3.1178 \(\int \frac{(A+B x) (b x+c x^2)^{3/2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=426 \[ -\frac{\sqrt{b x+c x^2} \left (A e \left (-b^2 e^2-6 b c d e+8 c^2 d^2\right )-2 c e x (B d (8 c d-7 b e)-A e (2 c d-b e))-B d \left (5 b^2 e^2-36 b c d e+32 c^2 d^2\right )\right )}{8 d e^4 (d+e x) (c d-b e)}+\frac{\left (B d \left (60 b^2 c d e^2-5 b^3 e^3-120 b c^2 d^2 e+64 c^3 d^3\right )-A e \left (6 b^2 c d e^2+b^3 e^3-24 b c^2 d^2 e+16 c^3 d^3\right )\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{16 d^{3/2} e^5 (c d-b e)^{3/2}}-\frac{\left (b x+c x^2\right )^{3/2} (3 e x (B d (4 c d-3 b e)-A e (2 c d-b e))+d (B d (8 c d-5 b e)-A e (b e+2 c d)))}{12 d e^2 (d+e x)^3 (c d-b e)}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) (-2 A c e-3 b B e+8 B c d)}{e^5} \]

[Out]

-((A*e*(8*c^2*d^2 - 6*b*c*d*e - b^2*e^2) - B*d*(32*c^2*d^2 - 36*b*c*d*e + 5*b^2*e^2) - 2*c*e*(B*d*(8*c*d - 7*b
*e) - A*e*(2*c*d - b*e))*x)*Sqrt[b*x + c*x^2])/(8*d*e^4*(c*d - b*e)*(d + e*x)) - ((d*(B*d*(8*c*d - 5*b*e) - A*
e*(2*c*d + b*e)) + 3*e*(B*d*(4*c*d - 3*b*e) - A*e*(2*c*d - b*e))*x)*(b*x + c*x^2)^(3/2))/(12*d*e^2*(c*d - b*e)
*(d + e*x)^3) - (Sqrt[c]*(8*B*c*d - 3*b*B*e - 2*A*c*e)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/e^5 + ((B*d*(64
*c^3*d^3 - 120*b*c^2*d^2*e + 60*b^2*c*d*e^2 - 5*b^3*e^3) - A*e*(16*c^3*d^3 - 24*b*c^2*d^2*e + 6*b^2*c*d*e^2 +
b^3*e^3))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(16*d^(3/2)*e^5*(c*d
- b*e)^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.529857, antiderivative size = 426, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {810, 812, 843, 620, 206, 724} \[ -\frac{\sqrt{b x+c x^2} \left (A e \left (-b^2 e^2-6 b c d e+8 c^2 d^2\right )-2 c e x (B d (8 c d-7 b e)-A e (2 c d-b e))-B d \left (5 b^2 e^2-36 b c d e+32 c^2 d^2\right )\right )}{8 d e^4 (d+e x) (c d-b e)}+\frac{\left (B d \left (60 b^2 c d e^2-5 b^3 e^3-120 b c^2 d^2 e+64 c^3 d^3\right )-A e \left (6 b^2 c d e^2+b^3 e^3-24 b c^2 d^2 e+16 c^3 d^3\right )\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{16 d^{3/2} e^5 (c d-b e)^{3/2}}-\frac{\left (b x+c x^2\right )^{3/2} (3 e x (B d (4 c d-3 b e)-A e (2 c d-b e))+d (B d (8 c d-5 b e)-A e (b e+2 c d)))}{12 d e^2 (d+e x)^3 (c d-b e)}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) (-2 A c e-3 b B e+8 B c d)}{e^5} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^(3/2))/(d + e*x)^4,x]

[Out]

-((A*e*(8*c^2*d^2 - 6*b*c*d*e - b^2*e^2) - B*d*(32*c^2*d^2 - 36*b*c*d*e + 5*b^2*e^2) - 2*c*e*(B*d*(8*c*d - 7*b
*e) - A*e*(2*c*d - b*e))*x)*Sqrt[b*x + c*x^2])/(8*d*e^4*(c*d - b*e)*(d + e*x)) - ((d*(B*d*(8*c*d - 5*b*e) - A*
e*(2*c*d + b*e)) + 3*e*(B*d*(4*c*d - 3*b*e) - A*e*(2*c*d - b*e))*x)*(b*x + c*x^2)^(3/2))/(12*d*e^2*(c*d - b*e)
*(d + e*x)^3) - (Sqrt[c]*(8*B*c*d - 3*b*B*e - 2*A*c*e)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/e^5 + ((B*d*(64
*c^3*d^3 - 120*b*c^2*d^2*e + 60*b^2*c*d*e^2 - 5*b^3*e^3) - A*e*(16*c^3*d^3 - 24*b*c^2*d^2*e + 6*b^2*c*d*e^2 +
b^3*e^3))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(16*d^(3/2)*e^5*(c*d
- b*e)^(3/2))

