\(\int \frac {A+B x+C x^2}{(d+e x)^{7/2} (f+g x) (2+5 x+3 x^2)} \, dx\) [41]

Optimal result

Integrand size = 39, antiderivative size = 667 \[ \int \frac {A+B x+C x^2}{(d+e x)^{7/2} (f+g x) \left (2+5 x+3 x^2\right )} \, dx=-\frac {2 \left (C d^2-B d e+A e^2\right )}{5 \left (3 d^2-5 d e+2 e^2\right ) (e f-d g) (d+e x)^{5/2}}+\frac {2 \left (C d \left (4 e^3 f+3 d^3 g-d e^2 (5 f+2 g)\right )+A e^2 \left (9 d^2 g+e^2 (5 f+2 g)-2 d e (3 f+5 g)\right )-B e \left (2 e^3 f+6 d^3 g-d^2 e (3 f+5 g)\right )\right )}{3 \left (3 d^2-5 d e+2 e^2\right )^2 (e f-d g)^2 (d+e x)^{3/2}}-\frac {2 \left (C \left (4 e^6 f^2+9 d^6 g^2-9 d^4 e^2 g (5 f+2 g)-6 d^2 e^4 f (3 f+5 g)+d^3 e^3 \left (15 f^2+73 f g+10 g^2\right )\right )-B e \left (36 d^2 e^3 f g+27 d^5 g^2+2 e^5 f (5 f+2 g)-6 d e^4 f (3 f+5 g)-9 d^4 e g (3 f+5 g)+d^3 e^2 \left (9 f^2+15 f g+19 g^2\right )\right )+A e^2 \left (54 d^4 g^2-24 d^3 e g (3 f+5 g)-15 d e^3 \left (3 f^2+5 f g+2 g^2\right )+e^4 \left (19 f^2+10 f g+4 g^2\right )+3 d^2 e^2 \left (9 f^2+45 f g+31 g^2\right )\right )\right )}{\left (3 d^2-5 d e+2 e^2\right )^3 (e f-d g)^3 \sqrt {d+e x}}-\frac {2 g^{5/2} \left (C f^2-B f g+A g^2\right ) \arctan \left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e f-d g}}\right )}{(3 f-2 g) (f-g) (e f-d g)^{7/2}}-\frac {18 \sqrt {3} (9 A-6 B+4 C) \text {arctanh}\left (\frac {\sqrt {3} \sqrt {d+e x}}{\sqrt {3 d-2 e}}\right )}{(3 d-2 e)^{7/2} (3 f-2 g)}+\frac {2 (A-B+C) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d-e}}\right )}{(d-e)^{7/2} (f-g)} \] Output:

1/5*(-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)^(5/2 
)+2/3*(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)^(3/2)-2*(C*(4*e^6*f^2+9*d^6*g^2-9*d^4*e^2*g*(5*f+ 
2*g)-6*d^2*e^4*f*(3*f+5*g)+d^3*e^3*(15*f^2+73*f*g+10*g^2))-B*e*(36*d^2*e^3 
*f*g+27*d^5*g^2+2*e^5*f*(5*f+2*g)-6*d*e^4*f*(3*f+5*g)-9*d^4*e*g*(3*f+5*g)+ 
d^3*e^2*(9*f^2+15*f*g+19*g^2))+A*e^2*(54*d^4*g^2-24*d^3*e*g*(3*f+5*g)-15*d 
*e^3*(3*f^2+5*f*g+2*g^2)+e^4*(19*f^2+10*f*g+4*g^2)+3*d^2*e^2*(9*f^2+45*f*g 
+31*g^2)))/(3*d^2-5*d*e+2*e^2)^3/(-d*g+e*f)^3/(e*x+d)^(1/2)-2*g^(5/2)*(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)^(7/2)-18*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)^(7/2)/(3*f-2*g)+2*(A-B+C)*arctanh((e*x+d)^(1 
/2)/(d-e)^(1/2))/(d-e)^(7/2)/(f-g)
 

Mathematica [A] (verified)

