\(\int \frac {(f+g x)^2 (a+b x+c x^2)^{3/2}}{d+e x} \, dx\) [863]

Optimal result

Integrand size = 29, antiderivative size = 662 \[ \int \frac {(f+g x)^2 \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=\frac {\left (3 b^4 e^4 g^2+128 c^4 d^2 (e f-d g)^2-32 c^3 e (5 b d-4 a e) (e f-d g)^2-6 b^2 c e^3 g (2 b e f-b d g+2 a e g)+8 b c^2 e^2 \left (2 b (e f-d g)^2+3 a e g (2 e f-d g)\right )+2 c e \left (\left (16 c^2 d^2-3 b^2 e^2-4 c e (2 b d-3 a e)\right ) g (4 c e f-2 c d g-b e g)-8 c e (2 c d-b e) \left (2 c e f^2-b d g^2\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{128 c^3 e^5}-\frac {\left (3 b^2 e^2 g^2-16 c^2 (e f-d g)^2-6 b c e g (2 e f-d g)-6 c e g (4 c e f-2 c d g-b e g) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 c^2 e^3}+\frac {g^2 \left (a+b x+c x^2\right )^{5/2}}{5 c e}-\frac {\left (3 b^5 e^5 g^2+256 c^5 d^3 (e f-d g)^2-384 c^4 d e (b d-a e) (e f-d g)^2-6 b^3 c e^4 g (2 b e f-b d g+4 a e g)+16 b c^2 e^3 \left (3 a^2 e^2 g^2+b^2 (e f-d g)^2+3 a b e g (2 e f-d g)\right )+96 c^3 e^2 \left (b^2 d (e f-d g)^2-2 a b e (e f-d g)^2-a^2 e^2 g (2 e f-d g)\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{7/2} e^6}+\frac {\left (c d^2-b d e+a e^2\right )^{3/2} (e f-d g)^2 \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e^6} \]

[Out]

-1/48*(3*b^2*e^2*g^2-16*c^2*(-d*g+e*f)^2-6*b*c*e*g*(-d*g+2*e*f)-6*c*e*g*(-b*e*g-2*c*d*g+4*c*e*f)*x)*(c*x^2+b*x
+a)^(3/2)/c^2/e^3+1/5*g^2*(c*x^2+b*x+a)^(5/2)/c/e-1/256*(3*b^5*e^5*g^2+256*c^5*d^3*(-d*g+e*f)^2-384*c^4*d*e*(-
a*e+b*d)*(-d*g+e*f)^2-6*b^3*c*e^4*g*(4*a*e*g-b*d*g+2*b*e*f)+16*b*c^2*e^3*(3*a^2*e^2*g^2+b^2*(-d*g+e*f)^2+3*a*b
*e*g*(-d*g+2*e*f))+96*c^3*e^2*(b^2*d*(-d*g+e*f)^2-2*a*b*e*(-d*g+e*f)^2-a^2*e^2*g*(-d*g+2*e*f)))*arctanh(1/2*(2
*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(7/2)/e^6+(a*e^2-b*d*e+c*d^2)^(3/2)*(-d*g+e*f)^2*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^6+1/128*(3*b^4*e^4*g^2+128*c^4*d^2*(-d*g+e*f
)^2-32*c^3*e*(-4*a*e+5*b*d)*(-d*g+e*f)^2-6*b^2*c*e^3*g*(2*a*e*g-b*d*g+2*b*e*f)+8*b*c^2*e^2*(2*b*(-d*g+e*f)^2+3
*a*e*g*(-d*g+2*e*f))+2*c*e*((16*c^2*d^2-3*b^2*e^2-4*c*e*(-3*a*e+2*b*d))*g*(-b*e*g-2*c*d*g+4*c*e*f)-8*c*e*(-b*e
+2*c*d)*(-b*d*g^2+2*c*e*f^2))*x)*(c*x^2+b*x+a)^(1/2)/c^3/e^5

