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

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

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Rubi [A]  time = 1.12, antiderivative size = 703, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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 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 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 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]

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]

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*}

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Mathematica [A]  time = 2.47, size = 650, normalized size = 0.92 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\sqrt {c} \left (e \sqrt {x} \left (4 A c e \left (-45 b^4 e^4+30 b^3 c e^3 (e x-5 d)+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 )+B \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 )\right )-\frac {15360 c^3 d^{5/2} (B d-A e) (c d-b e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt {c d-b e}}{\sqrt {d} \sqrt {b+c x}}\right )}{\sqrt {b+c x}}\right )+\frac {15 \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ) \left (4 A c e \left (3 b^5 e^5+10 b^4 c d e^4+80 b^3 c^2 d^2 e^3-480 b^2 c^3 d^3 e^2+640 b c^4 d^4 e-256 c^5 d^5\right )+B \left (-5 b^6 e^6-12 b^5 c d e^5-40 b^4 c^2 d^2 e^4-320 b^3 c^3 d^3 e^3+1920 b^2 c^4 d^4 e^2-2560 b c^5 d^5 e+1024 c^6 d^6\right )\right )}{\sqrt {b} \sqrt {\frac {c x}{b}+1}}\right )}{7680 c^{7/2} e^7 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x*(b + c*x)]*((15*(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))*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c*x
)/b]) + Sqrt[c]*(e*Sqrt[x]*(4*A*c*e*(-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)) + B*(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 - 130*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^(5/2)*(B*d - A*e)*(c*d - b*e)^(
5/2)*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/Sqrt[b + c*x])))/(7680*c^(7/2)*e^7*Sqrt[x])

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IntegrateAlgebraic [A]  time = 11.42, size = 905, normalized size = 1.29 \begin {gather*} \frac {\sqrt {c x^2+b x} \left (-7680 B d^5 c^5+1280 B e^5 x^5 c^5+1536 A e^5 x^4 c^5-1536 B d e^4 x^4 c^5-1920 A d e^4 x^3 c^5+1920 B d^2 e^3 x^3 c^5+2560 A d^2 e^3 x^2 c^5-2560 B d^3 e^2 x^2 c^5+7680 A d^4 e c^5-3840 A d^3 e^2 x c^5+3840 B d^4 e x c^5+3200 b B e^5 x^4 c^4+4032 A b e^5 x^3 c^4-4032 b B d e^4 x^3 c^4-17280 A b d^3 e^2 c^4-5440 A b d e^4 x^2 c^4+5440 b B d^2 e^3 x^2 c^4+17280 b B d^4 e c^4+8320 A b d^2 e^3 x c^4-8320 b B d^3 e^2 x c^4+10560 A b^2 d^2 e^3 c^3+2160 b^2 B e^5 x^3 c^3-10560 b^2 B d^3 e^2 c^3+2976 A b^2 e^5 x^2 c^3-2976 b^2 B d e^4 x^2 c^3-4720 A b^2 d e^4 x c^3+4720 b^2 B d^2 e^3 x c^3-600 A b^3 d e^4 c^2+600 b^3 B d^2 e^3 c^2+40 b^3 B e^5 x^2 c^2+120 A b^3 e^5 x c^2-120 b^3 B d e^4 x c^2-180 A b^4 e^5 c+180 b^4 B d e^4 c-50 b^4 B e^5 x c+75 b^5 B e^5\right )}{7680 c^3 e^6}-\frac {2 \left (B c^2 \sqrt {c d-b e} d^{11/2}-A c^2 e \sqrt {c d-b e} d^{9/2}-2 b B c e \sqrt {c d-b e} d^{9/2}+b^2 B e^2 \sqrt {c d-b e} d^{7/2}+2 A b c e^2 \sqrt {c d-b e} d^{7/2}-A b^2 e^3 \sqrt {c d-b e} d^{5/2}\right ) \tanh ^{-1}\left (\frac {\sqrt {c} d+\sqrt {c} e x-e \sqrt {c x^2+b x}}{\sqrt {d} \sqrt {c d-b e}}\right )}{e^7}+\frac {\left (-1024 B d^6 c^6+1024 A d^5 e c^6-2560 A b d^4 e^2 c^5+2560 b B d^5 e c^5+1920 A b^2 d^3 e^3 c^4-1920 b^2 B d^4 e^2 c^4-320 A b^3 d^2 e^4 c^3+320 b^3 B d^3 e^3 c^3-40 A b^4 d e^5 c^2+40 b^4 B d^2 e^4 c^2-12 A b^5 e^6 c+12 b^5 B d e^5 c+5 b^6 B e^6\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {c x^2+b x}\right )}{1024 c^{7/2} e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b*x + c*x^2]*(-7680*B*c^5*d^5 + 17280*b*B*c^4*d^4*e + 7680*A*c^5*d^4*e - 10560*b^2*B*c^3*d^3*e^2 - 17280
*A*b*c^4*d^3*e^2 + 600*b^3*B*c^2*d^2*e^3 + 10560*A*b^2*c^3*d^2*e^3 + 180*b^4*B*c*d*e^4 - 600*A*b^3*c^2*d*e^4 +
75*b^5*B*e^5 - 180*A*b^4*c*e^5 + 3840*B*c^5*d^4*e*x - 8320*b*B*c^4*d^3*e^2*x - 3840*A*c^5*d^3*e^2*x + 4720*b^
2*B*c^3*d^2*e^3*x + 8320*A*b*c^4*d^2*e^3*x - 120*b^3*B*c^2*d*e^4*x - 4720*A*b^2*c^3*d*e^4*x - 50*b^4*B*c*e^5*x
+ 120*A*b^3*c^2*e^5*x - 2560*B*c^5*d^3*e^2*x^2 + 5440*b*B*c^4*d^2*e^3*x^2 + 2560*A*c^5*d^2*e^3*x^2 - 2976*b^2
*B*c^3*d*e^4*x^2 - 5440*A*b*c^4*d*e^4*x^2 + 40*b^3*B*c^2*e^5*x^2 + 2976*A*b^2*c^3*e^5*x^2 + 1920*B*c^5*d^2*e^3
*x^3 - 4032*b*B*c^4*d*e^4*x^3 - 1920*A*c^5*d*e^4*x^3 + 2160*b^2*B*c^3*e^5*x^3 + 4032*A*b*c^4*e^5*x^3 - 1536*B*
c^5*d*e^4*x^4 + 3200*b*B*c^4*e^5*x^4 + 1536*A*c^5*e^5*x^4 + 1280*B*c^5*e^5*x^5))/(7680*c^3*e^6) - (2*(B*c^2*d^
(11/2)*Sqrt[c*d - b*e] - 2*b*B*c*d^(9/2)*e*Sqrt[c*d - b*e] - A*c^2*d^(9/2)*e*Sqrt[c*d - b*e] + b^2*B*d^(7/2)*e
^2*Sqrt[c*d - b*e] + 2*A*b*c*d^(7/2)*e^2*Sqrt[c*d - b*e] - A*b^2*d^(5/2)*e^3*Sqrt[c*d - b*e])*ArcTanh[(Sqrt[c]
*d + Sqrt[c]*e*x - e*Sqrt[b*x + c*x^2])/(Sqrt[d]*Sqrt[c*d - b*e])])/e^7 + ((-1024*B*c^6*d^6 + 2560*b*B*c^5*d^5
*e + 1024*A*c^6*d^5*e - 1920*b^2*B*c^4*d^4*e^2 - 2560*A*b*c^5*d^4*e^2 + 320*b^3*B*c^3*d^3*e^3 + 1920*A*b^2*c^4
*d^3*e^3 + 40*b^4*B*c^2*d^2*e^4 - 320*A*b^3*c^3*d^2*e^4 + 12*b^5*B*c*d*e^5 - 40*A*b^4*c^2*d*e^5 + 5*b^6*B*e^6
- 12*A*b^5*c*e^6)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[b*x + c*x^2]])/(1024*c^(7/2)*e^7)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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 >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Erro
r: Bad Argument Type

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maple [B]  time = 0.06, size = 4097, normalized size = 5.83 \begin {gather*} \text {output too large to display} \end {gather*}

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]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-2*c*d>0)', see `assume?` f
or more details)Is b*e-2*c*d zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x\right )}^{5/2}\,\left (A+B\,x\right )}{d+e\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (A + B x\right )}{d + e x}\, dx \end {gather*}

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]

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

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