3.1190 \(\int \frac{(A+B x) (b x+c x^2)^{5/2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=633 \[ -\frac{5 \left (b x+c x^2\right )^{3/2} \left (3 e x \left (A e \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )-B d \left (9 b^2 e^2-32 b c d e+24 c^2 d^2\right )\right )+d \left (A e \left (-b^2 e^2-12 b c d e+16 c^2 d^2\right )-B d \left (7 b^2 e^2-52 b c d e+48 c^2 d^2\right )\right )\right )}{96 d e^4 (d+e x)^3 (c d-b e)}-\frac{5 \sqrt{b x+c x^2} \left (-2 c e x \left (A e \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (17 b^2 e^2-64 b c d e+48 c^2 d^2\right )\right )-A e \left (16 b^2 c d e^2+b^3 e^3-80 b c^2 d^2 e+64 c^3 d^3\right )+B d \left (120 b^2 c d e^2-7 b^3 e^3-304 b c^2 d^2 e+192 c^3 d^3\right )\right )}{64 d e^6 (d+e x) (c d-b e)}-\frac{5 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (4 A c e (2 c d-b e)-B \left (3 b^2 e^2-20 b c d e+24 c^2 d^2\right )\right )}{4 e^7}+\frac{5 \left (A e \left (144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4-256 b c^3 d^3 e+128 c^4 d^4\right )-B d \left (672 b^2 c^2 d^2 e^2-168 b^3 c d e^3+7 b^4 e^4-896 b c^3 d^3 e+384 c^4 d^4\right )\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{128 d^{3/2} e^7 (c d-b e)^{3/2}}+\frac{\left (b x+c x^2\right )^{5/2} (-A e+3 B d+2 B e x)}{4 e^2 (d+e x)^4} \]

[Out]

