3.869 \(\int \frac{(a+b x+c x^2)^{5/2}}{(d+e x) (f+g x)} \, dx\)

Optimal. Leaf size=886 \[ \frac{\tanh ^{-1}\left (\frac{b d-2 a e+(2 c d-b e) x}{2 \sqrt{c d^2-b e d+a e^2} \sqrt{c x^2+b x+a}}\right ) \left (c d^2-b e d+a e^2\right )^{5/2}}{e^5 (e f-d g)}+\frac{\left (c x^2+b x+a\right )^{3/2} \left (c d^2-b e d+a e^2\right )}{3 e^2 (e f-d g)}-\frac{(2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right ) \left (c d^2-b e d+a e^2\right )}{16 c^{3/2} e^5 (e f-d g)}+\frac{\left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right ) \sqrt{c x^2+b x+a} \left (c d^2-b e d+a e^2\right )}{8 c e^4 (e f-d g)}-\frac{\left (8 c e f^2-g (11 b e f-3 b d g-8 a e g)-6 c g (e f-d g) x\right ) \left (c x^2+b x+a\right )^{3/2}}{24 e g^2 (e f-d g)}+\frac{\left (128 c^4 e f^5-320 c^3 e g (b f-a g) f^3-b^3 g^4 (5 b e f+3 b d g-8 a e g)+48 c^2 g^2 \left (5 b^2 e f^3-10 a b e g f^2+a^2 g^2 (5 e f-d g)\right )-8 b c g^3 \left (5 b^2 e f^2+12 a^2 e g^2-3 a b g (5 e f+d g)\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right )}{128 c^{3/2} e g^5 (e f-d g)}-\frac{\left (c f^2-b g f+a g^2\right )^{5/2} \tanh ^{-1}\left (\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right )}{g^5 (e f-d g)}-\frac{\left (64 c^3 e f^4-16 c^2 e g (9 b f-8 a g) f^2-b^2 g^3 (5 b e f+3 b d g-8 a e g)+4 c g^2 \left (22 b^2 e f^2+16 a^2 e g^2-3 a b g (13 e f-d g)\right )-2 c g \left (16 c^2 e f^3+b g^2 (5 b e f+3 b d g-8 a e g)-4 c g \left (6 b e f^2-a g (7 e f-3 d g)\right )\right ) x\right ) \sqrt{c x^2+b x+a}}{64 c e g^4 (e f-d g)} \]

[Out]

((c*d^2 - b*d*e + a*e^2)*(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^4*(e*f - d*g)) - ((64*c^3*e*f^4 - 16*c^2*e*f^2*g*(9*b*f - 8*a*g) - b^2*g^3*(5*b*e*f + 3*b*d*g -
8*a*e*g) + 4*c*g^2*(22*b^2*e*f^2 + 16*a^2*e*g^2 - 3*a*b*g*(13*e*f - d*g)) - 2*c*g*(16*c^2*e*f^3 + b*g^2*(5*b*
e*f + 3*b*d*g - 8*a*e*g) - 4*c*g*(6*b*e*f^2 - a*g*(7*e*f - 3*d*g)))*x)*Sqrt[a + b*x + c*x^2])/(64*c*e*g^4*(e*f
- d*g)) + ((c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^(3/2))/(3*e^2*(e*f - d*g)) - ((8*c*e*f^2 - g*(11*b*e*f -
3*b*d*g - 8*a*e*g) - 6*c*g*(e*f - d*g)*x)*(a + b*x + c*x^2)^(3/2))/(24*e*g^2*(e*f - d*g)) - ((2*c*d - b*e)*(c
*d^2 - b*d*e + a*e^2)*(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^5*(e*f - d*g)) + ((128*c^4*e*f^5 - 320*c^3*e*f^3*g*(b*f - a*g) - b^3*g^4*(5*b*e*f
+ 3*b*d*g - 8*a*e*g) + 48*c^2*g^2*(5*b^2*e*f^3 - 10*a*b*e*f^2*g + a^2*g^2*(5*e*f - d*g)) - 8*b*c*g^3*(5*b^2*e*
f^2 + 12*a^2*e*g^2 - 3*a*b*g*(5*e*f + d*g)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(128*c^(3
/2)*e*g^5*(e*f - d*g)) + ((c*d^2 - b*d*e + a*e^2)^(5/2)*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^5*(e*f - d*g)) - ((c*f^2 - b*f*g + a*g^2)^(5/2)*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^5*(e*f - d*g))

________________________________________________________________________________________

Rubi [A]  time = 1.80028, antiderivative size = 886, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {895, 734, 814, 843, 621, 206, 724} \[ \frac{\tanh ^{-1}\left (\frac{b d-2 a e+(2 c d-b e) x}{2 \sqrt{c d^2-b e d+a e^2} \sqrt{c x^2+b x+a}}\right ) \left (c d^2-b e d+a e^2\right )^{5/2}}{e^5 (e f-d g)}+\frac{\left (c x^2+b x+a\right )^{3/2} \left (c d^2-b e d+a e^2\right )}{3 e^2 (e f-d g)}-\frac{(2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right ) \left (c d^2-b e d+a e^2\right )}{16 c^{3/2} e^5 (e f-d g)}+\frac{\left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right ) \sqrt{c x^2+b x+a} \left (c d^2-b e d+a e^2\right )}{8 c e^4 (e f-d g)}-\frac{\left (8 c e f^2-g (11 b e f-3 b d g-8 a e g)-6 c g (e f-d g) x\right ) \left (c x^2+b x+a\right )^{3/2}}{24 e g^2 (e f-d g)}+\frac{\left (128 c^4 e f^5-320 c^3 e g (b f-a g) f^3-b^3 g^4 (5 b e f+3 b d g-8 a e g)+48 c^2 g^2 \left (5 b^2 e f^3-10 a b e g f^2+a^2 g^2 (5 e f-d g)\right )-8 b c g^3 \left (5 b^2 e f^2+12 a^2 e g^2-3 a b g (5 e f+d g)\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right )}{128 c^{3/2} e g^5 (e f-d g)}-\frac{\left (c f^2-b g f+a g^2\right )^{5/2} \tanh ^{-1}\left (\frac{b f-2 a g+(2 c f-b g) x}{2 \sqrt{c f^2-b g f+a g^2} \sqrt{c x^2+b x+a}}\right )}{g^5 (e f-d g)}-\frac{\left (64 c^3 e f^4-16 c^2 e g (9 b f-8 a g) f^2-b^2 g^3 (5 b e f+3 b d g-8 a e g)+4 c g^2 \left (22 b^2 e f^2+16 a^2 e g^2-3 a b g (13 e f-d g)\right )-2 c g \left (16 c^2 e f^3+b g^2 (5 b e f+3 b d g-8 a e g)-4 c g \left (6 b e f^2-a g (7 e f-3 d g)\right )\right ) x\right ) \sqrt{c x^2+b x+a}}{64 c e g^4 (e f-d g)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^(5/2)/((d + e*x)*(f + g*x)),x]

[Out]

((c*d^2 - b*d*e + a*e^2)*(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^4*(e*f - d*g)) - ((64*c^3*e*f^4 - 16*c^2*e*f^2*g*(9*b*f - 8*a*g) - b^2*g^3*(5*b*e*f + 3*b*d*g -
8*a*e*g) + 4*c*g^2*(22*b^2*e*f^2 + 16*a^2*e*g^2 - 3*a*b*g*(13*e*f - d*g)) - 2*c*g*(16*c^2*e*f^3 + b*g^2*(5*b*
e*f + 3*b*d*g - 8*a*e*g) - 4*c*g*(6*b*e*f^2 - a*g*(7*e*f - 3*d*g)))*x)*Sqrt[a + b*x + c*x^2])/(64*c*e*g^4*(e*f
- d*g)) + ((c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^(3/2))/(3*e^2*(e*f - d*g)) - ((8*c*e*f^2 - g*(11*b*e*f -
3*b*d*g - 8*a*e*g) - 6*c*g*(e*f - d*g)*x)*(a + b*x + c*x^2)^(3/2))/(24*e*g^2*(e*f - d*g)) - ((2*c*d - b*e)*(c
*d^2 - b*d*e + a*e^2)*(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^5*(e*f - d*g)) + ((128*c^4*e*f^5 - 320*c^3*e*f^3*g*(b*f - a*g) - b^3*g^4*(5*b*e*f
+ 3*b*d*g - 8*a*e*g) + 48*c^2*g^2*(5*b^2*e*f^3 - 10*a*b*e*f^2*g + a^2*g^2*(5*e*f - d*g)) - 8*b*c*g^3*(5*b^2*e*
f^2 + 12*a^2*e*g^2 - 3*a*b*g*(5*e*f + d*g)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(128*c^(3
/2)*e*g^5*(e*f - d*g)) + ((c*d^2 - b*d*e + a*e^2)^(5/2)*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^5*(e*f - d*g)) - ((c*f^2 - b*f*g + a*g^2)^(5/2)*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^5*(e*f - d*g))

Rule 895

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)/(((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))), x_Symbol] :> Dist[(c
*d^2 - b*d*e + a*e^2)/(e*(e*f - d*g)), Int[(a + b*x + c*x^2)^(p - 1)/(d + e*x), x], x] - Dist[1/(e*(e*f - d*g)
), Int[(Simp[c*d*f - b*e*f + a*e*g - c*(e*f - d*g)*x, x]*(a + b*x + c*x^2)^(p - 1))/(f + g*x), 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] && Fra
ctionQ[p] && GtQ[p, 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]

Rubi steps

\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{5/2}}{(d+e x) (f+g x)} \, dx &=-\frac{\int \frac{(c d f-b e f+a e g-c (e f-d g) x) \left (a+b x+c x^2\right )^{3/2}}{f+g x} \, dx}{e (e f-d g)}+\frac{\left (c d^2-b d e+a e^2\right ) \int \frac{\left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx}{e (e f-d g)}\\ &=\frac{\left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}{3 e^2 (e f-d g)}-\frac{\left (8 c e f^2-g (11 b e f-3 b d g-8 a e g)-6 c g (e f-d g) x\right ) \left (a+b x+c x^2\right )^{3/2}}{24 e g^2 (e f-d g)}-\frac{\left (c d^2-b d e+a e^2\right ) \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^2 (e f-d g)}+\frac{\int \frac{\left (\frac{1}{2} c \left (f \left (8 b c f-3 b^2 g-4 a c g\right ) (e f-d g)+8 g (b f-2 a g) (c d f-b e f+a e g)\right )+\frac{1}{2} c \left (16 c^2 e f^3+b g^2 (5 b e f+3 b d g-8 a e g)-4 c g \left (6 b e f^2-a g (7 e f-3 d g)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{f+g x} \, dx}{8 c e g^2 (e f-d g)}\\ &=\frac{\left (c d^2-b d e+a e^2\right ) \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^4 (e f-d g)}-\frac{\left (64 c^3 e f^4-16 c^2 e f^2 g (9 b f-8 a g)-b^2 g^3 (5 b e f+3 b d g-8 a e g)+4 c g^2 \left (22 b^2 e f^2+16 a^2 e g^2-3 a b g (13 e f-d g)\right )-2 c g \left (16 c^2 e f^3+b g^2 (5 b e f+3 b d g-8 a e g)-4 c g \left (6 b e f^2-a g (7 e f-3 d g)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{64 c e g^4 (e f-d g)}+\frac{\left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}{3 e^2 (e f-d g)}-\frac{\left (8 c e f^2-g (11 b e f-3 b d g-8 a e g)-6 c g (e f-d g) x\right ) \left (a+b x+c x^2\right )^{3/2}}{24 e g^2 (e f-d g)}+\frac{\left (c d^2-b d e+a e^2\right ) \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^4 (e f-d g)}-\frac{\int \frac{\frac{1}{4} c \left (4 c g (b f-2 a g) \left (f \left (8 b c f-3 b^2 g-4 a c g\right ) (e f-d g)+8 g (b f-2 a g) (c d f-b e f+a e g)\right )-f \left (4 b c f-b^2 g-4 a c g\right ) \left (16 c^2 e f^3+b g^2 (5 b e f+3 b d g-8 a e g)-4 c g \left (6 b e f^2-a g (7 e f-3 d g)\right )\right )\right )-\frac{1}{4} c \left (128 c^4 e f^5-320 c^3 e f^3 g (b f-a g)-b^3 g^4 (5 b e f+3 b d g-8 a e g)+48 c^2 g^2 \left (5 b^2 e f^3-10 a b e f^2 g+a^2 g^2 (5 e f-d g)\right )-8 b c g^3 \left (5 b^2 e f^2+12 a^2 e g^2-3 a b g (5 e f+d g)\right )\right ) x}{(f+g x) \sqrt{a+b x+c x^2}} \, dx}{32 c^2 e g^4 (e f-d g)}\\ &=\frac{\left (c d^2-b d e+a e^2\right ) \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^4 (e f-d g)}-\frac{\left (64 c^3 e f^4-16 c^2 e f^2 g (9 b f-8 a g)-b^2 g^3 (5 b e f+3 b d g-8 a e g)+4 c g^2 \left (22 b^2 e f^2+16 a^2 e g^2-3 a b g (13 e f-d g)\right )-2 c g \left (16 c^2 e f^3+b g^2 (5 b e f+3 b d g-8 a e g)-4 c g \left (6 b e f^2-a g (7 e f-3 d g)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{64 c e g^4 (e f-d g)}+\frac{\left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}{3 e^2 (e f-d g)}-\frac{\left (8 c e f^2-g (11 b e f-3 b d g-8 a e g)-6 c g (e f-d g) x\right ) \left (a+b x+c x^2\right )^{3/2}}{24 e g^2 (e f-d g)}+\frac{\left (c d^2-b d e+a e^2\right )^3 \int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{e^5 (e f-d g)}-\frac{\left ((2 c d-b e) \left (c d^2-b d e+a e^2\right ) \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^5 (e f-d g)}-\frac{\left (c f^2-b f g+a g^2\right )^3 \int \frac{1}{(f+g x) \sqrt{a+b x+c x^2}} \, dx}{g^5 (e f-d g)}+\frac{\left (128 c^4 e f^5-320 c^3 e f^3 g (b f-a g)-b^3 g^4 (5 b e f+3 b d g-8 a e g)+48 c^2 g^2 \left (5 b^2 e f^3-10 a b e f^2 g+a^2 g^2 (5 e f-d g)\right )-8 b c g^3 \left (5 b^2 e f^2+12 a^2 e g^2-3 a b g (5 e f+d g)\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{128 c e g^5 (e f-d g)}\\ &=\frac{\left (c d^2-b d e+a e^2\right ) \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^4 (e f-d g)}-\frac{\left (64 c^3 e f^4-16 c^2 e f^2 g (9 b f-8 a g)-b^2 g^3 (5 b e f+3 b d g-8 a e g)+4 c g^2 \left (22 b^2 e f^2+16 a^2 e g^2-3 a b g (13 e f-d g)\right )-2 c g \left (16 c^2 e f^3+b g^2 (5 b e f+3 b d g-8 a e g)-4 c g \left (6 b e f^2-a g (7 e f-3 d g)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{64 c e g^4 (e f-d g)}+\frac{\left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}{3 e^2 (e f-d g)}-\frac{\left (8 c e f^2-g (11 b e f-3 b d g-8 a e g)-6 c g (e f-d g) x\right ) \left (a+b x+c x^2\right )^{3/2}}{24 e g^2 (e f-d g)}-\frac{\left (2 \left (c d^2-b d e+a e^2\right )^3\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^5 (e f-d g)}-\frac{\left ((2 c d-b e) \left (c d^2-b d e+a e^2\right ) \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^5 (e f-d g)}+\frac{\left (2 \left (c f^2-b f g+a g^2\right )^3\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^5 (e f-d g)}+\frac{\left (128 c^4 e f^5-320 c^3 e f^3 g (b f-a g)-b^3 g^4 (5 b e f+3 b d g-8 a e g)+48 c^2 g^2 \left (5 b^2 e f^3-10 a b e f^2 g+a^2 g^2 (5 e f-d g)\right )-8 b c g^3 \left (5 b^2 e f^2+12 a^2 e g^2-3 a b g (5 e f+d 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 )}{64 c e g^5 (e f-d g)}\\ &=\frac{\left (c d^2-b d e+a e^2\right ) \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^4 (e f-d g)}-\frac{\left (64 c^3 e f^4-16 c^2 e f^2 g (9 b f-8 a g)-b^2 g^3 (5 b e f+3 b d g-8 a e g)+4 c g^2 \left (22 b^2 e f^2+16 a^2 e g^2-3 a b g (13 e f-d g)\right )-2 c g \left (16 c^2 e f^3+b g^2 (5 b e f+3 b d g-8 a e g)-4 c g \left (6 b e f^2-a g (7 e f-3 d g)\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{64 c e g^4 (e f-d g)}+\frac{\left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}{3 e^2 (e f-d g)}-\frac{\left (8 c e f^2-g (11 b e f-3 b d g-8 a e g)-6 c g (e f-d g) x\right ) \left (a+b x+c x^2\right )^{3/2}}{24 e g^2 (e f-d g)}-\frac{(2 c d-b e) \left (c d^2-b d e+a e^2\right ) \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^5 (e f-d g)}+\frac{\left (128 c^4 e f^5-320 c^3 e f^3 g (b f-a g)-b^3 g^4 (5 b e f+3 b d g-8 a e g)+48 c^2 g^2 \left (5 b^2 e f^3-10 a b e f^2 g+a^2 g^2 (5 e f-d g)\right )-8 b c g^3 \left (5 b^2 e f^2+12 a^2 e g^2-3 a b g (5 e f+d g)\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{3/2} e g^5 (e f-d g)}+\frac{\left (c d^2-b d e+a e^2\right )^{5/2} \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^5 (e f-d g)}-\frac{\left (c f^2-b f g+a g^2\right )^{5/2} \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^5 (e f-d g)}\\ \end{align*}

Mathematica [A]  time = 2.69505, size = 647, normalized size = 0.73 \[ \frac{3 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right ) \left (240 c^2 e^2 g^2 (e f-d g) \left (a^2 e^2 g^2-2 a b e g (d g+e f)+b^2 \left (d^2 g^2+d e f g+e^2 f^2\right )\right )-40 b^2 c e^3 g^3 (e f-d g) (-3 a e g+b d g+b e f)+320 c^3 e g \left (-a d^3 e g^4+a e^4 f^3 g+b d^4 g^4-b e^4 f^4\right )+5 b^4 e^4 g^4 (d g-e f)+128 c^4 \left (e^5 f^5-d^5 g^5\right )\right )+2 \sqrt{c} \left (-e g \sqrt{a+x (b+c x)} (d g-e f) \left (8 c^2 e g \left (a e g (-56 d g-56 e f+27 e g x)+b \left (54 d^2 g^2+2 d e g (27 f-13 g x)+e^2 \left (54 f^2-26 f g x+17 g^2 x^2\right )\right )\right )+2 b c e^2 g^2 (278 a e g+b (-132 d g-132 e f+59 e g x))+15 b^3 e^3 g^3-16 c^3 \left (-6 d^2 e g^2 (g x-2 f)+12 d^3 g^3+2 d e^2 g \left (6 f^2-3 f g x+2 g^2 x^2\right )+e^3 \left (-6 f^2 g x+12 f^3+4 f g^2 x^2-3 g^3 x^3\right )\right )\right )-192 c g^5 \left (e (a e-b d)+c d^2\right )^{5/2} \tanh ^{-1}\left (\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )+192 c e^5 \left (g (a g-b f)+c f^2\right )^{5/2} \tanh ^{-1}\left (\frac{2 a g-b f+b g x-2 c f x}{2 \sqrt{a+x (b+c x)} \sqrt{g (a g-b f)+c f^2}}\right )\right )}{384 c^{3/2} e^5 g^5 (e f-d g)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^(5/2)/((d + e*x)*(f + g*x)),x]

[Out]

(3*(5*b^4*e^4*g^4*(-(e*f) + d*g) - 40*b^2*c*e^3*g^3*(e*f - d*g)*(b*e*f + b*d*g - 3*a*e*g) + 320*c^3*e*g*(-(b*e
^4*f^4) + a*e^4*f^3*g + b*d^4*g^4 - a*d^3*e*g^4) + 128*c^4*(e^5*f^5 - d^5*g^5) + 240*c^2*e^2*g^2*(e*f - d*g)*(
a^2*e^2*g^2 - 2*a*b*e*g*(e*f + d*g) + b^2*(e^2*f^2 + d*e*f*g + d^2*g^2)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[
a + x*(b + c*x)])] + 2*Sqrt[c]*(-(e*g*(-(e*f) + d*g)*Sqrt[a + x*(b + c*x)]*(15*b^3*e^3*g^3 + 2*b*c*e^2*g^2*(27
8*a*e*g + b*(-132*e*f - 132*d*g + 59*e*g*x)) - 16*c^3*(12*d^3*g^3 - 6*d^2*e*g^2*(-2*f + g*x) + 2*d*e^2*g*(6*f^
2 - 3*f*g*x + 2*g^2*x^2) + e^3*(12*f^3 - 6*f^2*g*x + 4*f*g^2*x^2 - 3*g^3*x^3)) + 8*c^2*e*g*(a*e*g*(-56*e*f - 5
6*d*g + 27*e*g*x) + b*(54*d^2*g^2 + 2*d*e*g*(27*f - 13*g*x) + e^2*(54*f^2 - 26*f*g*x + 17*g^2*x^2))))) - 192*c
*(c*d^2 + e*(-(b*d) + a*e))^(5/2)*g^5*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)])] + 192*c*e^5*(c*f^2 + g*(-(b*f) + a*g))^(5/2)*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)])]))/(384*c^(3/2)*e^5*g^5*(e*f - d*g))

________________________________________________________________________________________

Maple [B]  time = 0.372, size = 9052, normalized size = 10.2 \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)^(5/2)/(e*x+d)/(g*x+f),x)

[Out]

result too large to display

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(5/2)/(e*x+d)/(g*x+f),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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)^(5/2)/(e*x+d)/(g*x+f),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)**(5/2)/(e*x+d)/(g*x+f),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)^(5/2)/(e*x+d)/(g*x+f),x, algorithm="giac")

[Out]

Exception raised: TypeError