Rule 810

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*((d*g - e*f*(m + 2))*(c*d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*
f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2
- b*d*e + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x
+ c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1)
- c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1
) - b*(d*g*(m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*
c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

Rule 812

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + b*x + c*x^2)^p)/(e^2*(m + 1)*(m
+ 2*p + 2)), x] + Dist[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(
b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p
+ 2))*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] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 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 620

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

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{(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx &=-\frac{(d (B d (8 c d-5 b e)-A e (2 c d+b e))+3 e (B d (4 c d-3 b e)-A e (2 c d-b e)) x) \left (b x+c x^2\right )^{3/2}}{12 d e^2 (c d-b e) (d+e x)^3}-\frac{\int \frac{\left (-\frac{1}{2} b (B d (8 c d-5 b e)-A e (2 c d+b e))-c (B d (8 c d-7 b e)-A e (2 c d-b e)) x\right ) \sqrt{b x+c x^2}}{(d+e x)^2} \, dx}{4 d e^2 (c d-b e)}\\ &=-\frac{\left (A e \left (8 c^2 d^2-6 b c d e-b^2 e^2\right )-B d \left (32 c^2 d^2-36 b c d e+5 b^2 e^2\right )-2 c e (B d (8 c d-7 b e)-A e (2 c d-b e)) x\right ) \sqrt{b x+c x^2}}{8 d e^4 (c d-b e) (d+e x)}-\frac{(d (B d (8 c d-5 b e)-A e (2 c d+b e))+3 e (B d (4 c d-3 b e)-A e (2 c d-b e)) x) \left (b x+c x^2\right )^{3/2}}{12 d e^2 (c d-b e) (d+e x)^3}+\frac{\int \frac{\frac{1}{2} b \left (A e \left (8 c^2 d^2-6 b c d e-b^2 e^2\right )-B d \left (32 c^2 d^2-36 b c d e+5 b^2 e^2\right )\right )-4 c d (c d-b e) (8 B c d-3 b B e-2 A c e) x}{(d+e x) \sqrt{b x+c x^2}} \, dx}{8 d e^4 (c d-b e)}\\ &=-\frac{\left (A e \left (8 c^2 d^2-6 b c d e-b^2 e^2\right )-B d \left (32 c^2 d^2-36 b c d e+5 b^2 e^2\right )-2 c e (B d (8 c d-7 b e)-A e (2 c d-b e)) x\right ) \sqrt{b x+c x^2}}{8 d e^4 (c d-b e) (d+e x)}-\frac{(d (B d (8 c d-5 b e)-A e (2 c d+b e))+3 e (B d (4 c d-3 b e)-A e (2 c d-b e)) x) \left (b x+c x^2\right )^{3/2}}{12 d e^2 (c d-b e) (d+e x)^3}-\frac{(c (8 B c d-3 b B e-2 A c e)) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{2 e^5}+\frac{\left (B d \left (64 c^3 d^3-120 b c^2 d^2 e+60 b^2 c d e^2-5 b^3 e^3\right )-A e \left (16 c^3 d^3-24 b c^2 d^2 e+6 b^2 c d e^2+b^3 e^3\right )\right ) \int \frac{1}{(d+e x) \sqrt{b x+c x^2}} \, dx}{16 d e^5 (c d-b e)}\\ &=-\frac{\left (A e \left (8 c^2 d^2-6 b c d e-b^2 e^2\right )-B d \left (32 c^2 d^2-36 b c d e+5 b^2 e^2\right )-2 c e (B d (8 c d-7 b e)-A e (2 c d-b e)) x\right ) \sqrt{b x+c x^2}}{8 d e^4 (c d-b e) (d+e x)}-\frac{(d (B d (8 c d-5 b e)-A e (2 c d+b e))+3 e (B d (4 c d-3 b e)-A e (2 c d-b e)) x) \left (b x+c x^2\right )^{3/2}}{12 d e^2 (c d-b e) (d+e x)^3}-\frac{(c (8 B c d-3 b B e-2 A c e)) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{e^5}-\frac{\left (B d \left (64 c^3 d^3-120 b c^2 d^2 e+60 b^2 c d e^2-5 b^3 e^3\right )-A e \left (16 c^3 d^3-24 b c^2 d^2 e+6 b^2 c d e^2+b^3 e^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac{-b d-(2 c d-b e) x}{\sqrt{b x+c x^2}}\right )}{8 d e^5 (c d-b e)}\\ &=-\frac{\left (A e \left (8 c^2 d^2-6 b c d e-b^2 e^2\right )-B d \left (32 c^2 d^2-36 b c d e+5 b^2 e^2\right )-2 c e (B d (8 c d-7 b e)-A e (2 c d-b e)) x\right ) \sqrt{b x+c x^2}}{8 d e^4 (c d-b e) (d+e x)}-\frac{(d (B d (8 c d-5 b e)-A e (2 c d+b e))+3 e (B d (4 c d-3 b e)-A e (2 c d-b e)) x) \left (b x+c x^2\right )^{3/2}}{12 d e^2 (c d-b e) (d+e x)^3}-\frac{\sqrt{c} (8 B c d-3 b B e-2 A c e) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{e^5}+\frac{\left (B d \left (64 c^3 d^3-120 b c^2 d^2 e+60 b^2 c d e^2-5 b^3 e^3\right )-A e \left (16 c^3 d^3-24 b c^2 d^2 e+6 b^2 c d e^2+b^3 e^3\right )\right ) \tanh ^{-1}\left (\frac{b d+(2 c d-b e) x}{2 \sqrt{d} \sqrt{c d-b e} \sqrt{b x+c x^2}}\right )}{16 d^{3/2} e^5 (c d-b e)^{3/2}}\\ \end{align*}

Mathematica [B]  time = 6.19357, size = 1549, normalized size = 3.64 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^(3/2))/(d + e*x)^4,x]

[Out]

((-(B*d) + A*e)*x*(b + c*x)*(x*(b + c*x))^(3/2))/(3*d*(-(c*d) + b*e)*(d + e*x)^3) + ((x*(b + c*x))^(3/2)*(((-2
*c*d*(B*d - A*e) + (e*(5*b*B*d - 6*A*c*d + A*b*e))/2)*x^(5/2)*(b + c*x)^(5/2))/(2*d*(-(c*d) + b*e)*(d + e*x)^2
) + ((((-3*c*d*(B*d*(4*c*d - 5*b*e) + A*e*(2*c*d - b*e)))/2 + (e*(-5*b^2*B*d*e + A*(24*c^2*d^2 - 18*b*c*d*e -
b^2*e^2)))/4)*x^(5/2)*(b + c*x)^(5/2))/(d*(-(c*d) + b*e)*(d + e*x)) + (((-48*A*c^3*d^3 - 2*b^2*c*d*e*(70*B*d -
13*A*e) + 24*b*c^2*d^2*(5*B*d + A*e) + 3*b^3*e^2*(5*B*d + A*e))*((2*b*x^(3/2)*Sqrt[b + c*x]*(1 + (c*x)/b)^2*(
(3/(4*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1))/2 + (3*b^2*((2*c*x)/b - (2*Sqrt[c]*Sqrt[x]*ArcSinh[(Sqrt[c]*Sqrt[
x])/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c*x)/b])))/(32*c^2*x^2*(1 + (c*x)/b)^2)))/(3*e) - (d*((2*b*Sqrt[x]*Sqrt[b + c
*x]*(1 + (c*x)/b)^2*((3/(2*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1))/4 + (3*Sqrt[b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqr
t[b]])/(8*Sqrt[c]*Sqrt[x]*(1 + (c*x)/b)^(5/2))))/e - (d*((2*c*Sqrt[x]*Sqrt[b + c*x]*(1 + (c*x)/b)*(1/(2*(1 + (
c*x)/b)) + (Sqrt[b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(2*Sqrt[c]*Sqrt[x]*(1 + (c*x)/b)^(3/2))))/e - ((c*d -
b*e)*((2*Sqrt[b]*Sqrt[c]*Sqrt[1 + (c*x)/b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(e*Sqrt[b + c*x]) - (2*(c*d - b
*e)*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(Sqrt[d]*e*Sqrt[-(c*d) + b*e])))/e))/e))/e))
/8 - c*(A*e*(12*c^2*d^2 - 12*b*c*d*e - b^2*e^2) - B*d*(24*c^2*d^2 - 30*b*c*d*e + 5*b^2*e^2))*((2*b*x^(5/2)*Sqr
t[b + c*x]*(1 + (c*x)/b)^2*((5*(1/(2*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1)))/8 - (15*b^3*((2*c*x)/b - (4*c^2*x
^2)/(3*b^2) - (2*Sqrt[c]*Sqrt[x]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c*x)/b])))/(256*c^3*x^
3*(1 + (c*x)/b)^2)))/(5*e) - (d*((2*b*x^(3/2)*Sqrt[b + c*x]*(1 + (c*x)/b)^2*((3/(4*(1 + (c*x)/b)^2) + (1 + (c*
x)/b)^(-1))/2 + (3*b^2*((2*c*x)/b - (2*Sqrt[c]*Sqrt[x]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (
c*x)/b])))/(32*c^2*x^2*(1 + (c*x)/b)^2)))/(3*e) - (d*((2*b*Sqrt[x]*Sqrt[b + c*x]*(1 + (c*x)/b)^2*((3/(2*(1 + (
c*x)/b)^2) + (1 + (c*x)/b)^(-1))/4 + (3*Sqrt[b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(8*Sqrt[c]*Sqrt[x]*(1 + (c
*x)/b)^(5/2))))/e - (d*((2*c*Sqrt[x]*Sqrt[b + c*x]*(1 + (c*x)/b)*(1/(2*(1 + (c*x)/b)) + (Sqrt[b]*ArcSinh[(Sqrt
[c]*Sqrt[x])/Sqrt[b]])/(2*Sqrt[c]*Sqrt[x]*(1 + (c*x)/b)^(3/2))))/e - ((c*d - b*e)*((2*Sqrt[b]*Sqrt[c]*Sqrt[1 +
(c*x)/b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(e*Sqrt[b + c*x]) - (2*(c*d - b*e)*ArcTan[(Sqrt[-(c*d) + b*e]*Sq
rt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(Sqrt[d]*e*Sqrt[-(c*d) + b*e])))/e))/e))/e))/e))/(d*(-(c*d) + b*e)))/(2*d*(-(
c*d) + b*e))))/(3*d*(-(c*d) + b*e)*x^(3/2)*(b + c*x)^(3/2))

________________________________________________________________________________________

Maple [B]  time = 0.016, size = 11396, normalized size = 26.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d)^4,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((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 31.3493, size = 10426, normalized size = 24.47 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

[-1/48*(24*(8*B*c^3*d^8 - (19*B*b*c^2 + 2*A*c^3)*d^7*e + 2*(7*B*b^2*c + 2*A*b*c^2)*d^6*e^2 - (3*B*b^3 + 2*A*b^
2*c)*d^5*e^3 + (8*B*c^3*d^5*e^3 - (19*B*b*c^2 + 2*A*c^3)*d^4*e^4 + 2*(7*B*b^2*c + 2*A*b*c^2)*d^3*e^5 - (3*B*b^
3 + 2*A*b^2*c)*d^2*e^6)*x^3 + 3*(8*B*c^3*d^6*e^2 - (19*B*b*c^2 + 2*A*c^3)*d^5*e^3 + 2*(7*B*b^2*c + 2*A*b*c^2)*
d^4*e^4 - (3*B*b^3 + 2*A*b^2*c)*d^3*e^5)*x^2 + 3*(8*B*c^3*d^7*e - (19*B*b*c^2 + 2*A*c^3)*d^6*e^2 + 2*(7*B*b^2*
c + 2*A*b*c^2)*d^5*e^3 - (3*B*b^3 + 2*A*b^2*c)*d^4*e^4)*x)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)
) - 3*(64*B*c^3*d^7 - A*b^3*d^3*e^4 - 8*(15*B*b*c^2 + 2*A*c^3)*d^6*e + 12*(5*B*b^2*c + 2*A*b*c^2)*d^5*e^2 - (5
*B*b^3 + 6*A*b^2*c)*d^4*e^3 + (64*B*c^3*d^4*e^3 - A*b^3*e^7 - 8*(15*B*b*c^2 + 2*A*c^3)*d^3*e^4 + 12*(5*B*b^2*c
+ 2*A*b*c^2)*d^2*e^5 - (5*B*b^3 + 6*A*b^2*c)*d*e^6)*x^3 + 3*(64*B*c^3*d^5*e^2 - A*b^3*d*e^6 - 8*(15*B*b*c^2 +
2*A*c^3)*d^4*e^3 + 12*(5*B*b^2*c + 2*A*b*c^2)*d^3*e^4 - (5*B*b^3 + 6*A*b^2*c)*d^2*e^5)*x^2 + 3*(64*B*c^3*d^6*
e - A*b^3*d^2*e^5 - 8*(15*B*b*c^2 + 2*A*c^3)*d^5*e^2 + 12*(5*B*b^2*c + 2*A*b*c^2)*d^4*e^3 - (5*B*b^3 + 6*A*b^2
*c)*d^3*e^4)*x)*sqrt(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x
+ d)) - 2*(96*B*c^3*d^7*e - 3*A*b^3*d^3*e^5 - 12*(17*B*b*c^2 + 2*A*c^3)*d^6*e^2 + 3*(41*B*b^2*c + 14*A*b*c^2)
*d^5*e^3 - 15*(B*b^3 + A*b^2*c)*d^4*e^4 + 24*(B*c^3*d^4*e^4 - 2*B*b*c^2*d^3*e^5 + B*b^2*c*d^2*e^6)*x^3 + (176*
B*c^3*d^5*e^3 + 3*A*b^3*d*e^7 - 2*(191*B*b*c^2 + 22*A*c^3)*d^4*e^4 + (239*B*b^2*c + 88*A*b*c^2)*d^3*e^5 - (33*
B*b^3 + 47*A*b^2*c)*d^2*e^6)*x^2 + 2*(120*B*c^3*d^6*e^2 - 4*A*b^3*d^2*e^6 - (257*B*b*c^2 + 30*A*c^3)*d^5*e^3 +