(-5*(B*d*(192*c^3*d^3 - 304*b*c^2*d^2*e + 120*b^2*c*d*e^2 - 7*b^3*e^3) - A*e*(64*c^3*d^3 - 80*b*c^2*d^2*e + 16
*b^2*c*d*e^2 + b^3*e^3) - 2*c*e*(A*e*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2) - B*d*(48*c^2*d^2 - 64*b*c*d*e + 17*b
^2*e^2))*x)*Sqrt[b*x + c*x^2])/(64*d*e^6*(c*d - b*e)*(d + e*x)) - (5*(d*(A*e*(16*c^2*d^2 - 12*b*c*d*e - b^2*e^
2) - B*d*(48*c^2*d^2 - 52*b*c*d*e + 7*b^2*e^2)) + 3*e*(A*e*(8*c^2*d^2 - 8*b*c*d*e + b^2*e^2) - B*d*(24*c^2*d^2
- 32*b*c*d*e + 9*b^2*e^2))*x)*(b*x + c*x^2)^(3/2))/(96*d*e^4*(c*d - b*e)*(d + e*x)^3) + ((3*B*d - A*e + 2*B*e
*x)*(b*x + c*x^2)^(5/2))/(4*e^2*(d + e*x)^4) - (5*Sqrt[c]*(4*A*c*e*(2*c*d - b*e) - B*(24*c^2*d^2 - 20*b*c*d*e
+ 3*b^2*e^2))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(4*e^7) + (5*(A*e*(128*c^4*d^4 - 256*b*c^3*d^3*e + 144*b
^2*c^2*d^2*e^2 - 16*b^3*c*d*e^3 - b^4*e^4) - B*d*(384*c^4*d^4 - 896*b*c^3*d^3*e + 672*b^2*c^2*d^2*e^2 - 168*b^
3*c*d*e^3 + 7*b^4*e^4))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(128*d
^(3/2)*e^7*(c*d - b*e)^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.902376, antiderivative size = 633, 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, 810, 843, 620, 206, 724} \[ -\frac{5 \left (b x+c x^2\right )^{3/2} \left (3 e x \left (A e \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )-B d \left (9 b^2 e^2-32 b c d e+24 c^2 d^2\right )\right )+d \left (A e \left (-b^2 e^2-12 b c d e+16 c^2 d^2\right )-B d \left (7 b^2 e^2-52 b c d e+48 c^2 d^2\right )\right )\right )}{96 d e^4 (d+e x)^3 (c d-b e)}-\frac{5 \sqrt{b x+c x^2} \left (-2 c e x \left (A e \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (17 b^2 e^2-64 b c d e+48 c^2 d^2\right )\right )-A e \left (16 b^2 c d e^2+b^3 e^3-80 b c^2 d^2 e+64 c^3 d^3\right )+B d \left (120 b^2 c d e^2-7 b^3 e^3-304 b c^2 d^2 e+192 c^3 d^3\right )\right )}{64 d e^6 (d+e x) (c d-b e)}-\frac{5 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (4 A c e (2 c d-b e)-B \left (3 b^2 e^2-20 b c d e+24 c^2 d^2\right )\right )}{4 e^7}+\frac{5 \left (A e \left (144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4-256 b c^3 d^3 e+128 c^4 d^4\right )-B d \left (672 b^2 c^2 d^2 e^2-168 b^3 c d e^3+7 b^4 e^4-896 b c^3 d^3 e+384 c^4 d^4\right )\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{128 d^{3/2} e^7 (c d-b e)^{3/2}}+\frac{\left (b x+c x^2\right )^{5/2} (-A e+3 B d+2 B e x)}{4 e^2 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*(B*d*(192*c^3*d^3 - 304*b*c^2*d^2*e + 120*b^2*c*d*e^2 - 7*b^3*e^3) - A*e*(64*c^3*d^3 - 80*b*c^2*d^2*e + 16
*b^2*c*d*e^2 + b^3*e^3) - 2*c*e*(A*e*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2) - B*d*(48*c^2*d^2 - 64*b*c*d*e + 17*b
^2*e^2))*x)*Sqrt[b*x + c*x^2])/(64*d*e^6*(c*d - b*e)*(d + e*x)) - (5*(d*(A*e*(16*c^2*d^2 - 12*b*c*d*e - b^2*e^
2) - B*d*(48*c^2*d^2 - 52*b*c*d*e + 7*b^2*e^2)) + 3*e*(A*e*(8*c^2*d^2 - 8*b*c*d*e + b^2*e^2) - B*d*(24*c^2*d^2
- 32*b*c*d*e + 9*b^2*e^2))*x)*(b*x + c*x^2)^(3/2))/(96*d*e^4*(c*d - b*e)*(d + e*x)^3) + ((3*B*d - A*e + 2*B*e
*x)*(b*x + c*x^2)^(5/2))/(4*e^2*(d + e*x)^4) - (5*Sqrt[c]*(4*A*c*e*(2*c*d - b*e) - B*(24*c^2*d^2 - 20*b*c*d*e
+ 3*b^2*e^2))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(4*e^7) + (5*(A*e*(128*c^4*d^4 - 256*b*c^3*d^3*e + 144*b
^2*c^2*d^2*e^2 - 16*b^3*c*d*e^3 - b^4*e^4) - B*d*(384*c^4*d^4 - 896*b*c^3*d^3*e + 672*b^2*c^2*d^2*e^2 - 168*b^
3*c*d*e^3 + 7*b^4*e^4))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(128*d
^(3/2)*e^7*(c*d - b*e)^(3/2))

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 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 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)^5} \, dx &=\frac{(3 B d-A e+2 B e x) \left (b x+c x^2\right )^{5/2}}{4 e^2 (d+e x)^4}-\frac{5 \int \frac{(2 b (3 B d-A e)+4 (3 B c d-b B e-A c e) x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx}{16 e^2}\\ &=-\frac{5 \left (d \left (A e \left (16 c^2 d^2-12 b c d e-b^2 e^2\right )-B d \left (48 c^2 d^2-52 b c d e+7 b^2 e^2\right )\right )+3 e \left (A e \left (8 c^2 d^2-8 b c d e+b^2 e^2\right )-B d \left (24 c^2 d^2-32 b c d e+9 b^2 e^2\right )\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{96 d e^4 (c d-b e) (d+e x)^3}+\frac{(3 B d-A e+2 B e x) \left (b x+c x^2\right )^{5/2}}{4 e^2 (d+e x)^4}+\frac{5 \int \frac{\left (b \left (A e \left (16 c^2 d^2-12 b c d e-b^2 e^2\right )-B d \left (48 c^2 d^2-52 b c d e+7 b^2 e^2\right )\right )+2 c \left (A e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right )-B d \left (48 c^2 d^2-64 b c d e+17 b^2 e^2\right )\right ) x\right ) \sqrt{b x+c x^2}}{(d+e x)^2} \, dx}{64 d e^4 (c d-b e)}\\ &=-\frac{5 \left (B d \left (192 c^3 d^3-304 b c^2 d^2 e+120 b^2 c d e^2-7 b^3 e^3\right )-A e \left (64 c^3 d^3-80 b c^2 d^2 e+16 b^2 c d e^2+b^3 e^3\right )-2 c e \left (A e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right )-B d \left (48 c^2 d^2-64 b c d e+17 b^2 e^2\right )\right ) x\right ) \sqrt{b x+c x^2}}{64 d e^6 (c d-b e) (d+e x)}-\frac{5 \left (d \left (A e \left (16 c^2 d^2-12 b c d e-b^2 e^2\right )-B d \left (48 c^2 d^2-52 b c d e+7 b^2 e^2\right )\right )+3 e \left (A e \left (8 c^2 d^2-8 b c d e+b^2 e^2\right )-B d \left (24 c^2 d^2-32 b c d e+9 b^2 e^2\right )\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{96 d e^4 (c d-b e) (d+e x)^3}+\frac{(3 B d-A e+2 B e x) \left (b x+c x^2\right )^{5/2}}{4 e^2 (d+e x)^4}-\frac{5 \int \frac{-b \left (B d \left (192 c^3 d^3-304 b c^2 d^2 e+120 b^2 c d e^2-7 b^3 e^3\right )-A e \left (64 c^3 d^3-80 b c^2 d^2 e+16 b^2 c d e^2+b^3 e^3\right )\right )+16 c d (c d-b e) \left (4 A c e (2 c d-b e)-B \left (24 c^2 d^2-20 b c d e+3 b^2 e^2\right )\right ) x}{(d+e x) \sqrt{b x+c x^2}} \, dx}{128 d e^6 (c d-b e)}\\ &=-\frac{5 \left (B d \left (192 c^3 d^3-304 b c^2 d^2 e+120 b^2 c d e^2-7 b^3 e^3\right )-A e \left (64 c^3 d^3-80 b c^2 d^2 e+16 b^2 c d e^2+b^3 e^3\right )-2 c e \left (A e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right )-B d \left (48 c^2 d^2-64 b c d e+17 b^2 e^2\right )\right ) x\right ) \sqrt{b x+c x^2}}{64 d e^6 (c d-b e) (d+e x)}-\frac{5 \left (d \left (A e \left (16 c^2 d^2-12 b c d e-b^2 e^2\right )-B d \left (48 c^2 d^2-52 b c d e+7 b^2 e^2\right )\right )+3 e \left (A e \left (8 c^2 d^2-8 b c d e+b^2 e^2\right )-B d \left (24 c^2 d^2-32 b c d e+9 b^2 e^2\right )\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{96 d e^4 (c d-b e) (d+e x)^3}+\frac{(3 B d-A e+2 B e x) \left (b x+c x^2\right )^{5/2}}{4 e^2 (d+e x)^4}-\frac{\left (5 c \left (4 A c e (2 c d-b e)-B \left (24 c^2 d^2-20 b c d e+3 b^2 e^2\right )\right )\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{8 e^7}+\frac{\left (5 \left (A e \left (128 c^4 d^4-256 b c^3 d^3 e+144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4\right )-B d \left (384 c^4 d^4-896 b c^3 d^3 e+672 b^2 c^2 d^2 e^2-168 b^3 c d e^3+7 b^4 e^4\right )\right )\right ) \int \frac{1}{(d+e x) \sqrt{b x+c x^2}} \, dx}{128 d e^7 (c d-b e)}\\ &=-\frac{5 \left (B d \left (192 c^3 d^3-304 b c^2 d^2 e+120 b^2 c d e^2-7 b^3 e^3\right )-A e \left (64 c^3 d^3-80 b c^2 d^2 e+16 b^2 c d e^2+b^3 e^3\right )-2 c e \left (A e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right )-B d \left (48 c^2 d^2-64 b c d e+17 b^2 e^2\right )\right ) x\right ) \sqrt{b x+c x^2}}{64 d e^6 (c d-b e) (d+e x)}-\frac{5 \left (d \left (A e \left (16 c^2 d^2-12 b c d e-b^2 e^2\right )-B d \left (48 c^2 d^2-52 b c d e+7 b^2 e^2\right )\right )+3 e \left (A e \left (8 c^2 d^2-8 b c d e+b^2 e^2\right )-B d \left (24 c^2 d^2-32 b c d e+9 b^2 e^2\right )\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{96 d e^4 (c d-b e) (d+e x)^3}+\frac{(3 B d-A e+2 B e x) \left (b x+c x^2\right )^{5/2}}{4 e^2 (d+e x)^4}-\frac{\left (5 c \left (4 A c e (2 c d-b e)-B \left (24 c^2 d^2-20 b c d e+3 b^2 e^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{4 e^7}-\frac{\left (5 \left (A e \left (128 c^4 d^4-256 b c^3 d^3 e+144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4\right )-B d \left (384 c^4 d^4-896 b c^3 d^3 e+672 b^2 c^2 d^2 e^2-168 b^3 c d e^3+7 b^4 e^4\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 )}{64 d e^7 (c d-b e)}\\ &=-\frac{5 \left (B d \left (192 c^3 d^3-304 b c^2 d^2 e+120 b^2 c d e^2-7 b^3 e^3\right )-A e \left (64 c^3 d^3-80 b c^2 d^2 e+16 b^2 c d e^2+b^3 e^3\right )-2 c e \left (A e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right )-B d \left (48 c^2 d^2-64 b c d e+17 b^2 e^2\right )\right ) x\right ) \sqrt{b x+c x^2}}{64 d e^6 (c d-b e) (d+e x)}-\frac{5 \left (d \left (A e \left (16 c^2 d^2-12 b c d e-b^2 e^2\right )-B d \left (48 c^2 d^2-52 b c d e+7 b^2 e^2\right )\right )+3 e \left (A e \left (8 c^2 d^2-8 b c d e+b^2 e^2\right )-B d \left (24 c^2 d^2-32 b c d e+9 b^2 e^2\right )\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{96 d e^4 (c d-b e) (d+e x)^3}+\frac{(3 B d-A e+2 B e x) \left (b x+c x^2\right )^{5/2}}{4 e^2 (d+e x)^4}-\frac{5 \sqrt{c} \left (4 A c e (2 c d-b e)-B \left (24 c^2 d^2-20 b c d e+3 b^2 e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 e^7}+\frac{5 \left (A e \left (128 c^4 d^4-256 b c^3 d^3 e+144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4\right )-B d \left (384 c^4 d^4-896 b c^3 d^3 e+672 b^2 c^2 d^2 e^2-168 b^3 c d e^3+7 b^4 e^4\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 )}{128 d^{3/2} e^7 (c d-b e)^{3/2}}\\ \end{align*}

Mathematica [B]  time = 6.28533, size = 2696, normalized size = 4.26 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-(B*d) + A*e)*x*(b + c*x)*(x*(b + c*x))^(5/2))/(4*d*(-(c*d) + b*e)*(d + e*x)^4) + ((x*(b + c*x))^(5/2)*(((-3
*c*d*(B*d - A*e) + (e*(7*b*B*d - 8*A*c*d + A*b*e))/2)*x^(7/2)*(b + c*x)^(7/2))/(3*d*(-(c*d) + b*e)*(d + e*x)^3
) + (((-2*c*d*(B*d*(6*c*d - 7*b*e) + A*e*(2*c*d - b*e)) + (e*(-7*b^2*B*d*e + A*(48*c^2*d^2 - 40*b*c*d*e - b^2*
e^2)))/4)*x^(7/2)*(b + c*x)^(7/2))/(2*d*(-(c*d) + b*e)*(d + e*x)^2) + ((((e*(-192*A*c^3*d^3 - 4*b^2*c*d*e*(91*
B*d - 17*A*e) + 3*b^3*e^2*(7*B*d + A*e) + 16*b*c^2*d^2*(21*B*d + 8*A*e)))/8 + (5*c*d*(A*e*(32*c^2*d^2 - 32*b*c
*d*e - b^2*e^2) - B*d*(48*c^2*d^2 - 56*b*c*d*e + 7*b^2*e^2)))/4)*x^(7/2)*(b + c*x)^(7/2))/(d*(-(c*d) + b*e)*(d
+ e*x)) + ((-(c*d*(-192*A*c^3*d^3 - 4*b^2*c*d*e*(91*B*d - 17*A*e) + 3*b^3*e^2*(7*B*d + A*e) + 16*b*c^2*d^2*(2
1*B*d + 8*A*e)))/8 + (b*e*(-192*A*c^3*d^3 - 4*b^2*c*d*e*(91*B*d - 17*A*e) + 3*b^3*e^2*(7*B*d + A*e) + 16*b*c^2
*d^2*(21*B*d + 8*A*e)))/8 - (7*b*((e*(-192*A*c^3*d^3 - 4*b^2*c*d*e*(91*B*d - 17*A*e) + 3*b^3*e^2*(7*B*d + A*e)
+ 16*b*c^2*d^2*(21*B*d + 8*A*e)))/8 + (5*c*d*(A*e*(32*c^2*d^2 - 32*b*c*d*e - b^2*e^2) - B*d*(48*c^2*d^2 - 56*
b*c*d*e + 7*b^2*e^2)))/4))/2)*((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]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(16*Sqr
t[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[(Sq
rt[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) - 6*c*((e*(-192*A*c^3*d
^3 - 4*b^2*c*d*e*(91*B*d - 17*A*e) + 3*b^3*e^2*(7*B*d + A*e) + 16*b*c^2*d^2*(21*B*d + 8*A*e)))/8 + (5*c*d*(A*e
*(32*c^2*d^2 - 32*b*c*d*e - b^2*e^2) - B*d*(48*c^2*d^2 - 56*b*c*d*e + 7*b^2*e^2)))/4)*((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])))/(51
2*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]*S
qrt[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*Sqr
t[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]*Sqr
t[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)))/(3*d*(-(c*d) + b*e))))/(4*d*(-(c*d) +
b*e)*x^(5/2)*(b + c*x)^(5/2))

________________________________________________________________________________________

Maple [B]  time = 0.026, size = 23819, normalized size = 37.6 \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)^5,x)

[Out]

result too large to display

________________________________________________________________________________________

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)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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((B*x+A)*(c*x^2+b*x)^(5/2)/(e*x+d)^5,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

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)**5,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [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)^5,x, algorithm="giac")

[Out]

Timed out