\(\int \frac {(f+g x)^2 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{d+e x} \, dx\) [268]

Optimal result

Integrand size = 44, antiderivative size = 449 \[ \int \frac {(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=-\frac {\left (c d^2-a e^2\right ) \left (3 a^2 e^4 g^2-6 a c d e^2 g (2 e f-d g)+c^2 d^2 \left (16 e^2 f^2-20 d e f g+7 d^2 g^2\right )\right ) \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^3 d^3 e^4}+\frac {(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 e}-\frac {\left (15 a^2 e^4 g^2-12 a c d e^2 g (5 e f-d g)-c^2 d^2 \left (32 e^2 f^2-100 d e f g+35 d^2 g^2\right )-6 c d e g \left (3 a e^2 g+c d (4 e f-7 d g)\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{240 c^2 d^2 e^3}+\frac {\left (c d^2-a e^2\right )^3 \left (3 a^2 e^4 g^2-6 a c d e^2 g (2 e f-d g)+c^2 d^2 \left (16 e^2 f^2-20 d e f g+7 d^2 g^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{128 c^{7/2} d^{7/2} e^{9/2}} \] Output:

-1/128*(-a*e^2+c*d^2)*(3*a^2*e^4*g^2-6*a*c*d*e^2*g*(-d*g+2*e*f)+c^2*d^2*(7 
*d^2*g^2-20*d*e*f*g+16*e^2*f^2))*(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^3/d^3/e^4+1/5*(g*x+f)^2*(a*d*e+(a*e^2+c*d^2)*x+c* 
d*e*x^2)^(3/2)/e-1/240*(15*a^2*e^4*g^2-12*a*c*d*e^2*g*(-d*g+5*e*f)-c^2*d^2 
*(35*d^2*g^2-100*d*e*f*g+32*e^2*f^2)-6*c*d*e*g*(3*a*e^2*g+c*d*(-7*d*g+4*e* 
f))*x)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^2/d^2/e^3+1/128*(-a*e^2+c 
*d^2)^3*(3*a^2*e^4*g^2-6*a*c*d*e^2*g*(-d*g+2*e*f)+c^2*d^2*(7*d^2*g^2-20*d* 
e*f*g+16*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^(7/2)/d^(7/2)/e^(9/2)
 

Mathematica [A] (verified)

Time = 2.34 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.10 \[ \int \frac {(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\frac {\left (c d^2-a e^2\right )^3 \sqrt {(a e+c d x) (d+e x)} \left (-\frac {\sqrt {c} \sqrt {d} \sqrt {e} \left (-45 a^4 e^8 g^2+30 a^3 c d e^6 g (6 e f+d g+e g x)+6 a^2 c^2 d^2 e^4 \left (6 d^2 g^2-3 d e g (10 f+g x)-4 e^2 \left (10 f^2+5 f g x+g^2 x^2\right )\right )+c^4 d^4 \left (105 d^4 g^2-10 d^3 e g (30 f+7 g x)-16 d e^3 x \left (10 f^2+10 f g x+3 g^2 x^2\right )-64 e^4 x^2 \left (10 f^2+15 f g x+6 g^2 x^2\right )+8 d^2 e^2 \left (30 f^2+25 f g x+7 g^2 x^2\right )\right )-2 a c^3 d^3 e^2 \left (95 d^3 g^2-d^2 e g (310 f+61 g x)+8 d e^2 \left (40 f^2+25 f g x+6 g^2 x^2\right )+8 e^3 x \left (70 f^2+90 f g x+33 g^2 x^2\right )\right )\right )}{\left (c d^2-a e^2\right )^3}+\frac {15 \left (3 a^2 e^4 g^2+6 a c d e^2 g (-2 e f+d g)+c^2 d^2 \left (16 e^2 f^2-20 d e f g+7 d^2 g^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{1920 c^{7/2} d^{7/2} e^{9/2}} \] Input:

