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

Optimal. Leaf size=1066 \[ \frac{(2 c f-b g) \left (8 c^2 f^2-b^2 g^2-4 c g (2 b f-3 a g)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right ) e^2}{16 c^{3/2} g^3 (e f-d g)^3}-\frac{\left (c f^2-b g f+a g^2\right )^{3/2} \tanh ^{-1}\left (\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right ) e^2}{g^3 (e f-d g)^3}-\frac{\left (8 c^2 f^2+b^2 g^2-2 c g (5 b f-4 a g)-2 c g (2 c f-b g) x\right ) \sqrt{c x^2+b x+a} e^2}{8 c g^2 (e f-d g)^3}+\frac{\left (c x^2+b x+a\right )^{3/2} e}{(e f-d g)^2 (f+g x)}-\frac{3 \left (8 c^2 f^2+b^2 g^2-4 c g (2 b f-a g)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right ) e}{8 \sqrt{c} g^3 (e f-d g)^2}+\frac{3 (2 c f-b g) \sqrt{c f^2-b g f+a g^2} \tanh ^{-1}\left (\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right ) e}{2 g^3 (e f-d g)^2}+\frac{3 (4 c f-3 b g-2 c g x) \sqrt{c x^2+b x+a} e}{4 g^2 (e f-d g)^2}+\frac{\left (c x^2+b x+a\right )^{3/2}}{2 (e f-d g) (f+g x)^2}+\frac{3 \sqrt{c} (2 c f-b g) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right )}{2 g^3 (e f-d g)}-\frac{3 \left (8 c^2 f^2+b^2 g^2-4 c g (2 b f-a g)\right ) \tanh ^{-1}\left (\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right )}{8 g^3 (e f-d g) \sqrt{c f^2-b g f+a g^2}}+\frac{\left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right ) \sqrt{c x^2+b x+a}}{8 c (e f-d g)^3}-\frac{3 (4 c f-b g+2 c g x) \sqrt{c x^2+b x+a}}{4 g^2 (e f-d g) (f+g x)}-\frac{(2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right )}{16 c^{3/2} (e f-d g)^3 e}+\frac{\left (c d^2-b e d+a e^2\right )^{3/2} \tanh ^{-1}\left (\frac{b d-2 a e+(2 c d-b e) x}{2 \sqrt{c d^2-b e d+a e^2} \sqrt{c x^2+b x+a}}\right )}{(e f-d g)^3 e} \]

[Out]

((8*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 4*a*e) - 2*c*e*(2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(8*c*(e*f - d*g)
^3) + (3*e*(4*c*f - 3*b*g - 2*c*g*x)*Sqrt[a + b*x + c*x^2])/(4*g^2*(e*f - d*g)^2) - (3*(4*c*f - b*g + 2*c*g*x)
*Sqrt[a + b*x + c*x^2])/(4*g^2*(e*f - d*g)*(f + g*x)) - (e^2*(8*c^2*f^2 + b^2*g^2 - 2*c*g*(5*b*f - 4*a*g) - 2*
c*g*(2*c*f - b*g)*x)*Sqrt[a + b*x + c*x^2])/(8*c*g^2*(e*f - d*g)^3) + (a + b*x + c*x^2)^(3/2)/(2*(e*f - d*g)*(
f + g*x)^2) + (e*(a + b*x + c*x^2)^(3/2))/((e*f - d*g)^2*(f + g*x)) - ((2*c*d - b*e)*(8*c^2*d^2 - b^2*e^2 - 4*
c*e*(2*b*d - 3*a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(3/2)*e*(e*f - d*g)^3) + (3
*Sqrt[c]*(2*c*f - b*g)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*g^3*(e*f - d*g)) + (e^2*(2*c
*f - b*g)*(8*c^2*f^2 - b^2*g^2 - 4*c*g*(2*b*f - 3*a*g))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])]
)/(16*c^(3/2)*g^3*(e*f - d*g)^3) - (3*e*(8*c^2*f^2 + b^2*g^2 - 4*c*g*(2*b*f - a*g))*ArcTanh[(b + 2*c*x)/(2*Sqr
t[c]*Sqrt[a + b*x + c*x^2])])/(8*Sqrt[c]*g^3*(e*f - d*g)^2) + ((c*d^2 - b*d*e + a*e^2)^(3/2)*ArcTanh[(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*(e*f - d*g)^3) + (3*e*(2*c*f
- b*g)*Sqrt[c*f^2 - b*f*g + a*g^2]*ArcTanh[(b*f - 2*a*g + (2*c*f - b*g)*x)/(2*Sqrt[c*f^2 - b*f*g + a*g^2]*Sqr
t[a + b*x + c*x^2])])/(2*g^3*(e*f - d*g)^2) - (e^2*(c*f^2 - b*f*g + a*g^2)^(3/2)*ArcTanh[(b*f - 2*a*g + (2*c*f
- b*g)*x)/(2*Sqrt[c*f^2 - b*f*g + a*g^2]*Sqrt[a + b*x + c*x^2])])/(g^3*(e*f - d*g)^3) - (3*(8*c^2*f^2 + b^2*g
^2 - 4*c*g*(2*b*f - a*g))*ArcTanh[(b*f - 2*a*g + (2*c*f - b*g)*x)/(2*Sqrt[c*f^2 - b*f*g + a*g^2]*Sqrt[a + b*x
+ c*x^2])])/(8*g^3*(e*f - d*g)*Sqrt[c*f^2 - b*f*g + a*g^2])

