\(\int \frac {(A+B x) (b x+c x^2)^{5/2}}{(d+e x)^2} \, dx\) [1187]

Optimal result

Integrand size = 26, antiderivative size = 574 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^2} \, dx=-\frac {\left (10 A c e \left (64 c^3 d^3-112 b c^2 d^2 e+48 b^2 c d e^2-b^3 e^3\right )-B \left (768 c^4 d^4-1408 b c^3 d^3 e+656 b^2 c^2 d^2 e^2-20 b^3 c d e^3-3 b^4 e^4\right )-2 c e \left (8 b c e (6 B d-5 A e) (2 c d-b e)-(12 B c d-b B e-10 A c e) \left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{128 c^2 e^6}-\frac {\left (10 A c e (8 c d-7 b e)-B \left (96 c^2 d^2-92 b c d e+3 b^2 e^2\right )+6 c e (12 B c d-b B e-10 A c e) x\right ) \left (b x+c x^2\right )^{3/2}}{48 c e^4}+\frac {(6 B d-5 A e+B e x) \left (b x+c x^2\right )^{5/2}}{5 e^2 (d+e x)}+\frac {\left (10 A c e \left (128 c^4 d^4-256 b c^3 d^3 e+144 b^2 c^2 d^2 e^2-16 b^3 c d e^3-b^4 e^4\right )-B \left (1536 c^5 d^5-3200 b c^4 d^4 e+1920 b^2 c^3 d^3 e^2-240 b^3 c^2 d^2 e^3-20 b^4 c d e^4-3 b^5 e^5\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{128 c^{5/2} e^7}+\frac {d^{3/2} (c d-b e)^{3/2} (B d (12 c d-7 b e)-5 A e (2 c d-b e)) \text {arctanh}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{2 e^7} \]

[Out]

-1/48*(10*A*c*e*(-7*b*e+8*c*d)-B*(3*b^2*e^2-92*b*c*d*e+96*c^2*d^2)+6*c*e*(-10*A*c*e-B*b*e+12*B*c*d)*x)*(c*x^2+
b*x)^(3/2)/c/e^4+1/5*(B*e*x-5*A*e+6*B*d)*(c*x^2+b*x)^(5/2)/e^2/(e*x+d)+1/128*(10*A*c*e*(-b^4*e^4-16*b^3*c*d*e^
3+144*b^2*c^2*d^2*e^2-256*b*c^3*d^3*e+128*c^4*d^4)-B*(-3*b^5*e^5-20*b^4*c*d*e^4-240*b^3*c^2*d^2*e^3+1920*b^2*c
^3*d^3*e^2-3200*b*c^4*d^4*e+1536*c^5*d^5))*arctanh(x*c^(1/2)/(c*x^2+b*x)^(1/2))/c^(5/2)/e^7+1/2*d^(3/2)*(-b*e+
c*d)^(3/2)*(B*d*(-7*b*e+12*c*d)-5*A*e*(-b*e+2*c*d))*arctanh(1/2*(b*d+(-b*e+2*c*d)*x)/d^(1/2)/(-b*e+c*d)^(1/2)/
(c*x^2+b*x)^(1/2))/e^7-1/128*(10*A*c*e*(-b^3*e^3+48*b^2*c*d*e^2-112*b*c^2*d^2*e+64*c^3*d^3)-B*(-3*b^4*e^4-20*b
^3*c*d*e^3+656*b^2*c^2*d^2*e^2-1408*b*c^3*d^3*e+768*c^4*d^4)-2*c*e*(8*b*c*e*(-5*A*e+6*B*d)*(-b*e+2*c*d)-(-10*A
*c*e-B*b*e+12*B*c*d)*(-3*b^2*e^2-8*b*c*d*e+16*c^2*d^2))*x)*(c*x^2+b*x)^(1/2)/c^2/e^6

