3.1188 \(\int \frac{(A+B x) (b x+c x^2)^{5/2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=508 \[ \frac{5 \sqrt{b x+c x^2} \left (-2 c e x \left (16 A c e (2 c d-b e)-B \left (b^2 e^2-32 b c d e+48 c^2 d^2\right )\right )+8 A c e \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right )-B \left (88 b^2 c d e^2-b^3 e^3-272 b c^2 d^2 e+192 c^3 d^3\right )\right )}{64 c e^6}-\frac{5 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (8 A c e \left (18 b^2 c d e^2-b^3 e^3-48 b c^2 d^2 e+32 c^3 d^3\right )-B \left (288 b^2 c^2 d^2 e^2-24 b^3 c d e^3-b^4 e^4-640 b c^3 d^3 e+384 c^4 d^4\right )\right )}{64 c^{3/2} e^7}+\frac{5 \sqrt{d} \sqrt{c d-b e} \left (A e \left (3 b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (7 b^2 e^2-28 b c d e+24 c^2 d^2\right )\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 e^7}+\frac{\left (b x+c x^2\right )^{5/2} (-2 A e+3 B d+B e x)}{4 e^2 (d+e x)^2}-\frac{5 \left (b x+c x^2\right )^{3/2} (e x (-4 A c e-b B e+6 B c d)-2 A e (8 c d-3 b e)+B d (24 c d-13 b e))}{24 e^4 (d+e x)} \]

[Out]

(5*(8*A*c*e*(16*c^2*d^2 - 20*b*c*d*e + 5*b^2*e^2) - B*(192*c^3*d^3 - 272*b*c^2*d^2*e + 88*b^2*c*d*e^2 - b^3*e^
3) - 2*c*e*(16*A*c*e*(2*c*d - b*e) - B*(48*c^2*d^2 - 32*b*c*d*e + b^2*e^2))*x)*Sqrt[b*x + c*x^2])/(64*c*e^6) -
(5*(B*d*(24*c*d - 13*b*e) - 2*A*e*(8*c*d - 3*b*e) + e*(6*B*c*d - b*B*e - 4*A*c*e)*x)*(b*x + c*x^2)^(3/2))/(24
*e^4*(d + e*x)) + ((3*B*d - 2*A*e + B*e*x)*(b*x + c*x^2)^(5/2))/(4*e^2*(d + e*x)^2) - (5*(8*A*c*e*(32*c^3*d^3
- 48*b*c^2*d^2*e + 18*b^2*c*d*e^2 - b^3*e^3) - B*(384*c^4*d^4 - 640*b*c^3*d^3*e + 288*b^2*c^2*d^2*e^2 - 24*b^3
*c*d*e^3 - b^4*e^4))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(64*c^(3/2)*e^7) + (5*Sqrt[d]*Sqrt[c*d - b*e]*(A*
e*(16*c^2*d^2 - 16*b*c*d*e + 3*b^2*e^2) - B*d*(24*c^2*d^2 - 28*b*c*d*e + 7*b^2*e^2))*ArcTanh[(b*d + (2*c*d - b
*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(8*e^7)

________________________________________________________________________________________

Rubi [A]  time = 0.747649, antiderivative size = 508, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {812, 814, 843, 620, 206, 724} \[ \frac{5 \sqrt{b x+c x^2} \left (-2 c e x \left (16 A c e (2 c d-b e)-B \left (b^2 e^2-32 b c d e+48 c^2 d^2\right )\right )+8 A c e \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right )-B \left (88 b^2 c d e^2-b^3 e^3-272 b c^2 d^2 e+192 c^3 d^3\right )\right )}{64 c e^6}-\frac{5 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (8 A c e \left (18 b^2 c d e^2-b^3 e^3-48 b c^2 d^2 e+32 c^3 d^3\right )-B \left (288 b^2 c^2 d^2 e^2-24 b^3 c d e^3-b^4 e^4-640 b c^3 d^3 e+384 c^4 d^4\right )\right )}{64 c^{3/2} e^7}+\frac{5 \sqrt{d} \sqrt{c d-b e} \left (A e \left (3 b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (7 b^2 e^2-28 b c d e+24 c^2 d^2\right )\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 e^7}+\frac{\left (b x+c x^2\right )^{5/2} (-2 A e+3 B d+B e x)}{4 e^2 (d+e x)^2}-\frac{5 \left (b x+c x^2\right )^{3/2} (e x (-4 A c e-b B e+6 B c d)-2 A e (8 c d-3 b e)+B d (24 c d-13 b e))}{24 e^4 (d+e x)} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^(5/2))/(d + e*x)^3,x]