________________________________________________________________________________________

Rubi [A]  time = 1.7059, antiderivative size = 1066, normalized size of antiderivative = 1., number of steps used = 30, number of rules used = 9, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.31, Rules used = {960, 734, 814, 843, 621, 206, 724, 732, 812} \[ \frac{(2 c f-b g) \left (8 c^2 f^2-b^2 g^2-4 c g (2 b f-3 a g)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right ) e^2}{16 c^{3/2} g^3 (e f-d g)^3}-\frac{\left (c f^2-b g f+a g^2\right )^{3/2} \tanh ^{-1}\left (\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right ) e^2}{g^3 (e f-d g)^3}-\frac{\left (8 c^2 f^2+b^2 g^2-2 c g (5 b f-4 a g)-2 c g (2 c f-b g) x\right ) \sqrt{c x^2+b x+a} e^2}{8 c g^2 (e f-d g)^3}+\frac{\left (c x^2+b x+a\right )^{3/2} e}{(e f-d g)^2 (f+g x)}-\frac{3 \left (8 c^2 f^2+b^2 g^2-4 c g (2 b f-a g)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right ) e}{8 \sqrt{c} g^3 (e f-d g)^2}+\frac{3 (2 c f-b g) \sqrt{c f^2-b g f+a g^2} \tanh ^{-1}\left (\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right ) e}{2 g^3 (e f-d g)^2}+\frac{3 (4 c f-3 b g-2 c g x) \sqrt{c x^2+b x+a} e}{4 g^2 (e f-d g)^2}+\frac{\left (c x^2+b x+a\right )^{3/2}}{2 (e f-d g) (f+g x)^2}+\frac{3 \sqrt{c} (2 c f-b g) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right )}{2 g^3 (e f-d g)}-\frac{3 \left (8 c^2 f^2+b^2 g^2-4 c g (2 b f-a g)\right ) \tanh ^{-1}\left (\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right )}{8 g^3 (e f-d g) \sqrt{c f^2-b g f+a g^2}}+\frac{\left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right ) \sqrt{c x^2+b x+a}}{8 c (e f-d g)^3}-\frac{3 (4 c f-b g+2 c g x) \sqrt{c x^2+b x+a}}{4 g^2 (e f-d g) (f+g x)}-\frac{(2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right )}{16 c^{3/2} (e f-d g)^3 e}+\frac{\left (c d^2-b e d+a e^2\right )^{3/2} \tanh ^{-1}\left (\frac{b d-2 a e+(2 c d-b e) x}{2 \sqrt{c d^2-b e d+a e^2} \sqrt{c x^2+b x+a}}\right )}{(e f-d g)^3 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((8*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 4*a*e) - 2*c*e*(2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(8*c*(e*f - d*g)
^3) + (3*e*(4*c*f - 3*b*g - 2*c*g*x)*Sqrt[a + b*x + c*x^2])/(4*g^2*(e*f - d*g)^2) - (3*(4*c*f - b*g + 2*c*g*x)
*Sqrt[a + b*x + c*x^2])/(4*g^2*(e*f - d*g)*(f + g*x)) - (e^2*(8*c^2*f^2 + b^2*g^2 - 2*c*g*(5*b*f - 4*a*g) - 2*
c*g*(2*c*f - b*g)*x)*Sqrt[a + b*x + c*x^2])/(8*c*g^2*(e*f - d*g)^3) + (a + b*x + c*x^2)^(3/2)/(2*(e*f - d*g)*(
f + g*x)^2) + (e*(a + b*x + c*x^2)^(3/2))/((e*f - d*g)^2*(f + g*x)) - ((2*c*d - b*e)*(8*c^2*d^2 - b^2*e^2 - 4*
c*e*(2*b*d - 3*a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(3/2)*e*(e*f - d*g)^3) + (3
*Sqrt[c]*(2*c*f - b*g)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*g^3*(e*f - d*g)) + (e^2*(2*c
*f - b*g)*(8*c^2*f^2 - b^2*g^2 - 4*c*g*(2*b*f - 3*a*g))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])]
)/(16*c^(3/2)*g^3*(e*f - d*g)^3) - (3*e*(8*c^2*f^2 + b^2*g^2 - 4*c*g*(2*b*f - a*g))*ArcTanh[(b + 2*c*x)/(2*Sqr
t[c]*Sqrt[a + b*x + c*x^2])])/(8*Sqrt[c]*g^3*(e*f - d*g)^2) + ((c*d^2 - b*d*e + a*e^2)^(3/2)*ArcTanh[(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*(e*f - d*g)^3) + (3*e*(2*c*f
- b*g)*Sqrt[c*f^2 - b*f*g + a*g^2]*ArcTanh[(b*f - 2*a*g + (2*c*f - b*g)*x)/(2*Sqrt[c*f^2 - b*f*g + a*g^2]*Sqr
t[a + b*x + c*x^2])])/(2*g^3*(e*f - d*g)^2) - (e^2*(c*f^2 - b*f*g + a*g^2)^(3/2)*ArcTanh[(b*f - 2*a*g + (2*c*f
- b*g)*x)/(2*Sqrt[c*f^2 - b*f*g + a*g^2]*Sqrt[a + b*x + c*x^2])])/(g^3*(e*f - d*g)^3) - (3*(8*c^2*f^2 + b^2*g
^2 - 4*c*g*(2*b*f - a*g))*ArcTanh[(b*f - 2*a*g + (2*c*f - b*g)*x)/(2*Sqrt[c*f^2 - b*f*g + a*g^2]*Sqrt[a + b*x
+ c*x^2])])/(8*g^3*(e*f - d*g)*Sqrt[c*f^2 - b*f*g + a*g^2])

