\(\int \frac {(d+e x)^2 (f+g x)^3}{(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [282]

Optimal result

Integrand size = 44, antiderivative size = 453 \[ \int \frac {(d+e x)^2 (f+g x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 (c d f-a e g)^3 (d+e x)}{c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {g \left (57 a^2 e^4 g^2-2 a c d e^2 g (63 e f+5 d g)+3 c^2 d^2 \left (24 e^2 f^2+6 d e f g-d^2 g^2\right )\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 c^4 d^4 e^2}-\frac {g^2 \left (11 a e^2 g-c d (18 e f+d g)\right ) x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^3 d^3 e}+\frac {g^3 x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2}-\frac {\left (35 a^3 e^6 g^3-15 a^2 c d e^4 g^2 (6 e f+d g)+3 a c^2 d^2 e^2 g \left (24 e^2 f^2+12 d e f g-d^2 g^2\right )-c^3 d^3 \left (16 e^3 f^3+24 d e^2 f^2 g-6 d^2 e f g^2+d^3 g^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 c^{9/2} d^{9/2} e^{5/2}} \] Output:

-2*(-a*e*g+c*d*f)^3*(e*x+d)/c^4/d^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2 
)+1/24*g*(57*a^2*e^4*g^2-2*a*c*d*e^2*g*(5*d*g+63*e*f)+3*c^2*d^2*(-d^2*g^2+ 
6*d*e*f*g+24*e^2*f^2))*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^4/d^4/e^2 
-1/12*g^2*(11*a*e^2*g-c*d*(d*g+18*e*f))*x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2 
)^(1/2)/c^3/d^3/e+1/3*g^3*x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^2/ 
d^2-1/8*(35*a^3*e^6*g^3-15*a^2*c*d*e^4*g^2*(d*g+6*e*f)+3*a*c^2*d^2*e^2*g*( 
-d^2*g^2+12*d*e*f*g+24*e^2*f^2)-c^3*d^3*(d^3*g^3-6*d^2*e*f*g^2+24*d*e^2*f^ 
2*g+16*e^3*f^3))*arctanh(c^(1/2)*d^(1/2)*(e*x+d)/e^(1/2)/(a*d*e+(a*e^2+c*d 
^2)*x+c*d*e*x^2)^(1/2))/c^(9/2)/d^(9/2)/e^(5/2)
 

Mathematica [A] (verified)

Time = 1.64 (sec) , antiderivative size = 393, normalized size of antiderivative = 0.87 \[ \int \frac {(d+e x)^2 (f+g x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {-\sqrt {c} \sqrt {d} \sqrt {e} (d+e x) \left (-105 a^3 e^5 g^3+5 a^2 c d e^3 g^2 (54 e f+2 d g-7 e g x)+a c^2 d^2 e g \left (3 d^2 g^2+2 d e g (-9 f+4 g x)+e^2 \left (-216 f^2+90 f g x+14 g^2 x^2\right )\right )+c^3 d^3 \left (3 d^2 g^3 x-2 d e g^2 x (9 f+g x)+4 e^2 \left (12 f^3-18 f^2 g x-9 f g^2 x^2-2 g^3 x^3\right )\right )\right )+3 \left (-35 a^3 e^6 g^3+15 a^2 c d e^4 g^2 (6 e f+d g)+3 a c^2 d^2 e^2 g \left (-24 e^2 f^2-12 d e f g+d^2 g^2\right )+c^3 d^3 \left (16 e^3 f^3+24 d e^2 f^2 g-6 d^2 e f g^2+d^3 g^3\right )\right ) \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{24 c^{9/2} d^{9/2} e^{5/2} \sqrt {(a e+c d x) (d+e x)}} \] Input:

Integrate[((d + e*x)^2*(f + g*x)^3)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2 
)^(3/2),x]
 

Output:

(-(Sqrt[c]*Sqrt[d]*Sqrt[e]*(d + e*x)*(-105*a^3*e^5*g^3 + 5*a^2*c*d*e^3*g^2 
*(54*e*f + 2*d*g - 7*e*g*x) + a*c^2*d^2*e*g*(3*d^2*g^2 + 2*d*e*g*(-9*f + 4 
*g*x) + e^2*(-216*f^2 + 90*f*g*x + 14*g^2*x^2)) + c^3*d^3*(3*d^2*g^3*x - 2 
*d*e*g^2*x*(9*f + g*x) + 4*e^2*(12*f^3 - 18*f^2*g*x - 9*f*g^2*x^2 - 2*g^3* 
x^3)))) + 3*(-35*a^3*e^6*g^3 + 15*a^2*c*d*e^4*g^2*(6*e*f + d*g) + 3*a*c^2* 
d^2*e^2*g*(-24*e^2*f^2 - 12*d*e*f*g + d^2*g^2) + c^3*d^3*(16*e^3*f^3 + 24* 
d*e^2*f^2*g - 6*d^2*e*f*g^2 + d^3*g^3))*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*Ar 
cTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x])])/(24*c^ 
(9/2)*d^(9/2)*e^(5/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])
 

Rubi [A] (verified)

Time = 1.59 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.205, Rules used = {1211, 25, 2192, 27, 2192, 27, 1160, 1092, 219}

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.

\(\displaystyle \int \frac {(d+e x)^2 (f+g x)^3}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 1211

\(\displaystyle \frac {\int -\frac {-c^3 d^3 g^3 x^3 e^6+c^2 d^2 g^2 \left (a e^2 g-c d (3 e f+d g)\right ) x^2 e^5+\left (a^3 g^3 e^4-a^2 c d g^2 (3 e f+d g) e^2+3 a c^2 d^2 f g (e f+d g) e-c^3 d^3 f^2 (e f+3 d g)\right ) e^5-c d g \left (a^2 g^2 e^3-a c d g (3 e f+d g) e+3 c^2 d^2 f (e f+d g)\right ) x e^5}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{c^4 d^4 e^5}-\frac {2 (d+e x) (c d f-a e g)^3}{c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int \frac {-c^3 d^3 g^3 x^3 e^6+c^2 d^2 g^2 \left (a e^2 g-c d (3 e f+d g)\right ) x^2 e^5+\left (a^3 g^3 e^4-a^2 c d g^2 (3 e f+d g) e^2+3 a c^2 d^2 f g (e f+d g) e-c^3 d^3 f^2 (e f+3 d g)\right ) e^5-c d g \left (a^2 g^2 e^3-a c d g (3 e f+d g) e+3 c^2 d^2 f (e f+d g)\right ) x e^5}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{c^4 d^4 e^5}-\frac {2 (d+e x) (c d f-a e g)^3}{c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 2192

\(\displaystyle -\frac {\frac {\int \frac {c^3 d^3 g^2 \left (11 a e^2 g-c d (18 e f+d g)\right ) x^2 e^6+6 c d \left (a^3 g^3 e^4-a^2 c d g^2 (3 e f+d g) e^2+3 a c^2 d^2 f g (e f+d g) e-c^3 d^3 f^2 (e f+3 d g)\right ) e^6-2 c^2 d^2 g \left (3 a^2 g^2 e^3-a c d g (9 e f+5 d g) e+9 c^2 d^2 f (e f+d g)\right ) x e^6}{2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{3 c d e}-\frac {1}{3} c^2 d^2 e^5 g^3 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^4 d^4 e^5}-\frac {2 (d+e x) (c d f-a e g)^3}{c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\int \frac {c^3 d^3 g^2 \left (11 a e^2 g-c d (18 e f+d g)\right ) x^2 e^6+6 c d \left (a^3 g^3 e^4-a^2 c d g^2 (3 e f+d g) e^2+3 a c^2 d^2 f g (e f+d g) e-c^3 d^3 f^2 (e f+3 d g)\right ) e^6-2 c^2 d^2 g \left (3 a^2 g^2 e^3-a c d g (9 e f+5 d g) e+9 c^2 d^2 f (e f+d g)\right ) x e^6}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{6 c d e}-\frac {1}{3} c^2 d^2 e^5 g^3 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^4 d^4 e^5}-\frac {2 (d+e x) (c d f-a e g)^3}{c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 2192

\(\displaystyle -\frac {\frac {\frac {\int \frac {c^2 d^2 e^6 \left (2 e \left (12 a^3 g^3 e^4-a^2 c d g^2 (36 e f+23 d g) e^2-12 c^3 d^3 f^2 (e f+3 d g)+a c^2 d^2 g \left (36 e^2 f^2+54 d e g f+d^2 g^2\right )\right )-c d g \left (57 a^2 g^2 e^4-2 a c d g (63 e f+5 d g) e^2+3 c^2 d^2 \left (24 e^2 f^2+6 d e g f-d^2 g^2\right )\right ) x\right )}{2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 c d e}+\frac {1}{2} c^2 d^2 e^5 g^2 x \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (11 a e^2 g-c d (d g+18 e f)\right )}{6 c d e}-\frac {1}{3} c^2 d^2 e^5 g^3 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^4 d^4 e^5}-\frac {2 (d+e x) (c d f-a e g)^3}{c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\frac {1}{4} c d e^5 \int \frac {2 e \left (12 a^3 g^3 e^4-a^2 c d g^2 (36 e f+23 d g) e^2-12 c^3 d^3 f^2 (e f+3 d g)+a c^2 d^2 g \left (36 e^2 f^2+54 d e g f+d^2 g^2\right )\right )-c d g \left (57 a^2 g^2 e^4-2 a c d g (63 e f+5 d g) e^2+3 c^2 d^2 \left (24 e^2 f^2+6 d e g f-d^2 g^2\right )\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx+\frac {1}{2} c^2 d^2 e^5 g^2 x \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (11 a e^2 g-c d (d g+18 e f)\right )}{6 c d e}-\frac {1}{3} c^2 d^2 e^5 g^3 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^4 d^4 e^5}-\frac {2 (d+e x) (c d f-a e g)^3}{c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 1160

\(\displaystyle -\frac {\frac {\frac {1}{4} c d e^5 \left (\frac {3 \left (35 a^3 e^6 g^3-15 a^2 c d e^4 g^2 (d g+6 e f)+3 a c^2 d^2 e^2 g \left (-d^2 g^2+12 d e f g+24 e^2 f^2\right )-c^3 d^3 \left (d^3 g^3-6 d^2 e f g^2+24 d e^2 f^2 g+16 e^3 f^3\right )\right ) \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 e}-\frac {g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (57 a^2 e^4 g^2-2 a c d e^2 g (5 d g+63 e f)+3 c^2 d^2 \left (-d^2 g^2+6 d e f g+24 e^2 f^2\right )\right )}{e}\right )+\frac {1}{2} c^2 d^2 e^5 g^2 x \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (11 a e^2 g-c d (d g+18 e f)\right )}{6 c d e}-\frac {1}{3} c^2 d^2 e^5 g^3 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^4 d^4 e^5}-\frac {2 (d+e x) (c d f-a e g)^3}{c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 1092

\(\displaystyle -\frac {\frac {\frac {1}{4} c d e^5 \left (\frac {3 \left (35 a^3 e^6 g^3-15 a^2 c d e^4 g^2 (d g+6 e f)+3 a c^2 d^2 e^2 g \left (-d^2 g^2+12 d e f g+24 e^2 f^2\right )-c^3 d^3 \left (d^3 g^3-6 d^2 e f g^2+24 d e^2 f^2 g+16 e^3 f^3\right )\right ) \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{e}-\frac {g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (57 a^2 e^4 g^2-2 a c d e^2 g (5 d g+63 e f)+3 c^2 d^2 \left (-d^2 g^2+6 d e f g+24 e^2 f^2\right )\right )}{e}\right )+\frac {1}{2} c^2 d^2 e^5 g^2 x \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (11 a e^2 g-c d (d g+18 e f)\right )}{6 c d e}-\frac {1}{3} c^2 d^2 e^5 g^3 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^4 d^4 e^5}-\frac {2 (d+e x) (c d f-a e g)^3}{c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\frac {\frac {1}{4} c d e^5 \left (\frac {3 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right ) \left (35 a^3 e^6 g^3-15 a^2 c d e^4 g^2 (d g+6 e f)+3 a c^2 d^2 e^2 g \left (-d^2 g^2+12 d e f g+24 e^2 f^2\right )-c^3 d^3 \left (d^3 g^3-6 d^2 e f g^2+24 d e^2 f^2 g+16 e^3 f^3\right )\right )}{2 \sqrt {c} \sqrt {d} e^{3/2}}-\frac {g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (57 a^2 e^4 g^2-2 a c d e^2 g (5 d g+63 e f)+3 c^2 d^2 \left (-d^2 g^2+6 d e f g+24 e^2 f^2\right )\right )}{e}\right )+\frac {1}{2} c^2 d^2 e^5 g^2 x \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (11 a e^2 g-c d (d g+18 e f)\right )}{6 c d e}-\frac {1}{3} c^2 d^2 e^5 g^3 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^4 d^4 e^5}-\frac {2 (d+e x) (c d f-a e g)^3}{c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

Input:

Int[((d + e*x)^2*(f + g*x)^3)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2 
),x]
 

Output:

(-2*(c*d*f - a*e*g)^3*(d + e*x))/(c^4*d^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + 
 c*d*e*x^2]) - (-1/3*(c^2*d^2*e^5*g^3*x^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + 
 c*d*e*x^2]) + ((c^2*d^2*e^5*g^2*(11*a*e^2*g - c*d*(18*e*f + d*g))*x*Sqrt[ 
a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/2 + (c*d*e^5*(-((g*(57*a^2*e^4*g^2 
 - 2*a*c*d*e^2*g*(63*e*f + 5*d*g) + 3*c^2*d^2*(24*e^2*f^2 + 6*d*e*f*g - d^ 
2*g^2))*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/e) + (3*(35*a^3*e^6*g 
^3 - 15*a^2*c*d*e^4*g^2*(6*e*f + d*g) + 3*a*c^2*d^2*e^2*g*(24*e^2*f^2 + 12 
*d*e*f*g - d^2*g^2) - c^3*d^3*(16*e^3*f^3 + 24*d*e^2*f^2*g - 6*d^2*e*f*g^2 
 + d^3*g^3))*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e 
]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(2*Sqrt[c]*Sqrt[d]*e^(3/2 
))))/4)/(6*c*d*e))/(c^4*d^4*e^5)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1160

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol 
] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b 
*e)/(2*c)   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] 
 && NeQ[p, -1]
 

rule 1211

Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* 
(x_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[-2*(2*c*d - b*e)^(m - 2)*(c*( 
e*f + d*g) - b*e*g)^n*((d + e*x)/(c^(m + n - 1)*e^(n - 1)*Sqrt[a + b*x + c* 
x^2])), x] + Simp[1/(c^(m + n - 1)*e^(n - 2))   Int[ExpandToSum[((2*c*d - b 
*e)^(m - 1)*(c*(e*f + d*g) - b*e*g)^n - c^(m + n - 1)*e^n*(d + e*x)^(m - 1) 
*(f + g*x)^n)/(c*d - b*e - c*e*x), x]/Sqrt[a + b*x + c*x^2], x], x] /; Free 
Q[{a, b, c, d, e, f, g}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[m, 0] 
&& IGtQ[n, 0]
 

rule 2192

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = 
Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + 
 c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1))   Int[(a 
+ b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b 
*e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c 
, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3170\) vs. \(2(423)=846\).

Time = 3.31 (sec) , antiderivative size = 3171, normalized size of antiderivative = 7.00

method	result	size
default	\(\text {Expression too large to display}\)	\(3171\)

Input:

int((e*x+d)^2*(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2),x,method=_ 
RETURNVERBOSE)
 

Output:

2*d^2*f^3*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+( 
a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)+e*g^2*(2*d*g+3*e*f)*(1/2*x^3/d/e/c/(a*d*e+ 
(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-5/4*(a*e^2+c*d^2)/d/e/c*(x^2/d/e/c/(a*d*e 
+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-3/2*(a*e^2+c*d^2)/d/e/c*(-x/d/e/c/(a*d*e 
+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-1/2*(a*e^2+c*d^2)/d/e/c*(-1/d/e/c/(a*d*e 
+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-(a*e^2+c*d^2)/d/e/c*(2*c*d*e*x+a*e^2+c*d 
^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2 
))+1/d/e/c*ln((1/2*a*e^2+1/2*c*d^2+c*d*x*e)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c* 
d^2)*x+c*d*x^2*e)^(1/2))/(d*e*c)^(1/2))-2*a/c*(-1/d/e/c/(a*d*e+(a*e^2+c*d^ 
2)*x+c*d*x^2*e)^(1/2)-(a*e^2+c*d^2)/d/e/c*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d 
^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)))-3/2*a/c* 
(-x/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-1/2*(a*e^2+c*d^2)/d/e/c* 
(-1/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-(a*e^2+c*d^2)/d/e/c*(2*c 
*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x 
+c*d*x^2*e)^(1/2))+1/d/e/c*ln((1/2*a*e^2+1/2*c*d^2+c*d*x*e)/(d*e*c)^(1/2)+ 
(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))/(d*e*c)^(1/2)))+d*f^2*(3*d*g+2*e* 
f)*(-1/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-(a*e^2+c*d^2)/d/e/c*( 
2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2 
)*x+c*d*x^2*e)^(1/2))+g*(d^2*g^2+6*d*e*f*g+3*e^2*f^2)*(x^2/d/e/c/(a*d*e+(a 
*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-3/2*(a*e^2+c*d^2)/d/e/c*(-x/d/e/c/(a*d*e...
 

