C \,d^{2} e^{2} g -4 C d \,e^{3} f \right )}{\left (d g -e f \right )^{2} \left (3 d -2 e \right )^{2} \left (d -e \right )^{2} \sqrt {e x +d}}+\frac {2 \left (81 A -54 B +36 C \right ) \arctan \left (\frac {3 \sqrt {e x +d}}{\sqrt {-9 d +6 e}}\right )}{\left (3 f -2 g \right ) \left (3 d -2 e \right )^{2} \sqrt {-9 d +6 e}}\) | \(386\) |
Input:
int((C*x^2+B*x+A)/(e*x+d)^(5/2)/(g*x+f)/(3*x^2+5*x+2),x,method=_RETURNVERB OSE)
Output:
-2*(A-B+C)*arctan((e*x+d)^(1/2)/(-d+e)^(1/2))/(f-g)/(-d+e)^(5/2)-2*g^2*(A* g^2-B*f*g+C*f^2)/(f-g)/(3*f-2*g)/(d*g-e*f)^2/((d*g-e*f)*g)^(1/2)*arctanh(g *(e*x+d)^(1/2)/((d*g-e*f)*g)^(1/2))+2/9*(A*e^2-B*d*e+C*d^2)/(e*x+d)^(3/2)/ (d-e)/(d*g-e*f)/(d-2/3*e)+2/9*(3*C*d^4*g-6*B*d^3*e*g+9*d^2*((A+5/9*B-2/9*C )*g+1/3*(B-5/3*C)*f)*e^2-6*d*(5/3*A*g+f*(A-2/3*C))*e^3+5*e^4*(2/5*A*g+f*(A -2/5*B)))/(e*x+d)^(1/2)/(d-e)^2/(d*g-e*f)^2/(d-2/3*e)^2+(162*A-108*B+72*C) /(-9*d+6*e)^(1/2)*arctan(3*(e*x+d)^(1/2)/(-9*d+6*e)^(1/2))/(3*f-2*g)/(3*d- 2*e)^2
Timed out. \[ \int \frac {A+B x+C x^2}{(d+e x)^{5/2} (f+g x) \left (2+5 x+3 x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((C*x^2+B*x+A)/(e*x+d)^(5/2)/(g*x+f)/(3*x^2+5*x+2),x, algorithm=" fricas")
Output:
Timed out
Time = 61.07 (sec) , antiderivative size = 471, normalized size of antiderivative = 1.25 \[ \int \frac {A+B x+C x^2}{(d+e x)^{5/2} (f+g x) \left (2+5 x+3 x^2\right )} \, dx=\begin {cases} \frac {2 \left (\frac {e g \left (A g^{2} - B f g + C f^{2}\right ) \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- \frac {d g - e f}{g}}} \right )}}{\sqrt {- \frac {d g - e f}{g}} \left (f - g\right ) \left (3 f - 2 g\right ) \left (d g - e f\right )^{2}} + \frac {e \left (A e^{2} - B d e + C d^{2}\right )}{3 \left (d - e\right ) \left (d + e x\right )^{\frac {3}{2}} \cdot \left (3 d - 2 e\right ) \left (d g - e f\right )} + \frac {e \left (9 A d^{2} e^{2} g - 6 A d e^{3} f - 10 A d e^{3} g + 5 A e^{4} f + 2 A e^{4} g - 6 B d^{3} e g + 3 B d^{2} e^{2} f + 5 B d^{2} e^{2} g - 2 B e^{4} f + 3 C d^{4} g - 5 C d^{2} e^{2} f - 2 C d^{2} e^{2} g + 4 C d e^{3} f\right )}{\left (d - e\right )^{2} \sqrt {d + e x} \left (3 d - 2 e\right )^{2} \left (d g - e f\right )^{2}} - \frac {e \left (A - B + C\right ) \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d + e}} \right )}}{\sqrt {- d + e} \left (d - e\right )^{2} \left (f - g\right )} + \frac {3 e \left (9 A - 6 B + 4 C\right ) \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d + \frac {2 e}{3}}} \right )}}{\sqrt {- d + \frac {2 e}{3}} \left (3 d - 2 e\right )^{2} \cdot \left (3 f - 2 g\right )}\right )}{e} & \text {for}\: e \neq 0 \\\frac {\frac {\left (9 A - 6 B + 4 C\right ) \log {\left (3 x + 2 \right )}}{3 \cdot \left (3 f - 2 g\right )} - \frac {\left (A - B + C\right ) \log {\left (x + 1 \right )}}{f - g} + \frac {\left (A g^{2} - B f g + C f^{2}\right ) \left (\begin {cases} \frac {x}{f} & \text {for}\: g = 0 \\\frac {\log {\left (f + g x \right )}}{g} & \text {otherwise} \end {cases}\right )}{\left (f - g\right ) \left (3 f - 2 g\right )}}{d^{\frac {5}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate((C*x**2+B*x+A)/(e*x+d)**(5/2)/(g*x+f)/(3*x**2+5*x+2),x)
Output:
Piecewise((2*(e*g*(A*g**2 - B*f*g + C*f**2)*atan(sqrt(d + e*x)/sqrt(-(d*g - e*f)/g))/(sqrt(-(d*g - e*f)/g)*(f - g)*(3*f - 2*g)*(d*g - e*f)**2) + e*( A*e**2 - B*d*e + C*d**2)/(3*(d - e)*(d + e*x)**(3/2)*(3*d - 2*e)*(d*g - e* f)) + e*(9*A*d**2*e**2*g - 6*A*d*e**3*f - 10*A*d*e**3*g + 5*A*e**4*f + 2*A *e**4*g - 6*B*d**3*e*g + 3*B*d**2*e**2*f + 5*B*d**2*e**2*g - 2*B*e**4*f + 3*C*d**4*g - 5*C*d**2*e**2*f - 2*C*d**2*e**2*g + 4*C*d*e**3*f)/((d - e)**2 *sqrt(d + e*x)*(3*d - 2*e)**2*(d*g - e*f)**2) - e*(A - B + C)*atan(sqrt(d + e*x)/sqrt(-d + e))/(sqrt(-d + e)*(d - e)**2*(f - g)) + 3*e*(9*A - 6*B + 4*C)*atan(sqrt(d + e*x)/sqrt(-d + 2*e/3))/(sqrt(-d + 2*e/3)*(3*d - 2*e)**2 *(3*f - 2*g)))/e, Ne(e, 0)), (((9*A - 6*B + 4*C)*log(3*x + 2)/(3*(3*f - 2* g)) - (A - B + C)*log(x + 1)/(f - g) + (A*g**2 - B*f*g + C*f**2)*Piecewise ((x/f, Eq(g, 0)), (log(f + g*x)/g, True))/((f - g)*(3*f - 2*g)))/d**(5/2), True))
Exception generated. \[ \int \frac {A+B x+C x^2}{(d+e x)^{5/2} (f+g x) \left (2+5 x+3 x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((C*x^2+B*x+A)/(e*x+d)^(5/2)/(g*x+f)/(3*x^2+5*x+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*d-4*e>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 789 vs. \(2 (342) = 684\).
Time = 0.30 (sec) , antiderivative size = 789, normalized size of antiderivative = 2.10 \[ \int \frac {A+B x+C x^2}{(d+e x)^{5/2} (f+g x) \left (2+5 x+3 x^2\right )} \, dx =\text {Too large to display} \] Input:
integrate((C*x^2+B*x+A)/(e*x+d)^(5/2)/(g*x+f)/(3*x^2+5*x+2),x, algorithm=" giac")
Output:
6*sqrt(3)*(9*A - 6*B + 4*C)*arctan(sqrt(3)*sqrt(e*x + d)/sqrt(-3*d + 2*e)) /((27*d^2*f - 36*d*e*f + 12*e^2*f - 18*d^2*g + 24*d*e*g - 8*e^2*g)*sqrt(-3 *d + 2*e)) + 2*(C*f^2*g^2 - B*f*g^3 + A*g^4)*arctan(sqrt(e*x + d)*g/sqrt(e *f*g - d*g^2))/((3*e^2*f^4 - 6*d*e*f^3*g - 5*e^2*f^3*g + 3*d^2*f^2*g^2 + 1 0*d*e*f^2*g^2 + 2*e^2*f^2*g^2 - 5*d^2*f*g^3 - 4*d*e*f*g^3 + 2*d^2*g^4)*sqr t(e*f*g - d*g^2)) - 2*(A - B + C)*arctan(sqrt(e*x + d)/sqrt(-d + e))/((d^2 *f - 2*d*e*f + e^2*f - d^2*g + 2*d*e*g - e^2*g)*sqrt(-d + e)) - 2/3*(3*C*d ^4*e*f - 9*(e*x + d)*B*d^2*e^2*f + 15*(e*x + d)*C*d^2*e^2*f - 3*B*d^3*e^2* f - 5*C*d^3*e^2*f + 18*(e*x + d)*A*d*e^3*f - 12*(e*x + d)*C*d*e^3*f + 