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 574, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {826, 828, 857, 634, 212, 738} \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (10 A c e \left (-b^4 e^4-16 b^3 c d e^3+144 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right )-B \left (-3 b^5 e^5-20 b^4 c d e^4-240 b^3 c^2 d^2 e^3+1920 b^2 c^3 d^3 e^2-3200 b c^4 d^4 e+1536 c^5 d^5\right )\right )}{128 c^{5/2} e^7}+\frac {d^{3/2} (c d-b e)^{3/2} (B d (12 c d-7 b e)-5 A e (2 c d-b e)) \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{2 e^7}-\frac {\left (b x+c x^2\right )^{3/2} \left (6 c e x (-10 A c e-b B e+12 B c d)+10 A c e (8 c d-7 b e)-B \left (3 b^2 e^2-92 b c d e+96 c^2 d^2\right )\right )}{48 c e^4}-\frac {\sqrt {b x+c x^2} \left (-2 c e x \left (8 b c e (6 B d-5 A e) (2 c d-b e)-\left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right ) (-10 A c e-b B e+12 B c d)\right )+10 A c e \left (-b^3 e^3+48 b^2 c d e^2-112 b c^2 d^2 e+64 c^3 d^3\right )-B \left (-3 b^4 e^4-20 b^3 c d e^3+656 b^2 c^2 d^2 e^2-1408 b c^3 d^3 e+768 c^4 d^4\right )\right )}{128 c^2 e^6}+\frac {\left (b x+c x^2\right )^{5/2} (-5 A e+6 B d+B e x)}{5 e^2 (d+e x)} \]

[In]

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

[Out]

-1/128*((10*A*c*e*(64*c^3*d^3 - 112*b*c^2*d^2*e + 48*b^2*c*d*e^2 - b^3*e^3) - B*(768*c^4*d^4 - 1408*b*c^3*d^3*
e + 656*b^2*c^2*d^2*e^2 - 20*b^3*c*d*e^3 - 3*b^4*e^4) - 2*c*e*(8*b*c*e*(6*B*d - 5*A*e)*(2*c*d - b*e) - (12*B*c
*d - b*B*e - 10*A*c*e)*(16*c^2*d^2 - 8*b*c*d*e - 3*b^2*e^2))*x)*Sqrt[b*x + c*x^2])/(c^2*e^6) - ((10*A*c*e*(8*c
*d - 7*b*e) - B*(96*c^2*d^2 - 92*b*c*d*e + 3*b^2*e^2) + 6*c*e*(12*B*c*d - b*B*e - 10*A*c*e)*x)*(b*x + c*x^2)^(
3/2))/(48*c*e^4) + ((6*B*d - 5*A*e + B*e*x)*(b*x + c*x^2)^(5/2))/(5*e^2*(d + e*x)) + ((10*A*c*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*(1536*c^5*d^5 - 3200*b*c^4*d^4*e + 1920
*b^2*c^3*d^3*e^2 - 240*b^3*c^2*d^2*e^3 - 20*b^4*c*d*e^4 - 3*b^5*e^5))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/
(128*c^(5/2)*e^7) + (d^(3/2)*(c*d - b*e)^(3/2)*(B*d*(12*c*d - 7*b*e) - 5*A*e*(2*c*d - b*e))*ArcTanh[(b*d + (2*
c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(2*e^7)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 634

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 738

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]

Rule 826

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 828

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 857

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]

Rubi steps \begin{align*} \text {integral}& = \frac {(6 B d-5 A e+B e x) \left (b x+c x^2\right )^{5/2}}{5 e^2 (d+e x)}-\frac {\int \frac {(b (6 B d-5 A e)+(12 B c d-b B e-10 A c e) x) \left (b x+c x^2\right )^{3/2}}{d+e x} \, dx}{2 e^2} \\ & = -\frac {\left (10 A c e (8 c d-7 b e)-B \left (96 c^2 d^2-92 b c d e+3 b^2 e^2\right )+6 c e (12 B c d-b B e-10 A c e) x\right ) \left (b x+c x^2\right )^{3/2}}{48 c e^4}+\frac {(6 B d-5 A e+B e x) \left (b x+c x^2\right )^{5/2}}{5 e^2 (d+e x)}+\frac {\int \frac {\left (\frac {1}{2} b d \left (10 A c e (8 c d-7 b e)-2 B \left (48 c^2 d^2-46 b c d e+\frac {3 b^2 e^2}{2}\right )\right )+\frac {1}{2} \left (8 b c e (6 B d-5 A e) (2 c d-b e)-2 (12 B c d-b B e-10 A c e) \left (8 c^2 d^2-4 b c d e-\frac {3 b^2 e^2}{2}\right )\right ) x\right ) \sqrt {b x+c x^2}}{d+e x} \, dx}{16 c e^4} \\ & = -\frac {\left (10 A c e \left (64 c^3 d^3-112 b c^2 d^2 e+48 b^2 c d e^2-b^3 e^3\right )-B \left (768 c^4 d^4-1408 b c^3 d^3 e+656 b^2 c^2 d^2 e^2-20 b^3 c d e^3-3 b^4 e^4\right )-2 c e \left (8 b c e (6 B d-5 A e) (2 c d-b e)-(12 B c d-b B e-10 A c e) \left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{128 c^2 e^6}-\frac {\left (10 A c e (8 c d-7 b e)-B \left (96 c^2 d^2-92 b c d e+3 b^2 e^2\right )+6 c e (12 B c d-b B e-10 A c e) x\right ) \left (b x+c x^2\right )^{3/2}}{48 c e^4}+\frac {(6 B d-5 A e+B e x) \left (b x+c x^2\right )^{5/2}}{5 e^2 (d+e x)}-\frac {\int \frac {-\frac {1}{4} b d \left (10 A c e \left (64 c^3 d^3-112 b c^2 d^2 e+48 b^2 c d e^2-b^3 e^3\right )-B \left (768 c^4 d^4-1408 b c^3 d^3 e+656 b^2 c^2 d^2 e^2-20 b^3 c d e^3-3 b^4 e^4\right )\right )-\frac {1}{4} \left (10 A c 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 \left (1536 c^5 d^5-3200 b c^4 d^4 e+1920 b^2 c^3 d^3 e^2-240 b^3 c^2 d^2 e^3-20 b^4 c d e^4-3 b^5 e^5\right )\right ) x}{(d+e x) \sqrt {b x+c x^2}} \, dx}{64 c^2 e^6} \\ & = -\frac {\left (10 A c e \left (64 c^3 d^3-112 b c^2 d^2 e+48 b^2 c d e^2-b^3 e^3\right )-B \left (768 c^4 d^4-1408 b c^3 d^3 e+656 b^2 c^2 d^2 e^2-20 b^3 c d e^3-3 b^4 e^4\right )-2 c e \left (8 b c e (6 B d-5 A e) (2 c d-b e)-(12 B c d-b B e-10 A c e) \left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{128 c^2 e^6}-\frac {\left (10 A c e (8 c d-7 b e)-B \left (96 c^2 d^2-92 b c d e+3 b^2 e^2\right )+6 c e (12 B c d-b B e-10 A c e) x\right ) \left (b x+c x^2\right )^{3/2}}{48 c e^4}+\frac {(6 B d-5 A e+B e x) \left (b x+c x^2\right )^{5/2}}{5 e^2 (d+e x)}+\frac {\left (d^2 (c d-b e)^2 (B d (12 c d-7 b e)-5 A e (2 c d-b e))\right ) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{2 e^7}+\frac {\left (10 A c 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 \left (1536 c^5 d^5-3200 b c^4 d^4 e+1920 b^2 c^3 d^3 e^2-240 b^3 c^2 d^2 e^3-20 b^4 c d e^4-3 b^5 e^5\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{256 c^2 e^7} \\ & = -\frac {\left (10 A c e \left (64 c^3 d^3-112 b c^2 d^2 e+48 b^2 c d e^2-b^3 e^3\right )-B \left (768 c^4 d^4-1408 b c^3 d^3 e+656 b^2 c^2 d^2 e^2-20 b^3 c d e^3-3 b^4 e^4\right )-2 c e \left (8 b c e (6 B d-5 A e) (2 c d-b e)-(12 B c d-b B e-10 A c e) \left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{128 c^2 e^6}-\frac {\left (10 A c e (8 c d-7 b e)-B \left (96 c^2 d^2-92 b c d e+3 b^2 e^2\right )+6 c e (12 B c d-b B e-10 A c e) x\right ) \left (b x+c x^2\right )^{3/2}}{48 c e^4}+\frac {(6 B d-5 A e+B e x) \left (b x+c x^2\right )^{5/2}}{5 e^2 (d+e x)}-\frac {\left (d^2 (c d-b e)^2 (B d (12 c d-7 b e)-5 A e (2 c d-b e))\right ) \text {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 )}{e^7}+\frac {\left (10 A c 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 \left (1536 c^5 d^5-3200 b c^4 d^4 e+1920 b^2 c^3 d^3 e^2-240 b^3 c^2 d^2 e^3-20 b^4 c d e^4-3 b^5 e^5\right )\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{128 c^2 e^7} \\ & = -\frac {\left (10 A c e \left (64 c^3 d^3-112 b c^2 d^2 e+48 b^2 c d e^2-b^3 e^3\right )-B \left (768 c^4 d^4-1408 b c^3 d^3 e+656 b^2 c^2 d^2 e^2-20 b^3 c d e^3-3 b^4 e^4\right )-2 c e \left (8 b c e (6 B d-5 A e) (2 c d-b e)-(12 B c d-b B e-10 A c e) \left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{128 c^2 e^6}-\frac {\left (10 A c e (8 c d-7 b e)-B \left (96 c^2 d^2-92 b c d e+3 b^2 e^2\right )+6 c e (12 B c d-b B e-10 A c e) x\right ) \left (b x+c x^2\right )^{3/2}}{48 c e^4}+\frac {(6 B d-5 A e+B e x) \left (b x+c x^2\right )^{5/2}}{5 e^2 (d+e x)}+\frac {\left (10 A c 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 \left (1536 c^5 d^5-3200 b c^4 d^4 e+1920 b^2 c^3 d^3 e^2-240 b^3 c^2 d^2 e^3-20 b^4 c d e^4-3 b^5 e^5\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{128 c^{5/2} e^7}+\frac {d^{3/2} (c d-b e)^{3/2} (B d (12 c d-7 b e)-5 A e (2 c d-b e)) \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 )}{2 e^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 13.39 (sec) , antiderivative size = 855, normalized size of antiderivative = 1.49 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {(x (b+c x))^{5/2} \left (\frac {(-B d+A e) x^{7/2} (b+c x)}{d+e x}+\frac {e (7 b B d-2 A c d-5 A b e) \left (15 \left (-256 c^5 d^5+640 b c^4 d^4 e-480 b^2 c^3 d^3 e^2+80 b^3 c^2 d^2 e^3+10 b^4 c d e^4+3 b^5 e^5\right ) (b+c x) \text {arcsinh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )+\sqrt {b} \sqrt {c} \sqrt {1+\frac {c x}{b}} \left (e \sqrt {x} (b+c x) \left (-45 b^4 e^4+30 b^3 c e^3 (-5 d+e x)+4 b^2 c^2 e^2 \left (660 d^2-295 d e x+186 e^2 x^2\right )+16 b c^3 e \left (-270 d^3+130 d^2 e x-85 d e^2 x^2+63 e^3 x^3\right )+32 c^4 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )+3840 c^2 d^{5/2} (c d-b e)^{5/2} \sqrt {b+c x} \text {arctanh}\left (\frac {\sqrt {c d-b e} \sqrt {x}}{\sqrt {d} \sqrt {b+c x}}\right )\right )\right )+3 (B d-A e) \left (-15 \left (-1024 c^6 d^6+2560 b c^5 d^5 e-1920 b^2 c^4 d^4 e^2+320 b^3 c^3 d^3 e^3+40 b^4 c^2 d^2 e^4+12 b^5 c d e^5+5 b^6 e^6\right ) (b+c x) \text {arcsinh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )+\sqrt {b} \sqrt {c} \sqrt {1+\frac {c x}{b}} \left (e \sqrt {x} (b+c x) \left (75 b^5 e^5+10 b^4 c e^4 (18 d-5 e x)+40 b^3 c^2 e^3 \left (15 d^2-3 d e x+e^2 x^2\right )+16 b^2 c^3 e^2 \left (-660 d^3+295 d^2 e x-186 d e^2 x^2+135 e^3 x^3\right )+64 b c^4 e \left (270 d^4-130 d^3 e x+85 d^2 e^2 x^2-63 d e^3 x^3+50 e^4 x^4\right )-128 c^5 \left (60 d^5-30 d^4 e x+20 d^3 e^2 x^2-15 d^2 e^3 x^3+12 d e^4 x^4-10 e^5 x^5\right )\right )-15360 c^3 d^{7/2} (c d-b e)^{5/2} \sqrt {b+c x} \text {arctanh}\left (\frac {\sqrt {c d-b e} \sqrt {x}}{\sqrt {d} \sqrt {b+c x}}\right )\right )\right )}{3840 \sqrt {b} c^{5/2} e^7 (b+c x)^3 \sqrt {1+\frac {c x}{b}}}\right )}{d (-c d+b e) x^{5/2}} \]

