3.9.62 \(\int \frac {(f+g x)^3 (a+b x+c x^2)^{3/2}}{d+e x} \, dx\) [862]

Optimal. Leaf size=1098 \[ -\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} \]

[Out]

1/192*(7*b^3*e^3*g^3+64*c^3*(-d*g+e*f)^3-4*b*c*e^2*g^2*(a*e*g-3*b*d*g+9*b*e*f)+24*b*c^2*e*g*(d^2*g^2-3*d*e*f*g
+3*e^2*f^2)+2*c*e*g*(7*b^2*e^2*g^2-4*c*e*g*(a*e*g-3*b*d*g+9*b*e*f)+24*c^2*(d^2*g^2-3*d*e*f*g+3*e^2*f^2))*x)*(c
*x^2+b*x+a)^(3/2)/c^3/e^4+1/60*g^2*(-7*b*e*g-22*c*d*g+36*c*e*f)*(c*x^2+b*x+a)^(5/2)/c^2/e^2+1/6*g^3*(e*x+d)*(c
*x^2+b*x+a)^(5/2)/c/e^2+1/3072*(4*c*e*(-b*e+2*c*d)*(8*c*e*(-2*a*e+b*d)*(24*c^2*e^2*f^3+7*b^2*d*e*g^3-4*c*d*g^2
*(a*e*g-3*b*d*g+9*b*e*f))-d*(-4*a*c*e-3*b^2*e+8*b*c*d)*g*(7*b^2*e^2*g^2-4*c*e*g*(a*e*g-3*b*d*g+9*b*e*f)+24*c^2
*(d^2*g^2-3*d*e*f*g+3*e^2*f^2)))-2*(4*c^2*d^2-1/2*b^2*e^2-2*c*e*(-a*e+b*d))*(8*c*e*(-b*e+2*c*d)*(24*c^2*e^2*f^
3+7*b^2*d*e*g^3-4*c*d*g^2*(a*e*g-3*b*d*g+9*b*e*f))-2*(8*c^2*d^2-4*b*c*d*e-3/2*b^2*e^2+6*a*c*e^2)*g*(7*b^2*e^2*
g^2-4*c*e*g*(a*e*g-3*b*d*g+9*b*e*f)+24*c^2*(d^2*g^2-3*d*e*f*g+3*e^2*f^2))))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x
^2+b*x+a)^(1/2))/c^(9/2)/e^7+(a*e^2-b*d*e+c*d^2)^(3/2)*(-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^7-1/1536*(21*b^5*e^5*g^3-1536*c^5*d^2*(-d*g+e*f)^3+384*c^4*e*(-4
*a*e+5*b*d)*(-d*g+e*f)^3-12*b^3*c*e^4*g^2*(8*a*e*g-3*b*d*g+9*b*e*f)+24*b*c^2*e^3*g*(2*a^2*e^2*g^2+6*a*b*e*g*(-
d*g+3*e*f)+3*b^2*(d^2*g^2-3*d*e*f*g+3*e^2*f^2))-96*b*c^3*e^2*(2*b*(-d*g+e*f)^3+3*a*e*g*(d^2*g^2-3*d*e*f*g+3*e^
2*f^2))+2*c*e*(8*c*e*(-b*e+2*c*d)*(24*c^2*e^2*f^3+7*b^2*d*e*g^3-4*c*d*g^2*(a*e*g-3*b*d*g+9*b*e*f))-2*(8*c^2*d^
2-4*b*c*d*e-3/2*b^2*e^2+6*a*c*e^2)*g*(7*b^2*e^2*g^2-4*c*e*g*(a*e*g-3*b*d*g+9*b*e*f)+24*c^2*(d^2*g^2-3*d*e*f*g+
3*e^2*f^2)))*x)*(c*x^2+b*x+a)^(1/2)/c^4/e^6

________________________________________________________________________________________

Rubi [A]
time = 2.33, antiderivative size = 1098, normalized size of antiderivative = 1.00, 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 = {1667, 828, 857, 635, 212, 738} \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/1536*((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])/(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 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 828

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 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 1667

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]))

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 ) \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^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 ) \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 )}{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*}

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Mathematica [A]
time = 11.64, size = 743, normalized size = 0.68 \begin {gather*} \frac {5120 (e f-d g)^3 (a+x (b+c x))^{3/2}+\frac {1920 e g (e f-d g)^2 (b+2 c x) (a+x (b+c x))^{3/2}}{c}+\frac {3072 e^2 g^2 (e f-d g) (a+x (b+c x))^{5/2}}{c}+\frac {2560 e^3 g^2 (f+g x) (a+x (b+c x))^{5/2}}{c}+\frac {360 \left (b^2-4 a c\right ) e g (e f-d g)^2 \left (-2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}+\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 )\right )}{c^{5/2}}-\frac {60 e^2 g (-2 c f+b g) (e f-d g) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)} \left (-3 b^2+8 b c x+4 c \left (5 a+2 c x^2\right )\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 {e^3 g \left (1792 g (2 c f-b g) (a+x (b+c x))^{5/2}+5 \left (24 c^2 f^2+7 b^2 g^2-4 c g (6 b f+a g)\right ) \left (\frac {16 (b+2 c x) (a+x (b+c x))^{3/2}}{c}+\frac {3 \left (b^2-4 a c\right ) \left (-2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}+\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 )\right )}{c^{5/2}}\right )\right )}{c^2}+\frac {960 (e f-d g)^3 \left (-\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 ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )-2 \sqrt {c} \left (e \sqrt {a+x (b+c x)} \left (-b^2 e^2+4 c^2 d (-2 d+e x)-2 c e (-5 b d+4 a e+b e x)\right )+8 c \left (c d^2+e (-b d+a e)\right )^{3/2} \tanh ^{-1}\left (\frac {-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)}}\right )\right )\right )}{c^{3/2} e^3}}{15360 e^4} \end {gather*}

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 [A]
time = 0.17, size = 1409, normalized size = 1.28

method	result	size
default	\(\text {Expression too large to display}\)	\(1409\)
risch	\(\text {Expression too large to display}\)	\(6884\)

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,method=_RETURNVERBOSE)

[Out]

g/e^3*(g^2*e^2*(1/6*x*(c*x^2+b*x+a)^(5/2)/c-7/12*b/c*(1/5*(c*x^2+b*x+a)^(5/2)/c-1/2*b/c*(1/8*(2*c*x+b)/c*(c*x^
2+b*x+a)^(3/2)+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/
c^(1/2)+(c*x^2+b*x+a)^(1/2)))))-1/6*a/c*(1/8*(2*c*x+b)/c*(c*x^2+b*x+a)^(3/2)+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)
/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))+(-d*e*g^2+3*e^2*
f*g)*(1/5*(c*x^2+b*x+a)^(5/2)/c-1/2*b/c*(1/8*(2*c*x+b)/c*(c*x^2+b*x+a)^(3/2)+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)
/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))+d^2*g^2*(1/8*(2*
c*x+b)/c*(c*x^2+b*x+a)^(3/2)+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*l
n((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))-3*d*e*f*g*(1/8*(2*c*x+b)/c*(c*x^2+b*x+a)^(3/2)+3/16*(4*a*c-b^2)/c
*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))+3*
e^2*f^2*(1/8*(2*c*x+b)/c*(c*x^2+b*x+a)^(3/2)+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*
c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))+(-d^3*g^3+3*d^2*e*f*g^2-3*d*e^2*f^2*g+e^3*f^3)/e
^4*(1/3*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)+1/2*(b*e-2*c*d)/e*(1/4*(2*c*(x+d/e)+
(b*e-2*c*d)/e)/c*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2-b*d*e+c*d^2
)/e^2-(b*e-2*c*d)^2/e^2)/c^(3/2)*ln((1/2*(b*e-2*c*d)/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(
a*e^2-b*d*e+c*d^2)/e^2)^(1/2)))+(a*e^2-b*d*e+c*d^2)/e^2*((c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2
)/e^2)^(1/2)+1/2*(b*e-2*c*d)/e*ln((1/2*(b*e-2*c*d)/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*
e^2-b*d*e+c*d^2)/e^2)^(1/2))/c^(1/2)-(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)*(c*(x+d/e)^2+(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))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (f + g x\right )^{3} \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{d + e x}\, dx \end {gather*}

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]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (f+g\,x\right )}^3\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{d+e\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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