Integrate[((f + g*x)^2*(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]*(-45*a^4*e^8*g^2 + 30*a^3*c*d*e^6*g*(6*e*f + d*g + e*g*x) + 6*a^2*c^2*d 
^2*e^4*(6*d^2*g^2 - 3*d*e*g*(10*f + g*x) - 4*e^2*(10*f^2 + 5*f*g*x + g^2*x 
^2)) + c^4*d^4*(105*d^4*g^2 - 10*d^3*e*g*(30*f + 7*g*x) - 16*d*e^3*x*(10*f 
^2 + 10*f*g*x + 3*g^2*x^2) - 64*e^4*x^2*(10*f^2 + 15*f*g*x + 6*g^2*x^2) + 
8*d^2*e^2*(30*f^2 + 25*f*g*x + 7*g^2*x^2)) - 2*a*c^3*d^3*e^2*(95*d^3*g^2 - 
 d^2*e*g*(310*f + 61*g*x) + 8*d*e^2*(40*f^2 + 25*f*g*x + 6*g^2*x^2) + 8*e^ 
3*x*(70*f^2 + 90*f*g*x + 33*g^2*x^2))))/(c*d^2 - a*e^2)^3) + (15*(3*a^2*e^ 
4*g^2 + 6*a*c*d*e^2*g*(-2*e*f + d*g) + c^2*d^2*(16*e^2*f^2 - 20*d*e*f*g + 
7*d^2*g^2))*ArcTanh[(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])))/(1920*c^(7/2)*d^(7/2)*e^(9/2))
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 430, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.159, Rules used = {1215, 1236, 27, 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.

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

\(\Big \downarrow \) 1215

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

\(\Big \downarrow \) 1236

\(\displaystyle \frac {\int -\frac {1}{2} c d (f+g x) \left (3 c f d^2-a e (7 e f-4 d g)-\left (3 a g e^2+c d (4 e f-7 d g)\right ) x\right ) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{5 c d e}+\frac {(f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 e}-\frac {\int (f+g x) \left (3 c f d^2-a e (7 e f-4 d g)-\left (3 a g e^2+c d (4 e f-7 d g)\right ) x\right ) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{10 e}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {(f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 e}-\frac {\frac {5 \left (c d^2-a e^2\right ) \left (3 a^2 e^4 g^2-6 a c d e^2 g (2 e f-d g)+c^2 d^2 \left (7 d^2 g^2-20 d e f g+16 e^2 f^2\right )\right ) \int \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{16 c^2 d^2 e^2}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \left (15 a^2 e^4 g^2-6 c d e g x \left (3 a e^2 g+c d (4 e f-7 d g)\right )-12 a c d e^2 g (5 e f-d g)-2 c^2 \left (\frac {35 d^4 g^2}{2}-50 d^3 e f g+16 d^2 e^2 f^2\right )\right )}{24 c^2 d^2 e^2}}{10 e}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {(f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 e}-\frac {\frac {5 \left (c d^2-a e^2\right ) \left (3 a^2 e^4 g^2-6 a c d e^2 g (2 e f-d g)+c^2 d^2 \left (7 d^2 g^2-20 d e f g+16 e^2 f^2\right )\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 c d e}\right )}{16 c^2 d^2 e^2}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \left (15 a^2 e^4 g^2-6 c d e g x \left (3 a e^2 g+c d (4 e f-7 d g)\right )-12 a c d e^2 g (5 e f-d g)-2 c^2 \left (\frac {35 d^4 g^2}{2}-50 d^3 e f g+16 d^2 e^2 f^2\right )\right )}{24 c^2 d^2 e^2}}{10 e}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {(f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 e}-\frac {\frac {5 \left (c d^2-a e^2\right ) \left (3 a^2 e^4 g^2-6 a c d e^2 g (2 e f-d g)+c^2 d^2 \left (7 d^2 g^2-20 d e f g+16 e^2 f^2\right )\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 c d e}\right )}{16 c^2 d^2 e^2}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \left (15 a^2 e^4 g^2-6 c d e g x \left (3 a e^2 g+c d (4 e f-7 d g)\right )-12 a c d e^2 g (5 e f-d g)-2 c^2 \left (\frac {35 d^4 g^2}{2}-50 d^3 e f g+16 d^2 e^2 f^2\right )\right )}{24 c^2 d^2 e^2}}{10 e}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {(f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 e}-\frac {\frac {5 \left (c d^2-a e^2\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{3/2} d^{3/2} e^{3/2}}\right ) \left (3 a^2 e^4 g^2-6 a c d e^2 g (2 e f-d g)+c^2 d^2 \left (7 d^2 g^2-20 d e f g+16 e^2 f^2\right )\right )}{16 c^2 d^2 e^2}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \left (15 a^2 e^4 g^2-6 c d e g x \left (3 a e^2 g+c d (4 e f-7 d g)\right )-12 a c d e^2 g (5 e f-d g)-2 c^2 \left (\frac {35 d^4 g^2}{2}-50 d^3 e f g+16 d^2 e^2 f^2\right )\right )}{24 c^2 d^2 e^2}}{10 e}\)

Input:

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

Output:

((f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(5*e) - (((15* 
a^2*e^4*g^2 - 12*a*c*d*e^2*g*(5*e*f - d*g) - 2*c^2*(16*d^2*e^2*f^2 - 50*d^ 
3*e*f*g + (35*d^4*g^2)/2) - 6*c*d*e*g*(3*a*e^2*g + c*d*(4*e*f - 7*d*g))*x) 
*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(24*c^2*d^2*e^2) + (5*(c*d 
^2 - a*e^2)*(3*a^2*e^4*g^2 - 6*a*c*d*e^2*g*(2*e*f - d*g) + c^2*d^2*(16*e^2 
*f^2 - 20*d*e*f*g + 7*d^2*g^2))*(((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^2*d^2*e^2))/ 
(10*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 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]
 

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. \(1068\) vs. \(2(421)=842\).

Time = 2.20 (sec) , antiderivative size = 1069, normalized size of antiderivative = 2.38

method	result	size
default	\(\text {Expression too large to display}\)	\(1069\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.58 (sec) , antiderivative size = 1592, normalized size of antiderivative = 3.55 \[ \int \frac {(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/7680*(15*sqrt(c*d*e)*(16*(c^5*d^8*e^2 - 3*a*c^4*d^6*e^4 + 3*a^2*c^3*d^ 
4*e^6 - a^3*c^2*d^2*e^8)*f^2 - 4*(5*c^5*d^9*e - 12*a*c^4*d^7*e^3 + 6*a^2*c 
^3*d^5*e^5 + 4*a^3*c^2*d^3*e^7 - 3*a^4*c*d*e^9)*f*g + (7*c^5*d^10 - 15*a*c 
^4*d^8*e^2 + 6*a^2*c^3*d^6*e^4 + 2*a^3*c^2*d^4*e^6 + 3*a^4*c*d^2*e^8 - 3*a 
^5*e^10)*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*(384*c^5*d^5*e^5*g^2*x^4 + 
48*(20*c^5*d^5*e^5*f*g + (c^5*d^6*e^4 + 11*a*c^4*d^4*e^6)*g^2)*x^3 - 80*(3 
*c^5*d^7*e^3 - 8*a*c^4*d^5*e^5 - 3*a^2*c^3*d^3*e^7)*f^2 + 20*(15*c^5*d^8*e 
^2 - 31*a*c^4*d^6*e^4 + 9*a^2*c^3*d^4*e^6 - 9*a^3*c^2*d^2*e^8)*f*g - (105* 
c^5*d^9*e - 190*a*c^4*d^7*e^3 + 36*a^2*c^3*d^5*e^5 + 30*a^3*c^2*d^3*e^7 - 
45*a^4*c*d*e^9)*g^2 + 8*(80*c^5*d^5*e^5*f^2 + 20*(c^5*d^6*e^4 + 9*a*c^4*d^ 
4*e^6)*f*g - (7*c^5*d^7*e^3 - 12*a*c^4*d^5*e^5 - 3*a^2*c^3*d^3*e^7)*g^2)*x 
^2 + 2*(80*(c^5*d^6*e^4 + 7*a*c^4*d^4*e^6)*f^2 - 20*(5*c^5*d^7*e^3 - 10*a* 
c^4*d^5*e^5 - 3*a^2*c^3*d^3*e^7)*f*g + (35*c^5*d^8*e^2 - 61*a*c^4*d^6*e^4 
+ 9*a^2*c^3*d^4*e^6 - 15*a^3*c^2*d^2*e^8)*g^2)*x)*sqrt(c*d*e*x^2 + a*d*e + 
 (c*d^2 + a*e^2)*x))/(c^4*d^4*e^5), -1/3840*(15*sqrt(-c*d*e)*(16*(c^5*d^8* 
e^2 - 3*a*c^4*d^6*e^4 + 3*a^2*c^3*d^4*e^6 - a^3*c^2*d^2*e^8)*f^2 - 4*(5*c^ 
5*d^9*e - 12*a*c^4*d^7*e^3 + 6*a^2*c^3*d^5*e^5 + 4*a^3*c^2*d^3*e^7 - 3*a^4 
*c*d*e^9)*f*g + (7*c^5*d^10 - 15*a*c^4*d^8*e^2 + 6*a^2*c^3*d^6*e^4 + 2*...
 

Sympy [A] (verification not implemented)

Time = 66.02 (sec) , antiderivative size = 3509, normalized size of antiderivative = 7.82 \[ \int \frac {(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\text {Too large to display} \] Input:

integrate((g*x+f)**2*(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**2*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), Tru 
e)) + 2*a*e*f*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)*sqr 
t(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 
)) + a*e*g**2*Piecewise(((-a*(a*d*e/4 - (a*e**2/8 + c*d**2/8)*(5*a*e**2...
 

Maxima [F(-2)]

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

integrate((g*x+f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(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 = 0.35 (sec) , antiderivative size = 828, normalized size of antiderivative = 1.84 \[ \int \frac {(f+g x)^2 \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}{1920} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, c d g^{2} x + \frac {20 \, c^{5} d^{5} e^{4} f g + c^{5} d^{6} e^{3} g^{2} + 11 \, a c^{4} d^{4} e^{5} g^{2}}{c^{4} d^{4} e^{4}}\right )} x + \frac {80 \, c^{5} d^{5} e^{4} f^{2} + 20 \, c^{5} d^{6} e^{3} f g + 180 \, a c^{4} d^{4} e^{5} f g - 7 \, c^{5} d^{7} e^{2} g^{2} + 12 \, a c^{4} d^{5} e^{4} g^{2} + 3 \, a^{2} c^{3} d^{3} e^{6} g^{2}}{c^{4} d^{4} e^{4}}\right )} x + \frac {80 \, c^{5} d^{6} e^{3} f^{2} + 560 \, a c^{4} d^{4} e^{5} f^{2} - 100 \, c^{5} d^{7} e^{2} f g + 200 \, a c^{4} d^{5} e^{4} f g + 60 \, a^{2} c^{3} d^{3} e^{6} f g + 35 \, c^{5} d^{8} e g^{2} - 61 \, a c^{4} d^{6} e^{3} g^{2} + 9 \, a^{2} c^{3} d^{4} e^{5} g^{2} - 15 \, a^{3} c^{2} d^{2} e^{7} g^{2}}{c^{4} d^{4} e^{4}}\right )} x - \frac {240 \, c^{5} d^{7} e^{2} f^{2} - 640 \, a c^{4} d^{5} e^{4} f^{2} - 240 \, a^{2} c^{3} d^{3} e^{6} f^{2} - 300 \, c^{5} d^{8} e f g + 620 \, a c^{4} d^{6} e^{3} f g - 180 \, a^{2} c^{3} d^{4} e^{5} f g + 180 \, a^{3} c^{2} d^{2} e^{7} f g + 105 \, c^{5} d^{9} g^{2} - 190 \, a c^{4} d^{7} e^{2} g^{2} + 36 \, a^{2} c^{3} d^{5} e^{4} g^{2} + 30 \, a^{3} c^{2} d^{3} e^{6} g^{2} - 45 \, a^{4} c d e^{8} g^{2}}{c^{4} d^{4} e^{4}}\right )} - \frac {{\left (16 \, c^{5} d^{8} e^{2} f^{2} - 48 \, a c^{4} d^{6} e^{4} f^{2} + 48 \, a^{2} c^{3} d^{4} e^{6} f^{2} - 16 \, a^{3} c^{2} d^{2} e^{8} f^{2} - 20 \, c^{5} d^{9} e f g + 48 \, a c^{4} d^{7} e^{3} f g - 24 \, a^{2} c^{3} d^{5} e^{5} f g - 16 \, a^{3} c^{2} d^{3} e^{7} f g + 12 \, a^{4} c d e^{9} f g + 7 \, c^{5} d^{10} g^{2} - 15 \, a c^{4} d^{8} e^{2} g^{2} + 6 \, a^{2} c^{3} d^{6} e^{4} g^{2} + 2 \, a^{3} c^{2} d^{4} e^{6} g^{2} + 3 \, a^{4} c d^{2} e^{8} g^{2} - 3 \, a^{5} e^{10} 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 )}{256 \, \sqrt {c d e} c^{3} d^{3} e^{4}} \] Input:

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

Output:

1/1920*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*(6*(8*c*d*g^2*x + 
 (20*c^5*d^5*e^4*f*g + c^5*d^6*e^3*g^2 + 11*a*c^4*d^4*e^5*g^2)/(c^4*d^4*e^ 
4))*x + (80*c^5*d^5*e^4*f^2 + 20*c^5*d^6*e^3*f*g + 180*a*c^4*d^4*e^5*f*g - 
 7*c^5*d^7*e^2*g^2 + 12*a*c^4*d^5*e^4*g^2 + 3*a^2*c^3*d^3*e^6*g^2)/(c^4*d^ 
4*e^4))*x + (80*c^5*d^6*e^3*f^2 + 560*a*c^4*d^4*e^5*f^2 - 100*c^5*d^7*e^2* 
f*g + 200*a*c^4*d^5*e^4*f*g + 60*a^2*c^3*d^3*e^6*f*g + 35*c^5*d^8*e*g^2 - 
61*a*c^4*d^6*e^3*g^2 + 9*a^2*c^3*d^4*e^5*g^2 - 15*a^3*c^2*d^2*e^7*g^2)/(c^ 
4*d^4*e^4))*x - (240*c^5*d^7*e^2*f^2 - 640*a*c^4*d^5*e^4*f^2 - 240*a^2*c^3 
*d^3*e^6*f^2 - 300*c^5*d^8*e*f*g + 620*a*c^4*d^6*e^3*f*g - 180*a^2*c^3*d^4 
*e^5*f*g + 180*a^3*c^2*d^2*e^7*f*g + 105*c^5*d^9*g^2 - 190*a*c^4*d^7*e^2*g 
^2 + 36*a^2*c^3*d^5*e^4*g^2 + 30*a^3*c^2*d^3*e^6*g^2 - 45*a^4*c*d*e^8*g^2) 
/(c^4*d^4*e^4)) - 1/256*(16*c^5*d^8*e^2*f^2 - 48*a*c^4*d^6*e^4*f^2 + 48*a^ 
2*c^3*d^4*e^6*f^2 - 16*a^3*c^2*d^2*e^8*f^2 - 20*c^5*d^9*e*f*g + 48*a*c^4*d 
^7*e^3*f*g - 24*a^2*c^3*d^5*e^5*f*g - 16*a^3*c^2*d^3*e^7*f*g + 12*a^4*c*d* 
e^9*f*g + 7*c^5*d^10*g^2 - 15*a*c^4*d^8*e^2*g^2 + 6*a^2*c^3*d^6*e^4*g^2 + 
2*a^3*c^2*d^4*e^6*g^2 + 3*a^4*c*d^2*e^8*g^2 - 3*a^5*e^10*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^3*d^3*e^4)
 

Mupad [F(-1)]

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

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

Reduce [F]

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

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

Output:

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