[Out]

(5*(8*A*c*e*(16*c^2*d^2 - 20*b*c*d*e + 5*b^2*e^2) - B*(192*c^3*d^3 - 272*b*c^2*d^2*e + 88*b^2*c*d*e^2 - b^3*e^
3) - 2*c*e*(16*A*c*e*(2*c*d - b*e) - B*(48*c^2*d^2 - 32*b*c*d*e + b^2*e^2))*x)*Sqrt[b*x + c*x^2])/(64*c*e^6) -
(5*(B*d*(24*c*d - 13*b*e) - 2*A*e*(8*c*d - 3*b*e) + e*(6*B*c*d - b*B*e - 4*A*c*e)*x)*(b*x + c*x^2)^(3/2))/(24
*e^4*(d + e*x)) + ((3*B*d - 2*A*e + B*e*x)*(b*x + c*x^2)^(5/2))/(4*e^2*(d + e*x)^2) - (5*(8*A*c*e*(32*c^3*d^3
- 48*b*c^2*d^2*e + 18*b^2*c*d*e^2 - b^3*e^3) - B*(384*c^4*d^4 - 640*b*c^3*d^3*e + 288*b^2*c^2*d^2*e^2 - 24*b^3
*c*d*e^3 - b^4*e^4))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(64*c^(3/2)*e^7) + (5*Sqrt[d]*Sqrt[c*d - b*e]*(A*
e*(16*c^2*d^2 - 16*b*c*d*e + 3*b^2*e^2) - B*d*(24*c^2*d^2 - 28*b*c*d*e + 7*b^2*e^2))*ArcTanh[(b*d + (2*c*d - b
*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(8*e^7)