Time = 17.53 (sec) , antiderivative size = 755, normalized size of antiderivative = 1.13 \[ \int \frac {A+B x+C x^2}{(d+e x)^{7/2} (f+g x) \left (2+5 x+3 x^2\right )} \, dx=2 \left (-\frac {C d^2-B d e+A e^2}{5 (3 d-2 e) (d-e) (e f-d g) (d+e x)^{5/2}}+\frac {C d \left (4 e^3 f+3 d^3 g-d e^2 (5 f+2 g)\right )+A e^2 \left (9 d^2 g+e^2 (5 f+2 g)-2 d e (3 f+5 g)\right )-B e \left (2 e^3 f+6 d^3 g-d^2 e (3 f+5 g)\right )}{3 (3 d-2 e)^2 (d-e)^2 (e f-d g)^2 (d+e x)^{3/2}}-\frac {9 B d^3 e^3 f^2-15 C d^3 e^3 f^2-27 A d^2 e^4 f^2+18 C d^2 e^4 f^2+45 A d e^5 f^2-18 B d e^5 f^2-19 A e^6 f^2+10 B e^6 f^2-4 C e^6 f^2-27 B d^4 e^2 f g+45 C d^4 e^2 f g+72 A d^3 e^3 f g+15 B d^3 e^3 f g-73 C d^3 e^3 f g-135 A d^2 e^4 f g+36 B d^2 e^4 f g+30 C d^2 e^4 f g+75 A d e^5 f g-30 B d e^5 f g-10 A e^6 f g+4 B e^6 f g-9 C d^6 g^2+27 B d^5 e g^2-54 A d^4 e^2 g^2-45 B d^4 e^2 g^2+18 C d^4 e^2 g^2+120 A d^3 e^3 g^2+19 B d^3 e^3 g^2-10 C d^3 e^3 g^2-93 A d^2 e^4 g^2+30 A d e^5 g^2-4 A e^6 g^2}{(3 d-2 e)^3 (d-e)^3 (-e f+d g)^3 \sqrt {d+e x}}-\frac {g^{5/2} \left (C f^2-g (B f-A g)\right ) \arctan \left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e f-d g}}\right )}{(3 f-2 g) (f-g) (e f-d g)^{7/2}}-\frac {9 \sqrt {3} (9 A-6 B+4 C) \text {arctanh}\left (\frac {\sqrt {3} \sqrt {d+e x}}{\sqrt {3 d-2 e}}\right )}{(3 d-2 e)^{7/2} (3 f-2 g)}+\frac {(A-B+C) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d-e}}\right )}{(d-e)^{7/2} (f-g)}\right ) \] Input:

Integrate[(A + B*x + C*x^2)/((d + e*x)^(7/2)*(f + g*x)*(2 + 5*x + 3*x^2)), 
x]
 

Output:

2*(-1/5*(C*d^2 - B*d*e + A*e^2)/((3*d - 2*e)*(d - e)*(e*f - d*g)*(d + e*x) 
^(5/2)) + (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*(3*d - 2*e)^2*(d - e)^2*(e*f - d*g)^2*(d + e*x)^(3/2)) - (9*B 
*d^3*e^3*f^2 - 15*C*d^3*e^3*f^2 - 27*A*d^2*e^4*f^2 + 18*C*d^2*e^4*f^2 + 45 
*A*d*e^5*f^2 - 18*B*d*e^5*f^2 - 19*A*e^6*f^2 + 10*B*e^6*f^2 - 4*C*e^6*f^2 
- 27*B*d^4*e^2*f*g + 45*C*d^4*e^2*f*g + 72*A*d^3*e^3*f*g + 15*B*d^3*e^3*f* 
g - 73*C*d^3*e^3*f*g - 135*A*d^2*e^4*f*g + 36*B*d^2*e^4*f*g + 30*C*d^2*e^4 
*f*g + 75*A*d*e^5*f*g - 30*B*d*e^5*f*g - 10*A*e^6*f*g + 4*B*e^6*f*g - 9*C* 
d^6*g^2 + 27*B*d^5*e*g^2 - 54*A*d^4*e^2*g^2 - 45*B*d^4*e^2*g^2 + 18*C*d^4* 
e^2*g^2 + 120*A*d^3*e^3*g^2 + 19*B*d^3*e^3*g^2 - 10*C*d^3*e^3*g^2 - 93*A*d 
^2*e^4*g^2 + 30*A*d*e^5*g^2 - 4*A*e^6*g^2)/((3*d - 2*e)^3*(d - e)^3*(-(e*f 
) + d*g)^3*Sqrt[d + e*x]) - (g^(5/2)*(C*f^2 - g*(B*f - A*g))*ArcTan[(Sqrt[ 
g]*Sqrt[d + e*x])/Sqrt[e*f - d*g]])/((3*f - 2*g)*(f - g)*(e*f - d*g)^(7/2) 
) - (9*Sqrt[3]*(9*A - 6*B + 4*C)*ArcTanh[(Sqrt[3]*Sqrt[d + e*x])/Sqrt[3*d 
- 2*e]])/((3*d - 2*e)^(7/2)*(3*f - 2*g)) + ((A - B + C)*ArcTanh[Sqrt[d + e 
*x]/Sqrt[d - e]])/((d - e)^(7/2)*(f - g)))
 

Rubi [A] (verified)

Time = 1.64 (sec) , antiderivative size = 571, normalized size of antiderivative = 0.86, 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.

\(\displaystyle \int \frac {A+B x+C x^2}{\left (3 x^2+5 x+2\right ) (d+e x)^{7/2} (f+g x)} \, dx\)

\(\Big \downarrow \) 2153

\(\displaystyle \int \left (\frac {A g^2-B f g+C f^2}{(3 f-2 g) (f-g) (d+e x)^{7/2} (f+g x)}+\frac {-A+B-C}{(x+1) (f-g) (d+e x)^{7/2}}+\frac {9 A-6 B+4 C}{(3 x+2) (3 f-2 g) (d+e x)^{7/2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 g^{5/2} \left (A g^2-B f g+C f^2\right ) \arctan \left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e f-d g}}\right )}{(3 f-2 g) (f-g) (e f-d g)^{7/2}}-\frac {18 \sqrt {3} (9 A-6 B+4 C) \text {arctanh}\left (\frac {\sqrt {3} \sqrt {d+e x}}{\sqrt {3 d-2 e}}\right )}{(3 d-2 e)^{7/2} (3 f-2 g)}+\frac {2 (A-B+C) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d-e}}\right )}{(d-e)^{7/2} (f-g)}-\frac {2 g^2 \left (A g^2-B f g+C f^2\right )}{(3 f-2 g) (f-g) \sqrt {d+e x} (e f-d g)^3}+\frac {2 g \left (A g^2-B f g+C f^2\right )}{3 (3 f-2 g) (f-g) (d+e x)^{3/2} (e f-d g)^2}-\frac {2 \left (A g^2-B f g+C f^2\right )}{5 (3 f-2 g) (f-g) (d+e x)^{5/2} (e f-d g)}+\frac {18 (9 A-6 B+4 C)}{(3 d-2 e)^3 (3 f-2 g) \sqrt {d+e x}}-\frac {2 (A-B+C)}{(d-e)^3 (f-g) \sqrt {d+e x}}+\frac {2 (9 A-6 B+4 C)}{(3 d-2 e)^2 (3 f-2 g) (d+e x)^{3/2}}-\frac {2 (A-B+C)}{3 (d-e)^2 (f-g) (d+e x)^{3/2}}+\frac {2 (9 A-6 B+4 C)}{5 (3 d-2 e) (3 f-2 g) (d+e x)^{5/2}}-\frac {2 (A-B+C)}{5 (d-e) (f-g) (d+e x)^{5/2}}\)

Input:

Int[(A + B*x + C*x^2)/((d + e*x)^(7/2)*(f + g*x)*(2 + 5*x + 3*x^2)),x]
 

Output:

