\(\int (d+e x) (f+g x) (a+b x+c x^2)^{3/4} \, dx\) [1103]

Optimal result

Integrand size = 25, antiderivative size = 226 \[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^{3/4} \, dx=\frac {\left (36 c^2 d f+11 b^2 e g-8 a c e g-18 b c (e f+d g)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{180 c^3}-\frac {(11 b e g-18 c (e f+d g)-14 c e g x) \left (a+b x+c x^2\right )^{7/4}}{63 c^2}-\frac {\left (b^2-4 a c\right )^{3/2} \left (36 c^2 d f+11 b^2 e g-8 a c e g-18 b c (e f+d g)\right ) \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\right |2\right )}{120 \sqrt {2} c^4 \sqrt [4]{a+b x+c x^2}} \] Output:

1/180*(36*c^2*d*f+11*b^2*e*g-8*a*c*e*g-18*b*c*(d*g+e*f))*(2*c*x+b)*(c*x^2+ 
b*x+a)^(3/4)/c^3-1/63*(11*b*e*g-18*c*(d*g+e*f)-14*c*e*g*x)*(c*x^2+b*x+a)^( 
7/4)/c^2-1/240*(-4*a*c+b^2)^(3/2)*(36*c^2*d*f+11*b^2*e*g-8*a*c*e*g-18*b*c* 
(d*g+e*f))*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/4)*EllipticE(sin(1/2*arcsin( 
(2*c*x+b)/(-4*a*c+b^2)^(1/2))),2^(1/2))*2^(1/2)/c^4/(c*x^2+b*x+a)^(1/4)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.37 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.85 \[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^{3/4} \, dx=\frac {(a+x (b+c x))^{7/4} (-11 b e g+2 c (9 e f+9 d g+7 e g x))+\frac {7 \left (9 c^2 d f+\frac {11}{4} b^2 e g-2 a c e g-\frac {9}{2} b c (e f+d g)\right ) (b+2 c x) \left (8 c (a+x (b+c x))-3 \sqrt {2} \left (b^2-4 a c\right ) \sqrt [4]{\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )\right )}{40 c^2 \sqrt [4]{a+x (b+c x)}}}{63 c^2} \] Input:

Integrate[(d + e*x)*(f + g*x)*(a + b*x + c*x^2)^(3/4),x]
 

Output:

((a + x*(b + c*x))^(7/4)*(-11*b*e*g + 2*c*(9*e*f + 9*d*g + 7*e*g*x)) + (7* 
(9*c^2*d*f + (11*b^2*e*g)/4 - 2*a*c*e*g - (9*b*c*(e*f + d*g))/2)*(b + 2*c* 
x)*(8*c*(a + x*(b + c*x)) - 3*Sqrt[2]*(b^2 - 4*a*c)*((c*(a + x*(b + c*x))) 
/(-b^2 + 4*a*c))^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, (b + 2*c*x)^2/(b^2 
 - 4*a*c)]))/(40*c^2*(a + x*(b + c*x))^(1/4)))/(63*c^2)
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(661\) vs. \(2(226)=452\).

Time = 1.04 (sec) , antiderivative size = 661, normalized size of antiderivative = 2.92, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1225, 1087, 1094, 834, 761, 1510}

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.

\(\displaystyle \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^{3/4} \, dx\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {\left (-8 a c e g+11 b^2 e g-18 b c d g-18 b c e f+36 c^2 d f\right ) \int \left (c x^2+b x+a\right )^{3/4}dx}{36 c^2}-\frac {\left (a+b x+c x^2\right )^{7/4} (11 b e g-18 c (d g+e f)-14 c e g x)}{63 c^2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\left (-8 a c e g+11 b^2 e g-18 b c d g-18 b c e f+36 c^2 d f\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{5 c}-\frac {3 \left (b^2-4 a c\right ) \int \frac {1}{\sqrt [4]{c x^2+b x+a}}dx}{20 c}\right )}{36 c^2}-\frac {\left (a+b x+c x^2\right )^{7/4} (11 b e g-18 c (d g+e f)-14 c e g x)}{63 c^2}\)

\(\Big \downarrow \) 1094

\(\displaystyle \frac {\left (-8 a c e g+11 b^2 e g-18 b c d g-18 b c e f+36 c^2 d f\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{5 c}-\frac {3 \left (b^2-4 a c\right ) \sqrt {(b+2 c x)^2} \int \frac {\sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{5 c (b+2 c x)}\right )}{36 c^2}-\frac {\left (a+b x+c x^2\right )^{7/4} (11 b e g-18 c (d g+e f)-14 c e g x)}{63 c^2}\)

