3.878 \(\int \frac{(f+g x)^4}{(d+e x) (a+b x+c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=496 \[ -\frac{2 \left (x \left (2 c^2 g^2 \left (a^2 (-g) (4 e f-d g)-3 a b f (e f-2 d g)+3 b^2 d f^2\right )-b c g^3 \left (-3 a^2 e g-4 a b (e f-d g)+4 b^2 d f\right )+b^3 g^4 (b d-a e)+c^3 f^2 (4 a g (2 e f-3 d g)-b f (4 d g+e f))+2 c^4 d f^4\right )-b^2 \left (a^2 e g^4+4 a c d f g^3+c^2 e f^4\right )+b c \left (a^2 g^3 (4 e f-3 d g)+2 a c f^2 g (3 d g+2 e f)+c^2 d f^4\right )+2 a c \left (a^2 e g^4-2 a c f g^2 (3 e f-2 d g)+c^2 f^3 (e f-4 d g)\right )+a b^3 d g^4\right )}{c^2 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}+\frac{g^3 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) (-3 b e g-2 c d g+8 c e f)}{2 c^{5/2} e^2}+\frac{g^4 \sqrt{a+b x+c x^2}}{c^2 e}+\frac{(e f-d g)^4 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^2 \left (a e^2-b d e+c d^2\right )^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.19943, antiderivative size = 496, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {1646, 1653, 843, 621, 206, 724} \[ -\frac{2 \left (x \left (2 c^2 g^2 \left (a^2 (-g) (4 e f-d g)-3 a b f (e f-2 d g)+3 b^2 d f^2\right )-b c g^3 \left (-3 a^2 e g-4 a b (e f-d g)+4 b^2 d f\right )+b^3 g^4 (b d-a e)+c^3 f^2 (4 a g (2 e f-3 d g)-b f (4 d g+e f))+2 c^4 d f^4\right )-b^2 \left (a^2 e g^4+4 a c d f g^3+c^2 e f^4\right )+b c \left (a^2 g^3 (4 e f-3 d g)+2 a c f^2 g (3 d g+2 e f)+c^2 d f^4\right )+2 a c \left (a^2 e g^4-2 a c f g^2 (3 e f-2 d g)+c^2 f^3 (e f-4 d g)\right )+a b^3 d g^4\right )}{c^2 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}+\frac{g^3 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) (-3 b e g-2 c d g+8 c e f)}{2 c^{5/2} e^2}+\frac{g^4 \sqrt{a+b x+c x^2}}{c^2 e}+\frac{(e f-d g)^4 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^2 \left (a e^2-b d e+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(f + g*x)^4/((d + e*x)*(a + b*x + c*x^2)^(3/2)),x]

[Out]

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

Rule 1646

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[(d + e*x)^m*Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2,
x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2
*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*f - b*
g))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]

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) \left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 \left (a b^3 d g^4-b^2 \left (c^2 e f^4+4 a c d f g^3+a^2 e g^4\right )+2 a c \left (a^2 e g^4+c^2 f^3 (e f-4 d g)-2 a c f g^2 (3 e f-2 d g)\right )+b c \left (c^2 d f^4+a^2 g^3 (4 e f-3 d g)+2 a c f^2 g (2 e f+3 d g)\right )+\left (2 c^4 d f^4+b^3 (b d-a e) g^4-b c g^3 \left (4 b^2 d f-3 a^2 e g-4 a b (e f-d g)\right )+2 c^2 g^2 \left (3 b^2 d f^2-3 a b f (e f-2 d g)-a^2 g (4 e f-d g)\right )+c^3 f^2 (4 a g (2 e f-3 d g)-b f (e f+4 d g))\right ) x\right )}{c^2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt{a+b x+c x^2}}-\frac{2 \int \frac{-\frac{\left (b^2-4 a c\right ) \left (b d (b d-a e) g^4-c d g^3 (4 b d f-4 a e f+a d g)+c^2 f^2 \left (e^2 f^2-4 d e f g+6 d^2 g^2\right )\right )}{2 c^2 \left (c d^2-b d e+a e^2\right )}-\frac{\left (b^2-4 a c\right ) g^3 (4 c f-b g) x}{2 c^2}-\frac{\left (b^2-4 a c\right ) g^4 x^2}{2 c}}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{b^2-4 a c}\\ &=-\frac{2 \left (a b^3 d g^4-b^2 \left (c^2 e f^4+4 a c d f g^3+a^2 e g^4\right )+2 a c \left (a^2 e g^4+c^2 f^3 (e f-4 d g)-2 a c f g^2 (3 e f-2 d g)\right )+b c \left (c^2 d f^4+a^2 g^3 (4 e f-3 d g)+2 a c f^2 g (2 e f+3 d g)\right )+\left (2 c^4 d f^4+b^3 (b d-a e) g^4-b c g^3 \left (4 b^2 d f-3 a^2 e g-4 a b (e f-d g)\right )+2 c^2 g^2 \left (3 b^2 d f^2-3 a b f (e f-2 d g)-a^2 g (4 e f-d g)\right )+c^3 f^2 (4 a g (2 e f-3 d g)-b f (e f+4 d g))\right ) x\right )}{c^2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt{a+b x+c x^2}}+\frac{g^4 \sqrt{a+b x+c x^2}}{c^2 e}-\frac{2 \int \frac{-\frac{\left (b^2-4 a c\right ) e \left (2 c^2 e^3 f^4-8 c^2 d e^2 f^3 g+12 c^2 d^2 e f^2 g^2-8 b c d^2 e f g^3+8 a c d e^2 f g^3-b c d^3 g^4+3 b^2 d^2 e g^4-2 a c d^2 e g^4-3 a b d e^2 g^4\right )}{4 c \left (c d^2-b d e+a e^2\right )}-\frac{\left (b^2-4 a c\right ) e g^3 (8 c e f-2 c d g-3 b e g) x}{4 c}}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{c \left (b^2-4 a c\right ) e^2}\\ &=-\frac{2 \left (a b^3 d g^4-b^2 \left (c^2 e f^4+4 a c d f g^3+a^2 e g^4\right )+2 a c \left (a^2 e g^4+c^2 f^3 (e f-4 d g)-2 a c f g^2 (3 e f-2 d g)\right )+b c \left (c^2 d f^4+a^2 g^3 (4 e f-3 d g)+2 a c f^2 g (2 e f+3 d g)\right )+\left (2 c^4 d f^4+b^3 (b d-a e) g^4-b c g^3 \left (4 b^2 d f-3 a^2 e g-4 a b (e f-d g)\right )+2 c^2 g^2 \left (3 b^2 d f^2-3 a b f (e f-2 d g)-a^2 g (4 e f-d g)\right )+c^3 f^2 (4 a g (2 e f-3 d g)-b f (e f+4 d g))\right ) x\right )}{c^2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt{a+b x+c x^2}}+\frac{g^4 \sqrt{a+b x+c x^2}}{c^2 e}+\frac{(e f-d g)^4 \int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{e^2 \left (c d^2-b d e+a e^2\right )}+\frac{\left (g^3 (8 c e f-2 c d g-3 b e g)\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{2 c^2 e^2}\\ &=-\frac{2 \left (a b^3 d g^4-b^2 \left (c^2 e f^4+4 a c d f g^3+a^2 e g^4\right )+2 a c \left (a^2 e g^4+c^2 f^3 (e f-4 d g)-2 a c f g^2 (3 e f-2 d g)\right )+b c \left (c^2 d f^4+a^2 g^3 (4 e f-3 d g)+2 a c f^2 g (2 e f+3 d g)\right )+\left (2 c^4 d f^4+b^3 (b d-a e) g^4-b c g^3 \left (4 b^2 d f-3 a^2 e g-4 a b (e f-d g)\right )+2 c^2 g^2 \left (3 b^2 d f^2-3 a b f (e f-2 d g)-a^2 g (4 e f-d g)\right )+c^3 f^2 (4 a g (2 e f-3 d g)-b f (e f+4 d g))\right ) x\right )}{c^2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt{a+b x+c x^2}}+\frac{g^4 \sqrt{a+b x+c x^2}}{c^2 e}-\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^2 \left (c d^2-b d e+a e^2\right )}+\frac{\left (g^3 (8 c e f-2 c d g-3 b e g)\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 )}{c^2 e^2}\\ &=-\frac{2 \left (a b^3 d g^4-b^2 \left (c^2 e f^4+4 a c d f g^3+a^2 e g^4\right )+2 a c \left (a^2 e g^4+c^2 f^3 (e f-4 d g)-2 a c f g^2 (3 e f-2 d g)\right )+b c \left (c^2 d f^4+a^2 g^3 (4 e f-3 d g)+2 a c f^2 g (2 e f+3 d g)\right )+\left (2 c^4 d f^4+b^3 (b d-a e) g^4-b c g^3 \left (4 b^2 d f-3 a^2 e g-4 a b (e f-d g)\right )+2 c^2 g^2 \left (3 b^2 d f^2-3 a b f (e f-2 d g)-a^2 g (4 e f-d g)\right )+c^3 f^2 (4 a g (2 e f-3 d g)-b f (e f+4 d g))\right ) x\right )}{c^2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt{a+b x+c x^2}}+\frac{g^4 \sqrt{a+b x+c x^2}}{c^2 e}+\frac{g^3 (8 c e f-2 c d g-3 b e g) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 c^{5/2} e^2}+\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^2 \left (c d^2-b d e+a e^2\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 2.9499, size = 587, normalized size = 1.18 \[ \frac{-\frac{2 e \left (b^2 \left (3 a^2 e^2 g^4+a c g^3 \left (d^2 g+4 d e (2 f+3 g x)+e^2 x (g x-8 f)\right )+c^2 \left (d^2 g^4 x^2-12 d e f^2 g^2 x+2 e^2 f^4\right )\right )-2 b c \left (a^2 e g^3 (-5 d g+4 e f+5 e g x)+2 a c g \left (d^2 g^3 x+d e g \left (3 f^2+6 f g x-g^2 x^2\right )+e^2 f^2 (2 f-3 g x)\right )+c^2 e f^3 (d (f-4 g x)-e f x)\right )-4 c \left (a^2 c g^2 \left (d^2 g^2+d e g (4 f+g x)+e^2 \left (-6 f^2-4 f g x+g^2 x^2\right )\right )+2 a^3 e^2 g^4+a c^2 \left (d^2 g^4 x^2-2 d e f^2 g (2 f+3 g x)+e^2 f^3 (f+4 g x)\right )+c^3 d e f^4 x\right )+b^3 g^3 (3 a e g (e x-d)+c d x (d g+8 e f-e g x))-3 b^4 d e g^4 x\right )}{c^2 \left (b^2-4 a c\right ) \sqrt{a+x (b+c x)} \left (e (b d-a e)-c d^2\right )}+\frac{g^3 \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right ) (-3 b e g-2 c d g+8 c e f)}{c^{5/2}}-\frac{2 (e f-d g)^4 \log \left (2 \sqrt{a+x (b+c x)} \sqrt{a e^2-b d e+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\left (e (a e-b d)+c d^2\right )^{3/2}}+\frac{2 (e f-d g)^4 \log (d+e x)}{\left (e (a e-b d)+c d^2\right )^{3/2}}}{2 e^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)^4/((d + e*x)*(a + b*x + c*x^2)^(3/2)),x]

[Out]

((-2*e*(-3*b^4*d*e*g^4*x + b^3*g^3*(3*a*e*g*(-d + e*x) + c*d*x*(8*e*f + d*g - e*g*x)) + b^2*(3*a^2*e^2*g^4 + c
^2*(2*e^2*f^4 - 12*d*e*f^2*g^2*x + d^2*g^4*x^2) + a*c*g^3*(d^2*g + e^2*x*(-8*f + g*x) + 4*d*e*(2*f + 3*g*x)))
- 2*b*c*(a^2*e*g^3*(4*e*f - 5*d*g + 5*e*g*x) + c^2*e*f^3*(-(e*f*x) + d*(f - 4*g*x)) + 2*a*c*g*(d^2*g^3*x + e^2
*f^2*(2*f - 3*g*x) + d*e*g*(3*f^2 + 6*f*g*x - g^2*x^2))) - 4*c*(2*a^3*e^2*g^4 + c^3*d*e*f^4*x + a*c^2*(d^2*g^4
*x^2 - 2*d*e*f^2*g*(2*f + 3*g*x) + e^2*f^3*(f + 4*g*x)) + a^2*c*g^2*(d^2*g^2 + d*e*g*(4*f + g*x) + e^2*(-6*f^2
- 4*f*g*x + g^2*x^2)))))/(c^2*(b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*Sqrt[a + x*(b + c*x)]) + (2*(e*f - d*g
)^4*Log[d + e*x])/(c*d^2 + e*(-(b*d) + a*e))^(3/2) + (g^3*(8*c*e*f - 2*c*d*g - 3*b*e*g)*Log[b + 2*c*x + 2*Sqrt
[c]*Sqrt[a + x*(b + c*x)]])/c^(5/2) - (2*(e*f - d*g)^4*Log[-(b*d) + 2*a*e - 2*c*d*x + b*e*x + 2*Sqrt[c*d^2 - b
*d*e + a*e^2]*Sqrt[a + x*(b + c*x)]])/(c*d^2 + e*(-(b*d) + a*e))^(3/2))/(2*e^2)