[In]

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

[Out]

((x*(b + c*x))^(5/2)*(((-(B*d) + A*e)*x^(7/2)*(b + c*x))/(d + e*x) + (e*(7*b*B*d - 2*A*c*d - 5*A*b*e)*(15*(-25
6*c^5*d^5 + 640*b*c^4*d^4*e - 480*b^2*c^3*d^3*e^2 + 80*b^3*c^2*d^2*e^3 + 10*b^4*c*d*e^4 + 3*b^5*e^5)*(b + c*x)
*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]] + Sqrt[b]*Sqrt[c]*Sqrt[1 + (c*x)/b]*(e*Sqrt[x]*(b + c*x)*(-45*b^4*e^4 + 30
*b^3*c*e^3*(-5*d + e*x) + 4*b^2*c^2*e^2*(660*d^2 - 295*d*e*x + 186*e^2*x^2) + 16*b*c^3*e*(-270*d^3 + 130*d^2*e
*x - 85*d*e^2*x^2 + 63*e^3*x^3) + 32*c^4*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4)) +
3840*c^2*d^(5/2)*(c*d - b*e)^(5/2)*Sqrt[b + c*x]*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])]))
+ 3*(B*d - A*e)*(-15*(-1024*c^6*d^6 + 2560*b*c^5*d^5*e - 1920*b^2*c^4*d^4*e^2 + 320*b^3*c^3*d^3*e^3 + 40*b^4*
c^2*d^2*e^4 + 12*b^5*c*d*e^5 + 5*b^6*e^6)*(b + c*x)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]] + Sqrt[b]*Sqrt[c]*Sqrt[
1 + (c*x)/b]*(e*Sqrt[x]*(b + c*x)*(75*b^5*e^5 + 10*b^4*c*e^4*(18*d - 5*e*x) + 40*b^3*c^2*e^3*(15*d^2 - 3*d*e*x
+ e^2*x^2) + 16*b^2*c^3*e^2*(-660*d^3 + 295*d^2*e*x - 186*d*e^2*x^2 + 135*e^3*x^3) + 64*b*c^4*e*(270*d^4 - 13
0*d^3*e*x + 85*d^2*e^2*x^2 - 63*d*e^3*x^3 + 50*e^4*x^4) - 128*c^5*(60*d^5 - 30*d^4*e*x + 20*d^3*e^2*x^2 - 15*d
^2*e^3*x^3 + 12*d*e^4*x^4 - 10*e^5*x^5)) - 15360*c^3*d^(7/2)*(c*d - b*e)^(5/2)*Sqrt[b + c*x]*ArcTanh[(Sqrt[c*d
- b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])))/(3840*Sqrt[b]*c^(5/2)*e^7*(b + c*x)^3*Sqrt[1 + (c*x)/b])))/(d*(-(
c*d) + b*e)*x^(5/2))

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 1051, normalized size of antiderivative = 1.83

method	result	size
risch	\(\text {Expression too large to display}\)	\(1051\)
default	\(\text {Expression too large to display}\)	\(2390\)

[In]

int((B*x+A)*(c*x^2+b*x)^(5/2)/(e*x+d)^2,x,method=_RETURNVERBOSE)

[Out]

1/1920/c^2*(384*B*c^4*e^4*x^4+480*A*c^4*e^4*x^3+1008*B*b*c^3*e^4*x^3-960*B*c^4*d*e^3*x^3+1360*A*b*c^3*e^4*x^2-
1280*A*c^4*d*e^3*x^2+744*B*b^2*c^2*e^4*x^2-2720*B*b*c^3*d*e^3*x^2+1920*B*c^4*d^2*e^2*x^2+1180*A*b^2*c^2*e^4*x-
4160*A*b*c^3*d*e^3*x+2880*A*c^4*d^2*e^2*x+30*B*b^3*c*e^4*x-2360*B*b^2*c^2*d*e^3*x+6240*B*b*c^3*d^2*e^2*x-3840*
B*c^4*d^3*e*x+150*A*b^3*c*e^4-5280*A*b^2*c^2*d*e^3+12960*A*b*c^3*d^2*e^2-7680*A*c^4*d^3*e-45*B*b^4*e^4-300*B*b
^3*c*d*e^3+7920*B*b^2*c^2*d^2*e^2-17280*B*b*c^3*d^3*e+9600*B*c^4*d^4)*x*(c*x+b)/e^6/(x*(c*x+b))^(1/2)-1/256/e^
6/c^2*(256*c^2*d^3*(A*b^3*e^4-3*A*b^2*c*d*e^3+3*A*b*c^2*d^2*e^2-A*c^3*d^3*e-B*b^3*d*e^3+3*B*b^2*c*d^2*e^2-3*B*
b*c^2*d^3*e+B*c^3*d^4)/e^3*(1/d/(b*e-c*d)*e^2/(x+d/e)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2
)-1/2*(b*e-2*c*d)*e/d/(b*e-c*d)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b
*e-c*d)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e)))+256*c^2*d^2/e^2*(3*A*b
^3*e^4-12*A*b^2*c*d*e^3+15*A*b*c^2*d^2*e^2-6*A*c^3*d^3*e-4*B*b^3*d*e^3+15*B*b^2*c*d^2*e^2-18*B*b*c^2*d^3*e+7*B
*c^3*d^4)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x
+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))+(10*A*b^4*c*e^5+160*A*b^3*c^2*d*e^4-1440*A*b^
2*c^3*d^2*e^3+2560*A*b*c^4*d^3*e^2-1280*A*c^5*d^4*e-3*B*b^5*e^5-20*B*b^4*c*d*e^4-240*B*b^3*c^2*d^2*e^3+1920*B*
b^2*c^3*d^3*e^2-3200*B*b*c^4*d^4*e+1536*B*c^5*d^5)/e*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))/c^(1/2))