Rule 960

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && (IntegerQ[p] || (ILtQ[m, 0] &&
ILtQ[n, 0])) &&  !(IGtQ[m, 0] || IGtQ[n, 0])

Rule 734

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + 2*p + 1)), x] - Dist[p/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*
d - b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ
[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || Lt
Q[m, 1]) &&  !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 814

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*(a + b*x + c*x^
2)^p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0
] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 843

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 621

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 724

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 732

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + 1)), x] - Dist[p/(e*(m + 1)), Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^
(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ
[2*c*d - b*e, 0] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m + 2*p + 1, 0] && IntQua
draticQ[a, b, c, d, e, m, p, x]

Rule 812

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + b*x + c*x^2)^p)/(e^2*(m + 1)*(m
+ 2*p + 2)), x] + Dist[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(
b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p
+ 2))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rubi steps

\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{3/2}}{(d+e x) (f+g x)^3} \, dx &=\int \left (\frac{e^3 \left (a+b x+c x^2\right )^{3/2}}{(e f-d g)^3 (d+e x)}-\frac{g \left (a+b x+c x^2\right )^{3/2}}{(e f-d g) (f+g x)^3}-\frac{e g \left (a+b x+c x^2\right )^{3/2}}{(e f-d g)^2 (f+g x)^2}-\frac{e^2 g \left (a+b x+c x^2\right )^{3/2}}{(e f-d g)^3 (f+g x)}\right ) \, dx\\ &=\frac{e^3 \int \frac{\left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx}{(e f-d g)^3}-\frac{\left (e^2 g\right ) \int \frac{\left (a+b x+c x^2\right )^{3/2}}{f+g x} \, dx}{(e f-d g)^3}-\frac{(e g) \int \frac{\left (a+b x+c x^2\right )^{3/2}}{(f+g x)^2} \, dx}{(e f-d g)^2}-\frac{g \int \frac{\left (a+b x+c x^2\right )^{3/2}}{(f+g x)^3} \, dx}{e f-d g}\\ &=\frac{\left (a+b x+c x^2\right )^{3/2}}{2 (e f-d g) (f+g x)^2}+\frac{e \left (a+b x+c x^2\right )^{3/2}}{(e f-d g)^2 (f+g x)}-\frac{e^2 \int \frac{(b d-2 a e+(2 c d-b e) x) \sqrt{a+b x+c x^2}}{d+e x} \, dx}{2 (e f-d g)^3}+\frac{e^2 \int \frac{(b f-2 a g+(2 c f-b g) x) \sqrt{a+b x+c x^2}}{f+g x} \, dx}{2 (e f-d g)^3}-\frac{(3 e) \int \frac{(b+2 c x) \sqrt{a+b x+c x^2}}{f+g x} \, dx}{2 (e f-d g)^2}-\frac{3 \int \frac{(b+2 c x) \sqrt{a+b x+c x^2}}{(f+g x)^2} \, dx}{4 (e f-d g)}\\ &=\frac{\left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right ) \sqrt{a+b x+c x^2}}{8 c (e f-d g)^3}+\frac{3 e (4 c f-3 b g-2 c g x) \sqrt{a+b x+c x^2}}{4 g^2 (e f-d g)^2}-\frac{3 (4 c f-b g+2 c g x) \sqrt{a+b x+c x^2}}{4 g^2 (e f-d g) (f+g x)}-\frac{e^2 \left (8 c^2 f^2+b^2 g^2-2 