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Maple [B]  time = 0.277, size = 4453, normalized size = 9. \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)^4/(e*x+d)/(c*x^2+b*x+a)^(3/2),x)

[Out]

4/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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)*x*c^2*d*f^
4+2/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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)*b*c*d*f^
4+4/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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)*b^2*d*f^
3*g-1/e^3/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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)*b^
2*d^4*g^4+4/e^2/(a*e^2-b*d*e+c*d^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^3*f*g^3-6/e/(a*e^2-b*d*e+c*d^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^2*f^2*g^2+8*g^3/e^3*d^2*f/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*b-12*g^2/e^2*d*f^2/(4*a*c-b^2)/(
c*x^2+b*x+a)^(1/2)*b-2*g^4/e^3*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*d^2-12*g^2/e*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1
/2)*x*f^2-g^4/e^3*b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*d^2-6*g^2/e*b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*f^2+
4/e^2/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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)*b^2*d^
3*f*g^3+2/e^4/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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
)*b*c*d^5*g^4-12/e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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)*x*b*c*d^2*f^2*g^2+8/e^2/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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)*x*b*c*d^3*f*g^3-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^2-16/e/(a
*e^2-b*d*e+c*d^2)/(4*a*c-b^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)*x*c^2*d^2*f^3*
g-8/e^3/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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)*b*c*
d^4*f*g^3+12/e^2/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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)*b*c*d^3*f^2*g^2-8/e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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)*b*c*d^2*f^3*g-2/e^3/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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)*x*b*c*d^4*g^4-16/e^3/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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)*x*c^2*d^4*f*g^3+24/e^2/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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)*x*c^2*d^3*f^2*g^2+8/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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)*x*b*c*d*f^3*g-3/4*g^4/e*b^2/c^3/(c*x^2+b*x+a)^(1/2)+1/e^3/(a*
e^2-b*d*e+c*d^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^4*g^4-e/(a*e^2-b*d*e+c*d^
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))*f^4-4/(a*e^2-b*d*e+c*d^
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*f^3*g-g^4/e^2/c^(3/2)*ln((1/2*b+c*x)/c^
(1/2)+(c*x^2+b*x+a)^(1/2))*d+4*g^3/e/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*f-3/2*g^4/e*b/c^(5/2)
*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+2*g^4/e*a/c^2/(c*x^2+b*x+a)^(1/2)+g^4/e*x^2/c/(c*x^2+b*x+a)^(1/2)
-2*e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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)*x*b*c*f
^4-6/e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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)*b^2*d
^2*f^2*g^2+4/e^4/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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)*x*c^2*d^5*g^4+8*g^3/e^2*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*d*f+4*g^3/e^2*b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)
^(1/2)*d*f+16*g^3/e^3*d^2*f/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*c-24*g^2/e^2*d*f^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/
2)*x*c+4*g^3/e*b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*f+4*g^4/e*a/c*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x-g^4/e
^2*b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*d+e/(a*e^2-b*d*e+c*d^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)*f^4-1/2*g^4/e^2*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*d+2*g^3/e*b^3/c^2/(4*a*c-b^2)/(
c*x^2+b*x+a)^(1/2)*f-3/2*g^4/e*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+2*g^4/e*a/c^2*b^2/(4*a*c-b^2)/(c*x^2+
b*x+a)^(1/2)-4*g^4/e^4*d^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*c+16*g/e*f^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*c-
1/2*g^4/e^2*b/c^2/(c*x^2+b*x+a)^(1/2)*d+2*g^3/e*b/c^2/(c*x^2+b*x+a)^(1/2)*f+g^4/e^2*x/c/(c*x^2+b*x+a)^(1/2)*d-
4*g^3/e*x/c/(c*x^2+b*x+a)^(1/2)*f-2*g^4/e^4*d^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*b+4/(a*e^2-b*d*e+c*d^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+6/e/(a*e^2-b*d*e+c*d^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^2*f^2*g^2-e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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)*b^2*f^4-1/e^3/(a*e^2-b*d*e+c*d^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^4*g^4-4/e^2/(a*e^2-b*d*e+c*d^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^3*f*g^3+8*g/e*f^3/(4*a*c-b^2)/(c*x^2+b*x+a
)^(1/2)*b+3/2*g^4/e*b/c^2*x/(c*x^2+b*x+a)^(1/2)-3/4*g^4/e*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+4*g^3/e^2/c/
(c*x^2+b*x+a)^(1/2)*d*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)^(3/2),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)^(3/2),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)**4/(e*x+d)/(c*x**2+b*x+a)**(3/2),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)^4/(e*x+d)/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError