\(\int \frac {(f+g x)^3 \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{d+e x} \, dx\) [259]

Optimal result

Integrand size = 44, antiderivative size = 527 \[ \int \frac {(f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\frac {1}{24} \left (\frac {a g}{c d}+\frac {6 e f-7 d g}{e^2}\right ) (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}+\frac {(f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e}+\frac {\left (15 a^3 e^6 g^3-a^2 c d e^4 g^2 (72 e f-17 d g)+a c^2 d^2 e^2 g \left (136 e^2 f^2-96 d e f g+25 d^2 g^2\right )+c^3 d^3 \left (96 e^3 f^3-376 d e^2 f^2 g+360 d^2 e f g^2-105 d^3 g^3\right )-2 c d e g \left (5 a^2 e^4 g^2-2 a c d e^2 g (8 e f-3 d g)-c^2 d^2 \left (24 e^2 f^2-64 d e f g+35 d^2 g^2\right )\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 c^3 d^3 e^4}+\frac {\left (c d^2-a e^2\right ) \left (5 a^3 e^6 g^3-3 a^2 c d e^4 g^2 (8 e f-3 d g)+3 a c^2 d^2 e^2 g \left (16 e^2 f^2-16 d e f g+5 d^2 g^2\right )-c^3 d^3 \left (64 e^3 f^3-144 d e^2 f^2 g+120 d^2 e f g^2-35 d^3 g^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{64 c^{7/2} d^{7/2} e^{9/2}} \] Output:

1/24*(a*g/c/d+(-7*d*g+6*e*f)/e^2)*(g*x+f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x 
^2)^(1/2)+1/4*(g*x+f)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/e+1/192*(1 
5*a^3*e^6*g^3-a^2*c*d*e^4*g^2*(-17*d*g+72*e*f)+a*c^2*d^2*e^2*g*(25*d^2*g^2 
-96*d*e*f*g+136*e^2*f^2)+c^3*d^3*(-105*d^3*g^3+360*d^2*e*f*g^2-376*d*e^2*f 
^2*g+96*e^3*f^3)-2*c*d*e*g*(5*a^2*e^4*g^2-2*a*c*d*e^2*g*(-3*d*g+8*e*f)-c^2 
*d^2*(35*d^2*g^2-64*d*e*f*g+24*e^2*f^2))*x)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x 
^2)^(1/2)/c^3/d^3/e^4+1/64*(-a*e^2+c*d^2)*(5*a^3*e^6*g^3-3*a^2*c*d*e^4*g^2 
*(-3*d*g+8*e*f)+3*a*c^2*d^2*e^2*g*(5*d^2*g^2-16*d*e*f*g+16*e^2*f^2)-c^3*d^ 
3*(-35*d^3*g^3+120*d^2*e*f*g^2-144*d*e^2*f^2*g+64*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^(7/2) 
/d^(7/2)/e^(9/2)
 

Mathematica [A] (verified)

Time = 13.95 (sec) , antiderivative size = 472, normalized size of antiderivative = 0.90 \[ \int \frac {(f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\sqrt {c} \sqrt {d} \sqrt {e} \left (15 a^3 e^6 g^3+a^2 c d e^4 g^2 (17 d g-2 e (36 f+5 g x))+a c^2 d^2 e^2 g \left (25 d^2 g^2-12 d e g (8 f+g x)+8 e^2 \left (18 f^2+6 f g x+g^2 x^2\right )\right )+c^3 d^3 \left (-105 d^3 g^3+10 d^2 e g^2 (36 f+7 g x)-8 d e^2 g \left (54 f^2+30 f g x+7 g^2 x^2\right )+48 e^3 \left (4 f^3+6 f^2 g x+4 f g^2 x^2+g^3 x^3\right )\right )\right )+\frac {3 \sqrt {c d} \sqrt {c d^2-a e^2} \left (5 a^3 e^6 g^3+3 a^2 c d e^4 g^2 (-8 e f+3 d g)+3 a c^2 d^2 e^2 g \left (16 e^2 f^2-16 d e f g+5 d^2 g^2\right )+c^3 d^3 \left (-64 e^3 f^3+144 d e^2 f^2 g-120 d^2 e f g^2+35 d^3 g^3\right )\right ) \text {arcsinh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d} \sqrt {c d^2-a e^2}}\right )}{\sqrt {a e+c d x} \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}}\right )}{192 c^{7/2} d^{7/2} e^{9/2}} \] Input:

Integrate[((f + g*x)^3*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)]*(Sqrt[c]*Sqrt[d]*Sqrt[e]*(15*a^3*e^6*g^3 + 
a^2*c*d*e^4*g^2*(17*d*g - 2*e*(36*f + 5*g*x)) + a*c^2*d^2*e^2*g*(25*d^2*g^ 
2 - 12*d*e*g*(8*f + g*x) + 8*e^2*(18*f^2 + 6*f*g*x + g^2*x^2)) + c^3*d^3*( 
-105*d^3*g^3 + 10*d^2*e*g^2*(36*f + 7*g*x) - 8*d*e^2*g*(54*f^2 + 30*f*g*x 
+ 7*g^2*x^2) + 48*e^3*(4*f^3 + 6*f^2*g*x + 4*f*g^2*x^2 + g^3*x^3))) + (3*S 
qrt[c*d]*Sqrt[c*d^2 - a*e^2]*(5*a^3*e^6*g^3 + 3*a^2*c*d*e^4*g^2*(-8*e*f + 
3*d*g) + 3*a*c^2*d^2*e^2*g*(16*e^2*f^2 - 16*d*e*f*g + 5*d^2*g^2) + c^3*d^3 
*(-64*e^3*f^3 + 144*d*e^2*f^2*g - 120*d^2*e*f*g^2 + 35*d^3*g^3))*ArcSinh[( 
Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2]) 
])/(Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])))/(192*c^(7/2 
)*d^(7/2)*e^(9/2))
 

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 562, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1215, 1236, 27, 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.

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

\(\Big \downarrow \) 1215

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

\(\Big \downarrow \) 1236

\(\displaystyle \frac {\int -\frac {c d (f+g x)^2 \left (c f d^2-a e (7 e f-6 d g)-\left (a g e^2+c d (6 e f-7 d g)\right ) x\right )}{2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 c d e}+\frac {(f+g x)^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(f+g x)^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e}-\frac {\int \frac {(f+g x)^2 \left (c f d^2-a e (7 e f-6 d g)-\left (a g e^2+c d (6 e f-7 d g)\right ) x\right )}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 e}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {(f+g x)^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e}-\frac {\frac {\int \frac {(f+g x) \left (c^2 f (12 e f-7 d g) d^3-2 a c e \left (18 e^2 f^2-27 d e g f+14 d^2 g^2\right ) d+a^2 e^3 g (e f+4 d g)+\left (5 a^2 g^2 e^4-2 a c d g (8 e f-3 d g) e^2-c^2 d^2 \left (24 e^2 f^2-64 d e g f+35 d^2 g^2\right )\right ) x\right )}{2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{3 c d e}-\frac {1}{3} (f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (\frac {a e g}{c d}-\frac {7 d g}{e}+6 f\right )}{8 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(f+g x)^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e}-\frac {\frac {\int \frac {(f+g x) \left (c^2 f (12 e f-7 d g) d^3-2 a c e \left (18 e^2 f^2-27 d e g f+14 d^2 g^2\right ) d+a^2 e^3 g (e f+4 d g)+\left (5 a^2 g^2 e^4-2 a c d g (8 e f-3 d g) e^2-c^2 d^2 \left (24 e^2 f^2-64 d e g f+35 d^2 g^2\right )\right ) x\right )}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{6 c d e}-\frac {1}{3} (f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (\frac {a e g}{c d}-\frac {7 d g}{e}+6 f\right )}{8 e}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {(f+g x)^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e}-\frac {\frac {-\frac {3 \left (c d^2-a e^2\right ) \left (5 a^3 e^6 g^3-3 a^2 c d e^4 g^2 (8 e f-3 d g)+3 a c^2 d^2 e^2 g \left (5 d^2 g^2-16 d e f g+16 e^2 f^2\right )-c^3 d^3 \left (-35 d^3 g^3+120 d^2 e f g^2-144 d e^2 f^2 g+64 e^3 f^3\right )\right ) \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 c^2 d^2 e^2}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (15 a^3 e^6 g^3-2 c d e g x \left (5 a^2 e^4 g^2-2 a c d e^2 g (8 e f-3 d g)-c^2 d^2 \left (35 d^2 g^2-64 d e f g+24 e^2 f^2\right )\right )-a^2 c d e^4 g^2 (72 e f-17 d g)+a c^2 d^2 e^2 g \left (25 d^2 g^2-96 d e f g+136 e^2 f^2\right )+c^3 d^3 \left (-105 d^3 g^3+360 d^2 e f g^2-376 d e^2 f^2 g+96 e^3 f^3\right )\right )}{4 c^2 d^2 e^2}}{6 c d e}-\frac {1}{3} (f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (\frac {a e g}{c d}-\frac {7 d g}{e}+6 f\right )}{8 e}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {(f+g x)^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e}-\frac {\frac {-\frac {3 \left (c d^2-a e^2\right ) \left (5 a^3 e^6 g^3-3 a^2 c d e^4 g^2 (8 e f-3 d g)+3 a c^2 d^2 e^2 g \left (5 d^2 g^2-16 d e f g+16 e^2 f^2\right )-c^3 d^3 \left (-35 d^3 g^3+120 d^2 e f g^2-144 d e^2 f^2 g+64 e^3 f^3\right )\right ) \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 c^2 d^2 e^2}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (15 a^3 e^6 g^3-2 c d e g x \left (5 a^2 e^4 g^2-2 a c d e^2 g (8 e f-3 d g)-c^2 d^2 \left (35 d^2 g^2-64 d e f g+24 e^2 f^2\right )\right )-a^2 c d e^4 g^2 (72 e f-17 d g)+a c^2 d^2 e^2 g \left (25 d^2 g^2-96 d e f g+136 e^2 f^2\right )+c^3 d^3 \left (-105 d^3 g^3+360 d^2 e f g^2-376 d e^2 f^2 g+96 e^3 f^3\right )\right )}{4 c^2 d^2 e^2}}{6 c d e}-\frac {1}{3} (f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (\frac {a e g}{c d}-\frac {7 d g}{e}+6 f\right )}{8 e}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {(f+g x)^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e}-\frac {\frac {-\frac {3 \left (c d^2-a e^2\right ) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right ) \left (5 a^3 e^6 g^3-3 a^2 c d e^4 g^2 (8 e f-3 d g)+3 a c^2 d^2 e^2 g \left (5 d^2 g^2-16 d e f g+16 e^2 f^2\right )-c^3 d^3 \left (-35 d^3 g^3+120 d^2 e f g^2-144 d e^2 f^2 g+64 e^3 f^3\right )\right )}{8 c^{5/2} d^{5/2} e^{5/2}}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (15 a^3 e^6 g^3-2 c d e g x \left (5 a^2 e^4 g^2-2 a c d e^2 g (8 e f-3 d g)-c^2 d^2 \left (35 d^2 g^2-64 d e f g+24 e^2 f^2\right )\right )-a^2 c d e^4 g^2 (72 e f-17 d g)+a c^2 d^2 e^2 g \left (25 d^2 g^2-96 d e f g+136 e^2 f^2\right )+c^3 d^3 \left (-105 d^3 g^3+360 d^2 e f g^2-376 d e^2 f^2 g+96 e^3 f^3\right )\right )}{4 c^2 d^2 e^2}}{6 c d e}-\frac {1}{3} (f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (\frac {a e g}{c d}-\frac {7 d g}{e}+6 f\right )}{8 e}\)

Input:

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

Output:

((f + g*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*e) - (-1/3*(( 
6*f - (7*d*g)/e + (a*e*g)/(c*d))*(f + g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)* 
x + c*d*e*x^2]) + (-1/4*((15*a^3*e^6*g^3 - a^2*c*d*e^4*g^2*(72*e*f - 17*d* 
g) + a*c^2*d^2*e^2*g*(136*e^2*f^2 - 96*d*e*f*g + 25*d^2*g^2) + c^3*d^3*(96 
*e^3*f^3 - 376*d*e^2*f^2*g + 360*d^2*e*f*g^2 - 105*d^3*g^3) - 2*c*d*e*g*(5 
*a^2*e^4*g^2 - 2*a*c*d*e^2*g*(8*e*f - 3*d*g) - c^2*d^2*(24*e^2*f^2 - 64*d* 
e*f*g + 35*d^2*g^2))*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(c^2* 
d^2*e^2) - (3*(c*d^2 - a*e^2)*(5*a^3*e^6*g^3 - 3*a^2*c*d*e^4*g^2*(8*e*f - 
3*d*g) + 3*a*c^2*d^2*e^2*g*(16*e^2*f^2 - 16*d*e*f*g + 5*d^2*g^2) - c^3*d^3 
*(64*e^3*f^3 - 144*d*e^2*f^2*g + 120*d^2*e*f*g^2 - 35*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])])/(8*c^(5/2)*d^(5/2)*e^(5/2)))/(6*c*d*e))/(8*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. \(1322\) vs. \(2(499)=998\).

Time = 2.26 (sec) , antiderivative size = 1323, normalized size of antiderivative = 2.51

method	result	size
default	\(\text {Expression too large to display}\)	\(1323\)

Input:

int((g*x+f)^3*(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:

g/e^3*(d^2*g^2*(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*(d*g-3*e*f)*(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*l 
n((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^2*g^2*(1/4*x*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2 
*e)^(3/2)/d/e/c-5/8*(a*e^2+c*d^2)/d/e/c*(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)))-1/4*a/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)))+3*e^2*f^2*(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))-3*d*e*f*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*(...
 

Fricas [A] (verification not implemented)

Time = 0.64 (sec) , antiderivative size = 1274, normalized size of antiderivative = 2.42 \[ \int \frac {(f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\text {Too large to display} \] Input:

integrate((g*x+f)^3*(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:

[1/768*(3*(64*(c^4*d^5*e^3 - a*c^3*d^3*e^5)*f^3 - 48*(3*c^4*d^6*e^2 - 2*a* 
c^3*d^4*e^4 - a^2*c^2*d^2*e^6)*f^2*g + 24*(5*c^4*d^7*e - 3*a*c^3*d^5*e^3 - 
 a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*f*g^2 - (35*c^4*d^8 - 20*a*c^3*d^6*e^2 - 6 
*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 - 5*a^4*e^8)*g^3)*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*(48*c^4*d^4*e^4*g^3*x^3 + 192*c^4*d^4*e^4*f^3 - 14 
4*(3*c^4*d^5*e^3 - a*c^3*d^3*e^5)*f^2*g + 24*(15*c^4*d^6*e^2 - 4*a*c^3*d^4 
*e^4 - 3*a^2*c^2*d^2*e^6)*f*g^2 - (105*c^4*d^7*e - 25*a*c^3*d^5*e^3 - 17*a 
^2*c^2*d^3*e^5 - 15*a^3*c*d*e^7)*g^3 + 8*(24*c^4*d^4*e^4*f*g^2 - (7*c^4*d^ 
5*e^3 - a*c^3*d^3*e^5)*g^3)*x^2 + 2*(144*c^4*d^4*e^4*f^2*g - 24*(5*c^4*d^5 
*e^3 - a*c^3*d^3*e^5)*f*g^2 + (35*c^4*d^6*e^2 - 6*a*c^3*d^4*e^4 - 5*a^2*c^ 
2*d^2*e^6)*g^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^4*d^4*e 
^5), 1/384*(3*(64*(c^4*d^5*e^3 - a*c^3*d^3*e^5)*f^3 - 48*(3*c^4*d^6*e^2 - 
2*a*c^3*d^4*e^4 - a^2*c^2*d^2*e^6)*f^2*g + 24*(5*c^4*d^7*e - 3*a*c^3*d^5*e 
^3 - a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*f*g^2 - (35*c^4*d^8 - 20*a*c^3*d^6*e^2 
 - 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 - 5*a^4*e^8)*g^3)*sqrt(-c*d*e)*arct 
an(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^3*x^3 + 192*c^4*d^4*e^4*f^3 - 144*(3*c^4*d^5...
 

Sympy [F]

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

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

Maxima [F(-2)]