3*A* d^2*e^3*f + 5*B*d^2*e^3*f + 2*C*d^2*e^3*f - 15*(e*x + d)*A*e^4*f + 6*(e*x + d)*B*e^4*f - 5*A*d*e^4*f - 2*B*d*e^4*f + 2*A*e^5*f - 9*(e*x + d)*C*d^4*g - 3*C*d^5*g + 18*(e*x + d)*B*d^3*e*g + 3*B*d^4*e*g + 5*C*d^4*e*g - 27*(e* x + d)*A*d^2*e^2*g - 15*(e*x + d)*B*d^2*e^2*g + 6*(e*x + d)*C*d^2*e^2*g - 3*A*d^3*e^2*g - 5*B*d^3*e^2*g - 2*C*d^3*e^2*g + 30*(e*x + d)*A*d*e^3*g + 5 *A*d^2*e^3*g + 2*B*d^2*e^3*g - 6*(e*x + d)*A*e^4*g - 2*A*d*e^4*g)/((9*d^4* e^2*f^2 - 30*d^3*e^3*f^2 + 37*d^2*e^4*f^2 - 20*d*e^5*f^2 + 4*e^6*f^2 - 18* d^5*e*f*g + 60*d^4*e^2*f*g - 74*d^3*e^3*f*g + 40*d^2*e^4*f*g - 8*d*e^5*f*g + 9*d^6*g^2 - 30*d^5*e*g^2 + 37*d^4*e^2*g^2 - 20*d^3*e^3*g^2 + 4*d^2*e^4* g^2)*(e*x + d)^(3/2))
Timed out. \[ \int \frac {A+B x+C x^2}{(d+e x)^{5/2} (f+g x) \left (2+5 x+3 x^2\right )} \, dx=\text {Hanged} \] Input:
int((A + B*x + C*x^2)/((f + g*x)*(d + e*x)^(5/2)*(5*x + 3*x^2 + 2)),x)
Output:
\text{Hanged}
Time = 0.27 (sec) , antiderivative size = 26571, normalized size of antiderivative = 70.67 \[ \int \frac {A+B x+C x^2}{(d+e x)^{5/2} (f+g x) \left (2+5 x+3 x^2\right )} \, dx =\text {Too large to display} \] Input:
int((C*x^2+B*x+A)/(e*x+d)^(5/2)/(g*x+f)/(3*x^2+5*x+2),x)
Output:
( - 162*sqrt(g)*sqrt(d + e*x)*sqrt( - d*g + e*f)*atan((sqrt(d + e*x)*g)/(s qrt(g)*sqrt( - d*g + e*f)))*a*d**7*g**3 - 162*sqrt(g)*sqrt(d + e*x)*sqrt( - d*g + e*f)*atan((sqrt(d + e*x)*g)/(sqrt(g)*sqrt( - d*g + e*f)))*a*d**6*e *g**3*x + 810*sqrt(g)*sqrt(d + e*x)*sqrt( - d*g + e*f)*atan((sqrt(d + e*x) *g)/(sqrt(g)*sqrt( - d*g + e*f)))*a*d**6*e*g**3 + 810*sqrt(g)*sqrt(d + e*x )*sqrt( - d*g + e*f)*atan((sqrt(d + e*x)*g)/(sqrt(g)*sqrt( - d*g + e*f)))* a*d**5*e**2*g**3*x - 1674*sqrt(g)*sqrt(d + e*x)*sqrt( - d*g + e*f)*atan((s qrt(d + e*x)*g)/(sqrt(g)*sqrt( - d*g + e*f)))*a*d**5*e**2*g**3 - 1674*sqrt (g)*sqrt(d + e*x)*sqrt( - d*g + e*f)*atan((sqrt(d + e*x)*g)/(sqrt(g)*sqrt( - d*g + e*f)))*a*d**4*e**3*g**3*x + 1830*sqrt(g)*sqrt(d + e*x)*sqrt( - d* g + e*f)*atan((sqrt(d + e*x)*g)/(sqrt(g)*sqrt( - d*g + e*f)))*a*d**4*e**3* g**3 + 1830*sqrt(g)*sqrt(d + e*x)*sqrt( - d*g + e*f)*atan((sqrt(d + e*x)*g )/(sqrt(g)*sqrt( - d*g + e*f)))*a*d**3*e**4*g**3*x - 1116*sqrt(g)*sqrt(d + e*x)*sqrt( - d*g + e*f)*atan((sqrt(d + e*x)*g)/(sqrt(g)*sqrt( - d*g + e*f )))*a*d**3*e**4*g**3 - 1116*sqrt(g)*sqrt(d + e*x)*sqrt( - d*g + e*f)*atan( (sqrt(d + e*x)*g)/(sqrt(g)*sqrt( - d*g + e*f)))*a*d**2*e**5*g**3*x + 360*s qrt(g)*sqrt(d + e*x)*sqrt( - d*g + e*f)*atan((sqrt(d + e*x)*g)/(sqrt(g)*sq rt( - d*g + e*f)))*a*d**2*e**5*g**3 + 360*sqrt(g)*sqrt(d + e*x)*sqrt( - d* g + e*f)*atan((sqrt(d + e*x)*g)/(sqrt(g)*sqrt( - d*g + e*f)))*a*d*e**6*g** 3*x - 48*sqrt(g)*sqrt(d + e*x)*sqrt( - d*g + e*f)*atan((sqrt(d + e*x)*g...