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 662, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1667, 828, 857, 635, 212, 738} \[ \int \frac {(f+g x)^2 \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (96 c^3 e^2 \left (-a^2 e^2 g (2 e f-d g)-2 a b e (e f-d g)^2+b^2 d (e f-d g)^2\right )+16 b c^2 e^3 \left (3 a^2 e^2 g^2+3 a b e g (2 e f-d g)+b^2 (e f-d g)^2\right )-6 b^3 c e^4 g (4 a e g-b d g+2 b e f)-384 c^4 d e (b d-a e) (e f-d g)^2+3 b^5 e^5 g^2+256 c^5 d^3 (e f-d g)^2\right )}{256 c^{7/2} e^6}+\frac {(e f-d g)^2 \left (a e^2-b d e+c d^2\right )^{3/2} \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e^6}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (3 b^2 e^2 g^2-6 c e g x (-b e g-2 c d g+4 c e f)-6 b c e g (2 e f-d g)-16 c^2 (e f-d g)^2\right )}{48 c^2 e^3}+\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (g \left (-4 c e (2 b d-3 a e)-3 b^2 e^2+16 c^2 d^2\right ) (-b e g-2 c d g+4 c e f)-8 c e (2 c d-b e) \left (2 c e f^2-b d g^2\right )\right )-6 b^2 c e^3 g (2 a e g-b d g+2 b e f)-32 c^3 e (5 b d-4 a e) (e f-d g)^2+8 b c^2 e^2 \left (3 a e g (2 e f-d g)+2 b (e f-d g)^2\right )+3 b^4 e^4 g^2+128 c^4 d^2 (e f-d g)^2\right )}{128 c^3 e^5}+\frac {g^2 \left (a+b x+c x^2\right )^{5/2}}{5 c e} \]

[In]

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

[Out]

((3*b^4*e^4*g^2 + 128*c^4*d^2*(e*f - d*g)^2 - 32*c^3*e*(5*b*d - 4*a*e)*(e*f - d*g)^2 - 6*b^2*c*e^3*g*(2*b*e*f
- b*d*g + 2*a*e*g) + 8*b*c^2*e^2*(2*b*(e*f - d*g)^2 + 3*a*e*g*(2*e*f - d*g)) + 2*c*e*((16*c^2*d^2 - 3*b^2*e^2
- 4*c*e*(2*b*d - 3*a*e))*g*(4*c*e*f - 2*c*d*g - b*e*g) - 8*c*e*(2*c*d - b*e)*(2*c*e*f^2 - b*d*g^2))*x)*Sqrt[a
+ b*x + c*x^2])/(128*c^3*e^5) - ((3*b^2*e^2*g^2 - 16*c^2*(e*f - d*g)^2 - 6*b*c*e*g*(2*e*f - d*g) - 6*c*e*g*(4*
c*e*f - 2*c*d*g - b*e*g)*x)*(a + b*x + c*x^2)^(3/2))/(48*c^2*e^3) + (g^2*(a + b*x + c*x^2)^(5/2))/(5*c*e) - ((
3*b^5*e^5*g^2 + 256*c^5*d^3*(e*f - d*g)^2 - 384*c^4*d*e*(b*d - a*e)*(e*f - d*g)^2 - 6*b^3*c*e^4*g*(2*b*e*f - b
*d*g + 4*a*e*g) + 16*b*c^2*e^3*(3*a^2*e^2*g^2 + b^2*(e*f - d*g)^2 + 3*a*b*e*g*(2*e*f - d*g)) + 96*c^3*e^2*(b^2
*d*(e*f - d*g)^2 - 2*a*b*e*(e*f - d*g)^2 - a^2*e^2*g*(2*e*f - d*g)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b
*x + c*x^2])])/(256*c^(7/2)*e^6) + ((c*d^2 - b*d*e + a*e^2)^(3/2)*(e*f - d*g)^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^6

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*} \text {integral}& = \frac {g^2 \left (a+b x+c x^2\right )^{5/2}}{5 c e}+\frac {\int \frac {\left (\frac {5}{2} e \left (2 c e f^2-b d g^2\right )+\frac {5}{2} e g (4 c e f-2 c d g-b e g) x\right ) \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx}{5 c e^2} \\ & = -\frac {\left (3 b^2 e^2 g^2-16 c^2 (e f-d g)^2-6 b c e g (2 e f-d g)-6 c e g (4 c e f-2 c d g-b e g) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 c^2 e^3}+\frac {g^2 \left (a+b x+c x^2\right )^{5/2}}{5 c e}-\frac {\int \frac {\left (-\frac {5}{4} e \left (d \left (8 b c d-3 b^2 e-4 a c e\right ) g (4 c e f-2 c d g-b e g)-8 c e (b d-2 a e) \left (2 c e f^2-b d g^2\right )\right )-\frac {5}{4} e \left (\left (16 c^2 d^2-3 b^2 e^2-4 c e (2 b d-3 a e)\right ) g (4 c e f-2 c d g-b e g)-8 c e (2 c d-b e) \left (2 c e f^2-b d g^2\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{d+e x} \, dx}{40 c^2 e^4} \\ & = \frac {\left (3 b^4 e^4 g^2+128 c^4 d^2 (e f-d g)^2-32 c^3 e (5 b d-4 a e) (e f-d g)^2-6 b^2 c e^3 g (2 b e f-b d g+2 a e g)+8 b c^2 e^2 \left (2 b (e f-d g)^2+3 a e g (2 e f-d g)\right )+2 c e \left (\left (16 c^2 d^2-3 b^2 e^2-4 c e (2 b d-3 a e)\right ) g (4 c e f-2 c d g-b e g)-8 c e (2 c d-b e) \left (2 c e f^2-b d g^2\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{128 c^3 e^5}-\frac {\left (3 b^2 e^2 g^2-16 c^2 (e f-d g)^2-6 b c e g (2 e f-d g)-6 c e g (4 c e f-2 c d g-b e g) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 c^2 e^3}+\frac {g^2 \left (a+b x+c x^2\right )^{5/2}}{5 c e}+\frac {\int \frac {-\frac {5}{8} e \left (4 c e (b d-2 a e) \left (d \left (8 b c d-3 b^2 e-4 a c e\right ) g (4 c e f-2 c d g-b e g)-8 c e (b d-2 a e) \left (2 c e f^2-b d g^2\right )\right )-d \left (4 b c d-b^2 e-4 a c e\right ) \left (\left (16 c^2 d^2-3 b^2 e^2-4 c e (2 b d-3 a e)\right ) g (4 c e f-2 c d g-b e g)-8 c e (2 c d-b e) \left (2 c e f^2-b d g^2\right )\right )\right )-\frac {5}{8} e \left (4 c e (2 c d-b e) \left (d \left (8 b c d-3 b^2 e-4 a c e\right ) g (4 c e f-2 c d g-b e g)-8 c e (b d-2 a e) \left (2 c e f^2-b d 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 (\left (16 c^2 d^2-3 b^2 e^2-4 c e (2 b d-3 a e)\right ) g (4 c e f-2 c d g-b e g)-8 c e (2 c d-b e) \left (2 c e f^2-b d g^2\right )\right )\right ) x}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{160 c^3 e^6} \\ & = \frac {\left (3 b^4 e^4 g^2+128 c^4 d^2 (e f-d g)^2-32 c^3 e (5 b d-4 a e) (e f-d g)^2-6 b^2 c e^3 g (2 b e f-b d g+2 a e g)+8 b c^2 e^2 \left (2 b (e f-d g)^2+3 a e g (2 e f-d g)\right )+2 c e \left (\left (16 c^2 d^2-3 b^2 e^2-4 c e (2 b d-3 a e)\right ) g (4 c e f-2 c d g-b e g)-8 c e (2 c d-b e) \left (2 c e f^2-b d g^2\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{128 c^3 e^5}-\frac {\left (3 b^2 e^2 g^2-16 c^2 (e f-d g)^2-6 b c e g (2 e f-d g)-6 c e g (4 c e f-2 c d g-b e g) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 c^2 e^3}+\frac {g^2 \left (a+b x+c x^2\right )^{5/2}}{5 c e}+\frac {\left (\left (c d^2-b d e+a e^2\right )^2 (e f-d g)^2\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{e^6}-\frac {\left (3 b^5 e^5 g^2+256 c^5 d^3 (e f-d g)^2-384 c^4 d e (b d-a e) (e f-d g)^2-6 b^3 c e^4 g (2 b e f-b d g+4 a e g)+16 b c^2 e^3 \left (3 a^2 e^2 g^2+b^2 (e f-d g)^2+3 a b e g (2 e f-d g)\right )+96 c^3 e^2 \left (b^2 d (e f-d g)^2-2 a b e (e f-d g)^2-a^2 e^2 g (2 e f-d g)\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{256 c^3 e^6} \\ & = \frac {\left (3 b^4 e^4 g^2+128 c^4 d^2 (e f-d g)^2-32 c^3 e (5 b d-4 a e) (e f-d g)^2-6 b^2 c e^3 g (2 b e f-b d g+2 a e g)+8 b c^2 e^2 \left (2 b (e f-d g)^2+3 a e g (2 e f-d g)\right )+2 c e \left (\left (16 c^2 d^2-3 b^2 e^2-4 c e (2 b d-3 a e)\right ) g (4 c e f-2 c d g-b e g)-8 c e (2 c d-b e) \left (2 c e f^2-b d g^2\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{128 c^3 e^5}-\frac {\left (3 b^2 e^2 g^2-16 c^2 (e f-d g)^2-6 b c e g (2 e f-d g)-6 c e g (4 c e f-2 c d g-b e g) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 c^2 e^3}+\frac {g^2 \left (a+b x+c x^2\right )^{5/2}}{5 c e}-\frac {\left (2 \left (c d^2-b d e+a e^2\right )^2 (e f-d g)^2\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^6}-\frac {\left (3 b^5 e^5 g^2+256 c^5 d^3 (e f-d g)^2-384 c^4 d e (b d-a e) (e f-d g)^2-6 b^3 c e^4 g (2 b e f-b d g+4 a e g)+16 b c^2 e^3 \left (3 a^2 e^2 g^2+b^2 (e f-d g)^2+3 a b e g (2 e f-d g)\right )+96 c^3 e^2 \left (b^2 d (e f-d g)^2-2 a b e (e f-d g)^2-a^2 e^2 g (2 e f-d g)\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 )}{128 c^3 e^6} \\ & = \frac {\left (3 b^4 e^4 g^2+128 c^4 d^2 (e f-d g)^2-32 c^3 e (5 b d-4 a e) (e f-d g)^2-6 b^2 c e^3 g (2 b e f-b d g+2 a e g)+8 b c^2 e^2 \left (2 b (e f-d g)^2+3 a e g (2 e f-d g)\right )+2 c e \left (\left (16 c^2 d^2-3 b^2 e^2-4 c e (2 b d-3 a e)\right ) g (4 c e f-2 c d g-b e g)-8 c e (2 c d-b e) \left (2 c e f^2-b d g^2\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{128 c^3 e^5}-\frac {\left (3 b^2 e^2 g^2-16 c^2 (e f-d g)^2-6 b c e g (2 e f-d g)-6 c e g (4 c e f-2 c d g-b e g) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 c^2 e^3}+\frac {g^2 \left (a+b x+c x^2\right )^{5/2}}{5 c e}-\frac {\left (3 b^5 e^5 g^2+256 c^5 d^3 (e f-d g)^2-384 c^4 d e (b d-a e) (e f-d g)^2-6 b^3 c e^4 g (2 b e f-b d g+4 a e g)+16 b c^2 e^3 \left (3 a^2 e^2 g^2+b^2 (e f-d g)^2+3 a b e g (2 e f-d g)\right )+96 c^3 e^2 \left (b^2 d (e f-d g)^2-2 a b e (e f-d g)^2-a^2 e^2 g (2 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 )}{256 c^{7/2} e^6}+\frac {\left (c d^2-b d e+a e^2\right )^{3/2} (e f-d g)^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^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.88 (sec) , antiderivative size = 536, normalized size of antiderivative = 0.81 \[ \int \frac {(f+g x)^2 \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=\frac {1280 (e f-d g)^2 (a+x (b+c x))^{3/2}+\frac {480 e g (e f-d g) (b+2 c x) (a+x (b+c x))^{3/2}}{c}+\frac {768 e^2 g^2 (a+x (b+c x))^{5/2}}{c}+\frac {90 \left (b^2-4 a c\right ) e g (e f-d g) \left (-2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}+\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{c^{5/2}}+\frac {15 e^2 g (2 c f-b g) \left (\frac {16 (b+2 c x) (a+x (b+c x))^{3/2}}{c}+\frac {3 \left (b^2-4 a c\right ) \left (-2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}+\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{c^{5/2}}\right )}{c}+\frac {240 (e f-d g)^2 \left (-\left ((2 c d-b e) \left (8 c^2 d^2-b^2 e^2+4 c e (-2 b d+3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )-2 \sqrt {c} \left (e \sqrt {a+x (b+c x)} \left (-b^2 e^2+4 c^2 d (-2 d+e x)-2 c e (-5 b d+4 a e+b e x)\right )+8 c \left (c d^2+e (-b d+a e)\right )^{3/2} \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )\right )\right )}{c^{3/2} e^3}}{3840 e^3} \]

[In]

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

[Out]

(1280*(e*f - d*g)^2*(a + x*(b + c*x))^(3/2) + (480*e*g*(e*f - d*g)*(b + 2*c*x)*(a + x*(b + c*x))^(3/2))/c + (7
68*e^2*g^2*(a + x*(b + c*x))^(5/2))/c + (90*(b^2 - 4*a*c)*e*g*(e*f - d*g)*(-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) + (15*e^2*g*(2*c*f
- b*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 + (240*(e*f - d*g)^
2*(-((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))/(3840*e^3)

Maple [A] (verified)

Time = 0.89 (sec) , antiderivative size = 998, normalized size of antiderivative = 1.51

method	result	size
default	\(-\frac {g \left (d g \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )-2 e f \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )-e g \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )\right )}{e^{2}}+\frac {\left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) \left (\frac {\left (\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}{3}+\frac {\left (b e -2 c d \right ) \left (\frac {\left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{4 c}+\frac {\left (\frac {4 c \left (e^{2} a -b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\right )}{8 c^{\frac {3}{2}}}\right )}{2 e}+\frac {\left (e^{2} a -b d e +c \,d^{2}\right ) \left (\sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\right )}{2 e \sqrt {c}}-\frac {\left (e^{2} a -b d e +c \,d^{2}\right ) \ln \left (\frac {\frac {2 e^{2} a -2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}\right )}{e^{2}}\right )}{e^{3}}\)	\(998\)
risch	\(\text {Expression too large to display}\)	\(1395\)

[In]

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

[Out]

-g/e^2*(d*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))))-2*e*f*(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))))-e*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^2-2*d*e*f*g+e^2*f^2)/e^3*(1/3*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)+1/2*(b*e-
2*c*d)/e*(1/4*(2*c*(x+d/e)+(b*e-2*c*d)/e)/c*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+
1/8*(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)+((x+d/e)^
2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)))+(a*e^2-b*d*e+c*d^2)/e^2*(((x+d/e)^2*c+(b*e-2*c*d)/e
*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/2*(b*e-2*c*d)/e*ln((1/2*(b*e-2*c*d)/e+c*(x+d/e))/c^(1/2)+((x+d/e)^2*
c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/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)*((x+d/e)^2*c+(
b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))))

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((g*x+f)^2*(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(e>0)', see `assume?` for more
details)Is e

Giac [F(-2)]

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

[In]

integrate((g*x+f)^2*(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

Mupad [F(-1)]

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

[In]

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

[Out]

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