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

integrate((g*x+f)^3*(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 = 0.34 (sec) , antiderivative size = 635, normalized size of antiderivative = 1.20 \[ \int \frac {(f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\frac {1}{192} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, {\left (\frac {6 \, g^{3} x}{e} + \frac {24 \, c^{3} d^{3} e^{3} f g^{2} - 7 \, c^{3} d^{4} e^{2} g^{3} + a c^{2} d^{2} e^{4} g^{3}}{c^{3} d^{3} e^{4}}\right )} x + \frac {144 \, c^{3} d^{3} e^{3} f^{2} g - 120 \, c^{3} d^{4} e^{2} f g^{2} + 24 \, a c^{2} d^{2} e^{4} f g^{2} + 35 \, c^{3} d^{5} e g^{3} - 6 \, a c^{2} d^{3} e^{3} g^{3} - 5 \, a^{2} c d e^{5} g^{3}}{c^{3} d^{3} e^{4}}\right )} x + \frac {192 \, c^{3} d^{3} e^{3} f^{3} - 432 \, c^{3} d^{4} e^{2} f^{2} g + 144 \, a c^{2} d^{2} e^{4} f^{2} g + 360 \, c^{3} d^{5} e f g^{2} - 96 \, a c^{2} d^{3} e^{3} f g^{2} - 72 \, a^{2} c d e^{5} f g^{2} - 105 \, c^{3} d^{6} g^{3} + 25 \, a c^{2} d^{4} e^{2} g^{3} + 17 \, a^{2} c d^{2} e^{4} g^{3} + 15 \, a^{3} e^{6} g^{3}}{c^{3} d^{3} e^{4}}\right )} + \frac {{\left (64 \, c^{4} d^{5} e^{3} f^{3} - 64 \, a c^{3} d^{3} e^{5} f^{3} - 144 \, c^{4} d^{6} e^{2} f^{2} g + 96 \, a c^{3} d^{4} e^{4} f^{2} g + 48 \, a^{2} c^{2} d^{2} e^{6} f^{2} g + 120 \, c^{4} d^{7} e f g^{2} - 72 \, a c^{3} d^{5} e^{3} f g^{2} - 24 \, a^{2} c^{2} d^{3} e^{5} f g^{2} - 24 \, a^{3} c d e^{7} f g^{2} - 35 \, c^{4} d^{8} g^{3} + 20 \, a c^{3} d^{6} e^{2} g^{3} + 6 \, a^{2} c^{2} d^{4} e^{4} g^{3} + 4 \, a^{3} c d^{2} e^{6} g^{3} + 5 \, a^{4} e^{8} g^{3}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{128 \, \sqrt {c d e} c^{3} d^{3} e^{4}} \] Input:

integrate((g*x+f)^3*(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:

1/192*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*(6*g^3*x/e + (24*c 
^3*d^3*e^3*f*g^2 - 7*c^3*d^4*e^2*g^3 + a*c^2*d^2*e^4*g^3)/(c^3*d^3*e^4))*x 
 + (144*c^3*d^3*e^3*f^2*g - 120*c^3*d^4*e^2*f*g^2 + 24*a*c^2*d^2*e^4*f*g^2 
 + 35*c^3*d^5*e*g^3 - 6*a*c^2*d^3*e^3*g^3 - 5*a^2*c*d*e^5*g^3)/(c^3*d^3*e^ 
4))*x + (192*c^3*d^3*e^3*f^3 - 432*c^3*d^4*e^2*f^2*g + 144*a*c^2*d^2*e^4*f 
^2*g + 360*c^3*d^5*e*f*g^2 - 96*a*c^2*d^3*e^3*f*g^2 - 72*a^2*c*d*e^5*f*g^2 
 - 105*c^3*d^6*g^3 + 25*a*c^2*d^4*e^2*g^3 + 17*a^2*c*d^2*e^4*g^3 + 15*a^3* 
e^6*g^3)/(c^3*d^3*e^4)) + 1/128*(64*c^4*d^5*e^3*f^3 - 64*a*c^3*d^3*e^5*f^3 
 - 144*c^4*d^6*e^2*f^2*g + 96*a*c^3*d^4*e^4*f^2*g + 48*a^2*c^2*d^2*e^6*f^2 
*g + 120*c^4*d^7*e*f*g^2 - 72*a*c^3*d^5*e^3*f*g^2 - 24*a^2*c^2*d^3*e^5*f*g 
^2 - 24*a^3*c*d*e^7*f*g^2 - 35*c^4*d^8*g^3 + 20*a*c^3*d^6*e^2*g^3 + 6*a^2* 
c^2*d^4*e^4*g^3 + 4*a^3*c*d^2*e^6*g^3 + 5*a^4*e^8*g^3)*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^3*d^3*e^4)
 

Mupad [F(-1)]

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

int(((f + g*x)^3*(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)^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x), 
 x)
 

Reduce [F]

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

int((g*x+f)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d),x)
 

Output:

int((g*x+f)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d),x)