(157*B*b^2*c + 53*A*b*c^2)*d^4*e^4 - (20*B*b^3 + 19*A*b^2*c)*d^3*e^5)*x)*sqrt(c*x^2 + b*x))/(c^2*d^7*e^5 - 2*
b*c*d^6*e^6 + b^2*d^5*e^7 + (c^2*d^4*e^8 - 2*b*c*d^3*e^9 + b^2*d^2*e^10)*x^3 + 3*(c^2*d^5*e^7 - 2*b*c*d^4*e^8
+ b^2*d^3*e^9)*x^2 + 3*(c^2*d^6*e^6 - 2*b*c*d^5*e^7 + b^2*d^4*e^8)*x), 1/24*(3*(64*B*c^3*d^7 - A*b^3*d^3*e^4 -
8*(15*B*b*c^2 + 2*A*c^3)*d^6*e + 12*(5*B*b^2*c + 2*A*b*c^2)*d^5*e^2 - (5*B*b^3 + 6*A*b^2*c)*d^4*e^3 + (64*B*c
^3*d^4*e^3 - A*b^3*e^7 - 8*(15*B*b*c^2 + 2*A*c^3)*d^3*e^4 + 12*(5*B*b^2*c + 2*A*b*c^2)*d^2*e^5 - (5*B*b^3 + 6*
A*b^2*c)*d*e^6)*x^3 + 3*(64*B*c^3*d^5*e^2 - A*b^3*d*e^6 - 8*(15*B*b*c^2 + 2*A*c^3)*d^4*e^3 + 12*(5*B*b^2*c + 2
*A*b*c^2)*d^3*e^4 - (5*B*b^3 + 6*A*b^2*c)*d^2*e^5)*x^2 + 3*(64*B*c^3*d^6*e - A*b^3*d^2*e^5 - 8*(15*B*b*c^2 + 2
*A*c^3)*d^5*e^2 + 12*(5*B*b^2*c + 2*A*b*c^2)*d^4*e^3 - (5*B*b^3 + 6*A*b^2*c)*d^3*e^4)*x)*sqrt(-c*d^2 + b*d*e)*
arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) - 12*(8*B*c^3*d^8 - (19*B*b*c^2 + 2*A*c^3)*d^7
*e + 2*(7*B*b^2*c + 2*A*b*c^2)*d^6*e^2 - (3*B*b^3 + 2*A*b^2*c)*d^5*e^3 + (8*B*c^3*d^5*e^3 - (19*B*b*c^2 + 2*A*
c^3)*d^4*e^4 + 2*(7*B*b^2*c + 2*A*b*c^2)*d^3*e^5 - (3*B*b^3 + 2*A*b^2*c)*d^2*e^6)*x^3 + 3*(8*B*c^3*d^6*e^2 - (
19*B*b*c^2 + 2*A*c^3)*d^5*e^3 + 2*(7*B*b^2*c + 2*A*b*c^2)*d^4*e^4 - (3*B*b^3 + 2*A*b^2*c)*d^3*e^5)*x^2 + 3*(8*
B*c^3*d^7*e - (19*B*b*c^2 + 2*A*c^3)*d^6*e^2 + 2*(7*B*b^2*c + 2*A*b*c^2)*d^5*e^3 - (3*B*b^3 + 2*A*b^2*c)*d^4*e
^4)*x)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) + (96*B*c^3*d^7*e - 3*A*b^3*d^3*e^5 - 12*(17*B*b*c
^2 + 2*A*c^3)*d^6*e^2 + 3*(41*B*b^2*c + 14*A*b*c^2)*d^5*e^3 - 15*(B*b^3 + A*b^2*c)*d^4*e^4 + 24*(B*c^3*d^4*e^4
- 2*B*b*c^2*d^3*e^5 + B*b^2*c*d^2*e^6)*x^3 + (176*B*c^3*d^5*e^3 + 3*A*b^3*d*e^7 - 2*(191*B*b*c^2 + 22*A*c^3)*
d^4*e^4 + (239*B*b^2*c + 88*A*b*c^2)*d^3*e^5 - (33*B*b^3 + 47*A*b^2*c)*d^2*e^6)*x^2 + 2*(120*B*c^3*d^6*e^2 - 4
*A*b^3*d^2*e^6 - (257*B*b*c^2 + 30*A*c^3)*d^5*e^3 + (157*B*b^2*c + 53*A*b*c^2)*d^4*e^4 - (20*B*b^3 + 19*A*b^2*
c)*d^3*e^5)*x)*sqrt(c*x^2 + b*x))/(c^2*d^7*e^5 - 2*b*c*d^6*e^6 + b^2*d^5*e^7 + (c^2*d^4*e^8 - 2*b*c*d^3*e^9 +
b^2*d^2*e^10)*x^3 + 3*(c^2*d^5*e^7 - 2*b*c*d^4*e^8 + b^2*d^3*e^9)*x^2 + 3*(c^2*d^6*e^6 - 2*b*c*d^5*e^7 + b^2*d
^4*e^8)*x), 1/48*(48*(8*B*c^3*d^8 - (19*B*b*c^2 + 2*A*c^3)*d^7*e + 2*(7*B*b^2*c + 2*A*b*c^2)*d^6*e^2 - (3*B*b^
3 + 2*A*b^2*c)*d^5*e^3 + (8*B*c^3*d^5*e^3 - (19*B*b*c^2 + 2*A*c^3)*d^4*e^4 + 2*(7*B*b^2*c + 2*A*b*c^2)*d^3*e^5
- (3*B*b^3 + 2*A*b^2*c)*d^2*e^6)*x^3 + 3*(8*B*c^3*d^6*e^2 - (19*B*b*c^2 + 2*A*c^3)*d^5*e^3 + 2*(7*B*b^2*c + 2
*A*b*c^2)*d^4*e^4 - (3*B*b^3 + 2*A*b^2*c)*d^3*e^5)*x^2 + 3*(8*B*c^3*d^7*e - (19*B*b*c^2 + 2*A*c^3)*d^6*e^2 + 2
*(7*B*b^2*c + 2*A*b*c^2)*d^5*e^3 - (3*B*b^3 + 2*A*b^2*c)*d^4*e^4)*x)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c
)/(c*x)) + 3*(64*B*c^3*d^7 - A*b^3*d^3*e^4 - 8*(15*B*b*c^2 + 2*A*c^3)*d^6*e + 12*(5*B*b^2*c + 2*A*b*c^2)*d^5*e
^2 - (5*B*b^3 + 6*A*b^2*c)*d^4*e^3 + (64*B*c^3*d^4*e^3 - A*b^3*e^7 - 8*(15*B*b*c^2 + 2*A*c^3)*d^3*e^4 + 12*(5*
B*b^2*c + 2*A*b*c^2)*d^2*e^5 - (5*B*b^3 + 6*A*b^2*c)*d*e^6)*x^3 + 3*(64*B*c^3*d^5*e^2 - A*b^3*d*e^6 - 8*(15*B*
b*c^2 + 2*A*c^3)*d^4*e^3 + 12*(5*B*b^2*c + 2*A*b*c^2)*d^3*e^4 - (5*B*b^3 + 6*A*b^2*c)*d^2*e^5)*x^2 + 3*(64*B*c
^3*d^6*e - A*b^3*d^2*e^5 - 8*(15*B*b*c^2 + 2*A*c^3)*d^5*e^2 + 12*(5*B*b^2*c + 2*A*b*c^2)*d^4*e^3 - (5*B*b^3 +
6*A*b^2*c)*d^3*e^4)*x)*sqrt(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x
))/(e*x + d)) + 2*(96*B*c^3*d^7*e - 3*A*b^3*d^3*e^5 - 12*(17*B*b*c^2 + 2*A*c^3)*d^6*e^2 + 3*(41*B*b^2*c + 14*A
*b*c^2)*d^5*e^3 - 15*(B*b^3 + A*b^2*c)*d^4*e^4 + 24*(B*c^3*d^4*e^4 - 2*B*b*c^2*d^3*e^5 + B*b^2*c*d^2*e^6)*x^3
+ (176*B*c^3*d^5*e^3 + 3*A*b^3*d*e^7 - 2*(191*B*b*c^2 + 22*A*c^3)*d^4*e^4 + (239*B*b^2*c + 88*A*b*c^2)*d^3*e^5
- (33*B*b^3 + 47*A*b^2*c)*d^2*e^6)*x^2 + 2*(120*B*c^3*d^6*e^2 - 4*A*b^3*d^2*e^6 - (257*B*b*c^2 + 30*A*c^3)*d^
5*e^3 + (157*B*b^2*c + 53*A*b*c^2)*d^4*e^4 - (20*B*b^3 + 19*A*b^2*c)*d^3*e^5)*x)*sqrt(c*x^2 + b*x))/(c^2*d^7*e
^5 - 2*b*c*d^6*e^6 + b^2*d^5*e^7 + (c^2*d^4*e^8 - 2*b*c*d^3*e^9 + b^2*d^2*e^10)*x^3 + 3*(c^2*d^5*e^7 - 2*b*c*d
^4*e^8 + b^2*d^3*e^9)*x^2 + 3*(c^2*d^6*e^6 - 2*b*c*d^5*e^7 + b^2*d^4*e^8)*x), 1/24*(3*(64*B*c^3*d^7 - A*b^3*d^
3*e^4 - 8*(15*B*b*c^2 + 2*A*c^3)*d^6*e + 12*(5*B*b^2*c + 2*A*b*c^2)*d^5*e^2 - (5*B*b^3 + 6*A*b^2*c)*d^4*e^3 +
(64*B*c^3*d^4*e^3 - A*b^3*e^7 - 8*(15*B*b*c^2 + 2*A*c^3)*d^3*e^4 + 12*(5*B*b^2*c + 2*A*b*c^2)*d^2*e^5 - (5*B*b
^3 + 6*A*b^2*c)*d*e^6)*x^3 + 3*(64*B*c^3*d^5*e^2 - A*b^3*d*e^6 - 8*(15*B*b*c^2 + 2*A*c^3)*d^4*e^3 + 12*(5*B*b^
2*c + 2*A*b*c^2)*d^3*e^4 - (5*B*b^3 + 6*A*b^2*c)*d^2*e^5)*x^2 + 3*(64*B*c^3*d^6*e - A*b^3*d^2*e^5 - 8*(15*B*b*
c^2 + 2*A*c^3)*d^5*e^2 + 12*(5*B*b^2*c + 2*A*b*c^2)*d^4*e^3 - (5*B*b^3 + 6*A*b^2*c)*d^3*e^4)*x)*sqrt(-c*d^2 +
b*d*e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) + 24*(8*B*c^3*d^8 - (19*B*b*c^2 + 2*A*c
^3)*d^7*e + 2*(7*B*b^2*c + 2*A*b*c^2)*d^6*e^2 - (3*B*b^3 + 2*A*b^2*c)*d^5*e^3 + (8*B*c^3*d^5*e^3 - (19*B*b*c^2
+ 2*A*c^3)*d^4*e^4 + 2*(7*B*b^2*c + 2*A*b*c^2)*d^3*e^5 - (3*B*b^3 + 2*A*b^2*c)*d^2*e^6)*x^3 + 3*(8*B*c^3*d^6*
e^2 - (19*B*b*c^2 + 2*A*c^3)*d^5*e^3 + 2*(7*B*b^2*c + 2*A*b*c^2)*d^4*e^4 - (3*B*b^3 + 2*A*b^2*c)*d^3*e^5)*x^2
+ 3*(8*B*c^3*d^7*e - (19*B*b*c^2 + 2*A*c^3)*d^6*e^2 + 2*(7*B*b^2*c + 2*A*b*c^2)*d^5*e^3 - (3*B*b^3 + 2*A*b^2*c
)*d^4*e^4)*x)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (96*B*c^3*d^7*e - 3*A*b^3*d^3*e^5 - 12*(17*B
*b*c^2 + 2*A*c^3)*d^6*e^2 + 3*(41*B*b^2*c + 14*A*b*c^2)*d^5*e^3 - 15*(B*b^3 + A*b^2*c)*d^4*e^4 + 24*(B*c^3*d^4
*e^4 - 2*B*b*c^2*d^3*e^5 + B*b^2*c*d^2*e^6)*x^3 + (176*B*c^3*d^5*e^3 + 3*A*b^3*d*e^7 - 2*(191*B*b*c^2 + 22*A*c
^3)*d^4*e^4 + (239*B*b^2*c + 88*A*b*c^2)*d^3*e^5 - (33*B*b^3 + 47*A*b^2*c)*d^2*e^6)*x^2 + 2*(120*B*c^3*d^6*e^2
- 4*A*b^3*d^2*e^6 - (257*B*b*c^2 + 30*A*c^3)*d^5*e^3 + (157*B*b^2*c + 53*A*b*c^2)*d^4*e^4 - (20*B*b^3 + 19*A*
b^2*c)*d^3*e^5)*x)*sqrt(c*x^2 + b*x))/(c^2*d^7*e^5 - 2*b*c*d^6*e^6 + b^2*d^5*e^7 + (c^2*d^4*e^8 - 2*b*c*d^3*e^
9 + b^2*d^2*e^10)*x^3 + 3*(c^2*d^5*e^7 - 2*b*c*d^4*e^8 + b^2*d^3*e^9)*x^2 + 3*(c^2*d^6*e^6 - 2*b*c*d^5*e^7 + b
^2*d^4*e^8)*x)]

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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((B*x+A)*(c*x**2+b*x)**(3/2)/(e*x+d)**4,x)

[Out]

Timed out

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Giac [B]  time = 1.9583, size = 2414, normalized size = 5.67 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^(3/2)/(e*x+d)^4,x, algorithm="giac")

[Out]

sqrt(c*x^2 + b*x)*B*c*e^(-4) + 1/2*(8*B*c^2*d - 3*B*b*c*e - 2*A*c^2*e)*e^(-5)*log(abs(2*(sqrt(c)*x - sqrt(c*x^
2 + b*x))*sqrt(c) + b))/sqrt(c) + 1/8*(64*B*c^3*d^4 - 120*B*b*c^2*d^3*e - 16*A*c^3*d^3*e + 60*B*b^2*c*d^2*e^2
+ 24*A*b*c^2*d^2*e^2 - 5*B*b^3*d*e^3 - 6*A*b^2*c*d*e^3 - A*b^3*e^4)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x))*e
+ sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))/((c*d^2*e^5 - b*d*e^6)*sqrt(-c*d^2 + b*d*e)) + 1/24*(960*(sqrt(c)*x - sqrt
(c*x^2 + b*x))^4*B*c^4*d^5*e + 832*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*c^(9/2)*d^6 + 288*(sqrt(c)*x - sqrt(c*x
^2 + b*x))^5*B*c^(7/2)*d^4*e^2 - 400*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b*c^(7/2)*d^5*e - 352*(sqrt(c)*x - sq
rt(c*x^2 + b*x))^3*A*c^(9/2)*d^5*e + 1248*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b*c^4*d^6 - 1464*(sqrt(c)*x - sq
rt(c*x^2 + b*x))^4*B*b*c^3*d^4*e^2 - 432*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*c^4*d^4*e^2 - 1656*(sqrt(c)*x - s
qrt(c*x^2 + b*x))^2*B*b^2*c^3*d^5*e - 528*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b*c^4*d^5*e + 624*(sqrt(c)*x - s
qrt(c*x^2 + b*x))*B*b^2*c^(7/2)*d^6 - 504*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b*c^(5/2)*d^3*e^3 - 144*(sqrt(c)
*x - sqrt(c*x^2 + b*x))^5*A*c^(7/2)*d^3*e^3 - 840*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^2*c^(5/2)*d^4*e^2 + 16
*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b*c^(7/2)*d^4*e^2 - 876*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^3*c^(5/2)*d^5
*e - 264*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^2*c^(7/2)*d^5*e + 104*B*b^3*c^3*d^6 + 540*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^4*B*b^2*c^2*d^3*e^3 + 504*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b*c^3*d^3*e^3 + 414*(sqrt(c)*x - sqrt(c
*x^2 + b*x))^2*B*b^3*c^2*d^4*e^2 + 516*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^2*c^3*d^4*e^2 - 134*B*b^4*c^2*d^5
*e - 44*A*b^3*c^3*d^5*e + 252*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^2*c^(3/2)*d^2*e^4 + 216*(sqrt(c)*x - sqrt(
c*x^2 + b*x))^5*A*b*c^(5/2)*d^2*e^4 + 478*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^3*c^(3/2)*d^3*e^3 + 420*(sqrt(
c)*x - sqrt(c*x^2 + b*x))^3*A*b^2*c^(5/2)*d^3*e^3 + 282*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^4*c^(3/2)*d^4*e^2
+ 288*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^3*c^(5/2)*d^4*e^2 - 21*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^3*c*d^2
*e^4 - 54*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^2*c^2*d^2*e^4 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^4*c*d
^3*e^3 + 6*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^3*c^2*d^3*e^3 + 33*B*b^5*c*d^4*e^2 + 44*A*b^4*c^2*d^4*e^2 - 3
3*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^3*sqrt(c)*d*e^5 - 78*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^2*c^(3/2)*d
*e^5 - 40*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^4*sqrt(c)*d^2*e^4 - 106*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^
3*c^(3/2)*d^2*e^4 - 15*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^5*sqrt(c)*d^3*e^3 - 36*(sqrt(c)*x - sqrt(c*x^2 + b*
x))*A*b^4*c^(3/2)*d^3*e^3 - 33*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^3*c*d*e^5 - 24*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^2*A*b^4*c*d^2*e^4 - 3*A*b^5*c*d^3*e^3 + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^3*sqrt(c)*e^6 - 8*(sqrt(
c)*x - sqrt(c*x^2 + b*x))^3*A*b^4*sqrt(c)*d*e^5 - 3*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^5*sqrt(c)*d^2*e^4)/((c
^(3/2)*d^2*e^5 - b*sqrt(c)*d*e^6)*((sqrt(c)*x - sqrt(c*x^2 + b*x))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqr
t(c)*d + b*d)^3)