Fricas [A] (verification not implemented)

none

Time = 45.17 (sec) , antiderivative size = 3709, normalized size of antiderivative = 6.46 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/3840*(15*(1536*B*c^5*d^6 - 640*(5*B*b*c^4 + 2*A*c^5)*d^5*e + 640*(3*B*b^2*c^3 + 4*A*b*c^4)*d^4*e^2 - 240*(B
*b^3*c^2 + 6*A*b^2*c^3)*d^3*e^3 - 20*(B*b^4*c - 8*A*b^3*c^2)*d^2*e^4 - (3*B*b^5 - 10*A*b^4*c)*d*e^5 + (1536*B*
c^5*d^5*e - 640*(5*B*b*c^4 + 2*A*c^5)*d^4*e^2 + 640*(3*B*b^2*c^3 + 4*A*b*c^4)*d^3*e^3 - 240*(B*b^3*c^2 + 6*A*b
^2*c^3)*d^2*e^4 - 20*(B*b^4*c - 8*A*b^3*c^2)*d*e^5 - (3*B*b^5 - 10*A*b^4*c)*e^6)*x)*sqrt(c)*log(2*c*x + b - 2*
sqrt(c*x^2 + b*x)*sqrt(c)) - 1920*(12*B*c^5*d^5 - 5*A*b^2*c^3*d^2*e^3 - (19*B*b*c^4 + 10*A*c^5)*d^4*e + (7*B*b
^2*c^3 + 15*A*b*c^4)*d^3*e^2 + (12*B*c^5*d^4*e - 5*A*b^2*c^3*d*e^4 - (19*B*b*c^4 + 10*A*c^5)*d^3*e^2 + (7*B*b^
2*c^3 + 15*A*b*c^4)*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*(384*B*c^5*e^6*x^5 + 11520*B*c^5*d^5*e - 1920*(11*B*b*c^4 + 5*A*c^5)*d^4*e^2 + 240*
(41*B*b^2*c^3 + 70*A*b*c^4)*d^3*e^3 - 300*(B*b^3*c^2 + 24*A*b^2*c^3)*d^2*e^4 - 15*(3*B*b^4*c - 10*A*b^3*c^2)*d
*e^5 - 48*(12*B*c^5*d*e^5 - (21*B*b*c^4 + 10*A*c^5)*e^6)*x^4 + 8*(120*B*c^5*d^2*e^4 - 2*(107*B*b*c^4 + 50*A*c^
5)*d*e^5 + (93*B*b^2*c^3 + 170*A*b*c^4)*e^6)*x^3 - 2*(960*B*c^5*d^3*e^3 - 160*(11*B*b*c^4 + 5*A*c^5)*d^2*e^4 +
8*(101*B*b^2*c^3 + 175*A*b*c^4)*d*e^5 - 5*(3*B*b^3*c^2 + 118*A*b^2*c^3)*e^6)*x^2 + 5*(1152*B*c^5*d^4*e^2 - 96
*(23*B*b*c^4 + 10*A*c^5)*d^3*e^3 + 8*(139*B*b^2*c^3 + 220*A*b*c^4)*d^2*e^4 - 2*(27*B*b^3*c^2 + 410*A*b^2*c^3)*
d*e^5 - 3*(3*B*b^4*c - 10*A*b^3*c^2)*e^6)*x)*sqrt(c*x^2 + b*x))/(c^3*e^8*x + c^3*d*e^7), 1/3840*(3840*(12*B*c^
5*d^5 - 5*A*b^2*c^3*d^2*e^3 - (19*B*b*c^4 + 10*A*c^5)*d^4*e + (7*B*b^2*c^3 + 15*A*b*c^4)*d^3*e^2 + (12*B*c^5*d
^4*e - 5*A*b^2*c^3*d*e^4 - (19*B*b*c^4 + 10*A*c^5)*d^3*e^2 + (7*B*b^2*c^3 + 15*A*b*c^4)*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*(1536*B*c^5*d^6 - 640*(5*B*b*c
^4 + 2*A*c^5)*d^5*e + 640*(3*B*b^2*c^3 + 4*A*b*c^4)*d^4*e^2 - 240*(B*b^3*c^2 + 6*A*b^2*c^3)*d^3*e^3 - 20*(B*b^
4*c - 8*A*b^3*c^2)*d^2*e^4 - (3*B*b^5 - 10*A*b^4*c)*d*e^5 + (1536*B*c^5*d^5*e - 640*(5*B*b*c^4 + 2*A*c^5)*d^4*
e^2 + 640*(3*B*b^2*c^3 + 4*A*b*c^4)*d^3*e^3 - 240*(B*b^3*c^2 + 6*A*b^2*c^3)*d^2*e^4 - 20*(B*b^4*c - 8*A*b^3*c^
2)*d*e^5 - (3*B*b^5 - 10*A*b^4*c)*e^6)*x)*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x^2 + b*x)*sqrt(c)) + 2*(384*B*c^5*
e^6*x^5 + 11520*B*c^5*d^5*e - 1920*(11*B*b*c^4 + 5*A*c^5)*d^4*e^2 + 240*(41*B*b^2*c^3 + 70*A*b*c^4)*d^3*e^3 -
300*(B*b^3*c^2 + 24*A*b^2*c^3)*d^2*e^4 - 15*(3*B*b^4*c - 10*A*b^3*c^2)*d*e^5 - 48*(12*B*c^5*d*e^5 - (21*B*b*c^
4 + 10*A*c^5)*e^6)*x^4 + 8*(120*B*c^5*d^2*e^4 - 2*(107*B*b*c^4 + 50*A*c^5)*d*e^5 + (93*B*b^2*c^3 + 170*A*b*c^4
)*e^6)*x^3 - 2*(960*B*c^5*d^3*e^3 - 160*(11*B*b*c^4 + 5*A*c^5)*d^2*e^4 + 8*(101*B*b^2*c^3 + 175*A*b*c^4)*d*e^5
- 5*(3*B*b^3*c^2 + 118*A*b^2*c^3)*e^6)*x^2 + 5*(1152*B*c^5*d^4*e^2 - 96*(23*B*b*c^4 + 10*A*c^5)*d^3*e^3 + 8*(
139*B*b^2*c^3 + 220*A*b*c^4)*d^2*e^4 - 2*(27*B*b^3*c^2 + 410*A*b^2*c^3)*d*e^5 - 3*(3*B*b^4*c - 10*A*b^3*c^2)*e
^6)*x)*sqrt(c*x^2 + b*x))/(c^3*e^8*x + c^3*d*e^7), 1/1920*(15*(1536*B*c^5*d^6 - 640*(5*B*b*c^4 + 2*A*c^5)*d^5*
e + 640*(3*B*b^2*c^3 + 4*A*b*c^4)*d^4*e^2 - 240*(B*b^3*c^2 + 6*A*b^2*c^3)*d^3*e^3 - 20*(B*b^4*c - 8*A*b^3*c^2)
*d^2*e^4 - (3*B*b^5 - 10*A*b^4*c)*d*e^5 + (1536*B*c^5*d^5*e - 640*(5*B*b*c^4 + 2*A*c^5)*d^4*e^2 + 640*(3*B*b^2
*c^3 + 4*A*b*c^4)*d^3*e^3 - 240*(B*b^3*c^2 + 6*A*b^2*c^3)*d^2*e^4 - 20*(B*b^4*c - 8*A*b^3*c^2)*d*e^5 - (3*B*b^
5 - 10*A*b^4*c)*e^6)*x)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - 960*(12*B*c^5*d^5 - 5*A*b^2*c^3*d^
2*e^3 - (19*B*b*c^4 + 10*A*c^5)*d^4*e + (7*B*b^2*c^3 + 15*A*b*c^4)*d^3*e^2 + (12*B*c^5*d^4*e - 5*A*b^2*c^3*d*e
^4 - (19*B*b*c^4 + 10*A*c^5)*d^3*e^2 + (7*B*b^2*c^3 + 15*A*b*c^4)*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)) + (384*B*c^5*e^6*x^5 + 11520*B*c^5*d^5*e
- 1920*(11*B*b*c^4 + 5*A*c^5)*d^4*e^2 + 240*(41*B*b^2*c^3 + 70*A*b*c^4)*d^3*e^3 - 300*(B*b^3*c^2 + 24*A*b^2*c^
3)*d^2*e^4 - 15*(3*B*b^4*c - 10*A*b^3*c^2)*d*e^5 - 48*(12*B*c^5*d*e^5 - (21*B*b*c^4 + 10*A*c^5)*e^6)*x^4 + 8*(
120*B*c^5*d^2*e^4 - 2*(107*B*b*c^4 + 50*A*c^5)*d*e^5 + (93*B*b^2*c^3 + 170*A*b*c^4)*e^6)*x^3 - 2*(960*B*c^5*d^
3*e^3 - 160*(11*B*b*c^4 + 5*A*c^5)*d^2*e^4 + 8*(101*B*b^2*c^3 + 175*A*b*c^4)*d*e^5 - 5*(3*B*b^3*c^2 + 118*A*b^
2*c^3)*e^6)*x^2 + 5*(1152*B*c^5*d^4*e^2 - 96*(23*B*b*c^4 + 10*A*c^5)*d^3*e^3 + 8*(139*B*b^2*c^3 + 220*A*b*c^4)
*d^2*e^4 - 2*(27*B*b^3*c^2 + 410*A*b^2*c^3)*d*e^5 - 3*(3*B*b^4*c - 10*A*b^3*c^2)*e^6)*x)*sqrt(c*x^2 + b*x))/(c
^3*e^8*x + c^3*d*e^7), 1/1920*(1920*(12*B*c^5*d^5 - 5*A*b^2*c^3*d^2*e^3 - (19*B*b*c^4 + 10*A*c^5)*d^4*e + (7*B
*b^2*c^3 + 15*A*b*c^4)*d^3*e^2 + (12*B*c^5*d^4*e - 5*A*b^2*c^3*d*e^4 - (19*B*b*c^4 + 10*A*c^5)*d^3*e^2 + (7*B*
b^2*c^3 + 15*A*b*c^4)*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*(1536*B*c^5*d^6 - 640*(5*B*b*c^4 + 2*A*c^5)*d^5*e + 640*(3*B*b^2*c^3 + 4*A*b*c^4)*d^4*e^2 - 240*
(B*b^3*c^2 + 6*A*b^2*c^3)*d^3*e^3 - 20*(B*b^4*c - 8*A*b^3*c^2)*d^2*e^4 - (3*B*b^5 - 10*A*b^4*c)*d*e^5 + (1536*
B*c^5*d^5*e - 640*(5*B*b*c^4 + 2*A*c^5)*d^4*e^2 + 640*(3*B*b^2*c^3 + 4*A*b*c^4)*d^3*e^3 - 240*(B*b^3*c^2 + 6*A
*b^2*c^3)*d^2*e^4 - 20*(B*b^4*c - 8*A*b^3*c^2)*d*e^5 - (3*B*b^5 - 10*A*b^4*c)*e^6)*x)*sqrt(-c)*arctan(sqrt(c*x
^2 + b*x)*sqrt(-c)/(c*x)) + (384*B*c^5*e^6*x^5 + 11520*B*c^5*d^5*e - 1920*(11*B*b*c^4 + 5*A*c^5)*d^4*e^2 + 240
*(41*B*b^2*c^3 + 70*A*b*c^4)*d^3*e^3 - 300*(B*b^3*c^2 + 24*A*b^2*c^3)*d^2*e^4 - 15*(3*B*b^4*c - 10*A*b^3*c^2)*
d*e^5 - 48*(12*B*c^5*d*e^5 - (21*B*b*c^4 + 10*A*c^5)*e^6)*x^4 + 8*(120*B*c^5*d^2*e^4 - 2*(107*B*b*c^4 + 50*A*c
^5)*d*e^5 + (93*B*b^2*c^3 + 170*A*b*c^4)*e^6)*x^3 - 2*(960*B*c^5*d^3*e^3 - 160*(11*B*b*c^4 + 5*A*c^5)*d^2*e^4
+ 8*(101*B*b^2*c^3 + 175*A*b*c^4)*d*e^5 - 5*(3*B*b^3*c^2 + 118*A*b^2*c^3)*e^6)*x^2 + 5*(1152*B*c^5*d^4*e^2 - 9
6*(23*B*b*c^4 + 10*A*c^5)*d^3*e^3 + 8*(139*B*b^2*c^3 + 220*A*b*c^4)*d^2*e^4 - 2*(27*B*b^3*c^2 + 410*A*b^2*c^3)
*d*e^5 - 3*(3*B*b^4*c - 10*A*b^3*c^2)*e^6)*x)*sqrt(c*x^2 + b*x))/(c^3*e^8*x + c^3*d*e^7)]

Sympy [F]

\[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^2} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (A + B x\right )}{\left (d + e x\right )^{2}}\, dx \]

[In]

integrate((B*x+A)*(c*x**2+b*x)**(5/2)/(e*x+d)**2,x)

[Out]

Integral((x*(b + c*x))**(5/2)*(A + B*x)/(d + e*x)**2, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^2} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for
more detail

Giac [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^2} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{5/2}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^2} \,d x \]

[In]

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

[Out]

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