(2*(9*A - 6*B + 4*C))/(5*(3*d - 2*e)*(3*f - 2*g)*(d + e*x)^(5/2)) - (2*(A 
- B + C))/(5*(d - e)*(f - g)*(d + e*x)^(5/2)) - (2*(C*f^2 - B*f*g + A*g^2) 
)/(5*(3*f - 2*g)*(f - g)*(e*f - d*g)*(d + e*x)^(5/2)) + (2*(9*A - 6*B + 4* 
C))/((3*d - 2*e)^2*(3*f - 2*g)*(d + e*x)^(3/2)) - (2*(A - B + C))/(3*(d - 
e)^2*(f - g)*(d + e*x)^(3/2)) + (2*g*(C*f^2 - B*f*g + A*g^2))/(3*(3*f - 2* 
g)*(f - g)*(e*f - d*g)^2*(d + e*x)^(3/2)) + (18*(9*A - 6*B + 4*C))/((3*d - 
 2*e)^3*(3*f - 2*g)*Sqrt[d + e*x]) - (2*(A - B + C))/((d - e)^3*(f - g)*Sq 
rt[d + e*x]) - (2*g^2*(C*f^2 - B*f*g + A*g^2))/((3*f - 2*g)*(f - g)*(e*f - 
 d*g)^3*Sqrt[d + e*x]) - (2*g^(5/2)*(C*f^2 - B*f*g + A*g^2)*ArcTan[(Sqrt[g 
]*Sqrt[d + e*x])/Sqrt[e*f - d*g]])/((3*f - 2*g)*(f - g)*(e*f - d*g)^(7/2)) 
 - (18*Sqrt[3]*(9*A - 6*B + 4*C)*ArcTanh[(Sqrt[3]*Sqrt[d + e*x])/Sqrt[3*d 
- 2*e]])/((3*d - 2*e)^(7/2)*(3*f - 2*g)) + (2*(A - B + C)*ArcTanh[Sqrt[d + 
 e*x]/Sqrt[d - e]])/((d - e)^(7/2)*(f - g))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2153

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])
 

Maple [A] (verified)

Time = 211.90 (sec) , antiderivative size = 563, normalized size of antiderivative = 0.84

