3.1186 \(\int \frac{(A+B x) (b x+c x^2)^{5/2}}{d+e x} \, dx\)

Optimal. Leaf size=703 \[ \frac{\left (b x+c x^2\right )^{3/2} \left (-2 c e x \left (12 A c e (2 c d-b e)-B \left (-5 b^2 e^2-12 b c d e+24 c^2 d^2\right )\right )+4 A c e \left (3 b^2 e^2-22 b c d e+16 c^2 d^2\right )-B \left (12 b^2 c d e^2+5 b^3 e^3-88 b c^2 d^2 e+64 c^3 d^3\right )\right )}{192 c^2 e^4}+\frac{\sqrt{b x+c x^2} \left (3 \left (4 A c e \left (176 b^2 c^2 d^2 e^2-10 b^3 c d e^3-3 b^4 e^4-288 b c^3 d^3 e+128 c^4 d^4\right )-B \left (704 b^2 c^3 d^3 e^2-40 b^3 c^2 d^2 e^3-12 b^4 c d e^4-5 b^5 e^5-1152 b c^4 d^4 e+512 c^5 d^5\right )\right )-2 c e x \left (\left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right ) \left (12 A c e (2 c d-b e)-B \left (-5 b^2 e^2-12 b c d e+24 c^2 d^2\right )\right )+8 b c d e (2 c d-b e) (-12 A c e-5 b B e+12 B c d)\right )\right )}{1536 c^3 e^6}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (4 A c e \left (480 b^2 c^3 d^3 e^2-80 b^3 c^2 d^2 e^3-10 b^4 c d e^4-3 b^5 e^5-640 b c^4 d^4 e+256 c^5 d^5\right )-B \left (1920 b^2 c^4 d^4 e^2-320 b^3 c^3 d^3 e^3-40 b^4 c^2 d^2 e^4-12 b^5 c d e^5-5 b^6 e^6-2560 b c^5 d^5 e+1024 c^6 d^6\right )\right )}{512 c^{7/2} e^7}-\frac{d^{5/2} (B d-A e) (c d-b e)^{5/2} \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{e^7}-\frac{\left (b x+c x^2\right )^{5/2} (-12 A c e-5 b B e+12 B c d-10 B c e x)}{60 c e^2} \]

[Out]

((3*(4*A*c*e*(128*c^4*d^4 - 288*b*c^3*d^3*e + 176*b^2*c^2*d^2*e^2 - 10*b^3*c*d*e^3 - 3*b^4*e^4) - B*(512*c^5*d
^5 - 1152*b*c^4*d^4*e + 704*b^2*c^3*d^3*e^2 - 40*b^3*c^2*d^2*e^3 - 12*b^4*c*d*e^4 - 5*b^5*e^5)) - 2*c*e*(8*b*c
*d*e*(2*c*d - b*e)*(12*B*c*d - 5*b*B*e - 12*A*c*e) + (16*c^2*d^2 - 8*b*c*d*e - 3*b^2*e^2)*(12*A*c*e*(2*c*d - b
*e) - B*(24*c^2*d^2 - 12*b*c*d*e - 5*b^2*e^2)))*x)*Sqrt[b*x + c*x^2])/(1536*c^3*e^6) + ((4*A*c*e*(16*c^2*d^2 -
22*b*c*d*e + 3*b^2*e^2) - B*(64*c^3*d^3 - 88*b*c^2*d^2*e + 12*b^2*c*d*e^2 + 5*b^3*e^3) - 2*c*e*(12*A*c*e*(2*c
*d - b*e) - B*(24*c^2*d^2 - 12*b*c*d*e - 5*b^2*e^2))*x)*(b*x + c*x^2)^(3/2))/(192*c^2*e^4) - ((12*B*c*d - 5*b*
B*e - 12*A*c*e - 10*B*c*e*x)*(b*x + c*x^2)^(5/2))/(60*c*e^2) - ((4*A*c*e*(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) - B*(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))*ArcTanh[(Sqrt[c]*x)
/Sqrt[b*x + c*x^2]])/(512*c^(7/2)*e^7) - (d^(5/2)*(B*d - A*e)*(c*d - b*e)^(5/2)*ArcTanh[(b*d + (2*c*d - b*e)*x
)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/e^7

