3.870 \(\int \frac{(f+g x)^4}{(d+e x) \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=431 \[ \frac{g^2 \sqrt{a+b x+c x^2} \left (-4 c e g (4 a e g-7 b d g+18 b e f)+15 b^2 e^2 g^2+4 c^2 \left (11 d^2 g^2-36 d e f g+36 e^2 f^2\right )\right )}{24 c^3 e^3}-\frac{g \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (8 c^2 e g \left (a e g (4 e f-d g)+b \left (d^2 g^2-4 d e f g+6 e^2 f^2\right )\right )-6 b c e^2 g^2 (2 a e g-b d g+4 b e f)+5 b^3 e^3 g^3-16 c^3 \left (4 d^2 e f g^2-d^3 g^3-6 d e^2 f^2 g+4 e^3 f^3\right )\right )}{16 c^{7/2} e^4}+\frac{g^3 (d+e x) \sqrt{a+b x+c x^2} (-5 b e g-14 c d g+24 c e f)}{12 c^2 e^3}+\frac{(e f-d g)^4 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^4 \sqrt{a e^2-b d e+c d^2}}+\frac{g^4 (d+e x)^2 \sqrt{a+b x+c x^2}}{3 c e^3} \]

[Out]

(g^2*(15*b^2*e^2*g^2 - 4*c*e*g*(18*b*e*f - 7*b*d*g + 4*a*e*g) + 4*c^2*(36*e^2*f^2 - 36*d*e*f*g + 11*d^2*g^2))*
Sqrt[a + b*x + c*x^2])/(24*c^3*e^3) + (g^3*(24*c*e*f - 14*c*d*g - 5*b*e*g)*(d + e*x)*Sqrt[a + b*x + c*x^2])/(1
2*c^2*e^3) + (g^4*(d + e*x)^2*Sqrt[a + b*x + c*x^2])/(3*c*e^3) - (g*(5*b^3*e^3*g^3 - 6*b*c*e^2*g^2*(4*b*e*f -
b*d*g + 2*a*e*g) - 16*c^3*(4*e^3*f^3 - 6*d*e^2*f^2*g + 4*d^2*e*f*g^2 - d^3*g^3) + 8*c^2*e*g*(a*e*g*(4*e*f - d*
g) + b*(6*e^2*f^2 - 4*d*e*f*g + d^2*g^2)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(7/2)
*e^4) + ((e*f - d*g)^4*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^4*Sqrt[c*d^2 - b*d*e + a*e^2])

________________________________________________________________________________________

Rubi [A]  time = 1.37034, antiderivative size = 431, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {1653, 843, 621, 206, 724} \[ \frac{g^2 \sqrt{a+b x+c x^2} \left (-4 c e g (4 a e g-7 b d g+18 b e f)+15 b^2 e^2 g^2+4 c^2 \left (11 d^2 g^2-36 d e f g+36 e^2 f^2\right )\right )}{24 c^3 e^3}-\frac{g \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (8 c^2 e g \left (a e g (4 e f-d g)+b \left (d^2 g^2-4 d e f g+6 e^2 f^2\right )\right )-6 b c e^2 g^2 (2 a e g-b d g+4 b e f)+5 b^3 e^3 g^3-16 c^3 \left (4 d^2 e f g^2-d^3 g^3-6 d e^2 f^2 g+4 e^3 f^3\right )\right )}{16 c^{7/2} e^4}+\frac{g^3 (d+e x) \sqrt{a+b x+c x^2} (-5 b e g-14 c d g+24 c e f)}{12 c^2 e^3}+\frac{(e f-d g)^4 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^4 \sqrt{a e^2-b d e+c d^2}}+\frac{g^4 (d+e x)^2 \sqrt{a+b x+c x^2}}{3 c e^3} \]

Antiderivative was successfully verified.