method	result	size
pseudoelliptic	\(\frac {\left (486 A -324 B +216 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 )^{3}}-\frac {2 g^{3} \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 )^{3} \sqrt {\left (d g -e f \right ) g}}+\frac {\frac {2}{15} A \,e^{2}-\frac {2}{15} B d e +\frac {2}{15} C \,d^{2}}{\left (e x +d \right )^{\frac {5}{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}{9}-\frac {4 B \,d^{3} e g}{9}+\frac {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}}{3}-\frac {4 d \left (\frac {5 A g}{3}+f \left (A -\frac {2 C}{3}\right )\right ) e^{3}}{9}+\frac {10 e^{4} \left (\frac {2 A g}{5}+f \left (A -\frac {2 B}{5}\right )\right )}{27}}{\left (e x +d \right )^{\frac {3}{2}} \left (d -\frac {2 e}{3}\right )^{2} \left (d g -e f \right )^{2} \left (d -e \right )^{2}}+\frac {\frac {2 \left (4 A \,g^{2}+10 f \left (A -\frac {2 B}{5}\right ) g +19 \left (A -\frac {10 B}{19}+\frac {4 C}{19}\right ) f^{2}\right ) e^{6}}{27}-\frac {10 d \left (\frac {2 A \,g^{2}}{3}+\frac {5 f \left (A -\frac {2 B}{5}\right ) g}{3}+f^{2} \left (A -\frac {2 B}{5}\right )\right ) e^{5}}{3}+2 \left (\frac {31 A \,g^{2}}{9}+5 \left (A -\frac {4 B}{15}-\frac {2 C}{9}\right ) f g +f^{2} \left (A -\frac {2 C}{3}\right )\right ) d^{2} e^{4}-\frac {16 \left (\frac {\left (5 A +\frac {19 B}{24}-\frac {5 C}{12}\right ) g^{2}}{3}+f \left (A +\frac {5 B}{24}-\frac {73 C}{72}\right ) g +\frac {\left (B -\frac {5 C}{3}\right ) f^{2}}{8}\right ) d^{3} e^{3}}{3}+4 \left (\left (A +\frac {5 B}{6}-\frac {C}{3}\right ) g +\frac {\left (B -\frac {5 C}{3}\right ) f}{2}\right ) d^{4} g \,e^{2}-2 B \,d^{5} e \,g^{2}+\frac {2 C \,d^{6} g^{2}}{3}}{\left (d -e \right )^{3} \sqrt {e x +d}\, \left (d g -e f \right )^{3} \left (d -\frac {2 e}{3}\right )^{3}}+\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 {7}{2}}}\)	\(563\)
derivativedivides	\(\frac {2 \left (243 A -162 B +108 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 )^{3} \sqrt {-9 d +6 e}}-\frac {2 g^{3} \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 )^{3} \sqrt {\left (d g -e f \right ) g}}-\frac {2 \left (-A \,e^{2}+B d e -C \,d^{2}\right )}{5 \left (d g -e f \right ) \left (3 d -2 e \right ) \left (d -e \right ) \left (e x +d \right )^{\frac {5}{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 )}{3 \left (d g -e f \right )^{2} \left (3 d -2 e \right )^{2} \left (d -e \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (-54 A \,d^{4} e^{2} g^{2}+72 A \,d^{3} e^{3} f g +120 A \,d^{3} e^{3} g^{2}-27 A \,d^{2} e^{4} f^{2}-135 A \,d^{2} e^{4} f g -93 A \,d^{2} e^{4} g^{2}+45 A d \,e^{5} f^{2}+75 A d \,e^{5} f g +30 A d \,e^{5} g^{2}-19 A \,e^{6} f^{2}-10 A \,e^{6} f g -4 A \,e^{6} g^{2}+27 B \,d^{5} e \,g^{2}-27 B \,d^{4} e^{2} f g -45 B \,d^{4} e^{2} g^{2}+9 B \,d^{3} e^{3} f^{2}+15 B \,d^{3} e^{3} f g +19 B \,d^{3} e^{3} g^{2}+36 B \,d^{2} e^{4} f g -18 B d \,e^{5} f^{2}-30 B d \,e^{5} f g +10 B \,e^{6} f^{2}+4 B \,e^{6} f g -9 C \,d^{6} g^{2}+45 C \,d^{4} e^{2} f g +18 C \,d^{4} e^{2} g^{2}-15 C \,d^{3} e^{3} f^{2}-73 C \,d^{3} e^{3} f g -10 C \,d^{3} e^{3} g^{2}+18 C \,d^{2} e^{4} f^{2}+30 C \,d^{2} e^{4} f g -4 C \,e^{6} f^{2}\right )}{\left (d g -e f \right )^{3} \left (3 d -2 e \right )^{3} \left (d -e \right )^{3} \sqrt {e x +d}}+\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 )^{3} \sqrt {-d +e}}\)	\(761\)
default	\(\frac {2 \left (243 A -162 B +108 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 )^{3} \sqrt {-9 d +6 e}}-\frac {2 g^{3} \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 )^{3} \sqrt {\left (d g -e f \right ) g}}-\frac {2 \left (-A \,e^{2}+B d e -C \,d^{2}\right )}{5 \left (d g -e f \right ) \left (3 d -2 e \right ) \left (d -e \right ) \left (e x +d \right )^{\frac {5}{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 )}{3 \left (d g -e f \right )^{2} \left (3 d -2 e \right )^{2} \left (d -e \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (-54 A \,d^{4} e^{2} g^{2}+72 A \,d^{3} e^{3} f g +120 A \,d^{3} e^{3} g^{2}-27 A \,d^{2} e^{4} f^{2}-135 A \,d^{2} e^{4} f g -93 A \,d^{2} e^{4} g^{2}+45 A d \,e^{5} f^{2}+75 A d \,e^{5} f g +30 A d \,e^{5} g^{2}-19 A \,e^{6} f^{2}-10 A \,e^{6} f g -4 A \,e^{6} g^{2}+27 B \,d^{5} e \,g^{2}-27 B \,d^{4} e^{2} f g -45 B \,d^{4} e^{2} g^{2}+9 B \,d^{3} e^{3} f^{2}+15 B \,d^{3} e^{3} f g +19 B \,d^{3} e^{3} g^{2}+36 B \,d^{2} e^{4} f g -18 B d \,e^{5} f^{2}-30 B d \,e^{5} f g +10 B \,e^{6} f^{2}+4 B \,e^{6} f g -9 C \,d^{6} g^{2}+45 C \,d^{4} e^{2} f g +18 C \,d^{4} e^{2} g^{2}-15 C \,d^{3} e^{3} f^{2}-73 C \,d^{3} e^{3} f g -10 C \,d^{3} e^{3} g^{2}+18 C \,d^{2} e^{4} f^{2}+30 C \,d^{2} e^{4} f g -4 C \,e^{6} f^{2}\right )}{\left (d g -e f \right )^{3} \left (3 d -2 e \right )^{3} \left (d -e \right )^{3} \sqrt {e x +d}}+\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 )^{3} \sqrt {-d +e}}\)	\(761\)

Input:

int((C*x^2+B*x+A)/(e*x+d)^(7/2)/(g*x+f)/(3*x^2+5*x+2),x,method=_RETURNVERB 
OSE)
 

Output:

(486*A-324*B+216*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)^3-2*g^3*(A*g^2-B*f*g+C*f^2)/(f-g)/(3*f-2*g)/(d*g-e 
*f)^3/((d*g-e*f)*g)^(1/2)*arctanh(g*(e*x+d)^(1/2)/((d*g-e*f)*g)^(1/2))+2/1 
5*(A*e^2-B*d*e+C*d^2)/(e*x+d)^(5/2)/(d-e)/(d*g-e*f)/(d-2/3*e)+2/3*(1/3*C*d 
^4*g-2/3*B*d^3*e*g+d^2*((A+5/9*B-2/9*C)*g+1/3*(B-5/3*C)*f)*e^2-2/3*d*(5/3* 
A*g+f*(A-2/3*C))*e^3+5/9*e^4*(2/5*A*g+f*(A-2/5*B)))/(e*x+d)^(3/2)/(d-2/3*e 
)^2/(d*g-e*f)^2/(d-e)^2+2/27*((4*A*g^2+10*f*(A-2/5*B)*g+19*(A-10/19*B+4/19 
*C)*f^2)*e^6-45*d*(2/3*A*g^2+5/3*f*(A-2/5*B)*g+f^2*(A-2/5*B))*e^5+27*(31/9 
*A*g^2+5*(A-4/15*B-2/9*C)*f*g+f^2*(A-2/3*C))*d^2*e^4-72*(1/3*(5*A+19/24*B- 
5/12*C)*g^2+f*(A+5/24*B-73/72*C)*g+1/8*(B-5/3*C)*f^2)*d^3*e^3+54*((A+5/6*B 
-1/3*C)*g+1/2*(B-5/3*C)*f)*d^4*g*e^2-27*B*d^5*e*g^2+9*C*d^6*g^2)/(d-e)^3/( 
e*x+d)^(1/2)/(d*g-e*f)^3/(d-2/3*e)^3+2*(A-B+C)*arctan((e*x+d)^(1/2)/(-d+e) 
^(1/2))/(f-g)/(-d+e)^(7/2)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2}{(d+e x)^{7/2} (f+g x) \left (2+5 x+3 x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((C*x^2+B*x+A)/(e*x+d)^(7/2)/(g*x+f)/(3*x^2+5*x+2),x, algorithm=" 
fricas")
 

Output:

Timed out
 

Sympy [A] (verification not implemented)

Time = 96.53 (sec) , antiderivative size = 910, normalized size of antiderivative = 1.36 \[ \int \frac {A+B x+C x^2}{(d+e x)^{7/2} (f+g x) \left (2+5 x+3 x^2\right )} \, dx =\text {Too large to display} \] Input:

integrate((C*x**2+B*x+A)/(e*x+d)**(7/2)/(g*x+f)/(3*x**2+5*x+2),x)
 

Output:

