\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x) (f+g x)} \, dx\) [271]

Optimal result

Integrand size = 44, antiderivative size = 294 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x) (f+g x)} \, dx=\frac {\left (5 a e^2 g-c d (4 e f-d g)+2 c d e g x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e g^2}+\frac {\left (3 a^2 e^4 g^2-6 a c d e^2 g (2 e f-d g)+c^2 d^2 \left (8 e^2 f^2-4 d e f g-d^2 g^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 \sqrt {c} \sqrt {d} e^{3/2} g^3}-\frac {2 \sqrt {e f-d g} (c d f-a e g)^{3/2} \text {arctanh}\left (\frac {\sqrt {c d f-a e g} (d+e x)}{\sqrt {e f-d g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{g^3} \] Output:

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

Mathematica [A] (verified)

Time = 1.21 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x) (f+g x)} \, dx=\frac {\sqrt {a e+c d x} \sqrt {d+e x} \left (\sqrt {c} \sqrt {d} \sqrt {e} \left (g \sqrt {a e+c d x} \sqrt {d+e x} \left (5 a e^2 g+c d (-4 e f+d g+2 e g x)\right )-8 e \sqrt {-e f+d g} (c d f-a e g)^{3/2} \arctan \left (\frac {\sqrt {c d f-a e g} \sqrt {d+e x}}{\sqrt {-e f+d g} \sqrt {a e+c d x}}\right )\right )+\left (3 a^2 e^4 g^2+6 a c d e^2 g (-2 e f+d g)-c^2 d^2 \left (-8 e^2 f^2+4 d e f g+d^2 g^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )\right )}{4 \sqrt {c} \sqrt {d} e^{3/2} g^3 \sqrt {(a e+c d x) (d+e x)}} \] Input:

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

Output:

(Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(Sqrt[c]*Sqrt[d]*Sqrt[e]*(g*Sqrt[a*e + c* 
d*x]*Sqrt[d + e*x]*(5*a*e^2*g + c*d*(-4*e*f + d*g + 2*e*g*x)) - 8*e*Sqrt[- 
(e*f) + d*g]*(c*d*f - a*e*g)^(3/2)*ArcTan[(Sqrt[c*d*f - a*e*g]*Sqrt[d + e* 
x])/(Sqrt[-(e*f) + d*g]*Sqrt[a*e + c*d*x])]) + (3*a^2*e^4*g^2 + 6*a*c*d*e^ 
2*g*(-2*e*f + d*g) - c^2*d^2*(-8*e^2*f^2 + 4*d*e*f*g + d^2*g^2))*ArcTanh[( 
Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x])]))/(4*Sqrt[c]*S 
qrt[d]*e^(3/2)*g^3*Sqrt[(a*e + c*d*x)*(d + e*x)])
 

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.22, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1215, 1231, 27, 1269, 1092, 219, 1154, 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.

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

\(\Big \downarrow \) 1215

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

\(\Big \downarrow \) 1231

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (5 a e^2 g-c d (4 e f-d g)+2 c d e g x\right )}{4 e g^2}-\frac {-\frac {16 e (e f-d g) (c d f-a e g)^2 \int \frac {1}{4 (e f-d g) (c d f-a e g)-\frac {\left (c f d^2+a e (e f-2 d g)-\left (a e^2 g-c d (2 e f-d g)\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\left (-\frac {c f d^2+a e (e f-2 d g)-\left (a e^2 g-c d (2 e f-d g)\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}\right )}{g}-\frac {\text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right ) \left (3 a^2 e^4 g^2-6 a c d e^2 g (2 e f-d g)+c^2 d^2 \left (-d^2 g^2-4 d e f g+8 e^2 f^2\right )\right )}{\sqrt {c} \sqrt {d} \sqrt {e} g}}{8 e g^2}\)

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

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 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, 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 1231

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

rule 1269

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1239\) vs. \(2(264)=528\).

Time = 2.29 (sec) , antiderivative size = 1240, normalized size of antiderivative = 4.22

method	result	size
default	\(\text {Expression too large to display}\)	\(1240\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)/(g*x+f),x, algor 
ithm="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(d*g-e*f>0)', see `assume?` for m 
ore detail
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.80 (sec) , antiderivative size = 1106, normalized size of antiderivative = 3.76 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x) (f+g x)} \, dx =\text {Too large to display} \] Input:

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

Output:

(4*sqrt(d*g - e*f)*sqrt(a*e*g - c*d*f)*log(sqrt(g)*sqrt(e)*sqrt(a*e + c*d* 
x) - sqrt(2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(d*g - e*f)*sqrt(a*e*g - c*d*f) + 
a*e**2*g + c*d**2*g - 2*c*d*e*f) + sqrt(g)*sqrt(d)*sqrt(c)*sqrt(d + e*x))* 
a*c*d*e**3*g - 4*sqrt(d*g - e*f)*sqrt(a*e*g - c*d*f)*log(sqrt(g)*sqrt(e)*s 
qrt(a*e + c*d*x) - sqrt(2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(d*g - e*f)*sqrt(a*e 
*g - c*d*f) + a*e**2*g + c*d**2*g - 2*c*d*e*f) + sqrt(g)*sqrt(d)*sqrt(c)*s 
qrt(d + e*x))*c**2*d**2*e**2*f + 4*sqrt(d*g - e*f)*sqrt(a*e*g - c*d*f)*log 
(sqrt(g)*sqrt(e)*sqrt(a*e + c*d*x) + sqrt(2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(d 
*g - e*f)*sqrt(a*e*g - c*d*f) + a*e**2*g + c*d**2*g - 2*c*d*e*f) + sqrt(g) 
*sqrt(d)*sqrt(c)*sqrt(d + e*x))*a*c*d*e**3*g - 4*sqrt(d*g - e*f)*sqrt(a*e* 
g - c*d*f)*log(sqrt(g)*sqrt(e)*sqrt(a*e + c*d*x) + sqrt(2*sqrt(e)*sqrt(d)* 
sqrt(c)*sqrt(d*g - e*f)*sqrt(a*e*g - c*d*f) + a*e**2*g + c*d**2*g - 2*c*d* 
e*f) + sqrt(g)*sqrt(d)*sqrt(c)*sqrt(d + e*x))*c**2*d**2*e**2*f - 4*sqrt(d* 
g - e*f)*sqrt(a*e*g - c*d*f)*log(2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(d*g - e*f) 
*sqrt(a*e*g - c*d*f) + 2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(d + e*x)*sqrt(a*e + 
c*d*x)*g + 2*c*d*e*f + 2*c*d*e*g*x)*a*c*d*e**3*g + 4*sqrt(d*g - e*f)*sqrt( 
a*e*g - c*d*f)*log(2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(d*g - e*f)*sqrt(a*e*g - 
c*d*f) + 2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(d + e*x)*sqrt(a*e + c*d*x)*g + 2*c 
*d*e*f + 2*c*d*e*g*x)*c**2*d**2*e**2*f + 5*sqrt(d + e*x)*sqrt(a*e + c*d*x) 
*a*c*d*e**3*g**2 + sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**2*d**3*e*g**2 - 4...