c g (5 b f-4 a g)-2 c g (2 c f-b g) x\right ) \sqrt{a+b x+c x^2}}{8 c g^2 (e f-d g)^3}+\frac{\left (a+b x+c x^2\right )^{3/2}}{2 (e f-d g) (f+g x)^2}+\frac{e \left (a+b x+c x^2\right )^{3/2}}{(e f-d g)^2 (f+g x)}+\frac{\int \frac{\frac{1}{2} \left (4 c e (b d-2 a e)^2-d (2 c d-b e) \left (4 b c d-b^2 e-4 a c e\right )\right )-\frac{1}{2} (2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) x}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{8 c (e f-d g)^3}-\frac{e^2 \int \frac{\frac{1}{2} \left (4 c g (b f-2 a g)^2-f (2 c f-b g) \left (4 b c f-b^2 g-4 a c g\right )\right )-\frac{1}{2} (2 c f-b g) \left (8 c^2 f^2-b^2 g^2-4 c g (2 b f-3 a g)\right ) x}{(f+g x) \sqrt{a+b x+c x^2}} \, dx}{8 c g^2 (e f-d g)^3}+\frac{(3 e) \int \frac{c \left (3 b^2 f g+4 a c f g-4 b \left (c f^2+a g^2\right )\right )-c \left (8 c^2 f^2+b^2 g^2-4 c g (2 b f-a g)\right ) x}{(f+g x) \sqrt{a+b x+c x^2}} \, dx}{8 c g^2 (e f-d g)^2}+\frac{3 \int \frac{4 b c f-b^2 g-4 a c g+4 c (2 c f-b g) x}{(f+g x) \sqrt{a+b x+c x^2}} \, dx}{8 g^2 (e f-d g)}\\ &=\frac{\left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right ) \sqrt{a+b x+c x^2}}{8 c (e f-d g)^3}+\frac{3 e (4 c f-3 b g-2 c g x) \sqrt{a+b x+c x^2}}{4 g^2 (e f-d g)^2}-\frac{3 (4 c f-b g+2 c g x) \sqrt{a+b x+c x^2}}{4 g^2 (e f-d g) (f+g x)}-\frac{e^2 \left (8 c^2 f^2+b^2 g^2-2 c g (5 b f-4 a g)-2 c g (2 c f-b g) x\right ) \sqrt{a+b x+c x^2}}{8 c g^2 (e f-d g)^3}+\frac{\left (a+b x+c x^2\right )^{3/2}}{2 (e f-d g) (f+g x)^2}+\frac{e \left (a+b x+c x^2\right )^{3/2}}{(e f-d g)^2 (f+g x)}+\frac{\left (c d^2-b d e+a e^2\right )^2 \int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{e (e f-d g)^3}-\frac{\left ((2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{16 c e (e f-d g)^3}+\frac{(3 c (2 c f-b g)) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{2 g^3 (e f-d g)}+\frac{\left (3 e (2 c f-b g) \left (c f^2-b f g+a g^2\right )\right ) \int \frac{1}{(f+g x) \sqrt{a+b x+c x^2}} \, dx}{2 g^3 (e f-d g)^2}-\frac{\left (e^2 \left (c f^2-b f g+a g^2\right )^2\right ) \int \frac{1}{(f+g x) \sqrt{a+b x+c x^2}} \, dx}{g^3 (e f-d g)^3}+\frac{\left (e^2 (2 c f-b g) \left (8 c^2 f^2-b^2 g^2-4 c g (2 b f-3 a g)\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{16 c g^3 (e f-d g)^3}-\frac{\left (3 e \left (8 c^2 f^2+b^2 g^2-4 c g (2 b f-a g)\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{8 g^3 (e f-d g)^2}-\frac{\left (3 \left (8 c^2 f^2+b^2 g^2-4 c g (2 b f-a g)\right )\right ) \int \frac{1}{(f+g x) \sqrt{a+b x+c x^2}} \, dx}{8 g^3 (e f-d g)}\\ &=\frac{\left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right ) \sqrt{a+b x+c x^2}}{8 c (e f-d g)^3}+\frac{3 e (4 c f-3 b g-2 c g x) \sqrt{a+b x+c x^2}}{4 g^2 (e f-d g)^2}-\frac{3 (4 c f-b g+2 c g x) \sqrt{a+b x+c x^2}}{4 g^2 (e f-d g) (f+g x)}-\frac{e^2 \left (8 c^2 f^2+b^2 g^2-2 c g (5 b f-4 a g)-2 c g (2 c f-b g) x\right ) \sqrt{a+b x+c x^2}}{8 c g^2 (e f-d g)^3}+\frac{\left (a+b x+c x^2\right )^{3/2}}{2 (e f-d g) (f+g x)^2}+\frac{e \left (a+b x+c x^2\right )^{3/2}}{(e f-d g)^2 (f+g x)}-\frac{\left (2 \left (c d^2-b d e+a e^2\right )^2\right ) \operatorname{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 (e f-d g)^3}-\frac{\left ((2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{8 c e (e f-d g)^3}+\frac{(3 c (2 c f-b g)) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{g^3 (e f-d g)}-\frac{\left (3 e (2 c f-b g) \left (c f^2-b f g+a g^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c f^2-4 b f g+4 a g^2-x^2} \, dx,x,\frac{-b f+2 a g-(2 c f-b g) x}{\sqrt{a+b x+c x^2}}\right )}{g^3 (e f-d g)^2}+\frac{\left (2 e^2 \left (c f^2-b f g+a g^2\right )^2\right ) \operatorname{Subst}\left (\int \frac{1}{4 c f^2-4 b f g+4 a g^2-x^2} \, dx,x,\frac{-b f+2 a g-(2 c f-b g) x}{\sqrt{a+b x+c x^2}}\right )}{g^3 (e f-d g)^3}+\frac{\left (e^2 (2 c f-b g) \left (8 c^2 f^2-b^2 g^2-4 c g (2 b f-3 a g)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{8 c g^3 (e f-d g)^3}-\frac{\left (3 e \left (8 c^2 f^2+b^2 g^2-4 c g (2 b f-a g)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{4 g^3 (e f-d g)^2}+\frac{\left (3 \left (8 c^2 f^2+b^2 g^2-4 c g (2 b f-a g)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c f^2-4 b f g+4 a g^2-x^2} \, dx,x,\frac{-b f+2 a g-(2 c f-b g) x}{\sqrt{a+b x+c x^2}}\right )}{4 g^3 (e f-d g)}\\ &=\frac{\left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right ) \sqrt{a+b x+c x^2}}{8 c (e f-d g)^3}+\frac{3 e (4 c f-3 b g-2 c g x) \sqrt{a+b x+c x^2}}{4 g^2 (e f-d g)^2}-\frac{3 (4 c f-b g+2 c g x) \sqrt{a+b x+c x^2}}{4 g^2 (e f-d g) (f+g x)}-\frac{e^2 \left (8 c^2 f^2+b^2 g^2-2 c g (5 b f-4 a g)-2 c g (2 c f-b g) x\right ) \sqrt{a+b x+c x^2}}{8 c g^2 (e f-d g)^3}+\frac{\left (a+b x+c x^2\right )^{3/2}}{2 (e f-d g) (f+g x)^2}+\frac{e \left (a+b x+c x^2\right )^{3/2}}{(e f-d g)^2 (f+g x)}-\frac{(2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 c^{3/2} e (e f-d g)^3}+\frac{3 \sqrt{c} (2 c f-b g) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 g^3 (e f-d g)}+\frac{e^2 (2 c f-b g) \left (8 c^2 f^2-b^2 g^2-4 c g (2 b f-3 a g)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 c^{3/2} g^3 (e f-d g)^3}-\frac{3 e \left (8 c^2 f^2+b^2 g^2-4 c g (2 b f-a g)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 \sqrt{c} g^3 (e f-d g)^2}+\frac{\left (c d^2-b d e+a e^2\right )^{3/2} \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 (e f-d g)^3}+\frac{3 e (2 c f-b g) \sqrt{c f^2-b f g+a g^2} \tanh ^{-1}\left (\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b f g+a g^2} \sqrt{a+b x+c x^2}}\right )}{2 g^3 (e f-d g)^2}-\frac{e^2 \left (c f^2-b f g+a g^2\right )^{3/2} \tanh ^{-1}\left (\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b f g+a g^2} \sqrt{a+b x+c x^2}}\right )}{g^3 (e f-d g)^3}-\frac{3 \left (8 c^2 f^2+b^2 g^2-4 c g (2 b f-a g)\right ) \tanh ^{-1}\left (\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b f g+a g^2} \sqrt{a+b x+c x^2}}\right )}{8 g^3 (e f-d g) \sqrt{c f^2-b f g+a g^2}}\\ \end{align*}