Piecewise((2*(e*g**2*(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)**3) + 
e*(A*e**2 - B*d*e + C*d**2)/(5*(d - e)*(d + e*x)**(5/2)*(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)/(3*(d - 
e)**2*(d + e*x)**(3/2)*(3*d - 2*e)**2*(d*g - e*f)**2) + e*(54*A*d**4*e**2* 
g**2 - 72*A*d**3*e**3*f*g - 120*A*d**3*e**3*g**2 + 27*A*d**2*e**4*f**2 + 1 
35*A*d**2*e**4*f*g + 93*A*d**2*e**4*g**2 - 45*A*d*e**5*f**2 - 75*A*d*e**5* 
f*g - 30*A*d*e**5*g**2 + 19*A*e**6*f**2 + 10*A*e**6*f*g + 4*A*e**6*g**2 - 
27*B*d**5*e*g**2 + 27*B*d**4*e**2*f*g + 45*B*d**4*e**2*g**2 - 9*B*d**3*e** 
3*f**2 - 15*B*d**3*e**3*f*g - 19*B*d**3*e**3*g**2 - 36*B*d**2*e**4*f*g + 1 
8*B*d*e**5*f**2 + 30*B*d*e**5*f*g - 10*B*e**6*f**2 - 4*B*e**6*f*g + 9*C*d* 
*6*g**2 - 45*C*d**4*e**2*f*g - 18*C*d**4*e**2*g**2 + 15*C*d**3*e**3*f**2 + 
 73*C*d**3*e**3*f*g + 10*C*d**3*e**3*g**2 - 18*C*d**2*e**4*f**2 - 30*C*d** 
2*e**4*f*g + 4*C*e**6*f**2)/((d - e)**3*sqrt(d + e*x)*(3*d - 2*e)**3*(d*g 
- e*f)**3) - e*(A - B + C)*atan(sqrt(d + e*x)/sqrt(-d + e))/(sqrt(-d + e)* 
(d - e)**3*(f - g)) + 9*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)**3*(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 ...
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B x+C x^2}{(d+e x)^{7/2} (f+g x) \left (2+5 x+3 x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((C*x^2+B*x+A)/(e*x+d)^(7/2)/(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
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2365 vs. \(2 (630) = 1260\).

Time = 0.87 (sec) , antiderivative size = 2365, normalized size of antiderivative = 3.55 \[ \int \frac {A+B x+C x^2}{(d+e x)^{7/2} (f+g x) \left (2+5 x+3 x^2\right )} \, dx=\text {Too large to display} \] Input:

integrate((C*x^2+B*x+A)/(e*x+d)^(7/2)/(g*x+f)/(3*x^2+5*x+2),x, algorithm=" 
giac")
 

Output:

18*sqrt(3)*(9*A - 6*B + 4*C)*arctan(sqrt(3)*sqrt(e*x + d)/sqrt(-3*d + 2*e) 
)/((81*d^3*f - 162*d^2*e*f + 108*d*e^2*f - 24*e^3*f - 54*d^3*g + 108*d^2*e 
*g - 72*d*e^2*g + 16*e^3*g)*sqrt(-3*d + 2*e)) - 2*(C*f^2*g^3 - B*f*g^4 + A 
*g^5)*arctan(sqrt(e*x + d)*g/sqrt(e*f*g - d*g^2))/((3*e^3*f^5 - 9*d*e^2*f^ 
4*g - 5*e^3*f^4*g + 9*d^2*e*f^3*g^2 + 15*d*e^2*f^3*g^2 + 2*e^3*f^3*g^2 - 3 
*d^3*f^2*g^3 - 15*d^2*e*f^2*g^3 - 6*d*e^2*f^2*g^3 + 5*d^3*f*g^4 + 6*d^2*e* 
f*g^4 - 2*d^3*g^5)*sqrt(e*f*g - d*g^2)) - 2*(A - B + C)*arctan(sqrt(e*x + 
d)/sqrt(-d + e))/((d^3*f - 3*d^2*e*f + 3*d*e^2*f - e^3*f - d^3*g + 3*d^2*e 
*g - 3*d*e^2*g + e^3*g)*sqrt(-d + e)) - 2/15*(27*C*d^6*e^2*f^2 - 135*(e*x 
+ d)^2*B*d^3*e^3*f^2 + 225*(e*x + d)^2*C*d^3*e^3*f^2 - 45*(e*x + d)*B*d^4* 
e^3*f^2 + 75*(e*x + d)*C*d^4*e^3*f^2 - 27*B*d^5*e^3*f^2 - 90*C*d^5*e^3*f^2 
 + 405*(e*x + d)^2*A*d^2*e^4*f^2 - 270*(e*x + d)^2*C*d^2*e^4*f^2 + 90*(e*x 
 + d)*A*d^3*e^4*f^2 + 75*(e*x + d)*B*d^3*e^4*f^2 - 185*(e*x + d)*C*d^3*e^4 
*f^2 + 27*A*d^4*e^4*f^2 + 90*B*d^4*e^4*f^2 + 111*C*d^4*e^4*f^2 - 675*(e*x 
+ d)^2*A*d*e^5*f^2 + 270*(e*x + d)^2*B*d*e^5*f^2 - 225*(e*x + d)*A*d^2*e^5 
*f^2 + 150*(e*x + d)*C*d^2*e^5*f^2 - 90*A*d^3*e^5*f^2 - 111*B*d^3*e^5*f^2 
- 60*C*d^3*e^5*f^2 + 285*(e*x + d)^2*A*e^6*f^2 - 150*(e*x + d)^2*B*e^6*f^2 
 + 60*(e*x + d)^2*C*e^6*f^2 + 185*(e*x + d)*A*d*e^6*f^2 - 50*(e*x + d)*B*d 
*e^6*f^2 - 40*(e*x + d)*C*d*e^6*f^2 + 111*A*d^2*e^6*f^2 + 60*B*d^2*e^6*f^2 
 + 12*C*d^2*e^6*f^2 - 50*(e*x + d)*A*e^7*f^2 + 20*(e*x + d)*B*e^7*f^2 -...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2}{(d+e x)^{7/2} (f+g x) \left (2+5 x+3 x^2\right )} \, dx=\text {Hanged} \] Input:

int((A + B*x + C*x^2)/((f + g*x)*(d + e*x)^(7/2)*(5*x + 3*x^2 + 2)),x)
 

Output:

\text{Hanged}
 

Reduce [B] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 62380, normalized size of antiderivative = 93.52 \[ \int \frac {A+B x+C x^2}{(d+e x)^{7/2} (f+g x) \left (2+5 x+3 x^2\right )} \, dx =\text {Too large to display} \] Input:

int((C*x^2+B*x+A)/(e*x+d)^(7/2)/(g*x+f)/(3*x^2+5*x+2),x)
 

Output:

( - 2430*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**10*g**4 - 4860*sqrt(g)*sqrt(d + e*x)*sqr 
t( - d*g + e*f)*atan((sqrt(d + e*x)*g)/(sqrt(g)*sqrt( - d*g + e*f)))*a*d** 
9*e*g**4*x + 16200*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**9*e*g**4 - 2430*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**8*e**2*g**4*x**2 + 32400*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**8*e**2*g**4*x 
 - 46980*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**8*e**2*g**4 + 16200*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**7*e**3*g**4*x**2 - 93960*sqrt(g)*sqrt(d + e*x)*sqrt( - d*g + e*f)*ata 
n((sqrt(d + e*x)*g)/(sqrt(g)*sqrt( - d*g + e*f)))*a*d**7*e**3*g**4*x + 774 
00*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**7*e**3*g**4 - 46980*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**4*g**4*x**2 + 154800*sqrt(g)*sqrt(d + e*x)*sqrt( - d*g + e*f)*atan((sq 
rt(d + e*x)*g)/(sqrt(g)*sqrt( - d*g + e*f)))*a*d**6*e**4*g**4*x - 79230*sq 
rt(g)*sqrt(d + e*x)*sqrt( - d*g + e*f)*atan((sqrt(d + e*x)*g)/(sqrt(g)*sqr 
t( - d*g + e*f)))*a*d**6*e**4*g**4 + 77400*sqrt(g)*sqrt(d + e*x)*sqrt( ...