\(\int \frac {(f+g x)^3}{(d+e x) (a+b x+c x^2)^{3/2}} \, dx\) [879]

Optimal result

Integrand size = 29, antiderivative size = 357 \[ \int \frac {(f+g x)^3}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (b^2 \left (c e f^3+a d g^3\right )-2 a c \left (c f^2 (e f-3 d g)-a g^2 (3 e f-d g)\right )-b \left (c^2 d f^3+a^2 e g^3+3 a c f g (e f+d g)\right )-\left (2 c^3 d f^3-b^2 (b d-a e) g^3+c g^2 \left (3 b^2 d f-3 a b e f+3 a b d g-2 a^2 e g\right )+c^2 f (6 a g (e f-d g)-b f (e f+3 d g))\right ) x\right )}{c \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}+\frac {g^3 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{3/2} e}+\frac {(e f-d g)^3 \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e \left (c d^2-b d e+a e^2\right )^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1660, 857, 635, 212, 738} \[ \int \frac {(f+g x)^3}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (-x \left (c g^2 \left (-2 a^2 e g+3 a b d g-3 a b e f+3 b^2 d f\right )-b^2 g^3 (b d-a e)+c^2 f (6 a g (e f-d g)-b f (3 d g+e f))+2 c^3 d f^3\right )-b \left (a^2 e g^3+3 a c f g (d g+e f)+c^2 d f^3\right )+b^2 \left (a d g^3+c e f^3\right )-2 a c \left (c f^2 (e f-3 d g)-a g^2 (3 e f-d g)\right )\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}+\frac {g^3 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{3/2} e}+\frac {(e f-d g)^3 \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e \left (a e^2-b d e+c d^2\right )^{3/2}} \]

[In]

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

[Out]

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

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-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] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 1660

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[(d + e*x)^m*Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2,
x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*
c*f - b*g)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*f - b*
g))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (b^2 \left (c e f^3+a d g^3\right )-2 a c \left (c f^2 (e f-3 d g)-a g^2 (3 e f-d g)\right )-b \left (c^2 d f^3+a^2 e g^3+3 a c f g (e f+d g)\right )-\left (2 c^3 d f^3-b^2 (b d-a e) g^3+c g^2 \left (3 b^2 d f-3 a b e f+3 a b d g-2 a^2 e g\right )+c^2 f (6 a g (e f-d g)-b f (e f+3 d g))\right ) x\right )}{c \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}-\frac {2 \int \frac {\frac {\left (b^2-4 a c\right ) \left (d (b d-a e) g^3-c f \left (e^2 f^2-3 d e f g+3 d^2 g^2\right )\right )}{2 c \left (c d^2-b d e+a e^2\right )}-\frac {\left (b^2-4 a c\right ) g^3 x}{2 c}}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{b^2-4 a c} \\ & = \frac {2 \left (b^2 \left (c e f^3+a d g^3\right )-2 a c \left (c f^2 (e f-3 d g)-a g^2 (3 e f-d g)\right )-b \left (c^2 d f^3+a^2 e g^3+3 a c f g (e f+d g)\right )-\left (2 c^3 d f^3-b^2 (b d-a e) g^3+c g^2 \left (3 b^2 d f-3 a b e f+3 a b d g-2 a^2 e g\right )+c^2 f (6 a g (e f-d g)-b f (e f+3 d g))\right ) x\right )}{c \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}+\frac {g^3 \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{c e}+\frac {(e f-d g)^3 \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{e \left (c d^2-b d e+a e^2\right )} \\ & = \frac {2 \left (b^2 \left (c e f^3+a d g^3\right )-2 a c \left (c f^2 (e f-3 d g)-a g^2 (3 e f-d g)\right )-b \left (c^2 d f^3+a^2 e g^3+3 a c f g (e f+d g)\right )-\left (2 c^3 d f^3-b^2 (b d-a e) g^3+c g^2 \left (3 b^2 d f-3 a b e f+3 a b d g-2 a^2 e g\right )+c^2 f (6 a g (e f-d g)-b f (e f+3 d g))\right ) x\right )}{c \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}+\frac {\left (2 g^3\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{c e}-\frac {\left (2 (e f-d g)^3\right ) \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac {-b d+2 a e-(2 c d-b e) x}{\sqrt {a+b x+c x^2}}\right )}{e \left (c d^2-b d e+a e^2\right )} \\ & = \frac {2 \left (b^2 \left (c e f^3+a d g^3\right )-2 a c \left (c f^2 (e f-3 d g)-a g^2 (3 e f-d g)\right )-b \left (c^2 d f^3+a^2 e g^3+3 a c f g (e f+d g)\right )-\left (2 c^3 d f^3-b^2 (b d-a e) g^3+c g^2 \left (3 b^2 d f-3 a b e f+3 a b d g-2 a^2 e g\right )+c^2 f (6 a g (e f-d g)-b f (e f+3 d g))\right ) x\right )}{c \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}+\frac {g^3 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{3/2} e}+\frac {(e f-d g)^3 \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e \left (c d^2-b d e+a e^2\right )^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.95 (sec) , antiderivative size = 354, normalized size of antiderivative = 0.99 \[ \int \frac {(f+g x)^3}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (-b^3 d g^3 x+b^2 \left (a g^3 (-d+e x)+c \left (-e f^3+3 d f g^2 x\right )\right )+b \left (a^2 e g^3+c^2 f^2 (-e f x+d (f-3 g x))+3 a c g (e f (f-g x)+d g (f+g x))\right )+2 c \left (c^2 d f^3 x+a^2 g^2 (d g-e (3 f+g x))+a c f (-3 d g (f+g x)+e f (f+3 g x))\right )\right )}{c \left (-b^2+4 a c\right ) \left (c d^2+e (-b d+a e)\right ) \sqrt {a+x (b+c x)}}+\frac {2 (-e f+d g)^3 \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+e (b d-a e)}}\right )}{e \sqrt {-c d^2+e (b d-a e)} \left (c d^2+e (-b d+a e)\right )}-\frac {g^3 \log \left (c e \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{c^{3/2} e} \]

[In]

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

[Out]

(2*(-(b^3*d*g^3*x) + b^2*(a*g^3*(-d + e*x) + c*(-(e*f^3) + 3*d*f*g^2*x)) + b*(a^2*e*g^3 + c^2*f^2*(-(e*f*x) +
d*(f - 3*g*x)) + 3*a*c*g*(e*f*(f - g*x) + d*g*(f + g*x))) + 2*c*(c^2*d*f^3*x + a^2*g^2*(d*g - e*(3*f + g*x)) +
a*c*f*(-3*d*g*(f + g*x) + e*f*(f + 3*g*x)))))/(c*(-b^2 + 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[a + x*(b + c*
x)]) + (2*(-(e*f) + d*g)^3*ArcTan[(Sqrt[c]*(d + e*x) - e*Sqrt[a + x*(b + c*x)])/Sqrt[-(c*d^2) + e*(b*d - a*e)]
])/(e*Sqrt[-(c*d^2) + e*(b*d - a*e)]*(c*d^2 + e*(-(b*d) + a*e))) - (g^3*Log[c*e*(b + 2*c*x - 2*Sqrt[c]*Sqrt[a
+ x*(b + c*x)])])/(c^(3/2)*e)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(739\) vs. \(2(339)=678\).

Time = 0.84 (sec) , antiderivative size = 740, normalized size of antiderivative = 2.07

method	result	size
default	\(\frac {\left (-d^{3} g^{3}+3 d^{2} e f \,g^{2}-3 d \,e^{2} f^{2} g +e^{3} f^{3}\right ) \left (\frac {e^{2}}{\left (e^{2} a -b d e +c \,d^{2}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}-\frac {\left (b e -2 c d \right ) e \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{\left (e^{2} a -b d e +c \,d^{2}\right ) \left (\frac {4 c \left (e^{2} a -b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}-\frac {e^{2} \ln \left (\frac {\frac {2 e^{2} a -2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a -b d e +c \,d^{2}\right ) \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}\right )}{e^{4}}+\frac {g \left (\frac {2 d^{2} g^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+e^{2} g^{2} \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )+\frac {6 e^{2} f^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {6 d e f g \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\left (-d e \,g^{2}+3 e^{2} f g \right ) \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )\right )}{e^{3}}\)	\(740\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume((b/e-(2*c*d)/e^2)^2>0)', see `
assume?` for

Giac [F(-2)]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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