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

Optimal. Leaf size=1098 \[ \text{result too large to display} \]

[Out]

-((3*(7*b^5*e^5*g^3 - 512*c^5*d^2*(e*f - d*g)^3 + 128*c^4*e*(5*b*d - 4*a*e)*(e*f - d*g)^3 - 4*b^3*c*e^4*g^2*(9
*b*e*f - 3*b*d*g + 8*a*e*g) + 8*b*c^2*e^3*g*(2*a^2*e^2*g^2 + 6*a*b*e*g*(3*e*f - d*g) + 3*b^2*(3*e^2*f^2 - 3*d*
e*f*g + d^2*g^2)) - 32*b*c^3*e^2*(2*b*(e*f - d*g)^3 + 3*a*e*g*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2))) + 2*c*e*(8*c
*e*(2*c*d - b*e)*(24*c^2*e^2*f^3 + 7*b^2*d*e*g^3 - 4*c*d*g^2*(9*b*e*f - 3*b*d*g + a*e*g)) - 2*(8*c^2*d^2 - 4*b
*c*d*e - (3*b^2*e^2)/2 + 6*a*c*e^2)*g*(7*b^2*e^2*g^2 - 4*c*e*g*(9*b*e*f - 3*b*d*g + a*e*g) + 24*c^2*(3*e^2*f^2
- 3*d*e*f*g + d^2*g^2)))*x)*Sqrt[a + b*x + c*x^2])/(1536*c^4*e^6) + ((7*b^3*e^3*g^3 + 64*c^3*(e*f - d*g)^3 -
4*b*c*e^2*g^2*(9*b*e*f - 3*b*d*g + a*e*g) + 24*b*c^2*e*g*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2) + 2*c*e*g*(7*b^2*e^
2*g^2 - 4*c*e*g*(9*b*e*f - 3*b*d*g + a*e*g) + 24*c^2*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2))*x)*(a + b*x + c*x^2)^(
3/2))/(192*c^3*e^4) + (g^2*(36*c*e*f - 22*c*d*g - 7*b*e*g)*(a + b*x + c*x^2)^(5/2))/(60*c^2*e^2) + (g^3*(d + e
*x)*(a + b*x + c*x^2)^(5/2))/(6*c*e^2) + ((4*c*e*(2*c*d - b*e)*(8*c*e*(b*d - 2*a*e)*(24*c^2*e^2*f^3 + 7*b^2*d*
e*g^3 - 4*c*d*g^2*(9*b*e*f - 3*b*d*g + a*e*g)) - d*(8*b*c*d - 3*b^2*e - 4*a*c*e)*g*(7*b^2*e^2*g^2 - 4*c*e*g*(9
*b*e*f - 3*b*d*g + a*e*g) + 24*c^2*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2))) - 2*(4*c^2*d^2 - (b^2*e^2)/2 - 2*c*e*(b
*d - a*e))*(8*c*e*(2*c*d - b*e)*(24*c^2*e^2*f^3 + 7*b^2*d*e*g^3 - 4*c*d*g^2*(9*b*e*f - 3*b*d*g + a*e*g)) - 2*(
8*c^2*d^2 - 4*b*c*d*e - (3*b^2*e^2)/2 + 6*a*c*e^2)*g*(7*b^2*e^2*g^2 - 4*c*e*g*(9*b*e*f - 3*b*d*g + a*e*g) + 24
*c^2*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2))))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(3072*c^(9/2
)*e^7) + ((c*d^2 - b*d*e + a*e^2)^(3/2)*(e*f - d*g)^3*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^7

________________________________________________________________________________________

Rubi [A]  time = 3.86374, antiderivative size = 1098, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {1653, 814, 843, 621, 206, 724} \[ \frac{(d+e x) \left (c x^2+b x+a\right )^{5/2} g^3}{6 c e^2}+\frac{(36 c e f-22 c d g-7 b e g) \left (c x^2+b x+a\right )^{5/2} g^2}{60 c^2 e^2}+\frac{\left (7 b^3 e^3 g^3-4 b c e^2 (9 b e f-3 b d g+a e g) g^2+24 b c^2 e \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) g+2 c e \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right ) x g+64 c^3 (e f-d g)^3\right ) \left (c x^2+b x+a\right )^{3/2}}{192 c^3 e^4}+\frac{\left (4 c e (2 c d-b e) \left (8 c e (b d-2 a e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-d \left (-3 e b^2+8 c d b-4 a c e\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right )-2 \left (4 c^2 d^2-\frac{b^2 e^2}{2}-2 c e (b d-a e)\right ) \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c e d-\frac{3 b^2 e^2}{2}+6 a c e^2\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right )}{3072 c^{9/2} e^7}+\frac{\left (c d^2-b e d+a e^2\right )^{3/2} (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 e d+a e^2} \sqrt{c x^2+b x+a}}\right )}{e^7}-\frac{\left (3 \left (-512 d^2 (e f-d g)^3 c^5+128 e (5 b d-4 a e) (e f-d g)^3 c^4-32 b e^2 \left (2 b (e f-d g)^3+3 a e g \left (3 e^2 f^2-3 d e g f+d^2 g^2\right )\right ) c^3+8 b e^3 g \left (3 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) b^2+6 a e g (3 e f-d g) b+2 a^2 e^2 g^2\right ) c^2-4 b^3 e^4 g^2 (9 b e f-3 b d g+8 a e g) c+7 b^5 e^5 g^3\right )+2 c e \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c e d-\frac{3 b^2 e^2}{2}+6 a c e^2\right ) g \left (24 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (9 b e f-3 b d g+a e g) c+7 b^2 e^2 g^2\right )\right ) x\right ) \sqrt{c x^2+b x+a}}{1536 c^4 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((3*(7*b^5*e^5*g^3 - 512*c^5*d^2*(e*f - d*g)^3 + 128*c^4*e*(5*b*d - 4*a*e)*(e*f - d*g)^3 - 4*b^3*c*e^4*g^2*(9
*b*e*f - 3*b*d*g + 8*a*e*g) + 8*b*c^2*e^3*g*(2*a^2*e^2*g^2 + 6*a*b*e*g*(3*e*f - d*g) + 3*b^2*(3*e^2*f^2 - 3*d*
e*f*g + d^2*g^2)) - 32*b*c^3*e^2*(2*b*(e*f - d*g)^3 + 3*a*e*g*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2))) + 2*c*e*(8*c
*e*(2*c*d - b*e)*(24*c^2*e^2*f^3 + 7*b^2*d*e*g^3 - 4*c*d*g^2*(9*b*e*f - 3*b*d*g + a*e*g)) - 2*(8*c^2*d^2 - 4*b
*c*d*e - (3*b^2*e^2)/2 + 6*a*c*e^2)*g*(7*b^2*e^2*g^2 - 4*c*e*g*(9*b*e*f - 3*b*d*g + a*e*g) + 24*c^2*(3*e^2*f^2
- 3*d*e*f*g + d^2*g^2)))*x)*Sqrt[a + b*x + c*x^2])/(1536*c^4*e^6) + ((7*b^3*e^3*g^3 + 64*c^3*(e*f - d*g)^3 -
4*b*c*e^2*g^2*(9*b*e*f - 3*b*d*g + a*e*g) + 24*b*c^2*e*g*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2) + 2*c*e*g*(7*b^2*e^
2*g^2 - 4*c*e*g*(9*b*e*f - 3*b*d*g + a*e*g) + 24*c^2*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2))*x)*(a + b*x + c*x^2)^(
3/2))/(192*c^3*e^4) + (g^2*(36*c*e*f - 22*c*d*g - 7*b*e*g)*(a + b*x + c*x^2)^(5/2))/(60*c^2*e^2) + (g^3*(d + e
*x)*(a + b*x + c*x^2)^(5/2))/(6*c*e^2) + ((4*c*e*(2*c*d - b*e)*(8*c*e*(b*d - 2*a*e)*(24*c^2*e^2*f^3 + 7*b^2*d*
e*g^3 - 4*c*d*g^2*(9*b*e*f - 3*b*d*g + a*e*g)) - d*(8*b*c*d - 3*b^2*e - 4*a*c*e)*g*(7*b^2*e^2*g^2 - 4*c*e*g*(9
*b*e*f - 3*b*d*g + a*e*g) + 24*c^2*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2))) - 2*(4*c^2*d^2 - (b^2*e^2)/2 - 2*c*e*(b
*d - a*e))*(8*c*e*(2*c*d - b*e)*(24*c^2*e^2*f^3 + 7*b^2*d*e*g^3 - 4*c*d*g^2*(9*b*e*f - 3*b*d*g + a*e*g)) - 2*(
8*c^2*d^2 - 4*b*c*d*e - (3*b^2*e^2)/2 + 6*a*c*e^2)*g*(7*b^2*e^2*g^2 - 4*c*e*g*(9*b*e*f - 3*b*d*g + a*e*g) + 24
*c^2*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2))))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(3072*c^(9/2
)*e^7) + ((c*d^2 - b*d*e + a*e^2)^(3/2)*(e*f - d*g)^3*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^7

Rule 1653

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq
, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*e^(q - 1)*(
m + q + 2*p + 1)), x] + Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^
q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q
- 1) - c*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p +
1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] &&  !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))

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{(f+g x)^3 \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx &=\frac{g^3 (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}+\frac{\int \frac{\left (a+b x+c x^2\right )^{3/2} \left (\frac{1}{2} e \left (12 c e^2 f^3-d (5 b d+2 a e) g^3\right )-e g \left (e (6 b d+a e) g^2-c \left (18 e^2 f^2-5 d^2 g^2\right )\right ) x+\frac{1}{2} e^2 g^2 (36 c e f-22 c d g-7 b e g) x^2\right )}{d+e x} \, dx}{6 c e^3}\\ &=\frac{g^2 (36 c e f-22 c d g-7 b e g) \left (a+b x+c x^2\right )^{5/2}}{60 c^2 e^2}+\frac{g^3 (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}+\frac{\int \frac{\left (\frac{5}{4} e^3 \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )+\frac{5}{4} e^3 g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx}{30 c^2 e^5}\\ &=\frac{\left (7 b^3 e^3 g^3+64 c^3 (e f-d g)^3-4 b c e^2 g^2 (9 b e f-3 b d g+a e g)+24 b c^2 e g \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )+2 c e g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{192 c^3 e^4}+\frac{g^2 (36 c e f-22 c d g-7 b e g) \left (a+b x+c x^2\right )^{5/2}}{60 c^2 e^2}+\frac{g^3 (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}-\frac{\int \frac{\left (\frac{5}{8} e^3 \left (8 c e (b d-2 a e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )+2 \left (2 a c d e-b d \left (4 c d-\frac{3 b e}{2}\right )\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )+\frac{5}{8} e^3 \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-\frac{3 b^2 e^2}{2}-2 c e (2 b d-3 a e)\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{d+e x} \, dx}{240 c^3 e^7}\\ &=-\frac{\left (3 \left (7 b^5 e^5 g^3-512 c^5 d^2 (e f-d g)^3+128 c^4 e (5 b d-4 a e) (e f-d g)^3-4 b^3 c e^4 g^2 (9 b e f-3 b d g+8 a e g)+8 b c^2 e^3 g \left (2 a^2 e^2 g^2+6 a b e g (3 e f-d g)+3 b^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )-32 b c^3 e^2 \left (2 b (e f-d g)^3+3 a e g \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )+2 c e \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c d e-\frac{3 b^2 e^2}{2}+6 a c e^2\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{1536 c^4 e^6}+\frac{\left (7 b^3 e^3 g^3+64 c^3 (e f-d g)^3-4 b c e^2 g^2 (9 b e f-3 b d g+a e g)+24 b c^2 e g \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )+2 c e g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{192 c^3 e^4}+\frac{g^2 (36 c e f-22 c d g-7 b e g) \left (a+b x+c x^2\right )^{5/2}}{60 c^2 e^2}+\frac{g^3 (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}+\frac{\int \frac{\frac{5}{16} e^3 \left (4 c e (b d-2 a e) \left (8 c e (b d-2 a e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-d \left (8 b c d-3 b^2 e-4 a c e\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )-d \left (4 b c d-b^2 e-4 a c e\right ) \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c d e-\frac{3 b^2 e^2}{2}+6 a c e^2\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )\right )+\frac{5}{16} e^3 \left (4 c e (2 c d-b e) \left (8 c e (b d-2 a e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-d \left (8 b c d-3 b^2 e-4 a c e\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )-2 \left (4 c^2 d^2-\frac{b^2 e^2}{2}-2 c e (b d-a e)\right ) \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c d e-\frac{3 b^2 e^2}{2}+6 a c e^2\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )\right ) x}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{960 c^4 e^9}\\ &=-\frac{\left (3 \left (7 b^5 e^5 g^3-512 c^5 d^2 (e f-d g)^3+128 c^4 e (5 b d-4 a e) (e f-d g)^3-4 b^3 c e^4 g^2 (9 b e f-3 b d g+8 a e g)+8 b c^2 e^3 g \left (2 a^2 e^2 g^2+6 a b e g (3 e f-d g)+3 b^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )-32 b c^3 e^2 \left (2 b (e f-d g)^3+3 a e g \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )+2 c e \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c d e-\frac{3 b^2 e^2}{2}+6 a c e^2\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{1536 c^4 e^6}+\frac{\left (7 b^3 e^3 g^3+64 c^3 (e f-d g)^3-4 b c e^2 g^2 (9 b e f-3 b d g+a e g)+24 b c^2 e g \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )+2 c e g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{192 c^3 e^4}+\frac{g^2 (36 c e f-22 c d g-7 b e g) \left (a+b x+c x^2\right )^{5/2}}{60 c^2 e^2}+\frac{g^3 (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}+\frac{\left (\left (c d^2-b d e+a e^2\right )^2 (e f-d g)^3\right ) \int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{e^7}+\frac{\left (4 c e (2 c d-b e) \left (8 c e (b d-2 a e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-d \left (8 b c d-3 b^2 e-4 a c e\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )-2 \left (4 c^2 d^2-\frac{b^2 e^2}{2}-2 c e (b d-a e)\right ) \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c d e-\frac{3 b^2 e^2}{2}+6 a c e^2\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{3072 c^4 e^7}\\ &=-\frac{\left (3 \left (7 b^5 e^5 g^3-512 c^5 d^2 (e f-d g)^3+128 c^4 e (5 b d-4 a e) (e f-d g)^3-4 b^3 c e^4 g^2 (9 b e f-3 b d g+8 a e g)+8 b c^2 e^3 g \left (2 a^2 e^2 g^2+6 a b e g (3 e f-d g)+3 b^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )-32 b c^3 e^2 \left (2 b (e f-d g)^3+3 a e g \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )+2 c e \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c d e-\frac{3 b^2 e^2}{2}+6 a c e^2\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{1536 c^4 e^6}+\frac{\left (7 b^3 e^3 g^3+64 c^3 (e f-d g)^3-4 b c e^2 g^2 (9 b e f-3 b d g+a e g)+24 b c^2 e g \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )+2 c e g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{192 c^3 e^4}+\frac{g^2 (36 c e f-22 c d g-7 b e g) \left (a+b x+c x^2\right )^{5/2}}{60 c^2 e^2}+\frac{g^3 (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}-\frac{\left (2 \left (c d^2-b d e+a e^2\right )^2 (e f-d g)^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^7}+\frac{\left (4 c e (2 c d-b e) \left (8 c e (b d-2 a e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-d \left (8 b c d-3 b^2 e-4 a c e\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )-2 \left (4 c^2 d^2-\frac{b^2 e^2}{2}-2 c e (b d-a e)\right ) \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c d e-\frac{3 b^2 e^2}{2}+6 a c e^2\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\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 )}{1536 c^4 e^7}\\ &=-\frac{\left (3 \left (7 b^5 e^5 g^3-512 c^5 d^2 (e f-d g)^3+128 c^4 e (5 b d-4 a e) (e f-d g)^3-4 b^3 c e^4 g^2 (9 b e f-3 b d g+8 a e g)+8 b c^2 e^3 g \left (2 a^2 e^2 g^2+6 a b e g (3 e f-d g)+3 b^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )-32 b c^3 e^2 \left (2 b (e f-d g)^3+3 a e g \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )+2 c e \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c d e-\frac{3 b^2 e^2}{2}+6 a c e^2\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{1536 c^4 e^6}+\frac{\left (7 b^3 e^3 g^3+64 c^3 (e f-d g)^3-4 b c e^2 g^2 (9 b e f-3 b d g+a e g)+24 b c^2 e g \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )+2 c e g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{192 c^3 e^4}+\frac{g^2 (36 c e f-22 c d g-7 b e g) \left (a+b x+c x^2\right )^{5/2}}{60 c^2 e^2}+\frac{g^3 (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c e^2}+\frac{\left (4 c e (2 c d-b e) \left (8 c e (b d-2 a e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-d \left (8 b c d-3 b^2 e-4 a c e\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )-2 \left (4 c^2 d^2-\frac{b^2 e^2}{2}-2 c e (b d-a e)\right ) \left (8 c e (2 c d-b e) \left (24 c^2 e^2 f^3+7 b^2 d e g^3-4 c d g^2 (9 b e f-3 b d g+a e g)\right )-2 \left (8 c^2 d^2-4 b c d e-\frac{3 b^2 e^2}{2}+6 a c e^2\right ) g \left (7 b^2 e^2 g^2-4 c e g (9 b e f-3 b d g+a e g)+24 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{3072 c^{9/2} e^7}+\frac{\left (c d^2-b d e+a e^2\right )^{3/2} (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^7}\\ \end{align*}

Mathematica [A]  time = 2.48474, size = 743, normalized size = 0.68 \[ \frac{\frac{960 (e f-d g)^3 \left (-(2 c d-b e) \left (4 c e (3 a e-2 b d)-b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )-2 \sqrt{c} \left (e \sqrt{a+x (b+c x)} \left (-2 c e (4 a e-5 b d+b e x)-b^2 e^2+4 c^2 d (e x-2 d)\right )+8 c \left (e (a e-b d)+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )\right )\right )}{c^{3/2} e^3}-\frac{60 e^2 g (b g-2 c f) (e f-d g) \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left (4 c \left (5 a+2 c x^2\right )-3 b^2+8 b c x\right )+3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )}{c^{7/2}}+\frac{360 e g \left (b^2-4 a c\right ) (e f-d g)^2 \left (\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )-2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}\right )}{c^{5/2}}+\frac{e^3 g \left (5 \left (-4 c g (a g+6 b f)+7 b^2 g^2+24 c^2 f^2\right ) \left (\frac{3 \left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )-2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}\right )}{c^{5/2}}+\frac{16 (b+2 c x) (a+x (b+c x))^{3/2}}{c}\right )+1792 g (a+x (b+c x))^{5/2} (2 c f-b g)\right )}{c^2}+\frac{3072 e^2 g^2 (a+x (b+c x))^{5/2} (e f-d g)}{c}+\frac{1920 e g (b+2 c x) (a+x (b+c x))^{3/2} (e f-d g)^2}{c}+5120 (a+x (b+c x))^{3/2} (e f-d g)^3+\frac{2560 e^3 g^2 (f+g x) (a+x (b+c x))^{5/2}}{c}}{15360 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5120*(e*f - d*g)^3*(a + x*(b + c*x))^(3/2) + (1920*e*g*(e*f - d*g)^2*(b + 2*c*x)*(a + x*(b + c*x))^(3/2))/c +
(3072*e^2*g^2*(e*f - d*g)*(a + x*(b + c*x))^(5/2))/c + (2560*e^3*g^2*(f + g*x)*(a + x*(b + c*x))^(5/2))/c + (
360*(b^2 - 4*a*c)*e*g*(e*f - d*g)^2*(-2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] + (b^2 - 4*a*c)*ArcTanh[(b +
2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/c^(5/2) - (60*e^2*g*(-2*c*f + b*g)*(e*f - d*g)*(2*Sqrt[c]*(b + 2*
c*x)*Sqrt[a + x*(b + c*x)]*(-3*b^2 + 8*b*c*x + 4*c*(5*a + 2*c*x^2)) + 3*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x)/(2
*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/c^(7/2) + (e^3*g*(1792*g*(2*c*f - b*g)*(a + x*(b + c*x))^(5/2) + 5*(24*c^2*
f^2 + 7*b^2*g^2 - 4*c*g*(6*b*f + a*g))*((16*(b + 2*c*x)*(a + x*(b + c*x))^(3/2))/c + (3*(b^2 - 4*a*c)*(-2*Sqrt
[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] + (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))
/c^(5/2))))/c^2 + (960*(e*f - d*g)^3*(-((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)])])))/(c^(3/2)*e^3))/(15360*e^4)

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Maple [B]  time = 0.291, size = 10058, normalized size = 9.2 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

[Out]

Exception raised: ValueError

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

[Out]

Timed out

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

[Out]

Timed out

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

[Out]

Exception raised: TypeError