Mathematica [A]  time = 3.76584, size = 1036, normalized size = 0.97 \[ \frac{1}{4} \left (-\frac{\left ((2 c f-b g) \left (8 c^2 f^2-b^2 g^2+4 c g (3 a g-2 b f)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )+2 \sqrt{c} \left (8 c \tanh ^{-1}\left (\frac{-b f-2 c x f+2 a g+b g x}{2 \sqrt{c f^2+g (a g-b f)} \sqrt{a+x (b+c x)}}\right ) \left (c f^2+g (a g-b f)\right )^{3/2}+g \sqrt{a+x (b+c x)} \left (4 f (g x-2 f) c^2-2 g (-5 b f+4 a g+b g x) c-b^2 g^2\right )\right )\right ) e^2}{4 c^{3/2} g^3 (d g-e f)^3}+\frac{4 (a+x (b+c x))^{3/2} e}{(e f-d g)^2 (f+g x)}-\frac{3 \left (\left (8 c^2 f^2+b^2 g^2+4 c g (a g-2 b f)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )+2 \sqrt{c} \left (g \sqrt{a+x (b+c x)} (-4 c f+3 b g+2 c g x)+2 (2 c f-b g) \sqrt{c f^2+g (a g-b f)} \tanh ^{-1}\left (\frac{-b f-2 c x f+2 a g+b g x}{2 \sqrt{c f^2+g (a g-b f)} \sqrt{a+x (b+c x)}}\right )\right )\right ) e}{2 \sqrt{c} g^3 (e f-d g)^2}+\frac{2 (a+x (b+c x))^{3/2}}{(e f-d g) (f+g x)^2}+\frac{3 \left (\frac{(b g-2 c f) (a+x (b+c x))^{3/2}}{f+g x}-\frac{\left (2 f (2 f-g x) c^2+g (-5 b f+2 a g+b g x) c+b^2 g^2\right ) \sqrt{a+x (b+c x)}}{g^2}+\frac{4 \sqrt{c} (2 c f-b g) \left (c f^2+g (a g-b f)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )+\left (8 c^2 f^2+b^2 g^2+4 c g (a g-2 b f)\right ) \sqrt{c f^2+g (a g-b f)} \tanh ^{-1}\left (\frac{-b f-2 c x f+2 a g+b g x}{2 \sqrt{c f^2+g (a g-b f)} \sqrt{a+x (b+c x)}}\right )}{2 g^3}\right )}{(e f-d g) \left (c f^2+g (a g-b f)\right )}+\frac{-(2 c d-b e) \left (8 c^2 d^2-b^2 e^2+4 c e (3 a e-2 b d)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )-2 \sqrt{c} \left (8 c \tanh ^{-1}\left (\frac{-b d-2 c x d+2 a e+b e x}{2 \sqrt{c d^2+e (a e-b d)} \sqrt{a+x (b+c x)}}\right ) \left (c d^2+e (a e-b d)\right )^{3/2}+e \sqrt{a+x (b+c x)} \left (4 d (e x-2 d) c^2-2 e (-5 b d+4 a e+b e x) c-b^2 e^2\right )\right )}{4 c^{3/2} (e f-d g)^3 e}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*(a + x*(b + c*x))^(3/2))/((e*f - d*g)*(f + g*x)^2) + (4*e*(a + x*(b + c*x))^(3/2))/((e*f - d*g)^2*(f + g*x