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 814

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 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 )^{5/2}}{(d+e x)^3} \, dx &=\frac{(3 B d-2 A e+B e x) \left (b x+c x^2\right )^{5/2}}{4 e^2 (d+e x)^2}-\frac{5 \int \frac{(2 b (3 B d-2 A e)+2 (6 B c d-b B e-4 A c e) x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^2} \, dx}{16 e^2}\\ &=-\frac{5 (B d (24 c d-13 b e)-2 A e (8 c d-3 b e)+e (6 B c d-b B e-4 A c e) x) \left (b x+c x^2\right )^{3/2}}{24 e^4 (d+e x)}+\frac{(3 B d-2 A e+B e x) \left (b x+c x^2\right )^{5/2}}{4 e^2 (d+e x)^2}+\frac{5 \int \frac{\left (2 b (B d (24 c d-13 b e)-2 A e (8 c d-3 b e))-2 \left (16 A c e (2 c d-b e)-B \left (48 c^2 d^2-32 b c d e+b^2 e^2\right )\right ) x\right ) \sqrt{b x+c x^2}}{d+e x} \, dx}{32 e^4}\\ &=\frac{5 \left (8 A c e \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right )-B \left (192 c^3 d^3-272 b c^2 d^2 e+88 b^2 c d e^2-b^3 e^3\right )-2 c e \left (16 A c e (2 c d-b e)-B \left (48 c^2 d^2-32 b c d e+b^2 e^2\right )\right ) x\right ) \sqrt{b x+c x^2}}{64 c e^6}-\frac{5 (B d (24 c d-13 b e)-2 A e (8 c d-3 b e)+e (6 B c d-b B e-4 A c e) x) \left (b x+c x^2\right )^{3/2}}{24 e^4 (d+e x)}+\frac{(3 B d-2 A e+B e x) \left (b x+c x^2\right )^{5/2}}{4 e^2 (d+e x)^2}-\frac{5 \int \frac{b d \left (8 A c e \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right )-B \left (192 c^3 d^3-272 b c^2 d^2 e+88 b^2 c d e^2-b^3 e^3\right )\right )+\left (8 A c e \left (32 c^3 d^3-48 b c^2 d^2 e+18 b^2 c d e^2-b^3 e^3\right )-B \left (384 c^4 d^4-640 b c^3 d^3 e+288 b^2 c^2 d^2 e^2-24 b^3 c d e^3-b^4 e^4\right )\right ) x}{(d+e x) \sqrt{b x+c x^2}} \, dx}{128 c e^6}\\ &=\frac{5 \left (8 A c e \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right )-B \left (192 c^3 d^3-272 b c^2 d^2 e+88 b^2 c d e^2-b^3 e^3\right )-2 c e \left (16 A c e (2 c d-b e)-B \left (48 c^2 d^2-32 b c d e+b^2 e^2\right )\right ) x\right ) \sqrt{b x+c x^2}}{64 c e^6}-\frac{5 (B d (24 c d-13 b e)-2 A e (8 c d-3 b e)+e (6 B c d-b B e-4 A c e) x) \left (b x+c x^2\right )^{3/2}}{24 e^4 (d+e x)}+\frac{(3 B d-2 A e+B e x) \left (b x+c x^2\right )^{5/2}}{4 e^2 (d+e x)^2}+\frac{\left (5 d (c d-b e) \left (A e \left (16 c^2 d^2-16 b c d e+3 b^2 e^2\right )-B d \left (24 c^2 d^2-28 b c d e+7 b^2 e^2\right )\right )\right ) \int \frac{1}{(d+e x) \sqrt{b x+c x^2}} \, dx}{8 e^7}-\frac{\left (5 \left (8 A c e \left (32 c^3 d^3-48 b c^2 d^2 e+18 b^2 c d e^2-b^3 e^3\right )-B \left (384 c^4 d^4-640 b c^3 d^3 e+288 b^2 c^2 d^2 e^2-24 b^3 c d e^3-b^4 e^4\right )\right )\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{128 c e^7}\\ &=\frac{5 \left (8 A c e \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right )-B \left (192 c^3 d^3-272 b c^2 d^2 e+88 b^2 c d e^2-b^3 e^3\right )-2 c e \left (16 A c e (2 c d-b e)-B \left (48 c^2 d^2-32 b c d e+b^2 e^2\right )\right ) x\right ) \sqrt{b x+c x^2}}{64 c e^6}-\frac{5 (B d (24 c d-13 b e)-2 A e (8 c d-3 b e)+e (6 B c d-b B e-4 A c e) x) \left (b x+c x^2\right )^{3/2}}{24 e^4 (d+e x)}+\frac{(3 B d-2 A e+B e x) \left (b x+c x^2\right )^{5/2}}{4 e^2 (d+e x)^2}-\frac{\left (5 d (c d-b e) \left (A e \left (16 c^2 d^2-16 b c d e+3 b^2 e^2\right )-B d \left (24 c^2 d^2-28 b c d e+7 b^2 e^2\right )\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 )}{4 e^7}-\frac{\left (5 \left (8 A c e \left (32 c^3 d^3-48 b c^2 d^2 e+18 b^2 c d e^2-b^3 e^3\right )-B \left (384 c^4 d^4-640 b c^3 d^3 e+288 b^2 c^2 d^2 e^2-24 b^3 c d e^3-b^4 e^4\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{64 c e^7}\\ &=\frac{5 \left (8 A c e \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right )-B \left (192 c^3 d^3-272 b c^2 d^2 e+88 b^2 c d e^2-b^3 e^3\right )-2 c e \left (16 A c e (2 c d-b e)-B \left (48 c^2 d^2-32 b c d e+b^2 e^2\right )\right ) x\right ) \sqrt{b x+c x^2}}{64 c e^6}-\frac{5 (B d (24 c d-13 b e)-2 A e (8 c d-3 b e)+e (6 B c d-b B e-4 A c e) x) \left (b x+c x^2\right )^{3/2}}{24 e^4 (d+e x)}+\frac{(3 B d-2 A e+B e x) \left (b x+c x^2\right )^{5/2}}{4 e^2 (d+e x)^2}-\frac{5 \left (8 A c e \left (32 c^3 d^3-48 b c^2 d^2 e+18 b^2 c d e^2-b^3 e^3\right )-B \left (384 c^4 d^4-640 b c^3 d^3 e+288 b^2 c^2 d^2 e^2-24 b^3 c d e^3-b^4 e^4\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{3/2} e^7}+\frac{5 \sqrt{d} \sqrt{c d-b e} \left (A e \left (16 c^2 d^2-16 b c d e+3 b^2 e^2\right )-B d \left (24 c^2 d^2-28 b c d e+7 b^2 e^2\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 )}{8 e^7}\\ \end{align*}

Mathematica [B]  time = 6.17558, size = 2063, normalized size = 4.06 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^(5/2))/(d + e*x)^3,x]

[Out]

((-(B*d) + A*e)*x*(b + c*x)*(x*(b + c*x))^(5/2))/(2*d*(-(c*d) + b*e)*(d + e*x)^2) + ((x*(b + c*x))^(5/2)*(((-5
*c*d*(B*d - A*e) + (e*(7*b*B*d - 4*A*c*d - 3*A*b*e))/2)*x^(7/2)*(b + c*x)^(7/2))/(d*(-(c*d) + b*e)*(d + e*x))
+ (((8*A*c^2*d^2 + 4*b*c*d*(14*B*d - 11*A*e) - 5*b^2*e*(7*B*d - 3*A*e))*((2*b^2*x^(5/2)*Sqrt[b + c*x]*(1 + (c*
x)/b)^3*((5/(16*(1 + (c*x)/b)^3) + 5/(8*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1))/2 - (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])))/(512*c^3*
x^3*(1 + (c*x)/b)^3)))/(5*e) - (d*((2*b^2*x^(3/2)*Sqrt[b + c*x]*(1 + (c*x)/b)^3*((3*(5/(8*(1 + (c*x)/b)^3) + 5
/(6*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1)))/8 + (15*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])))/(256*c^2*x^2*(1 + (c*x)/b)^3)))/(3*e) - (d*((2*b^2*Sqrt[x]*Sqrt[b
+ c*x]*(1 + (c*x)/b)^3*((15/(8*(1 + (c*x)/b)^3) + 5/(4*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1))/6 + (5*Sqrt[b]*A
rcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(16*Sqrt[c]*Sqrt[x]*(1 + (c*x)/b)^(7/2))))/e - (d*((2*b*c*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])/Sq
rt[b]])/(8*Sqrt[c]*Sqrt[x]*(1 + (c*x)/b)^(5/2))))/e - ((c*d - b*e)*((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))/e))/e))/4 + 3*c*(B*d*(10*c*d - 7*b*e) - 3*A*e*(2*c*d - b*e))*((2*b^2*x^(7/2)*Sqrt[b + c*x]*(1 + (c
*x)/b)^3*((7*(3/(16*(1 + (c*x)/b)^3) + 1/(2*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1)))/12 + (35*b^4*((2*c*x)/b -
(4*c^2*x^2)/(3*b^2) + (16*c^3*x^3)/(15*b^3) - (2*Sqrt[c]*Sqrt[x]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*
Sqrt[1 + (c*x)/b])))/(2048*c^4*x^4*(1 + (c*x)/b)^3)))/(7*e) - (d*((2*b^2*x^(5/2)*Sqrt[b + c*x]*(1 + (c*x)/b)^3
*((5/(16*(1 + (c*x)/b)^3) + 5/(8*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1))/2 - (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])))/(512*c^3*x^3*(1
+ (c*x)/b)^3)))/(5*e) - (d*((2*b^2*x^(3/2)*Sqrt[b + c*x]*(1 + (c*x)/b)^3*((3*(5/(8*(1 + (c*x)/b)^3) + 5/(6*(1
+ (c*x)/b)^2) + (1 + (c*x)/b)^(-1)))/8 + (15*b^2*((2*c*x)/b - (2*Sqrt[c]*Sqrt[x]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqr
t[b]])/(Sqrt[b]*Sqrt[1 + (c*x)/b])))/(256*c^2*x^2*(1 + (c*x)/b)^3)))/(3*e) - (d*((2*b^2*Sqrt[x]*Sqrt[b + c*x]*
(1 + (c*x)/b)^3*((15/(8*(1 + (c*x)/b)^3) + 5/(4*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1))/6 + (5*Sqrt[b]*ArcSinh[
(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(16*Sqrt[c]*Sqrt[x]*(1 + (c*x)/b)^(7/2))))/e - (d*((2*b*c*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 - ((c*d - b*e)*((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))/e))/e))/e))/(d*(-(c*d) + b*e))))/(2*d*(-(c*d) + b*e)*x^(5/2)*(b + c*x)^(5/2))

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Maple [B]  time = 0.017, size = 11558, normalized size = 22.8 \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)^(5/2)/(e*x+d)^3,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)^(5/2)/(e*x+d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 53.4852, size = 8783, normalized size = 17.29 \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)^(5/2)/(e*x+d)^3,x, algorithm="fricas")

[Out]

[-1/384*(15*(384*B*c^4*d^6 - 128*(5*B*b*c^3 + 2*A*c^4)*d^5*e + 96*(3*B*b^2*c^2 + 4*A*b*c^3)*d^4*e^2 - 24*(B*b^
3*c + 6*A*b^2*c^2)*d^3*e^3 - (B*b^4 - 8*A*b^3*c)*d^2*e^4 + (384*B*c^4*d^4*e^2 - 128*(5*B*b*c^3 + 2*A*c^4)*d^3*
e^3 + 96*(3*B*b^2*c^2 + 4*A*b*c^3)*d^2*e^4 - 24*(B*b^3*c + 6*A*b^2*c^2)*d*e^5 - (B*b^4 - 8*A*b^3*c)*e^6)*x^2 +
2*(384*B*c^4*d^5*e - 128*(5*B*b*c^3 + 2*A*c^4)*d^4*e^2 + 96*(3*B*b^2*c^2 + 4*A*b*c^3)*d^3*e^3 - 24*(B*b^3*c +
6*A*b^2*c^2)*d^2*e^4 - (B*b^4 - 8*A*b^3*c)*d*e^5)*x)*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 2
40*(24*B*c^4*d^5 - 3*A*b^2*c^2*d^2*e^3 - 4*(7*B*b*c^3 + 4*A*c^4)*d^4*e + (7*B*b^2*c^2 + 16*A*b*c^3)*d^3*e^2 +
(24*B*c^4*d^3*e^2 - 3*A*b^2*c^2*e^5 - 4*(7*B*b*c^3 + 4*A*c^4)*d^2*e^3 + (7*B*b^2*c^2 + 16*A*b*c^3)*d*e^4)*x^2
+ 2*(24*B*c^4*d^4*e - 3*A*b^2*c^2*d*e^4 - 4*(7*B*b*c^3 + 4*A*c^4)*d^3*e^2 + (7*B*b^2*c^2 + 16*A*b*c^3)*d^2*e^3
)*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*
(48*B*c^4*e^6*x^5 - 2880*B*c^4*d^5*e + 240*(17*B*b*c^3 + 8*A*c^4)*d^4*e^2 - 120*(11*B*b^2*c^2 + 20*A*b*c^3)*d^
3*e^3 + 15*(B*b^3*c + 40*A*b^2*c^2)*d^2*e^4 - 8*(12*B*c^4*d*e^5 - (17*B*b*c^3 + 8*A*c^4)*e^6)*x^4 + 2*(120*B*c
^4*d^2*e^4 - 16*(11*B*b*c^3 + 5*A*c^4)*d*e^5 + (59*B*b^2*c^2 + 104*A*b*c^3)*e^6)*x^3 - (960*B*c^4*d^3*e^3 - 40
*(37*B*b*c^3 + 16*A*c^4)*d^2*e^4 + 4*(139*B*b^2*c^2 + 220*A*b*c^3)*d*e^5 - 3*(5*B*b^3*c + 88*A*b^2*c^2)*e^6)*x
^2 - 10*(432*B*c^4*d^4*e^2 - 48*(13*B*b*c^3 + 6*A*c^4)*d^3*e^3 + (209*B*b^2*c^2 + 368*A*b*c^3)*d^2*e^4 - 3*(B*
b^3*c + 32*A*b^2*c^2)*d*e^5)*x)*sqrt(c*x^2 + b*x))/(c^2*e^9*x^2 + 2*c^2*d*e^8*x + c^2*d^2*e^7), -1/384*(480*(2
4*B*c^4*d^5 - 3*A*b^2*c^2*d^2*e^3 - 4*(7*B*b*c^3 + 4*A*c^4)*d^4*e + (7*B*b^2*c^2 + 16*A*b*c^3)*d^3*e^2 + (24*B
*c^4*d^3*e^2 - 3*A*b^2*c^2*e^5 - 4*(7*B*b*c^3 + 4*A*c^4)*d^2*e^3 + (7*B*b^2*c^2 + 16*A*b*c^3)*d*e^4)*x^2 + 2*(
24*B*c^4*d^4*e - 3*A*b^2*c^2*d*e^4 - 4*(7*B*b*c^3 + 4*A*c^4)*d^3*e^2 + (7*B*b^2*c^2 + 16*A*b*c^3)*d^2*e^3)*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)) + 15*(384*B*c^4*d^6 - 128
*(5*B*b*c^3 + 2*A*c^4)*d^5*e + 96*(3*B*b^2*c^2 + 4*A*b*c^3)*d^4*e^2 - 24*(B*b^3*c + 6*A*b^2*c^2)*d^3*e^3 - (B*
b^4 - 8*A*b^3*c)*d^2*e^4 + (384*B*c^4*d^4*e^2 - 128*(5*B*b*c^3 + 2*A*c^4)*d^3*e^3 + 96*(3*B*b^2*c^2 + 4*A*b*c^
3)*d^2*e^4 - 24*(B*b^3*c + 6*A*b^2*c^2)*d*e^5 - (B*b^4 - 8*A*b^3*c)*e^6)*x^2 + 2*(384*B*c^4*d^5*e - 128*(5*B*b
*c^3 + 2*A*c^4)*d^4*e^2 + 96*(3*B*b^2*c^2 + 4*A*b*c^3)*d^3*e^3 - 24*(B*b^3*c + 6*A*b^2*c^2)*d^2*e^4 - (B*b^4 -
8*A*b^3*c)*d*e^5)*x)*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 2*(48*B*c^4*e^6*x^5 - 2880*B*c^4*
d^5*e + 240*(17*B*b*c^3 + 8*A*c^4)*d^4*e^2 - 120*(11*B*b^2*c^2 + 20*A*b*c^3)*d^3*e^3 + 15*(B*b^3*c + 40*A*b^2*
c^2)*d^2*e^4 - 8*(12*B*c^4*d*e^5 - (17*B*b*c^3 + 8*A*c^4)*e^6)*x^4 + 2*(120*B*c^4*d^2*e^4 - 16*(11*B*b*c^3 + 5
*A*c^4)*d*e^5 + (59*B*b^2*c^2 + 104*A*b*c^3)*e^6)*x^3 - (960*B*c^4*d^3*e^3 - 40*(37*B*b*c^3 + 16*A*c^4)*d^2*e^
4 + 4*(139*B*b^2*c^2 + 220*A*b*c^3)*d*e^5 - 3*(5*B*b^3*c + 88*A*b^2*c^2)*e^6)*x^2 - 10*(432*B*c^4*d^4*e^2 - 48
*(13*B*b*c^3 + 6*A*c^4)*d^3*e^3 + (209*B*b^2*c^2 + 368*A*b*c^3)*d^2*e^4 - 3*(B*b^3*c + 32*A*b^2*c^2)*d*e^5)*x)
*sqrt(c*x^2 + b*x))/(c^2*e^9*x^2 + 2*c^2*d*e^8*x + c^2*d^2*e^7), -1/192*(15*(384*B*c^4*d^6 - 128*(5*B*b*c^3 +
2*A*c^4)*d^5*e + 96*(3*B*b^2*c^2 + 4*A*b*c^3)*d^4*e^2 - 24*(B*b^3*c + 6*A*b^2*c^2)*d^3*e^3 - (B*b^4 - 8*A*b^3*
c)*d^2*e^4 + (384*B*c^4*d^4*e^2 - 128*(5*B*b*c^3 + 2*A*c^4)*d^3*e^3 + 96*(3*B*b^2*c^2 + 4*A*b*c^3)*d^2*e^4 - 2
4*(B*b^3*c + 6*A*b^2*c^2)*d*e^5 - (B*b^4 - 8*A*b^3*c)*e^6)*x^2 + 2*(384*B*c^4*d^5*e - 128*(5*B*b*c^3 + 2*A*c^4
)*d^4*e^2 + 96*(3*B*b^2*c^2 + 4*A*b*c^3)*d^3*e^3 - 24*(B*b^3*c + 6*A*b^2*c^2)*d^2*e^4 - (B*b^4 - 8*A*b^3*c)*d*
e^5)*x)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - 120*(24*B*c^4*d^5 - 3*A*b^2*c^2*d^2*e^3 - 4*(7*B*b
*c^3 + 4*A*c^4)*d^4*e + (7*B*b^2*c^2 + 16*A*b*c^3)*d^3*e^2 + (24*B*c^4*d^3*e^2 - 3*A*b^2*c^2*e^5 - 4*(7*B*b*c^
3 + 4*A*c^4)*d^2*e^3 + (7*B*b^2*c^2 + 16*A*b*c^3)*d*e^4)*x^2 + 2*(24*B*c^4*d^4*e - 3*A*b^2*c^2*d*e^4 - 4*(7*B*
b*c^3 + 4*A*c^4)*d^3*e^2 + (7*B*b^2*c^2 + 16*A*b*c^3)*d^2*e^3)*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)) - (48*B*c^4*e^6*x^5 - 2880*B*c^4*d^5*e + 240*(17*B*b*
c^3 + 8*A*c^4)*d^4*e^2 - 120*(11*B*b^2*c^2 + 20*A*b*c^3)*d^3*e^3 + 15*(B*b^3*c + 40*A*b^2*c^2)*d^2*e^4 - 8*(12
*B*c^4*d*e^5 - (17*B*b*c^3 + 8*A*c^4)*e^6)*x^4 + 2*(120*B*c^4*d^2*e^4 - 16*(11*B*b*c^3 + 5*A*c^4)*d*e^5 + (59*
B*b^2*c^2 + 104*A*b*c^3)*e^6)*x^3 - (960*B*c^4*d^3*e^3 - 40*(37*B*b*c^3 + 16*A*c^4)*d^2*e^4 + 4*(139*B*b^2*c^2
+ 220*A*b*c^3)*d*e^5 - 3*(5*B*b^3*c + 88*A*b^2*c^2)*e^6)*x^2 - 10*(432*B*c^4*d^4*e^2 - 48*(13*B*b*c^3 + 6*A*c
^4)*d^3*e^3 + (209*B*b^2*c^2 + 368*A*b*c^3)*d^2*e^4 - 3*(B*b^3*c + 32*A*b^2*c^2)*d*e^5)*x)*sqrt(c*x^2 + b*x))/
(c^2*e^9*x^2 + 2*c^2*d*e^8*x + c^2*d^2*e^7), -1/192*(240*(24*B*c^4*d^5 - 3*A*b^2*c^2*d^2*e^3 - 4*(7*B*b*c^3 +
4*A*c^4)*d^4*e + (7*B*b^2*c^2 + 16*A*b*c^3)*d^3*e^2 + (24*B*c^4*d^3*e^2 - 3*A*b^2*c^2*e^5 - 4*(7*B*b*c^3 + 4*A
*c^4)*d^2*e^3 + (7*B*b^2*c^2 + 16*A*b*c^3)*d*e^4)*x^2 + 2*(24*B*c^4*d^4*e - 3*A*b^2*c^2*d*e^4 - 4*(7*B*b*c^3 +
4*A*c^4)*d^3*e^2 + (7*B*b^2*c^2 + 16*A*b*c^3)*d^2*e^3)*x)*sqrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^2 + b*d*e)*s
qrt(c*x^2 + b*x)/((c*d - b*e)*x)) + 15*(384*B*c^4*d^6 - 128*(5*B*b*c^3 + 2*A*c^4)*d^5*e + 96*(3*B*b^2*c^2 + 4*
A*b*c^3)*d^4*e^2 - 24*(B*b^3*c + 6*A*b^2*c^2)*d^3*e^3 - (B*b^4 - 8*A*b^3*c)*d^2*e^4 + (384*B*c^4*d^4*e^2 - 128
*(5*B*b*c^3 + 2*A*c^4)*d^3*e^3 + 96*(3*B*b^2*c^2 + 4*A*b*c^3)*d^2*e^4 - 24*(B*b^3*c + 6*A*b^2*c^2)*d*e^5 - (B*
b^4 - 8*A*b^3*c)*e^6)*x^2 + 2*(384*B*c^4*d^5*e - 128*(5*B*b*c^3 + 2*A*c^4)*d^4*e^2 + 96*(3*B*b^2*c^2 + 4*A*b*c
^3)*d^3*e^3 - 24*(B*b^3*c + 6*A*b^2*c^2)*d^2*e^4 - (B*b^4 - 8*A*b^3*c)*d*e^5)*x)*sqrt(-c)*arctan(sqrt(c*x^2 +
b*x)*sqrt(-c)/(c*x)) - (48*B*c^4*e^6*x^5 - 2880*B*c^4*d^5*e + 240*(17*B*b*c^3 + 8*A*c^4)*d^4*e^2 - 120*(11*B*b
^2*c^2 + 20*A*b*c^3)*d^3*e^3 + 15*(B*b^3*c + 40*A*b^2*c^2)*d^2*e^4 - 8*(12*B*c^4*d*e^5 - (17*B*b*c^3 + 8*A*c^4
)*e^6)*x^4 + 2*(120*B*c^4*d^2*e^4 - 16*(11*B*b*c^3 + 5*A*c^4)*d*e^5 + (59*B*b^2*c^2 + 104*A*b*c^3)*e^6)*x^3 -
(960*B*c^4*d^3*e^3 - 40*(37*B*b*c^3 + 16*A*c^4)*d^2*e^4 + 4*(139*B*b^2*c^2 + 220*A*b*c^3)*d*e^5 - 3*(5*B*b^3*c
+ 88*A*b^2*c^2)*e^6)*x^2 - 10*(432*B*c^4*d^4*e^2 - 48*(13*B*b*c^3 + 6*A*c^4)*d^3*e^3 + (209*B*b^2*c^2 + 368*A
*b*c^3)*d^2*e^4 - 3*(B*b^3*c + 32*A*b^2*c^2)*d*e^5)*x)*sqrt(c*x^2 + b*x))/(c^2*e^9*x^2 + 2*c^2*d*e^8*x + c^2*d
^2*e^7)]

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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)**(5/2)/(e*x+d)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.90479, size = 1924, normalized size = 3.79 \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)^(5/2)/(e*x+d)^3,x, algorithm="giac")

[Out]

-5/4*(24*B*c^3*d^5 - 52*B*b*c^2*d^4*e - 16*A*c^3*d^4*e + 35*B*b^2*c*d^3*e^2 + 32*A*b*c^2*d^3*e^2 - 7*B*b^3*d^2
*e^3 - 19*A*b^2*c*d^2*e^3 + 3*A*b^3*d*e^4)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2
+ b*d*e))*e^(-7)/sqrt(-c*d^2 + b*d*e) + 1/192*sqrt(c*x^2 + b*x)*(2*(4*(6*B*c^2*x*e^(-3) - (24*B*c^5*d*e^21 -
17*B*b*c^4*e^22 - 8*A*c^5*e^22)*e^(-25)/c^3)*x + (288*B*c^5*d^2*e^20 - 312*B*b*c^4*d*e^21 - 144*A*c^5*d*e^21 +
59*B*b^2*c^3*e^22 + 104*A*b*c^4*e^22)*e^(-25)/c^3)*x - 3*(640*B*c^5*d^3*e^19 - 864*B*b*c^4*d^2*e^20 - 384*A*c
^5*d^2*e^20 + 264*B*b^2*c^3*d*e^21 + 432*A*b*c^4*d*e^21 - 5*B*b^3*c^2*e^22 - 88*A*b^2*c^3*e^22)*e^(-25)/c^3) -
5/128*(384*B*c^4*d^4 - 640*B*b*c^3*d^3*e - 256*A*c^4*d^3*e + 288*B*b^2*c^2*d^2*e^2 + 384*A*b*c^3*d^2*e^2 - 24
*B*b^3*c*d*e^3 - 144*A*b^2*c^2*d*e^3 - B*b^4*e^4 + 8*A*b^3*c*e^4)*e^(-7)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b
*x))*sqrt(c) + b))/c^(3/2) - 1/4*(48*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*c^(7/2)*d^5*e + 88*(sqrt(c)*x - sqrt(
c*x^2 + b*x))^2*B*c^4*d^6 - 156*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b*c^3*d^5*e - 72*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^2*A*c^4*d^5*e + 88*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b*c^(7/2)*d^6 - 100*(sqrt(c)*x - sqrt(c*x^2 + b*x)
)^3*B*b*c^(5/2)*d^4*e^2 - 40*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*c^(7/2)*d^4*e^2 - 160*(sqrt(c)*x - sqrt(c*x^2
+ b*x))*B*b^2*c^(5/2)*d^5*e - 72*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b*c^(7/2)*d^5*e + 22*B*b^2*c^3*d^6 + 75*(s
qrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^2*c^2*d^4*e^2 + 120*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b*c^3*d^4*e^2 - 35
*B*b^3*c^2*d^5*e - 18*A*b^2*c^3*d^5*e + 65*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^2*c^(3/2)*d^3*e^3 + 80*(sqrt(
c)*x - sqrt(c*x^2 + b*x))^3*A*b*c^(5/2)*d^3*e^3 + 83*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^3*c^(3/2)*d^4*e^2 + 1
24*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^2*c^(5/2)*d^4*e^2 - 7*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^3*c*d^3*e^3
- 51*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^2*c^2*d^3*e^3 + 13*B*b^4*c*d^4*e^2 + 27*A*b^3*c^2*d^4*e^2 - 13*(sq
rt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^3*sqrt(c)*d^2*e^4 - 49*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^2*c^(3/2)*d^2*
e^4 - 11*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^4*sqrt(c)*d^3*e^3 - 59*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^3*c^(3
/2)*d^3*e^3 + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^3*c*d^2*e^4 - 9*A*b^4*c*d^3*e^3 + 9*(sqrt(c)*x - sqrt(c*
x^2 + b*x))^3*A*b^3*sqrt(c)*d*e^5 + 7*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^4*sqrt(c)*d^2*e^4)*e^(-7)/(((sqrt(c)
*x - sqrt(c*x^2 + b*x))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c)*d + b*d)^2*sqrt(c))