________________________________________________________________________________________

Rubi [A]  time = 1.12496, antiderivative size = 703, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {814, 843, 620, 206, 724} \[ \frac{\left (b x+c x^2\right )^{3/2} \left (-2 c e x \left (12 A c e (2 c d-b e)-B \left (-5 b^2 e^2-12 b c d e+24 c^2 d^2\right )\right )+4 A c e \left (3 b^2 e^2-22 b c d e+16 c^2 d^2\right )-B \left (12 b^2 c d e^2+5 b^3 e^3-88 b c^2 d^2 e+64 c^3 d^3\right )\right )}{192 c^2 e^4}+\frac{\sqrt{b x+c x^2} \left (3 \left (4 A c e \left (176 b^2 c^2 d^2 e^2-10 b^3 c d e^3-3 b^4 e^4-288 b c^3 d^3 e+128 c^4 d^4\right )-B \left (704 b^2 c^3 d^3 e^2-40 b^3 c^2 d^2 e^3-12 b^4 c d e^4-5 b^5 e^5-1152 b c^4 d^4 e+512 c^5 d^5\right )\right )-2 c e x \left (\left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right ) \left (12 A c e (2 c d-b e)-B \left (-5 b^2 e^2-12 b c d e+24 c^2 d^2\right )\right )+8 b c d e (2 c d-b e) (-12 A c e-5 b B e+12 B c d)\right )\right )}{1536 c^3 e^6}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (4 A c e \left (480 b^2 c^3 d^3 e^2-80 b^3 c^2 d^2 e^3-10 b^4 c d e^4-3 b^5 e^5-640 b c^4 d^4 e+256 c^5 d^5\right )-B \left (1920 b^2 c^4 d^4 e^2-320 b^3 c^3 d^3 e^3-40 b^4 c^2 d^2 e^4-12 b^5 c d e^5-5 b^6 e^6-2560 b c^5 d^5 e+1024 c^6 d^6\right )\right )}{512 c^{7/2} e^7}-\frac{d^{5/2} (B d-A e) (c d-b e)^{5/2} \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{e^7}-\frac{\left (b x+c x^2\right )^{5/2} (-12 A c e-5 b B e+12 B c d-10 B c e x)}{60 c e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*(4*A*c*e*(128*c^4*d^4 - 288*b*c^3*d^3*e + 176*b^2*c^2*d^2*e^2 - 10*b^3*c*d*e^3 - 3*b^4*e^4) - B*(512*c^5*d
^5 - 1152*b*c^4*d^4*e + 704*b^2*c^3*d^3*e^2 - 40*b^3*c^2*d^2*e^3 - 12*b^4*c*d*e^4 - 5*b^5*e^5)) - 2*c*e*(8*b*c
*d*e*(2*c*d - b*e)*(12*B*c*d - 5*b*B*e - 12*A*c*e) + (16*c^2*d^2 - 8*b*c*d*e - 3*b^2*e^2)*(12*A*c*e*(2*c*d - b
*e) - B*(24*c^2*d^2 - 12*b*c*d*e - 5*b^2*e^2)))*x)*Sqrt[b*x + c*x^2])/(1536*c^3*e^6) + ((4*A*c*e*(16*c^2*d^2 -
22*b*c*d*e + 3*b^2*e^2) - B*(64*c^3*d^3 - 88*b*c^2*d^2*e + 12*b^2*c*d*e^2 + 5*b^3*e^3) - 2*c*e*(12*A*c*e*(2*c
*d - b*e) - B*(24*c^2*d^2 - 12*b*c*d*e - 5*b^2*e^2))*x)*(b*x + c*x^2)^(3/2))/(192*c^2*e^4) - ((12*B*c*d - 5*b*
B*e - 12*A*c*e - 10*B*c*e*x)*(b*x + c*x^2)^(5/2))/(60*c*e^2) - ((4*A*c*e*(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) - B*(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))*ArcTanh[(Sqrt[c]*x)
/Sqrt[b*x + c*x^2]])/(512*c^(7/2)*e^7) - (d^(5/2)*(B*d - A*e)*(c*d - b*e)^(5/2)*ArcTanh[(b*d + (2*c*d - b*e)*x
)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/e^7

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} \, dx &=-\frac{(12 B c d-5 b B e-12 A c e-10 B c e x) \left (b x+c x^2\right )^{5/2}}{60 c e^2}-\frac{\int \frac{\left (-\frac{1}{2} b d (12 B c d-5 b B e-12 A c e)+\frac{1}{2} \left (12 A c e (2 c d-b e)-B \left (24 c^2 d^2-12 b c d e-5 b^2 e^2\right )\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{d+e x} \, dx}{12 c e^2}\\ &=\frac{\left (4 A c e \left (16 c^2 d^2-22 b c d e+3 b^2 e^2\right )-B \left (64 c^3 d^3-88 b c^2 d^2 e+12 b^2 c d e^2+5 b^3 e^3\right )-2 c e \left (12 A c e (2 c d-b e)-B \left (24 c^2 d^2-12 b c d e-5 b^2 e^2\right )\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{192 c^2 e^4}-\frac{(12 B c d-5 b B e-12 A c e-10 B c e x) \left (b x+c x^2\right )^{5/2}}{60 c e^2}+\frac{\int \frac{\left (-\frac{3}{4} b d \left (4 A c e \left (16 c^2 d^2-22 b c d e+3 b^2 e^2\right )-B \left (64 c^3 d^3-88 b c^2 d^2 e+12 b^2 c d e^2+5 b^3 e^3\right )\right )-\frac{1}{4} \left (8 b c d e (2 c d-b e) (12 B c d-5 b B e-12 A c e)+\left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right ) \left (12 A c e (2 c d-b e)-B \left (24 c^2 d^2-12 b c d e-5 b^2 e^2\right )\right )\right ) x\right ) \sqrt{b x+c x^2}}{d+e x} \, dx}{96 c^2 e^4}\\ &=\frac{\left (3 \left (4 A c e \left (128 c^4 d^4-288 b c^3 d^3 e+176 b^2 c^2 d^2 e^2-10 b^3 c d e^3-3 b^4 e^4\right )-B \left (512 c^5 d^5-1152 b c^4 d^4 e+704 b^2 c^3 d^3 e^2-40 b^3 c^2 d^2 e^3-12 b^4 c d e^4-5 b^5 e^5\right )\right )-2 c e \left (8 b c d e (2 c d-b e) (12 B c d-5 b B e-12 A c e)+\left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right ) \left (12 A c e (2 c d-b e)-B \left (24 c^2 d^2-12 b c d e-5 b^2 e^2\right )\right )\right ) x\right ) \sqrt{b x+c x^2}}{1536 c^3 e^6}+\frac{\left (4 A c e \left (16 c^2 d^2-22 b c d e+3 b^2 e^2\right )-B \left (64 c^3 d^3-88 b c^2 d^2 e+12 b^2 c d e^2+5 b^3 e^3\right )-2 c e \left (12 A c e (2 c d-b e)-B \left (24 c^2 d^2-12 b c d e-5 b^2 e^2\right )\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{192 c^2 e^4}-\frac{(12 B c d-5 b B e-12 A c e-10 B c e x) \left (b x+c x^2\right )^{5/2}}{60 c e^2}-\frac{\int \frac{\frac{3}{8} b d \left (4 A c e \left (128 c^4 d^4-288 b c^3 d^3 e+176 b^2 c^2 d^2 e^2-10 b^3 c d e^3-3 b^4 e^4\right )-B \left (512 c^5 d^5-1152 b c^4 d^4 e+704 b^2 c^3 d^3 e^2-40 b^3 c^2 d^2 e^3-12 b^4 c d e^4-5 b^5 e^5\right )\right )+\frac{3}{8} \left (4 A c e \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 \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 )\right ) x}{(d+e x) \sqrt{b x+c x^2}} \, dx}{384 c^3 e^6}\\ &=\frac{\left (3 \left (4 A c e \left (128 c^4 d^4-288 b c^3 d^3 e+176 b^2 c^2 d^2 e^2-10 b^3 c d e^3-3 b^4 e^4\right )-B \left (512 c^5 d^5-1152 b c^4 d^4 e+704 b^2 c^3 d^3 e^2-40 b^3 c^2 d^2 e^3-12 b^4 c d e^4-5 b^5 e^5\right )\right )-2 c e \left (8 b c d e (2 c d-b e) (12 B c d-5 b B e-12 A c e)+\left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right ) \left (12 A c e (2 c d-b e)-B \left (24 c^2 d^2-12 b c d e-5 b^2 e^2\right )\right )\right ) x\right ) \sqrt{b x+c x^2}}{1536 c^3 e^6}+\frac{\left (4 A c e \left (16 c^2 d^2-22 b c d e+3 b^2 e^2\right )-B \left (64 c^3 d^3-88 b c^2 d^2 e+12 b^2 c d e^2+5 b^3 e^3\right )-2 c e \left (12 A c e (2 c d-b e)-B \left (24 c^2 d^2-12 b c d e-5 b^2 e^2\right )\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{192 c^2 e^4}-\frac{(12 B c d-5 b B e-12 A c e-10 B c e x) \left (b x+c x^2\right )^{5/2}}{60 c e^2}-\frac{\left (d^3 (B d-A e) (c d-b e)^3\right ) \int \frac{1}{(d+e x) \sqrt{b x+c x^2}} \, dx}{e^7}-\frac{\left (4 A c e \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 \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 )\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{1024 c^3 e^7}\\ &=\frac{\left (3 \left (4 A c e \left (128 c^4 d^4-288 b c^3 d^3 e+176 b^2 c^2 d^2 e^2-10 b^3 c d e^3-3 b^4 e^4\right )-B \left (512 c^5 d^5-1152 b c^4 d^4 e+704 b^2 c^3 d^3 e^2-40 b^3 c^2 d^2 e^3-12 b^4 c d e^4-5 b^5 e^5\right )\right )-2 c e \left (8 b c d e (2 c d-b e) (12 B c d-5 b B e-12 A c e)+\left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right ) \left (12 A c e (2 c d-b e)-B \left (24 c^2 d^2-12 b c d e-5 b^2 e^2\right )\right )\right ) x\right ) \sqrt{b x+c x^2}}{1536 c^3 e^6}+\frac{\left (4 A c e \left (16 c^2 d^2-22 b c d e+3 b^2 e^2\right )-B \left (64 c^3 d^3-88 b c^2 d^2 e+12 b^2 c d e^2+5 b^3 e^3\right )-2 c e \left (12 A c e (2 c d-b e)-B \left (24 c^2 d^2-12 b c d e-5 b^2 e^2\right )\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{192 c^2 e^4}-\frac{(12 B c d-5 b B e-12 A c e-10 B c e x) \left (b x+c x^2\right )^{5/2}}{60 c e^2}+\frac{\left (2 d^3 (B d-A e) (c d-b e)^3\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 )}{e^7}-\frac{\left (4 A c e \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 \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 )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{512 c^3 e^7}\\ &=\frac{\left (3 \left (4 A c e \left (128 c^4 d^4-288 b c^3 d^3 e+176 b^2 c^2 d^2 e^2-10 b^3 c d e^3-3 b^4 e^4\right )-B \left (512 c^5 d^5-1152 b c^4 d^4 e+704 b^2 c^3 d^3 e^2-40 b^3 c^2 d^2 e^3-12 b^4 c d e^4-5 b^5 e^5\right )\right )-2 c e \left (8 b c d e (2 c d-b e) (12 B c d-5 b B e-12 A c e)+\left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right ) \left (12 A c e (2 c d-b e)-B \left (24 c^2 d^2-12 b c d e-5 b^2 e^2\right )\right )\right ) x\right ) \sqrt{b x+c x^2}}{1536 c^3 e^6}+\frac{\left (4 A c e \left (16 c^2 d^2-22 b c d e+3 b^2 e^2\right )-B \left (64 c^3 d^3-88 b c^2 d^2 e+12 b^2 c d e^2+5 b^3 e^3\right )-2 c e \left (12 A c e (2 c d-b e)-B \left (24 c^2 d^2-12 b c d e-5 b^2 e^2\right )\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{192 c^2 e^4}-\frac{(12 B c d-5 b B e-12 A c e-10 B c e x) \left (b x+c x^2\right )^{5/2}}{60 c e^2}-\frac{\left (4 A c e \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 \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 )\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{512 c^{7/2} e^7}-\frac{d^{5/2} (B d-A e) (c d-b e)^{5/2} \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 )}{e^7}\\ \end{align*}

Mathematica [B]  time = 6.07871, size = 1869, normalized size = 2.66 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*(x*(b + c*x))^(5/2)*((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*Sqrt[c]*Sq
rt[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]*S
qrt[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))/(x^(5/2)*(b + c*x)^(5/2)) + (
B*(x*(b + c*x))^(5/2)*((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*Sq
rt[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])/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*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])/S
qrt[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]*ArcS
inh[(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))/(x^(5/2)*(b + c*x)^(5/2))

________________________________________________________________________________________

Maple [B]  time = 0.008, size = 4097, normalized size = 5.8 \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),x)

[Out]

1/5/e*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(5/2)*A-3/e^5*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))*b^2*c*A+3/e^6*d^5/(-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))*b^2*c*B+3/e^6*d^
5/(-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))*b*c^2*A-3/e^7*d^6/(-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))*b*c^2*B-3/4/e^4*d^3*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*b*c*B+3/64
/e^2/c*b^3*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*B*d+3/4/e^3*d^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*b*c*A+9/4/e^5*d^4*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/
2)*b*c*B-5/2/e^6*d^5*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c)/c^(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))*c^(3/2)*b*B-1/4/e^2*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*x*c*d*A+1/4/e^3*((x+d
/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*x*c*d^2*B-3/64/e/c*b^3*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)
-d*(b*e-c*d)/e^2)^(1/2)*x*A+5/16/e^3*d^2*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c)/c^(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))/c^(1/2)*b^3*A-5/16/e^4*d^3*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c)/c^(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))/c^(1/2)*b^3*B-5/96*B/e*b^2/c*(c*x^2+b*x)^(3/2)*x+5/256*B/e*b^4
/c^2*(c*x^2+b*x)^(1/2)*x-1/2/e^4*d^3*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*c^2*A+1/2/e^5
*d^4*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*c^2*B+1/e^4*d^3/(-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))*b^3*A-1/e^5*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))*b^3*B+3/128/e
^2/c^2*b^4*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*B*d-3/256/e^2/c^(5/2)*b^5*ln((1/2*(b*e-2*
c*d)/e+(x+d/e)*c)/c^(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))*B*d-1/8/e^2*((x+d/e)^2*c+
(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*x*b*B*d-1/16/e^2/c*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)
/e^2)^(3/2)*b^2*B*d-5/32/e^2*b^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*A+5/32/e^3*b^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^2*B-5/64/e^2/c*b^3*((x+d/e)^2*c+(b*e-2*c*d)/e*(x
+d/e)-d*(b*e-c*d)/e^2)^(1/2)*d*A+5/64/e^3/c*b^3*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*d^2*
B-1/e^7*d^6/(-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))*c^3*A+1/e^8*d^7/(-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))*c^3*B-15/8/e^4*d^3*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c)/c^(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))*c^(1/2)*b^2*A+15/8/e^5*d^4*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c)/c^(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))*c^(1/2)*b^2*B-9/4/e^4*d^3*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+
d/e)-d*(b*e-c*d)/e^2)^(1/2)*b*c*A+5/128/e^2*d/c^(3/2)*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c)/c^(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))*b^4*A-5/128/e^3*d^2/c^(3/2)*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c)/c^(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))*b^4*B+5/2/e^5*d^4*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*
c)/c^(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))*c^(3/2)*b*A-1/5/e^2*((x+d/e)^2*c+(b*e-2*
c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(5/2)*B*d+1/6*B/e*(c*x^2+b*x)^(5/2)*x+1/8/e*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)
-d*(b*e-c*d)/e^2)^(3/2)*x*b*A+1/12*B/e/c*(c*x^2+b*x)^(5/2)*b-5/192*B/e*b^3/c^2*(c*x^2+b*x)^(3/2)+5/512*B/e*b^5
/c^3*(c*x^2+b*x)^(1/2)-5/1024*B/e*b^6/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+11/24/e^3*((x+d/e)^2*c
+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*b*d^2*B-3/128/e/c^2*b^4*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*
e-c*d)/e^2)^(1/2)*A+3/256/e/c^(5/2)*b^5*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c)/c^(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))*A-1/e^6*d^5*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c)/c^(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))*c^(5/2)*A+1/e^7*d^6*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c)/c^(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))*c^(5/2)*B+1/3/e^3*d^2*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)
/e^2)^(3/2)*c*A-1/3/e^4*d^3*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*c*B+11/8/e^3*d^2*((x+d/e
)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*b^2*A-11/8/e^4*d^3*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*
e-c*d)/e^2)^(1/2)*b^2*B+1/e^5*d^4*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*c^2*A-1/e^6*d^5*((
x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*c^2*B-11/24/e^2*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(
b*e-c*d)/e^2)^(3/2)*b*d*A+1/16/e/c*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*b^2*A

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError