\(\int \frac {(f+g x) (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{d+e x} \, dx\) [269]

Optimal result

Integrand size = 42, antiderivative size = 267 \[ \int \frac {(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\frac {\left (c d^2-a e^2\right ) \left (3 a e^2 g-c d (8 e f-5 d g)\right ) \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c^2 d^2 e^3}+\frac {\left (3 a e^2 g+c d (8 e f-5 d g)+6 c d e g x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 c d e^2}-\frac {\left (c d^2-a e^2\right )^3 \left (3 a e^2 g-c d (8 e f-5 d g)\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{64 c^{5/2} d^{5/2} e^{7/2}} \] Output:

1/64*(-a*e^2+c*d^2)*(3*a*e^2*g-c*d*(-5*d*g+8*e*f))*(2*c*d*e*x+a*e^2+c*d^2) 
*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^2/d^2/e^3+1/24*(3*a*e^2*g+c*d*( 
-5*d*g+8*e*f)+6*c*d*e*g*x)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c/d/e^2 
-1/64*(-a*e^2+c*d^2)^3*(3*a*e^2*g-c*d*(-5*d*g+8*e*f))*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 = 1.06 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.12 \[ \int \frac {(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\frac {\left (c d^2-a e^2\right )^3 \sqrt {(a e+c d x) (d+e x)} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \left (-9 a^3 e^6 g+3 a^2 c d e^4 (8 e f+3 d g+2 e g x)+c^3 d^3 \left (15 d^3 g+8 d e^2 x (2 f+g x)+16 e^3 x^2 (4 f+3 g x)-2 d^2 e (12 f+5 g x)\right )+a c^2 d^2 e^2 \left (-31 d^2 g+4 d e (16 f+5 g x)+8 e^2 x (14 f+9 g x)\right )\right )}{\left (c d^2-a e^2\right )^3}-\frac {3 \left (3 a e^2 g+c d (-8 e f+5 d g)\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{192 c^{5/2} d^{5/2} e^{7/2}} \] Input:

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

Output:

((c*d^2 - a*e^2)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*((Sqrt[c]*Sqrt[d]*Sqrt[e] 
*(-9*a^3*e^6*g + 3*a^2*c*d*e^4*(8*e*f + 3*d*g + 2*e*g*x) + c^3*d^3*(15*d^3 
*g + 8*d*e^2*x*(2*f + g*x) + 16*e^3*x^2*(4*f + 3*g*x) - 2*d^2*e*(12*f + 5* 
g*x)) + a*c^2*d^2*e^2*(-31*d^2*g + 4*d*e*(16*f + 5*g*x) + 8*e^2*x*(14*f + 
9*g*x))))/(c*d^2 - a*e^2)^3 - (3*(3*a*e^2*g + c*d*(-8*e*f + 5*d*g))*ArcTan 
h[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x])])/(Sqrt[a*e 
+ c*d*x]*Sqrt[d + e*x])))/(192*c^(5/2)*d^(5/2)*e^(7/2))
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {1215, 1225, 1087, 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.

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

\(\Big \downarrow \) 1215

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

\(\Big \downarrow \) 1225

\(\displaystyle \frac {\left (c d^2-a e^2\right ) \left (3 a e^2 g-c d (8 e f-5 d g)\right ) \int \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{16 c d e^2}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \left (3 a e^2 g+c d (8 e f-5 d g)+6 c d e g x\right )}{24 c d e^2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\left (c d^2-a e^2\right ) \left (3 a e^2 g-c d (8 e f-5 d g)\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 c d e}\right )}{16 c d e^2}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \left (3 a e^2 g+c d (8 e f-5 d g)+6 c d e g x\right )}{24 c d e^2}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\left (c d^2-a e^2\right ) \left (3 a e^2 g-c d (8 e f-5 d g)\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 c d e}\right )}{16 c d e^2}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \left (3 a e^2 g+c d (8 e f-5 d g)+6 c d e g x\right )}{24 c d e^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\left (c d^2-a e^2\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{3/2} d^{3/2} e^{3/2}}\right ) \left (3 a e^2 g-c d (8 e f-5 d g)\right )}{16 c d e^2}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \left (3 a e^2 g+c d (8 e f-5 d g)+6 c d e g x\right )}{24 c d e^2}\)

Input:

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

Output:

((3*a*e^2*g + c*d*(8*e*f - 5*d*g) + 6*c*d*e*g*x)*(a*d*e + (c*d^2 + a*e^2)* 
x + c*d*e*x^2)^(3/2))/(24*c*d*e^2) + ((c*d^2 - a*e^2)*(3*a*e^2*g - c*d*(8* 
e*f - 5*d*g))*(((c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x 
 + c*d*e*x^2])/(4*c*d*e) - ((c*d^2 - a*e^2)^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^(3/2)*d^(3/2)*e^(3/2))))/(16*c*d*e^2)
 

Defintions of rubi rules used

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 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(490\) vs. \(2(243)=486\).

Time = 1.81 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.84

method	result	size
default	\(\frac {g \left (\frac {\left (2 c d x e +a \,e^{2}+c \,d^{2}\right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}{8 c d e}+\frac {3 \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 c d x e +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{8 d e c \sqrt {d e c}}\right )}{16 d e c}\right )}{e}-\frac {\left (d g -e f \right ) \left (\frac {\left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{3}+\frac {\left (a \,e^{2}-c \,d^{2}\right ) \left (\frac {\left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right ) \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{4 d e c}-\frac {\left (a \,e^{2}-c \,d^{2}\right )^{2} \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+d e c \left (x +\frac {d}{e}\right )}{\sqrt {d e c}}+\sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{8 d e c \sqrt {d e c}}\right )}{2}\right )}{e^{2}}\)	\(491\)

Input:

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

Output:

g/e*(1/8*(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)^(3/2)/c 
/d/e+3/16*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^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)))-(d*g-e*f)/e^2*(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)/d/e/c*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-1/8*( 
a*e^2-c*d^2)^2/d/e/c*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)))
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 960, normalized size of antiderivative = 3.60 \[ \int \frac {(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/768*(3*sqrt(c*d*e)*(8*(c^4*d^7*e - 3*a*c^3*d^5*e^3 + 3*a^2*c^2*d^3*e^5 
- a^3*c*d*e^7)*f - (5*c^4*d^8 - 12*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 4*a 
^3*c*d^2*e^6 - 3*a^4*e^8)*g)*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*(48*c^4*d^4* 
e^4*g*x^3 + 8*(8*c^4*d^4*e^4*f + (c^4*d^5*e^3 + 9*a*c^3*d^3*e^5)*g)*x^2 - 
8*(3*c^4*d^6*e^2 - 8*a*c^3*d^4*e^4 - 3*a^2*c^2*d^2*e^6)*f + (15*c^4*d^7*e 
- 31*a*c^3*d^5*e^3 + 9*a^2*c^2*d^3*e^5 - 9*a^3*c*d*e^7)*g + 2*(8*(c^4*d^5* 
e^3 + 7*a*c^3*d^3*e^5)*f - (5*c^4*d^6*e^2 - 10*a*c^3*d^4*e^4 - 3*a^2*c^2*d 
^2*e^6)*g)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^3*d^3*e^4), 
-1/384*(3*sqrt(-c*d*e)*(8*(c^4*d^7*e - 3*a*c^3*d^5*e^3 + 3*a^2*c^2*d^3*e^5 
 - a^3*c*d*e^7)*f - (5*c^4*d^8 - 12*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 4* 
a^3*c*d^2*e^6 - 3*a^4*e^8)*g)*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*(48*c^4*d^4*e^4*g*x^3 + 8*(8*c^ 
4*d^4*e^4*f + (c^4*d^5*e^3 + 9*a*c^3*d^3*e^5)*g)*x^2 - 8*(3*c^4*d^6*e^2 - 
8*a*c^3*d^4*e^4 - 3*a^2*c^2*d^2*e^6)*f + (15*c^4*d^7*e - 31*a*c^3*d^5*e^3 
+ 9*a^2*c^2*d^3*e^5 - 9*a^3*c*d*e^7)*g + 2*(8*(c^4*d^5*e^3 + 7*a*c^3*d^3*e 
^5)*f - (5*c^4*d^6*e^2 - 10*a*c^3*d^4*e^4 - 3*a^2*c^2*d^2*e^6)*g)*x)*sqrt( 
c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^3*d^3*e^4)]
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 8.11 (sec) , antiderivative size = 1853, normalized size of antiderivative = 6.94 \[ \int \frac {(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\text {Too large to display} \] Input:

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

Output:

a*e*f*Piecewise(((x/2 + (a*e**2/4 + c*d**2/4)/(c*d*e))*sqrt(a*d*e + c*d*e* 
x**2 + x*(a*e**2 + c*d**2)) + (a*d*e/2 - (a*e**2/4 + c*d**2/4)*(a*e**2 + c 
*d**2)/(2*c*d*e))*Piecewise((log(a*e**2 + c*d**2 + 2*c*d*e*x + 2*sqrt(c*d* 
e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)))/sqrt(c*d*e), Ne(a*d*e - 
 (a*e**2 + c*d**2)**2/(4*c*d*e), 0)), ((x - (-a*e**2 - c*d**2)/(2*c*d*e))* 
log(x - (-a*e**2 - c*d**2)/(2*c*d*e))/sqrt(c*d*e*(x - (-a*e**2 - c*d**2)/( 
2*c*d*e))**2), True)), Ne(c*d*e, 0)), (2*(a*d*e + x*(a*e**2 + c*d**2))**(3 
/2)/(3*(a*e**2 + c*d**2)), Ne(a*e**2 + c*d**2, 0)), (x*sqrt(a*d*e), True)) 
 + a*e*g*Piecewise(((-a*(a*e**2/6 + c*d**2/6)/(2*c) - (a*e**2 + c*d**2)*(a 
*d*e/3 - (a*e**2/6 + c*d**2/6)*(3*a*e**2/2 + 3*c*d**2/2)/(2*c*d*e))/(2*c*d 
*e))*Piecewise((log(a*e**2 + c*d**2 + 2*c*d*e*x + 2*sqrt(c*d*e)*sqrt(a*d*e 
 + c*d*e*x**2 + x*(a*e**2 + c*d**2)))/sqrt(c*d*e), Ne(a*d*e - (a*e**2 + c* 
d**2)**2/(4*c*d*e), 0)), ((x - (-a*e**2 - c*d**2)/(2*c*d*e))*log(x - (-a*e 
**2 - c*d**2)/(2*c*d*e))/sqrt(c*d*e*(x - (-a*e**2 - c*d**2)/(2*c*d*e))**2) 
, True)) + (x**2/3 + x*(a*e**2/6 + c*d**2/6)/(2*c*d*e) + (a*d*e/3 - (a*e** 
2/6 + c*d**2/6)*(3*a*e**2/2 + 3*c*d**2/2)/(2*c*d*e))/(c*d*e))*sqrt(a*d*e + 
 c*d*e*x**2 + x*(a*e**2 + c*d**2)), Ne(c*d*e, 0)), (2*(-a*d*e*(a*d*e + x*( 
a*e**2 + c*d**2))**(3/2)/3 + (a*d*e + x*(a*e**2 + c*d**2))**(5/2)/5)/(a*e* 
*2 + c*d**2)**2, Ne(a*e**2 + c*d**2, 0)), (x**2*sqrt(a*d*e)/2, True)) + c* 
d*f*Piecewise(((-a*(a*e**2/6 + c*d**2/6)/(2*c) - (a*e**2 + c*d**2)*(a*d...
 

Maxima [F(-2)]

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

integrate((g*x+f)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d),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(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.68 \[ \int \frac {(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\frac {1}{192} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, {\left (6 \, c d g x + \frac {8 \, c^{4} d^{4} e^{3} f + c^{4} d^{5} e^{2} g + 9 \, a c^{3} d^{3} e^{4} g}{c^{3} d^{3} e^{3}}\right )} x + \frac {8 \, c^{4} d^{5} e^{2} f + 56 \, a c^{3} d^{3} e^{4} f - 5 \, c^{4} d^{6} e g + 10 \, a c^{3} d^{4} e^{3} g + 3 \, a^{2} c^{2} d^{2} e^{5} g}{c^{3} d^{3} e^{3}}\right )} x - \frac {24 \, c^{4} d^{6} e f - 64 \, a c^{3} d^{4} e^{3} f - 24 \, a^{2} c^{2} d^{2} e^{5} f - 15 \, c^{4} d^{7} g + 31 \, a c^{3} d^{5} e^{2} g - 9 \, a^{2} c^{2} d^{3} e^{4} g + 9 \, a^{3} c d e^{6} g}{c^{3} d^{3} e^{3}}\right )} - \frac {{\left (8 \, c^{4} d^{7} e f - 24 \, a c^{3} d^{5} e^{3} f + 24 \, a^{2} c^{2} d^{3} e^{5} f - 8 \, a^{3} c d e^{7} f - 5 \, c^{4} d^{8} g + 12 \, a c^{3} d^{6} e^{2} g - 6 \, a^{2} c^{2} d^{4} e^{4} g - 4 \, a^{3} c d^{2} e^{6} g + 3 \, a^{4} e^{8} g\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{128 \, \sqrt {c d e} c^{2} d^{2} e^{3}} \] Input:

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

Output:

1/192*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*(6*c*d*g*x + (8*c^ 
4*d^4*e^3*f + c^4*d^5*e^2*g + 9*a*c^3*d^3*e^4*g)/(c^3*d^3*e^3))*x + (8*c^4 
*d^5*e^2*f + 56*a*c^3*d^3*e^4*f - 5*c^4*d^6*e*g + 10*a*c^3*d^4*e^3*g + 3*a 
^2*c^2*d^2*e^5*g)/(c^3*d^3*e^3))*x - (24*c^4*d^6*e*f - 64*a*c^3*d^4*e^3*f 
- 24*a^2*c^2*d^2*e^5*f - 15*c^4*d^7*g + 31*a*c^3*d^5*e^2*g - 9*a^2*c^2*d^3 
*e^4*g + 9*a^3*c*d*e^6*g)/(c^3*d^3*e^3)) - 1/128*(8*c^4*d^7*e*f - 24*a*c^3 
*d^5*e^3*f + 24*a^2*c^2*d^3*e^5*f - 8*a^3*c*d*e^7*f - 5*c^4*d^8*g + 12*a*c 
^3*d^6*e^2*g - 6*a^2*c^2*d^4*e^4*g - 4*a^3*c*d^2*e^6*g + 3*a^4*e^8*g)*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. \[ \int \frac {(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{d+e\,x} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 1003, normalized size of antiderivative = 3.76 \[ \int \frac {(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 9*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c*d*e**7*g + 9*sqrt(d + e*x)*sq 
rt(a*e + c*d*x)*a**2*c**2*d**3*e**5*g + 24*sqrt(d + e*x)*sqrt(a*e + c*d*x) 
*a**2*c**2*d**2*e**6*f + 6*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**2*d**2* 
e**6*g*x - 31*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**3*d**5*e**3*g + 64*sqrt 
(d + e*x)*sqrt(a*e + c*d*x)*a*c**3*d**4*e**4*f + 20*sqrt(d + e*x)*sqrt(a*e 
 + c*d*x)*a*c**3*d**4*e**4*g*x + 112*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c** 
3*d**3*e**5*f*x + 72*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**3*d**3*e**5*g*x* 
*2 + 15*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**4*d**7*e*g - 24*sqrt(d + e*x)*s 
qrt(a*e + c*d*x)*c**4*d**6*e**2*f - 10*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c** 
4*d**6*e**2*g*x + 16*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**4*d**5*e**3*f*x + 
8*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**4*d**5*e**3*g*x**2 + 64*sqrt(d + e*x) 
*sqrt(a*e + c*d*x)*c**4*d**4*e**4*f*x**2 + 48*sqrt(d + e*x)*sqrt(a*e + c*d 
*x)*c**4*d**4*e**4*g*x**3 + 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**4*e* 
*8*g - 12*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**3*c*d**2*e**6*g - 24*sqr 
t(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**3*c*d*e**7*f - 18*sqrt(e)*sqrt(d)*sqr 
t(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**2*d**4*e**4*g + 72*sqrt(e)*sqrt(d)*sqrt(c)*lo...