\(\Big \downarrow \) 834

\(\displaystyle \frac {\left (-8 a c e g+11 b^2 e g-18 b c d g-18 b c e f+36 c^2 d f\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{5 c}-\frac {3 \left (b^2-4 a c\right ) \sqrt {(b+2 c x)^2} \left (\frac {\sqrt {b^2-4 a c} \int \frac {1}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}-\frac {\sqrt {b^2-4 a c} \int \frac {1-\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}\right )}{5 c (b+2 c x)}\right )}{36 c^2}-\frac {\left (a+b x+c x^2\right )^{7/4} (11 b e g-18 c (d g+e f)-14 c e g x)}{63 c^2}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {\left (-8 a c e g+11 b^2 e g-18 b c d g-18 b c e f+36 c^2 d f\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{5 c}-\frac {3 \left (b^2-4 a c\right ) \sqrt {(b+2 c x)^2} \left (\frac {\left (b^2-4 a c\right )^{3/4} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}-\frac {\sqrt {b^2-4 a c} \int \frac {1-\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}\right )}{5 c (b+2 c x)}\right )}{36 c^2}-\frac {\left (a+b x+c x^2\right )^{7/4} (11 b e g-18 c (d g+e f)-14 c e g x)}{63 c^2}\)

\(\Big \downarrow \) 1510

\(\displaystyle \frac {\left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{5 c}-\frac {3 \left (b^2-4 a c\right ) \sqrt {(b+2 c x)^2} \left (\frac {\left (b^2-4 a c\right )^{3/4} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}-\frac {\sqrt {b^2-4 a c} \left (\frac {\sqrt [4]{b^2-4 a c} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}-\frac {\sqrt [4]{a+b x+c x^2} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )}\right )}{2 \sqrt {c}}\right )}{5 c (b+2 c x)}\right ) \left (-8 a c e g+11 b^2 e g-18 b c d g-18 b c e f+36 c^2 d f\right )}{36 c^2}-\frac {\left (a+b x+c x^2\right )^{7/4} (11 b e g-18 c (d g+e f)-14 c e g x)}{63 c^2}\)

Input:

Int[(d + e*x)*(f + g*x)*(a + b*x + c*x^2)^(3/4),x]
 

Output:

-1/63*((11*b*e*g - 18*c*(e*f + d*g) - 14*c*e*g*x)*(a + b*x + c*x^2)^(7/4)) 
/c^2 + ((36*c^2*d*f - 18*b*c*e*f - 18*b*c*d*g + 11*b^2*e*g - 8*a*c*e*g)*(( 
(b + 2*c*x)*(a + b*x + c*x^2)^(3/4))/(5*c) - (3*(b^2 - 4*a*c)*Sqrt[(b + 2* 
c*x)^2]*(-1/2*(Sqrt[b^2 - 4*a*c]*(-(((a + b*x + c*x^2)^(1/4)*Sqrt[b^2 - 4* 
a*c + 4*c*(a + b*x + c*x^2)])/((b^2 - 4*a*c)*(1 + (2*Sqrt[c]*Sqrt[a + b*x 
+ c*x^2])/Sqrt[b^2 - 4*a*c]))) + ((b^2 - 4*a*c)^(1/4)*(1 + (2*Sqrt[c]*Sqrt 
[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])*Sqrt[(b^2 - 4*a*c + 4*c*(a + b*x + c 
*x^2))/((b^2 - 4*a*c)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4* 
a*c])^2)]*EllipticE[2*ArcTan[(Sqrt[2]*c^(1/4)*(a + b*x + c*x^2)^(1/4))/(b^ 
2 - 4*a*c)^(1/4)], 1/2])/(Sqrt[2]*c^(1/4)*Sqrt[b^2 - 4*a*c + 4*c*(a + b*x 
+ c*x^2)])))/Sqrt[c] + ((b^2 - 4*a*c)^(3/4)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + 
 c*x^2])/Sqrt[b^2 - 4*a*c])*Sqrt[(b^2 - 4*a*c + 4*c*(a + b*x + c*x^2))/((b 
^2 - 4*a*c)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])^2)]* 
EllipticF[2*ArcTan[(Sqrt[2]*c^(1/4)*(a + b*x + c*x^2)^(1/4))/(b^2 - 4*a*c) 
^(1/4)], 1/2])/(4*Sqrt[2]*c^(3/4)*Sqrt[b^2 - 4*a*c + 4*c*(a + b*x + c*x^2) 
])))/(5*c*(b + 2*c*x))))/(36*c^2)
 

Defintions of rubi rules used

rule 761

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 
1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* 
EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 834

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, S 
imp[1/q   Int[1/Sqrt[a + b*x^4], x], x] - Simp[1/q   Int[(1 - q*x^2)/Sqrt[a 
 + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

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 1094

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[4*(Sqrt[(b 
+ 2*c*x)^2]/(b + 2*c*x))   Subst[Int[x^(4*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4 
*c*x^4], x], x, (a + b*x + c*x^2)^(1/4)], x] /; FreeQ[{a, b, c}, x] && Inte 
gerQ[4*p]
 

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 1510

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = 
 Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* 
(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E 
llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e 
}, x] && PosQ[c/a]
 

Maple [F]

\[\int \left (e x +d \right ) \left (g x +f \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{4}}d x\]

Input:

int((e*x+d)*(g*x+f)*(c*x^2+b*x+a)^(3/4),x)
 

Output:

int((e*x+d)*(g*x+f)*(c*x^2+b*x+a)^(3/4),x)
 

Fricas [F]

\[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^{3/4} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {3}{4}} {\left (e x + d\right )} {\left (g x + f\right )} \,d x } \] Input:

integrate((e*x+d)*(g*x+f)*(c*x^2+b*x+a)^(3/4),x, algorithm="fricas")
 

Output:

integral((e*g*x^2 + d*f + (e*f + d*g)*x)*(c*x^2 + b*x + a)^(3/4), x)
 

Sympy [F]

\[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^{3/4} \, dx=\int \left (d + e x\right ) \left (f + g x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{4}}\, dx \] Input:

integrate((e*x+d)*(g*x+f)*(c*x**2+b*x+a)**(3/4),x)
 

Output:

Integral((d + e*x)*(f + g*x)*(a + b*x + c*x**2)**(3/4), x)
 

Maxima [F]

\[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^{3/4} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {3}{4}} {\left (e x + d\right )} {\left (g x + f\right )} \,d x } \] Input:

integrate((e*x+d)*(g*x+f)*(c*x^2+b*x+a)^(3/4),x, algorithm="maxima")
 

Output:

integrate((c*x^2 + b*x + a)^(3/4)*(e*x + d)*(g*x + f), x)
 

Giac [F]

\[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^{3/4} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {3}{4}} {\left (e x + d\right )} {\left (g x + f\right )} \,d x } \] Input:

integrate((e*x+d)*(g*x+f)*(c*x^2+b*x+a)^(3/4),x, algorithm="giac")
 

Output:

integrate((c*x^2 + b*x + a)^(3/4)*(e*x + d)*(g*x + f), x)
 

Mupad [F(-1)]

Timed out. \[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^{3/4} \, dx=\int \left (f+g\,x\right )\,\left (d+e\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/4} \,d x \] Input:

int((f + g*x)*(d + e*x)*(a + b*x + c*x^2)^(3/4),x)
 

Output:

int((f + g*x)*(d + e*x)*(a + b*x + c*x^2)^(3/4), x)
 

Reduce [F]

\[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^{3/4} \, dx=\frac {-448 \left (c \,x^{2}+b x +a \right )^{\frac {3}{4}} a^{2} c e g +176 \left (c \,x^{2}+b x +a \right )^{\frac {3}{4}} a \,b^{2} e g -288 \left (c \,x^{2}+b x +a \right )^{\frac {3}{4}} a b c d g -288 \left (c \,x^{2}+b x +a \right )^{\frac {3}{4}} a b c e f +336 \left (c \,x^{2}+b x +a \right )^{\frac {3}{4}} a b c e g x +2016 \left (c \,x^{2}+b x +a \right )^{\frac {3}{4}} a \,c^{2} d f -132 \left (c \,x^{2}+b x +a \right )^{\frac {3}{4}} b^{3} e g x +216 \left (c \,x^{2}+b x +a \right )^{\frac {3}{4}} b^{2} c d g x +216 \left (c \,x^{2}+b x +a \right )^{\frac {3}{4}} b^{2} c e f x +120 \left (c \,x^{2}+b x +a \right )^{\frac {3}{4}} b^{2} c e g \,x^{2}+1008 \left (c \,x^{2}+b x +a \right )^{\frac {3}{4}} b \,c^{2} d f x +720 \left (c \,x^{2}+b x +a \right )^{\frac {3}{4}} b \,c^{2} d g \,x^{2}+720 \left (c \,x^{2}+b x +a \right )^{\frac {3}{4}} b \,c^{2} e f \,x^{2}+560 \left (c \,x^{2}+b x +a \right )^{\frac {3}{4}} b \,c^{2} e g \,x^{3}+672 \left (\int \frac {x}{\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}}}d x \right ) a^{2} c^{2} e g -1092 \left (\int \frac {x}{\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}}}d x \right ) a \,b^{2} c e g +1512 \left (\int \frac {x}{\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}}}d x \right ) a b \,c^{2} d g +1512 \left (\int \frac {x}{\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}}}d x \right ) a b \,c^{2} e f -3024 \left (\int \frac {x}{\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}}}d x \right ) a \,c^{3} d f +231 \left (\int \frac {x}{\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}}}d x \right ) b^{4} e g -378 \left (\int \frac {x}{\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}}}d x \right ) b^{3} c d g -378 \left (\int \frac {x}{\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}}}d x \right ) b^{3} c e f +756 \left (\int \frac {x}{\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}}}d x \right ) b^{2} c^{2} d f}{2520 b \,c^{2}} \] Input:

int((e*x+d)*(g*x+f)*(c*x^2+b*x+a)^(3/4),x)
                                                                                    
                                                                                    
 

Output:

( - 448*(a + b*x + c*x**2)**(3/4)*a**2*c*e*g + 176*(a + b*x + c*x**2)**(3/ 
4)*a*b**2*e*g - 288*(a + b*x + c*x**2)**(3/4)*a*b*c*d*g - 288*(a + b*x + c 
*x**2)**(3/4)*a*b*c*e*f + 336*(a + b*x + c*x**2)**(3/4)*a*b*c*e*g*x + 2016 
*(a + b*x + c*x**2)**(3/4)*a*c**2*d*f - 132*(a + b*x + c*x**2)**(3/4)*b**3 
*e*g*x + 216*(a + b*x + c*x**2)**(3/4)*b**2*c*d*g*x + 216*(a + b*x + c*x** 
2)**(3/4)*b**2*c*e*f*x + 120*(a + b*x + c*x**2)**(3/4)*b**2*c*e*g*x**2 + 1 
008*(a + b*x + c*x**2)**(3/4)*b*c**2*d*f*x + 720*(a + b*x + c*x**2)**(3/4) 
*b*c**2*d*g*x**2 + 720*(a + b*x + c*x**2)**(3/4)*b*c**2*e*f*x**2 + 560*(a 
+ b*x + c*x**2)**(3/4)*b*c**2*e*g*x**3 + 672*int(((a + b*x + c*x**2)**(3/4 
)*x)/(a + b*x + c*x**2),x)*a**2*c**2*e*g - 1092*int(((a + b*x + c*x**2)**( 
3/4)*x)/(a + b*x + c*x**2),x)*a*b**2*c*e*g + 1512*int(((a + b*x + c*x**2)* 
*(3/4)*x)/(a + b*x + c*x**2),x)*a*b*c**2*d*g + 1512*int(((a + b*x + c*x**2 
)**(3/4)*x)/(a + b*x + c*x**2),x)*a*b*c**2*e*f - 3024*int(((a + b*x + c*x* 
*2)**(3/4)*x)/(a + b*x + c*x**2),x)*a*c**3*d*f + 231*int(((a + b*x + c*x** 
2)**(3/4)*x)/(a + b*x + c*x**2),x)*b**4*e*g - 378*int(((a + b*x + c*x**2)* 
*(3/4)*x)/(a + b*x + c*x**2),x)*b**3*c*d*g - 378*int(((a + b*x + c*x**2)** 
(3/4)*x)/(a + b*x + c*x**2),x)*b**3*c*e*f + 756*int(((a + b*x + c*x**2)**( 
3/4)*x)/(a + b*x + c*x**2),x)*b**2*c**2*d*f)/(2520*b*c**2)