)) + (-((2*c*d - b*e)*(8*c^2*d^2 - b^2*e^2 + 4*c*e*(-2*b*d + 3*a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x
*(b + c*x)])]) - 2*Sqrt[c]*(e*Sqrt[a + x*(b + c*x)]*(-(b^2*e^2) + 4*c^2*d*(-2*d + e*x) - 2*c*e*(-5*b*d + 4*a*e
+ b*e*x)) + 8*c*(c*d^2 + e*(-(b*d) + a*e))^(3/2)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e
*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])]))/(4*c^(3/2)*e*(e*f - d*g)^3) - (3*e*((8*c^2*f^2 + b^2*g^2 + 4*c*g*(-
2*b*f + a*g))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + 2*Sqrt[c]*(g*(-4*c*f + 3*b*g + 2*c*g*x)
*Sqrt[a + x*(b + c*x)] + 2*(2*c*f - b*g)*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*ArcTanh[(-(b*f) + 2*a*g - 2*c*f*x + b*
g*x)/(2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a + x*(b + c*x)])])))/(2*Sqrt[c]*g^3*(e*f - d*g)^2) + (3*(((-2*c*f
+ b*g)*(a + x*(b + c*x))^(3/2))/(f + g*x) - (Sqrt[a + x*(b + c*x)]*(b^2*g^2 + 2*c^2*f*(2*f - g*x) + c*g*(-5*b
*f + 2*a*g + b*g*x)))/g^2 + (4*Sqrt[c]*(2*c*f - b*g)*(c*f^2 + g*(-(b*f) + a*g))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]
*Sqrt[a + x*(b + c*x)])] + (8*c^2*f^2 + b^2*g^2 + 4*c*g*(-2*b*f + a*g))*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*ArcTanh
[(-(b*f) + 2*a*g - 2*c*f*x + b*g*x)/(2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a + x*(b + c*x)])])/(2*g^3)))/((e*f
- d*g)*(c*f^2 + g*(-(b*f) + a*g))) - (e^2*((2*c*f - b*g)*(8*c^2*f^2 - b^2*g^2 + 4*c*g*(-2*b*f + 3*a*g))*ArcTa
nh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + 2*Sqrt[c]*(g*Sqrt[a + x*(b + c*x)]*(-(b^2*g^2) + 4*c^2*f*(
-2*f + g*x) - 2*c*g*(-5*b*f + 4*a*g + b*g*x)) + 8*c*(c*f^2 + g*(-(b*f) + a*g))^(3/2)*ArcTanh[(-(b*f) + 2*a*g -
2*c*f*x + b*g*x)/(2*Sqrt[c*f^2 + g*(-(b*f) + a*g)]*Sqrt[a + x*(b + c*x)])])))/(4*c^(3/2)*g^3*(-(e*f) + d*g)^3
))/4

________________________________________________________________________________________

Maple [B]  time = 0.356, size = 15927, normalized size = 14.9 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}{\left (g x + f\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError