\(\int \frac {(f+g x)^2 \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{d+e x} \, dx\) [260]

Optimal result

Integrand size = 44, antiderivative size = 314 \[ \int \frac {(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\frac {(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 e}-\frac {\left (3 a^2 e^4 g^2-4 a c d e^2 g (3 e f-d g)-c^2 d^2 \left (16 e^2 f^2-36 d e f g+15 d^2 g^2\right )-2 c d e g \left (a e^2 g+c d (4 e f-5 d g)\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 c^2 d^2 e^3}-\frac {\left (c d^2-a e^2\right ) \left (a^2 e^4 g^2-2 a c d e^2 g (2 e f-d g)+c^2 d^2 \left (8 e^2 f^2-12 d e f g+5 d^2 g^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 c^{5/2} d^{5/2} e^{7/2}} \] Output:

1/3*(g*x+f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/e-1/24*(3*a^2*e^4*g^ 
2-4*a*c*d*e^2*g*(-d*g+3*e*f)-c^2*d^2*(15*d^2*g^2-36*d*e*f*g+16*e^2*f^2)-2* 
c*d*e*g*(a*e^2*g+c*d*(-5*d*g+4*e*f))*x)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^ 
(1/2)/c^2/d^2/e^3-1/8*(-a*e^2+c*d^2)*(a^2*e^4*g^2-2*a*c*d*e^2*g*(-d*g+2*e* 
f)+c^2*d^2*(5*d^2*g^2-12*d*e*f*g+8*e^2*f^2))*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^(5/2)/d^(5/2)/e^(7/2 
)
 

Mathematica [A] (verified)

Time = 11.33 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.17 \[ \int \frac {(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (-\frac {g \left (3 a e^2 g+c d (-8 e f+5 d g)\right ) (a e+c d x)}{4 c d e}+g (a e+c d x) (f+g x)+\frac {3 \left (a^2 e^4 g^2+2 a c d e^2 g (-2 e f+d g)+c^2 d^2 \left (8 e^2 f^2-12 d e f g+5 d^2 g^2\right )\right ) \left (\sqrt {c d} \sqrt {e} \sqrt {c d^2-a e^2} (a e+c d x) \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}-\sqrt {c} \sqrt {d} \left (c d^2-a e^2\right ) \sqrt {a e+c d x} \text {arcsinh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d} \sqrt {c d^2-a e^2}}\right )\right )}{8 (c d)^{3/2} e^{5/2} \sqrt {c d^2-a e^2} (a e+c d x) \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}}\right )}{3 c d e} \] Input:

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

Output:

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-1/4*(g*(3*a*e^2*g + c*d*(-8*e*f + 5*d*g)) 
*(a*e + c*d*x))/(c*d*e) + g*(a*e + c*d*x)*(f + g*x) + (3*(a^2*e^4*g^2 + 2* 
a*c*d*e^2*g*(-2*e*f + d*g) + c^2*d^2*(8*e^2*f^2 - 12*d*e*f*g + 5*d^2*g^2)) 
*(Sqrt[c*d]*Sqrt[e]*Sqrt[c*d^2 - a*e^2]*(a*e + c*d*x)*Sqrt[(c*d*(d + e*x)) 
/(c*d^2 - a*e^2)] - Sqrt[c]*Sqrt[d]*(c*d^2 - a*e^2)*Sqrt[a*e + c*d*x]*ArcS 
inh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a* 
e^2])]))/(8*(c*d)^(3/2)*e^(5/2)*Sqrt[c*d^2 - a*e^2]*(a*e + c*d*x)*Sqrt[(c* 
d*(d + e*x))/(c*d^2 - a*e^2)])))/(3*c*d*e)
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1215, 1236, 27, 1225, 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.

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

\(\Big \downarrow \) 1215

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

\(\Big \downarrow \) 1236

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1225

\(\displaystyle \frac {(f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 e}-\frac {\frac {3 \left (c d^2-a e^2\right ) \left (a^2 e^4 g^2-2 a c d e^2 g (2 e f-d g)+c^2 d^2 \left (5 d^2 g^2-12 d e f g+8 e^2 f^2\right )\right ) \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 c^2 d^2 e^2}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (3 a^2 e^4 g^2-2 c d e g x \left (a e^2 g+c d (4 e f-5 d g)\right )-4 a c d e^2 g (3 e f-d g)-2 c^2 \left (\frac {15 d^4 g^2}{2}-18 d^3 e f g+8 d^2 e^2 f^2\right )\right )}{4 c^2 d^2 e^2}}{6 e}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {(f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 e}-\frac {\frac {3 \left (c d^2-a e^2\right ) \left (a^2 e^4 g^2-2 a c d e^2 g (2 e f-d g)+c^2 d^2 \left (5 d^2 g^2-12 d e f g+8 e^2 f^2\right )\right ) \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 c^2 d^2 e^2}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (3 a^2 e^4 g^2-2 c d e g x \left (a e^2 g+c d (4 e f-5 d g)\right )-4 a c d e^2 g (3 e f-d g)-2 c^2 \left (\frac {15 d^4 g^2}{2}-18 d^3 e f g+8 d^2 e^2 f^2\right )\right )}{4 c^2 d^2 e^2}}{6 e}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {(f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 e}-\frac {\frac {3 \left (c d^2-a e^2\right ) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right ) \left (a^2 e^4 g^2-2 a c d e^2 g (2 e f-d g)+c^2 d^2 \left (5 d^2 g^2-12 d e f g+8 e^2 f^2\right )\right )}{8 c^{5/2} d^{5/2} e^{5/2}}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (3 a^2 e^4 g^2-2 c d e g x \left (a e^2 g+c d (4 e f-5 d g)\right )-4 a c d e^2 g (3 e f-d g)-2 c^2 \left (\frac {15 d^4 g^2}{2}-18 d^3 e f g+8 d^2 e^2 f^2\right )\right )}{4 c^2 d^2 e^2}}{6 e}\)

Input:

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

Output:

((f + g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*e) - (((3*a^2 
*e^4*g^2 - 4*a*c*d*e^2*g*(3*e*f - d*g) - 2*c^2*(8*d^2*e^2*f^2 - 18*d^3*e*f 
*g + (15*d^4*g^2)/2) - 2*c*d*e*g*(a*e^2*g + c*d*(4*e*f - 5*d*g))*x)*Sqrt[a 
*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*c^2*d^2*e^2) + (3*(c*d^2 - a*e^2 
)*(a^2*e^4*g^2 - 2*a*c*d*e^2*g*(2*e*f - d*g) + c^2*d^2*(8*e^2*f^2 - 12*d*e 
*f*g + 5*d^2*g^2))*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])])/(8*c^(5/2)*d^(5/2)* 
e^(5/2)))/(6*e)
 

Defintions of rubi rules used

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 1215

Int[(((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_))/( 
(d_) + (e_.)*(x_)), x_Symbol] :> Int[(a/d + c*(x/e))*(f + g*x)^n*(a + b*x + 
 c*x^2)^(p - 1), x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[c*d^2 - 
 b*d*e + a*e^2, 0] && GtQ[p, 0]
 

rule 1225

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( 
x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 
 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), 
 x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p 
+ 3))/(2*c^2*(2*p + 3))   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, p}, x] &&  !LeQ[p, -1]
 

rule 1236

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 
1)/(c*(m + 2*p + 2))), x] + Simp[1/(c*(m + 2*p + 2))   Int[(d + e*x)^(m - 1 
)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m 
*(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ 
{a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege 
rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 0])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(690\) vs. \(2(290)=580\).

Time = 2.02 (sec) , antiderivative size = 691, normalized size of antiderivative = 2.20

method	result	size
default	\(\frac {\left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) \left (\sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}+\frac {\left (a \,e^{2}-c \,d^{2}\right ) \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+d e c \left (x +\frac {d}{e}\right )}{\sqrt {d e c}}+\sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{2 \sqrt {d e c}}\right )}{e^{3}}-\frac {g \left (d g \left (\frac {\left (2 c d x e +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{8 d e c \sqrt {d e c}}\right )-2 e f \left (\frac {\left (2 c d x e +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{8 d e c \sqrt {d e c}}\right )-e g \left (\frac {{\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}{3 d e c}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {\left (2 c d x e +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{8 d e c \sqrt {d e c}}\right )}{2 d e c}\right )\right )}{e^{2}}\)	\(691\)

Input:

int((g*x+f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/(e*x+d),x,method=_RE 
TURNVERBOSE)
 

Output:

(d^2*g^2-2*d*e*f*g+e^2*f^2)/e^3*((d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^( 
1/2)+1/2*(a*e^2-c*d^2)*ln((1/2*a*e^2-1/2*c*d^2+d*e*c*(x+d/e))/(d*e*c)^(1/2 
)+(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(d*e*c)^(1/2))-g/e^2*(d*g 
*(1/4*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/c/d/ 
e+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/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 
*e*f*(1/4*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/ 
c/d/e+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/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 
))-e*g*(1/3*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/d/e/c-1/2*(a*e^2+c*d^2 
)/d/e/c*(1/4*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/ 
2)/c/d/e+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/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))))
 

Fricas [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 812, normalized size of antiderivative = 2.59 \[ \int \frac {(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx =\text {Too large to display} \] Input:

integrate((g*x+f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d),x, alg 
orithm="fricas")
 

Output:

[-1/96*(3*sqrt(c*d*e)*(8*(c^3*d^4*e^2 - a*c^2*d^2*e^4)*f^2 - 4*(3*c^3*d^5* 
e - 2*a*c^2*d^3*e^3 - a^2*c*d*e^5)*f*g + (5*c^3*d^6 - 3*a*c^2*d^4*e^2 - a^ 
2*c*d^2*e^4 - a^3*e^6)*g^2)*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^3*d^3*e^ 
3*g^2*x^2 + 24*c^3*d^3*e^3*f^2 - 12*(3*c^3*d^4*e^2 - a*c^2*d^2*e^4)*f*g + 
(15*c^3*d^5*e - 4*a*c^2*d^3*e^3 - 3*a^2*c*d*e^5)*g^2 + 2*(12*c^3*d^3*e^3*f 
*g - (5*c^3*d^4*e^2 - a*c^2*d^2*e^4)*g^2)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d 
^2 + a*e^2)*x))/(c^3*d^3*e^4), 1/48*(3*sqrt(-c*d*e)*(8*(c^3*d^4*e^2 - a*c^ 
2*d^2*e^4)*f^2 - 4*(3*c^3*d^5*e - 2*a*c^2*d^3*e^3 - a^2*c*d*e^5)*f*g + (5* 
c^3*d^6 - 3*a*c^2*d^4*e^2 - a^2*c*d^2*e^4 - a^3*e^6)*g^2)*arctan(1/2*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)/(c^2*d^2*e^2*x^2 + a*c*d^2*e^2 + (c^2*d^3*e + a*c*d*e^3)*x)) + 2*(8* 
c^3*d^3*e^3*g^2*x^2 + 24*c^3*d^3*e^3*f^2 - 12*(3*c^3*d^4*e^2 - a*c^2*d^2*e 
^4)*f*g + (15*c^3*d^5*e - 4*a*c^2*d^3*e^3 - 3*a^2*c*d*e^5)*g^2 + 2*(12*c^3 
*d^3*e^3*f*g - (5*c^3*d^4*e^2 - a*c^2*d^2*e^4)*g^2)*x)*sqrt(c*d*e*x^2 + a* 
d*e + (c*d^2 + a*e^2)*x))/(c^3*d^3*e^4)]
 

Sympy [F]

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

integrate((g*x+f)**2*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d),x 
)
 

Output:

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

Maxima [F(-2)]

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

integrate((g*x+f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d),x, alg 
orithm="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(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.17 \[ \int \frac {(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\frac {1}{24} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (\frac {4 \, g^{2} x}{e} + \frac {12 \, c^{2} d^{2} e^{2} f g - 5 \, c^{2} d^{3} e g^{2} + a c d e^{3} g^{2}}{c^{2} d^{2} e^{3}}\right )} x + \frac {24 \, c^{2} d^{2} e^{2} f^{2} - 36 \, c^{2} d^{3} e f g + 12 \, a c d e^{3} f g + 15 \, c^{2} d^{4} g^{2} - 4 \, a c d^{2} e^{2} g^{2} - 3 \, a^{2} e^{4} g^{2}}{c^{2} d^{2} e^{3}}\right )} + \frac {{\left (8 \, c^{3} d^{4} e^{2} f^{2} - 8 \, a c^{2} d^{2} e^{4} f^{2} - 12 \, c^{3} d^{5} e f g + 8 \, a c^{2} d^{3} e^{3} f g + 4 \, a^{2} c d e^{5} f g + 5 \, c^{3} d^{6} g^{2} - 3 \, a c^{2} d^{4} e^{2} g^{2} - a^{2} c d^{2} e^{4} g^{2} - a^{3} e^{6} g^{2}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{16 \, \sqrt {c d e} c^{2} d^{2} e^{3}} \] Input:

integrate((g*x+f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d),x, alg 
orithm="giac")
 

Output:

1/24*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*g^2*x/e + (12*c^2*d 
^2*e^2*f*g - 5*c^2*d^3*e*g^2 + a*c*d*e^3*g^2)/(c^2*d^2*e^3))*x + (24*c^2*d 
^2*e^2*f^2 - 36*c^2*d^3*e*f*g + 12*a*c*d*e^3*f*g + 15*c^2*d^4*g^2 - 4*a*c* 
d^2*e^2*g^2 - 3*a^2*e^4*g^2)/(c^2*d^2*e^3)) + 1/16*(8*c^3*d^4*e^2*f^2 - 8* 
a*c^2*d^2*e^4*f^2 - 12*c^3*d^5*e*f*g + 8*a*c^2*d^3*e^3*f*g + 4*a^2*c*d*e^5 
*f*g + 5*c^3*d^6*g^2 - 3*a*c^2*d^4*e^2*g^2 - a^2*c*d^2*e^4*g^2 - a^3*e^6*g 
^2)*log(abs(-c*d^2 - a*e^2 - 2*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 
 + c*d^2*x + a*e^2*x + a*d*e))))/(sqrt(c*d*e)*c^2*d^2*e^3)
 

Mupad [F(-1)]

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

int(((f + g*x)^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x), 
x)
 

Output:

int(((f + g*x)^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x), 
 x)
 

Reduce [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 846, normalized size of antiderivative = 2.69 \[ \int \frac {(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx =\text {Too large to display} \] Input:

int((g*x+f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d),x)
 

Output:

( - 3*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c*d*e**5*g**2 - 4*sqrt(d + e*x) 
*sqrt(a*e + c*d*x)*a*c**2*d**3*e**3*g**2 + 12*sqrt(d + e*x)*sqrt(a*e + c*d 
*x)*a*c**2*d**2*e**4*f*g + 2*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**2*d**2*e 
**4*g**2*x + 15*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**3*d**5*e*g**2 - 36*sqrt 
(d + e*x)*sqrt(a*e + c*d*x)*c**3*d**4*e**2*f*g - 10*sqrt(d + e*x)*sqrt(a*e 
 + c*d*x)*c**3*d**4*e**2*g**2*x + 24*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**3* 
d**3*e**3*f**2 + 24*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**3*d**3*e**3*f*g*x + 
 8*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**3*d**3*e**3*g**2*x**2 + 3*sqrt(e)*sq 
rt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e* 
x))/sqrt(a*e**2 - c*d**2))*a**3*e**6*g**2 + 3*sqrt(e)*sqrt(d)*sqrt(c)*log( 
(sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - 
c*d**2))*a**2*c*d**2*e**4*g**2 - 12*sqrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*s 
qrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a 
**2*c*d*e**5*f*g + 9*sqrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x 
) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a*c**2*d**4*e**2 
*g**2 - 24*sqrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d 
)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a*c**2*d**3*e**3*f*g + 24* 
sqrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*s 
qrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a*c**2*d**2*e**4*f**2 - 15*sqrt(e)*sq 
rt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d +...