[In]

Int[(f + g*x)^4/((d + e*x)*Sqrt[a + b*x + c*x^2]),x]

[Out]

(g^2*(15*b^2*e^2*g^2 - 4*c*e*g*(18*b*e*f - 7*b*d*g + 4*a*e*g) + 4*c^2*(36*e^2*f^2 - 36*d*e*f*g + 11*d^2*g^2))*
Sqrt[a + b*x + c*x^2])/(24*c^3*e^3) + (g^3*(24*c*e*f - 14*c*d*g - 5*b*e*g)*(d + e*x)*Sqrt[a + b*x + c*x^2])/(1
2*c^2*e^3) + (g^4*(d + e*x)^2*Sqrt[a + b*x + c*x^2])/(3*c*e^3) - (g*(5*b^3*e^3*g^3 - 6*b*c*e^2*g^2*(4*b*e*f -
b*d*g + 2*a*e*g) - 16*c^3*(4*e^3*f^3 - 6*d*e^2*f^2*g + 4*d^2*e*f*g^2 - d^3*g^3) + 8*c^2*e*g*(a*e*g*(4*e*f - d*
g) + b*(6*e^2*f^2 - 4*d*e*f*g + d^2*g^2)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(7/2)
*e^4) + ((e*f - d*g)^4*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^4*Sqrt[c*d^2 - b*d*e + a*e^2])