Fricas [A] (verification not implemented)

Time = 1.65 (sec) , antiderivative size = 1402, normalized size of antiderivative = 3.09 \[ \int \frac {(d+e x)^2 (f+g x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^2*(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, a 
lgorithm="fricas")
 

Output:

[-1/96*(3*(16*a*c^3*d^3*e^4*f^3 + 24*(a*c^3*d^4*e^3 - 3*a^2*c^2*d^2*e^5)*f 
^2*g - 6*(a*c^3*d^5*e^2 + 6*a^2*c^2*d^3*e^4 - 15*a^3*c*d*e^6)*f*g^2 + (a*c 
^3*d^6*e + 3*a^2*c^2*d^4*e^3 + 15*a^3*c*d^2*e^5 - 35*a^4*e^7)*g^3 + (16*c^ 
4*d^4*e^3*f^3 + 24*(c^4*d^5*e^2 - 3*a*c^3*d^3*e^4)*f^2*g - 6*(c^4*d^6*e + 
6*a*c^3*d^4*e^3 - 15*a^2*c^2*d^2*e^5)*f*g^2 + (c^4*d^7 + 3*a*c^3*d^5*e^2 + 
 15*a^2*c^2*d^3*e^4 - 35*a^3*c*d*e^6)*g^3)*x)*sqrt(c*d*e)*log(8*c^2*d^2*e^ 
2*x^2 + c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4 - 4*sqrt(c*d*e*x^2 + a*d*e + (c* 
d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(c*d*e) + 8*(c^2*d^3*e + a 
*c*d*e^3)*x) - 4*(8*c^4*d^4*e^3*g^3*x^3 - 48*c^4*d^4*e^3*f^3 + 216*a*c^3*d 
^3*e^4*f^2*g + 18*(a*c^3*d^4*e^3 - 15*a^2*c^2*d^2*e^5)*f*g^2 - (3*a*c^3*d^ 
5*e^2 + 10*a^2*c^2*d^3*e^4 - 105*a^3*c*d*e^6)*g^3 + 2*(18*c^4*d^4*e^3*f*g^ 
2 + (c^4*d^5*e^2 - 7*a*c^3*d^3*e^4)*g^3)*x^2 + (72*c^4*d^4*e^3*f^2*g + 18* 
(c^4*d^5*e^2 - 5*a*c^3*d^3*e^4)*f*g^2 - (3*c^4*d^6*e + 8*a*c^3*d^4*e^3 - 3 
5*a^2*c^2*d^2*e^5)*g^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c 
^6*d^6*e^3*x + a*c^5*d^5*e^4), -1/48*(3*(16*a*c^3*d^3*e^4*f^3 + 24*(a*c^3* 
d^4*e^3 - 3*a^2*c^2*d^2*e^5)*f^2*g - 6*(a*c^3*d^5*e^2 + 6*a^2*c^2*d^3*e^4 
- 15*a^3*c*d*e^6)*f*g^2 + (a*c^3*d^6*e + 3*a^2*c^2*d^4*e^3 + 15*a^3*c*d^2* 
e^5 - 35*a^4*e^7)*g^3 + (16*c^4*d^4*e^3*f^3 + 24*(c^4*d^5*e^2 - 3*a*c^3*d^ 
3*e^4)*f^2*g - 6*(c^4*d^6*e + 6*a*c^3*d^4*e^3 - 15*a^2*c^2*d^2*e^5)*f*g^2 
+ (c^4*d^7 + 3*a*c^3*d^5*e^2 + 15*a^2*c^2*d^3*e^4 - 35*a^3*c*d*e^6)*g^3...
 

Sympy [F]

\[ \int \frac {(d+e x)^2 (f+g x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right )^{2} \left (f + g x\right )^{3}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((e*x+d)**2*(g*x+f)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2 
),x)
 

Output:

Integral((d + e*x)**2*(f + g*x)**3/((d + e*x)*(a*e + c*d*x))**(3/2), x)
 

Maxima [F(-2)]

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

integrate((e*x+d)^2*(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, a 
lgorithm="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(a*e^2-c*d^2>0)', see `assume?` f 
or more de
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^2 (f+g x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((e*x+d)^2*(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, a 
lgorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{%%{[%%%{1,[5,5,0]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[2,0, 
0,0]%%%}+
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^2 (f+g x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^3\,{\left (d+e\,x\right )}^2}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \] Input:

int(((f + g*x)^3*(d + e*x)^2)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2 
),x)
 

Output:

int(((f + g*x)^3*(d + e*x)^2)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2 
), x)
 

Reduce [F]

\[ \int \frac {(d+e x)^2 (f+g x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {\left (e x +d \right )^{2} \left (g x +f \right )^{3}}{{\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {3}{2}}}d x \] Input:

int((e*x+d)^2*(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)
 

Output:

int((e*x+d)^2*(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)