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 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)^4}{(d+e x) \sqrt{a+b x+c x^2}} \, dx &=\frac{g^4 (d+e x)^2 \sqrt{a+b x+c x^2}}{3 c e^3}+\frac{\int \frac{\frac{1}{2} e \left (6 c e^3 f^4-d^2 (b d+4 a e) g^4\right )-\frac{1}{2} e g \left (d e (7 b d+8 a e) g^3-c \left (24 e^3 f^3-2 d^3 g^3\right )\right ) x-\frac{1}{2} e^2 g^2 \left (e (11 b d+4 a e) g^2-c \left (36 e^2 f^2-10 d^2 g^2\right )\right ) x^2+\frac{1}{2} e^3 g^3 (24 c e f-14 c d g-5 b e g) x^3}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{3 c e^4}\\ &=\frac{g^3 (24 c e f-14 c d g-5 b e g) (d+e x) \sqrt{a+b x+c x^2}}{12 c^2 e^3}+\frac{g^4 (d+e x)^2 \sqrt{a+b x+c x^2}}{3 c e^3}+\frac{\int \frac{\frac{1}{4} e^4 \left (24 c^2 e^3 f^4+5 b d e (b d+2 a e) g^4-2 c d g^3 (b d (12 e f-5 d g)+6 a e (4 e f-d g))\right )+\frac{1}{2} e^4 g \left (5 b e^2 (2 b d+a e) g^3+2 c^2 \left (24 e^3 f^3-12 d^2 e f g^2+5 d^3 g^3\right )-c e g^2 (b d (48 e f-19 d g)+2 a e (12 e f+d g))\right ) x+\frac{1}{4} e^5 g^2 \left (15 b^2 e^2 g^2-4 c e g (18 b e f-7 b d g+4 a e g)+4 c^2 \left (36 e^2 f^2-36 d e f g+11 d^2 g^2\right )\right ) x^2}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{6 c^2 e^7}\\ &=\frac{g^2 \left (15 b^2 e^2 g^2-4 c e g (18 b e f-7 b d g+4 a e g)+4 c^2 \left (36 e^2 f^2-36 d e f g+11 d^2 g^2\right )\right ) \sqrt{a+b x+c x^2}}{24 c^3 e^3}+\frac{g^3 (24 c e f-14 c d g-5 b e g) (d+e x) \sqrt{a+b x+c x^2}}{12 c^2 e^3}+\frac{g^4 (d+e x)^2 \sqrt{a+b x+c x^2}}{3 c e^3}+\frac{\int \frac{\frac{3}{8} e^6 \left (16 c^3 e^3 f^4-5 b^3 d e^2 g^4+6 b c d e g^3 (4 b e f-b d g+2 a e g)-8 c^2 d g^2 \left (a e g (4 e f-d g)+b \left (6 e^2 f^2-4 d e f g+d^2 g^2\right )\right )\right )-\frac{3}{8} e^6 g \left (5 b^3 e^3 g^3-6 b c e^2 g^2 (4 b e f-b d g+2 a e g)-16 c^3 \left (4 e^3 f^3-6 d e^2 f^2 g+4 d^2 e f g^2-d^3 g^3\right )+8 c^2 e g \left (a e g (4 e f-d g)+b \left (6 e^2 f^2-4 d e f g+d^2 g^2\right )\right )\right ) x}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{6 c^3 e^9}\\ &=\frac{g^2 \left (15 b^2 e^2 g^2-4 c e g (18 b e f-7 b d g+4 a e g)+4 c^2 \left (36 e^2 f^2-36 d e f g+11 d^2 g^2\right )\right ) \sqrt{a+b x+c x^2}}{24 c^3 e^3}+\frac{g^3 (24 c e f-14 c d g-5 b e g) (d+e x) \sqrt{a+b x+c x^2}}{12 c^2 e^3}+\frac{g^4 (d+e x)^2 \sqrt{a+b x+c x^2}}{3 c e^3}+\frac{(e f-d g)^4 \int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{e^4}-\frac{\left (g \left (5 b^3 e^3 g^3-6 b c e^2 g^2 (4 b e f-b d g+2 a e g)-16 c^3 \left (4 e^3 f^3-6 d e^2 f^2 g+4 d^2 e f g^2-d^3 g^3\right )+8 c^2 e g \left (a e g (4 e f-d g)+b \left (6 e^2 f^2-4 d e f g+d^2 g^2\right )\right )\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{16 c^3 e^4}\\ &=\frac{g^2 \left (15 b^2 e^2 g^2-4 c e g (18 b e f-7 b d g+4 a e g)+4 c^2 \left (36 e^2 f^2-36 d e f g+11 d^2 g^2\right )\right ) \sqrt{a+b x+c x^2}}{24 c^3 e^3}+\frac{g^3 (24 c e f-14 c d g-5 b e g) (d+e x) \sqrt{a+b x+c x^2}}{12 c^2 e^3}+\frac{g^4 (d+e x)^2 \sqrt{a+b x+c x^2}}{3 c e^3}-\frac{\left (2 (e f-d g)^4\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^4}-\frac{\left (g \left (5 b^3 e^3 g^3-6 b c e^2 g^2 (4 b e f-b d g+2 a e g)-16 c^3 \left (4 e^3 f^3-6 d e^2 f^2 g+4 d^2 e f g^2-d^3 g^3\right )+8 c^2 e g \left (a e g (4 e f-d g)+b \left (6 e^2 f^2-4 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 )}{8 c^3 e^4}\\ &=\frac{g^2 \left (15 b^2 e^2 g^2-4 c e g (18 b e f-7 b d g+4 a e g)+4 c^2 \left (36 e^2 f^2-36 d e f g+11 d^2 g^2\right )\right ) \sqrt{a+b x+c x^2}}{24 c^3 e^3}+\frac{g^3 (24 c e f-14 c d g-5 b e g) (d+e x) \sqrt{a+b x+c x^2}}{12 c^2 e^3}+\frac{g^4 (d+e x)^2 \sqrt{a+b x+c x^2}}{3 c e^3}-\frac{g \left (5 b^3 e^3 g^3-6 b c e^2 g^2 (4 b e f-b d g+2 a e g)-16 c^3 \left (4 e^3 f^3-6 d e^2 f^2 g+4 d^2 e f g^2-d^3 g^3\right )+8 c^2 e g \left (a e g (4 e f-d g)+b \left (6 e^2 f^2-4 d e f g+d^2 g^2\right )\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 c^{7/2} e^4}+\frac{(e f-d g)^4 \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^4 \sqrt{c d^2-b d e+a e^2}}\\ \end{align*}

Mathematica [A]  time = 0.893065, size = 553, normalized size = 1.28 \[ \frac{\frac{6 e^2 g (e f-d g) \left (\left (-4 c g (a g+2 b f)+3 b^2 g^2+8 c^2 f^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )+6 \sqrt{c} g \sqrt{a+x (b+c x)} (2 c f-b g)\right )}{c^{5/2}}+\frac{e^3 g \left (\frac{3 (2 c f-b g) \left (-4 c g (3 a g+2 b f)+5 b^2 g^2+8 c^2 f^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{c^{5/2}}+\frac{2 g \sqrt{a+x (b+c x)} \left (-2 c g (8 a g+27 b f+5 b g x)+15 b^2 g^2+4 c^2 f (16 f+5 g x)\right )}{c^2}\right )}{c}+\frac{24 e g (2 c f-b g) (e f-d g)^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{c^{3/2}}+\frac{48 (e f-d g)^4 \tanh ^{-1}\left (\frac{-2 a e+b (d-e x)+2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )}{\sqrt{e (a e-b d)+c d^2}}+\frac{24 e^2 g^2 (f+g x) \sqrt{a+x (b+c x)} (e f-d g)}{c}+\frac{48 e g^2 \sqrt{a+x (b+c x)} (e f-d g)^2}{c}+\frac{48 g (e f-d g)^3 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{\sqrt{c}}+\frac{16 e^3 g^2 (f+g x)^2 \sqrt{a+x (b+c x)}}{c}}{48 e^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)^4/((d + e*x)*Sqrt[a + b*x + c*x^2]),x]

[Out]

((48*e*g^2*(e*f - d*g)^2*Sqrt[a + x*(b + c*x)])/c + (24*e^2*g^2*(e*f - d*g)*(f + g*x)*Sqrt[a + x*(b + c*x)])/c
+ (16*e^3*g^2*(f + g*x)^2*Sqrt[a + x*(b + c*x)])/c + (24*e*g*(2*c*f - b*g)*(e*f - d*g)^2*ArcTanh[(b + 2*c*x)/
(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/c^(3/2) + (48*g*(e*f - d*g)^3*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b
+ c*x)])])/Sqrt[c] + (6*e^2*g*(e*f - d*g)*(6*Sqrt[c]*g*(2*c*f - b*g)*Sqrt[a + x*(b + c*x)] + (8*c^2*f^2 + 3*b
^2*g^2 - 4*c*g*(2*b*f + a*g))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/c^(5/2) + (e^3*g*((2*g*
Sqrt[a + x*(b + c*x)]*(15*b^2*g^2 + 4*c^2*f*(16*f + 5*g*x) - 2*c*g*(27*b*f + 8*a*g + 5*b*g*x)))/c^2 + (3*(2*c*
f - b*g)*(8*c^2*f^2 + 5*b^2*g^2 - 4*c*g*(2*b*f + 3*a*g))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])
])/c^(5/2)))/c + (48*(e*f - d*g)^4*ArcTanh[(-2*a*e + 2*c*d*x + b*(d - e*x))/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*
Sqrt[a + x*(b + c*x)])])/Sqrt[c*d^2 + e*(-(b*d) + a*e)])/(48*e^4)

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Maple [B]  time = 0.304, size = 1597, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^4/(e*x+d)/(c*x^2+b*x+a)^(1/2),x)

[Out]

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

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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)^4/(e*x+d)/(c*x^2+b*x+a)^(1/2),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)^4/(e*x+d)/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (f + g x\right )^{4}}{\left (d + e x\right ) \sqrt{a + b x + c x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)**4/(e*x+d)/(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral((f + g*x)**4/((d + e*x)*sqrt(a + b*x + c*x**2)), x)

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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)^4/(e*x+d)/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError