3.13.54 \(\int \frac {A+B x}{(d+e x)^{5/2} (b x+c x^2)^3} \, dx\) [1254]

Optimal. Leaf size=644 \[ \frac {e \left (72 A c^4 d^4+5 b^4 e^3 (4 B d-7 A e)-9 b^3 c d e^2 (4 B d-5 A e)-36 b c^3 d^3 (B d+4 A e)+3 b^2 c^2 d^2 e (29 B d+9 A e)\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac {e \left (24 A c^5 d^5+8 b^4 c d e^3 (7 B d-10 A e)-5 b^5 e^4 (4 B d-7 A e)-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (5 B d+4 A e)-12 b c^4 d^4 (B d+5 A e)\right )}{4 b^4 d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-7 A e)-3 b c d (2 B d+A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (B d+3 A e)\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}-\frac {\left (48 A c^2 d^2-5 b^2 e (4 B d-7 A e)-12 b c d (2 B d-5 A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{9/2}}+\frac {c^{7/2} \left (48 A c^3 d^2-99 b^3 B e^2-12 b c^2 d (2 B d+13 A e)+11 b^2 c e (8 B d+13 A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{9/2}} \]

[Out]

1/12*e*(72*A*c^4*d^4+5*b^4*e^3*(-7*A*e+4*B*d)-9*b^3*c*d*e^2*(-5*A*e+4*B*d)-36*b*c^3*d^3*(4*A*e+B*d)+3*b^2*c^2*
d^2*e*(9*A*e+29*B*d))/b^4/d^3/(-b*e+c*d)^3/(e*x+d)^(3/2)+1/2*(-A*b*(-b*e+c*d)-c*(2*A*c*d-b*(A*e+B*d))*x)/b^2/d
/(-b*e+c*d)/(e*x+d)^(3/2)/(c*x^2+b*x)^2+1/4*(b*(-b*e+c*d)*(12*A*c^2*d^2+b^2*e*(-7*A*e+4*B*d)-3*b*c*d*(A*e+2*B*
d))+c*(24*A*c^3*d^3-b^3*e^2*(-7*A*e+4*B*d)+b^2*c*d*e*(-2*A*e+23*B*d)-12*b*c^2*d^2*(3*A*e+B*d))*x)/b^4/d^2/(-b*
e+c*d)^2/(e*x+d)^(3/2)/(c*x^2+b*x)-1/4*(48*A*c^2*d^2-5*b^2*e*(-7*A*e+4*B*d)-12*b*c*d*(-5*A*e+2*B*d))*arctanh((
e*x+d)^(1/2)/d^(1/2))/b^5/d^(9/2)+1/4*c^(7/2)*(48*A*c^3*d^2-99*b^3*B*e^2-12*b*c^2*d*(13*A*e+2*B*d)+11*b^2*c*e*
(13*A*e+8*B*d))*arctanh(c^(1/2)*(e*x+d)^(1/2)/(-b*e+c*d)^(1/2))/b^5/(-b*e+c*d)^(9/2)+1/4*e*(24*A*c^5*d^5+8*b^4
*c*d*e^3*(-10*A*e+7*B*d)-5*b^5*e^4*(-7*A*e+4*B*d)-6*b^3*c^2*d^2*e^2*(-3*A*e+4*B*d)+7*b^2*c^3*d^3*e*(4*A*e+5*B*
d)-12*b*c^4*d^4*(5*A*e+B*d))/b^4/d^4/(-b*e+c*d)^4/(e*x+d)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 1.13, antiderivative size = 644, 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 = {836, 842, 840, 1180, 214} \begin {gather*} -\frac {c x (2 A c d-b (A e+B d))+A b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 (d+e x)^{3/2} (c d-b e)}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (-5 b^2 e (4 B d-7 A e)-12 b c d (2 B d-5 A e)+48 A c^2 d^2\right )}{4 b^5 d^{9/2}}+\frac {c^{7/2} \left (11 b^2 c e (13 A e+8 B d)-12 b c^2 d (13 A e+2 B d)+48 A c^3 d^2-99 b^3 B e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{9/2}}+\frac {b (c d-b e) \left (b^2 e (4 B d-7 A e)-3 b c d (A e+2 B d)+12 A c^2 d^2\right )+c x \left (b^3 \left (-e^2\right ) (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )}{4 b^4 d^2 \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)^2}+\frac {e \left (5 b^4 e^3 (4 B d-7 A e)-9 b^3 c d e^2 (4 B d-5 A e)+3 b^2 c^2 d^2 e (9 A e+29 B d)-36 b c^3 d^3 (4 A e+B d)+72 A c^4 d^4\right )}{12 b^4 d^3 (d+e x)^{3/2} (c d-b e)^3}+\frac {e \left (-5 b^5 e^4 (4 B d-7 A e)+8 b^4 c d e^3 (7 B d-10 A e)-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (4 A e+5 B d)-12 b c^4 d^4 (5 A e+B d)+24 A c^5 d^5\right )}{4 b^4 d^4 \sqrt {d+e x} (c d-b e)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e*(72*A*c^4*d^4 + 5*b^4*e^3*(4*B*d - 7*A*e) - 9*b^3*c*d*e^2*(4*B*d - 5*A*e) - 36*b*c^3*d^3*(B*d + 4*A*e) + 3*
b^2*c^2*d^2*e*(29*B*d + 9*A*e)))/(12*b^4*d^3*(c*d - b*e)^3*(d + e*x)^(3/2)) + (e*(24*A*c^5*d^5 + 8*b^4*c*d*e^3
*(7*B*d - 10*A*e) - 5*b^5*e^4*(4*B*d - 7*A*e) - 6*b^3*c^2*d^2*e^2*(4*B*d - 3*A*e) + 7*b^2*c^3*d^3*e*(5*B*d + 4
*A*e) - 12*b*c^4*d^4*(B*d + 5*A*e)))/(4*b^4*d^4*(c*d - b*e)^4*Sqrt[d + e*x]) - (A*b*(c*d - b*e) + c*(2*A*c*d -
b*(B*d + A*e))*x)/(2*b^2*d*(c*d - b*e)*(d + e*x)^(3/2)*(b*x + c*x^2)^2) + (b*(c*d - b*e)*(12*A*c^2*d^2 + b^2*
e*(4*B*d - 7*A*e) - 3*b*c*d*(2*B*d + A*e)) + c*(24*A*c^3*d^3 - b^3*e^2*(4*B*d - 7*A*e) + b^2*c*d*e*(23*B*d - 2
*A*e) - 12*b*c^2*d^2*(B*d + 3*A*e))*x)/(4*b^4*d^2*(c*d - b*e)^2*(d + e*x)^(3/2)*(b*x + c*x^2)) - ((48*A*c^2*d^
2 - 5*b^2*e*(4*B*d - 7*A*e) - 12*b*c*d*(2*B*d - 5*A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5*d^(9/2)) + (c^(
7/2)*(48*A*c^3*d^2 - 99*b^3*B*e^2 - 12*b*c^2*d*(2*B*d + 13*A*e) + 11*b^2*c*e*(8*B*d + 13*A*e))*ArcTanh[(Sqrt[c
]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*(c*d - b*e)^(9/2))

Rule 214

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

Rule 836

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)
*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*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] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 840

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*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]

Rule 842

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e
*f - d*g)*((d + e*x)^(m + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[(d +
e*x)^(m + 1)*(Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x]/(a + b*x + c*x^2)), 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] && FractionQ[m] && LtQ[m, -1]

Rule 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {A+B x}{(d+e x)^{5/2} \left (b x+c x^2\right )^3} \, dx &=-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}-\frac {\int \frac {\frac {1}{2} \left (12 A c^2 d^2+b^2 e (4 B d-7 A e)-3 b c d (2 B d+A e)\right )-\frac {9}{2} c e (b B d-2 A c d+A b e) x}{(d+e x)^{5/2} \left (b x+c x^2\right )^2} \, dx}{2 b^2 d (c d-b e)}\\ &=-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-7 A e)-3 b c d (2 B d+A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (B d+3 A e)\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac {\int \frac {\frac {1}{4} (c d-b e)^2 \left (48 A c^2 d^2-5 b^2 e (4 B d-7 A e)-12 b c d (2 B d-5 A e)\right )+\frac {5}{4} c e \left (24 A c^3 d^3-b^3 e^2 (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (B d+3 A e)\right ) x}{(d+e x)^{5/2} \left (b x+c x^2\right )} \, dx}{2 b^4 d^2 (c d-b e)^2}\\ &=\frac {e \left (72 A c^4 d^4+5 b^4 e^3 (4 B d-7 A e)-9 b^3 c d e^2 (4 B d-5 A e)-36 b c^3 d^3 (B d+4 A e)+3 b^2 c^2 d^2 e (29 B d+9 A e)\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-7 A e)-3 b c d (2 B d+A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (B d+3 A e)\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac {\int \frac {\frac {1}{4} (c d-b e)^3 \left (48 A c^2 d^2-5 b^2 e (4 B d-7 A e)-12 b c d (2 B d-5 A e)\right )+\frac {1}{4} c e \left (72 A c^4 d^4+5 b^4 e^3 (4 B d-7 A e)-9 b^3 c d e^2 (4 B d-5 A e)-36 b c^3 d^3 (B d+4 A e)+3 b^2 c^2 d^2 e (29 B d+9 A e)\right ) x}{(d+e x)^{3/2} \left (b x+c x^2\right )} \, dx}{2 b^4 d^3 (c d-b e)^3}\\ &=\frac {e \left (72 A c^4 d^4+5 b^4 e^3 (4 B d-7 A e)-9 b^3 c d e^2 (4 B d-5 A e)-36 b c^3 d^3 (B d+4 A e)+3 b^2 c^2 d^2 e (29 B d+9 A e)\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac {e \left (24 A c^5 d^5+8 b^4 c d e^3 (7 B d-10 A e)-5 b^5 e^4 (4 B d-7 A e)-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (5 B d+4 A e)-12 b c^4 d^4 (B d+5 A e)\right )}{4 b^4 d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-7 A e)-3 b c d (2 B d+A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (B d+3 A e)\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac {\int \frac {\frac {1}{4} (c d-b e)^4 \left (48 A c^2 d^2-5 b^2 e (4 B d-7 A e)-12 b c d (2 B d-5 A e)\right )+\frac {1}{4} c e \left (24 A c^5 d^5+8 b^4 c d e^3 (7 B d-10 A e)-5 b^5 e^4 (4 B d-7 A e)-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (5 B d+4 A e)-12 b c^4 d^4 (B d+5 A e)\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 d^4 (c d-b e)^4}\\ &=\frac {e \left (72 A c^4 d^4+5 b^4 e^3 (4 B d-7 A e)-9 b^3 c d e^2 (4 B d-5 A e)-36 b c^3 d^3 (B d+4 A e)+3 b^2 c^2 d^2 e (29 B d+9 A e)\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac {e \left (24 A c^5 d^5+8 b^4 c d e^3 (7 B d-10 A e)-5 b^5 e^4 (4 B d-7 A e)-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (5 B d+4 A e)-12 b c^4 d^4 (B d+5 A e)\right )}{4 b^4 d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-7 A e)-3 b c d (2 B d+A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (B d+3 A e)\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac {\text {Subst}\left (\int \frac {\frac {1}{4} e (c d-b e)^4 \left (48 A c^2 d^2-5 b^2 e (4 B d-7 A e)-12 b c d (2 B d-5 A e)\right )-\frac {1}{4} c d e \left (24 A c^5 d^5+8 b^4 c d e^3 (7 B d-10 A e)-5 b^5 e^4 (4 B d-7 A e)-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (5 B d+4 A e)-12 b c^4 d^4 (B d+5 A e)\right )+\frac {1}{4} c e \left (24 A c^5 d^5+8 b^4 c d e^3 (7 B d-10 A e)-5 b^5 e^4 (4 B d-7 A e)-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (5 B d+4 A e)-12 b c^4 d^4 (B d+5 A e)\right ) x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{b^4 d^4 (c d-b e)^4}\\ &=\frac {e \left (72 A c^4 d^4+5 b^4 e^3 (4 B d-7 A e)-9 b^3 c d e^2 (4 B d-5 A e)-36 b c^3 d^3 (B d+4 A e)+3 b^2 c^2 d^2 e (29 B d+9 A e)\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac {e \left (24 A c^5 d^5+8 b^4 c d e^3 (7 B d-10 A e)-5 b^5 e^4 (4 B d-7 A e)-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (5 B d+4 A e)-12 b c^4 d^4 (B d+5 A e)\right )}{4 b^4 d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-7 A e)-3 b c d (2 B d+A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (B d+3 A e)\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac {\left (c \left (48 A c^2 d^2-5 b^2 e (4 B d-7 A e)-12 b c d (2 B d-5 A e)\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 b^5 d^4}-\frac {\left (c^4 \left (48 A c^3 d^2-99 b^3 B e^2-12 b c^2 d (2 B d+13 A e)+11 b^2 c e (8 B d+13 A e)\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 b^5 (c d-b e)^4}\\ &=\frac {e \left (72 A c^4 d^4+5 b^4 e^3 (4 B d-7 A e)-9 b^3 c d e^2 (4 B d-5 A e)-36 b c^3 d^3 (B d+4 A e)+3 b^2 c^2 d^2 e (29 B d+9 A e)\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac {e \left (24 A c^5 d^5+8 b^4 c d e^3 (7 B d-10 A e)-5 b^5 e^4 (4 B d-7 A e)-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (5 B d+4 A e)-12 b c^4 d^4 (B d+5 A e)\right )}{4 b^4 d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-7 A e)-3 b c d (2 B d+A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (B d+3 A e)\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}-\frac {\left (48 A c^2 d^2-5 b^2 e (4 B d-7 A e)-12 b c d (2 B d-5 A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{9/2}}+\frac {c^{7/2} \left (48 A c^3 d^2-99 b^3 B e^2-12 b c^2 d (2 B d+13 A e)+11 b^2 c e (8 B d+13 A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 6.02, size = 803, normalized size = 1.25 \begin {gather*} -\frac {\frac {b \left (b B d x \left (36 c^6 d^4 x^2 (d+e x)^2+3 b c^5 d^3 x (18 d-35 e x) (d+e x)^2+4 b^6 e^4 \left (3 d^2+20 d e x+15 e^2 x^2\right )+3 b^2 c^4 d^2 (d+e x)^2 \left (4 d^2-53 d e x+24 e^2 x^2\right )-8 b^5 c e^3 \left (6 d^3+25 d^2 e x+d e^2 x^2-15 e^3 x^3\right )+4 b^4 c^2 e^2 \left (18 d^4+12 d^3 e x-91 d^2 e^2 x^2-64 d e^3 x^3+15 e^4 x^4\right )-8 b^3 c^3 d e \left (6 d^4-6 d^3 e x-24 d^2 e^2 x^2+10 d e^3 x^3+21 e^4 x^4\right )\right )+A \left (-72 c^7 d^5 x^3 (d+e x)^2-36 b c^6 d^4 x^2 (3 d-5 e x) (d+e x)^2-3 b^2 c^5 d^3 x (d+e x)^2 \left (8 d^2-91 d e x+28 e^2 x^2\right )+3 b^3 c^4 d^2 (d+e x)^2 \left (2 d^3+21 d^2 e x-44 d e^2 x^2-18 e^3 x^3\right )-b^7 e^4 \left (-6 d^3+21 d^2 e x+140 d e^2 x^2+105 e^3 x^3\right )-2 b^6 c e^3 \left (12 d^4-30 d^3 e x-139 d^2 e^2 x^2+20 d e^3 x^3+105 e^4 x^4\right )+b^5 c^2 e^2 \left (36 d^5-30 d^4 e x+30 d^3 e^2 x^2+565 d^2 e^3 x^3+340 d e^4 x^4-105 e^5 x^5\right )-4 b^4 c^3 d e \left (6 d^5+15 d^4 e x+45 d^3 e^2 x^2+45 d^2 e^3 x^3-53 d e^4 x^4-60 e^5 x^5\right )\right )\right )}{d^4 (c d-b e)^4 x^2 (b+c x)^2 (d+e x)^{3/2}}+\frac {3 c^{7/2} \left (48 A c^3 d^2-99 b^3 B e^2-12 b c^2 d (2 B d+13 A e)+11 b^2 c e (8 B d+13 A e)\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{(-c d+b e)^{9/2}}+\frac {3 \left (48 A c^2 d^2+12 b c d (-2 B d+5 A e)+5 b^2 e (-4 B d+7 A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{9/2}}}{12 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/12*((b*(b*B*d*x*(36*c^6*d^4*x^2*(d + e*x)^2 + 3*b*c^5*d^3*x*(18*d - 35*e*x)*(d + e*x)^2 + 4*b^6*e^4*(3*d^2
+ 20*d*e*x + 15*e^2*x^2) + 3*b^2*c^4*d^2*(d + e*x)^2*(4*d^2 - 53*d*e*x + 24*e^2*x^2) - 8*b^5*c*e^3*(6*d^3 + 25
*d^2*e*x + d*e^2*x^2 - 15*e^3*x^3) + 4*b^4*c^2*e^2*(18*d^4 + 12*d^3*e*x - 91*d^2*e^2*x^2 - 64*d*e^3*x^3 + 15*e
^4*x^4) - 8*b^3*c^3*d*e*(6*d^4 - 6*d^3*e*x - 24*d^2*e^2*x^2 + 10*d*e^3*x^3 + 21*e^4*x^4)) + A*(-72*c^7*d^5*x^3
*(d + e*x)^2 - 36*b*c^6*d^4*x^2*(3*d - 5*e*x)*(d + e*x)^2 - 3*b^2*c^5*d^3*x*(d + e*x)^2*(8*d^2 - 91*d*e*x + 28
*e^2*x^2) + 3*b^3*c^4*d^2*(d + e*x)^2*(2*d^3 + 21*d^2*e*x - 44*d*e^2*x^2 - 18*e^3*x^3) - b^7*e^4*(-6*d^3 + 21*
d^2*e*x + 140*d*e^2*x^2 + 105*e^3*x^3) - 2*b^6*c*e^3*(12*d^4 - 30*d^3*e*x - 139*d^2*e^2*x^2 + 20*d*e^3*x^3 + 1
05*e^4*x^4) + b^5*c^2*e^2*(36*d^5 - 30*d^4*e*x + 30*d^3*e^2*x^2 + 565*d^2*e^3*x^3 + 340*d*e^4*x^4 - 105*e^5*x^
5) - 4*b^4*c^3*d*e*(6*d^5 + 15*d^4*e*x + 45*d^3*e^2*x^2 + 45*d^2*e^3*x^3 - 53*d*e^4*x^4 - 60*e^5*x^5))))/(d^4*
(c*d - b*e)^4*x^2*(b + c*x)^2*(d + e*x)^(3/2)) + (3*c^(7/2)*(48*A*c^3*d^2 - 99*b^3*B*e^2 - 12*b*c^2*d*(2*B*d +
13*A*e) + 11*b^2*c*e*(8*B*d + 13*A*e))*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/(-(c*d) + b*e)^(9/
2) + (3*(48*A*c^2*d^2 + 12*b*c*d*(-2*B*d + 5*A*e) + 5*b^2*e*(-4*B*d + 7*A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/
d^(9/2))/b^5

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Maple [A]
time = 1.23, size = 482, normalized size = 0.75 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(e*x+d)^(5/2)/(c*x^2+b*x)^3,x,method=_RETURNVERBOSE)

[Out]

2*e^4*(-1/d^4/e^4/b^5*((-1/8*e*b*(11*A*b*e+12*A*c*d-4*B*b*d)*(e*x+d)^(3/2)+(13/8*A*b^2*d*e^2+3/2*A*b*c*d^2*e-1
/2*B*b^2*d^2*e)*(e*x+d)^(1/2))/e^2/x^2+1/8*(35*A*b^2*e^2+60*A*b*c*d*e+48*A*c^2*d^2-20*B*b^2*d*e-24*B*b*c*d^2)/
d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2)))-1/3*(-A*e+B*d)/d^3/(b*e-c*d)^3/(e*x+d)^(3/2)-1/d^4/(b*e-c*d)^4*(-3*A*b
*e^2+6*A*c*d*e+2*B*b*d*e-5*B*c*d^2)/(e*x+d)^(1/2)-c^4/(b*e-c*d)^4/e^4/b^5*(((23/8*A*b^2*c^2*e^2-3/2*A*b*c^3*d*
e-19/8*B*b^3*c*e^2+B*b^2*c^2*d*e)*(e*x+d)^(3/2)+1/8*b*e*(25*A*b^2*c*e^2-37*A*b*c^2*d*e+12*A*c^3*d^2-21*B*b^3*e
^2+29*B*b^2*c*d*e-8*B*b*c^2*d^2)*(e*x+d)^(1/2))/(c*(e*x+d)+b*e-c*d)^2+1/8*(143*A*b^2*c*e^2-156*A*b*c^2*d*e+48*
A*c^3*d^2-99*B*b^3*e^2+88*B*b^2*c*d*e-24*B*b*c^2*d^2)/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)
^(1/2))))

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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)/(e*x+d)^(5/2)/(c*x^2+b*x)^3,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(c*d-%e*b>0)', see `assume?` fo
r more detai

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Fricas [F(-1)] Timed out
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)/(e*x+d)^(5/2)/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)] Timed out
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)/(e*x+d)**(5/2)/(c*x**2+b*x)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1600 vs. \(2 (642) = 1284\).
time = 1.58, size = 1600, normalized size = 2.48 \begin {gather*} \frac {{\left (24 \, B b c^{6} d^{2} - 48 \, A c^{7} d^{2} - 88 \, B b^{2} c^{5} d e + 156 \, A b c^{6} d e + 99 \, B b^{3} c^{4} e^{2} - 143 \, A b^{2} c^{5} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{4 \, {\left (b^{5} c^{4} d^{4} - 4 \, b^{6} c^{3} d^{3} e + 6 \, b^{7} c^{2} d^{2} e^{2} - 4 \, b^{8} c d e^{3} + b^{9} e^{4}\right )} \sqrt {-c^{2} d + b c e}} + \frac {2 \, {\left (15 \, {\left (x e + d\right )} B c d^{2} e^{4} + B c d^{3} e^{4} - 6 \, {\left (x e + d\right )} B b d e^{5} - 18 \, {\left (x e + d\right )} A c d e^{5} - B b d^{2} e^{5} - A c d^{2} e^{5} + 9 \, {\left (x e + d\right )} A b e^{6} + A b d e^{6}\right )}}{3 \, {\left (c^{4} d^{8} - 4 \, b c^{3} d^{7} e + 6 \, b^{2} c^{2} d^{6} e^{2} - 4 \, b^{3} c d^{5} e^{3} + b^{4} d^{4} e^{4}\right )} {\left (x e + d\right )}^{\frac {3}{2}}} - \frac {12 \, {\left (x e + d\right )}^{\frac {7}{2}} B b c^{6} d^{5} e - 24 \, {\left (x e + d\right )}^{\frac {7}{2}} A c^{7} d^{5} e - 36 \, {\left (x e + d\right )}^{\frac {5}{2}} B b c^{6} d^{6} e + 72 \, {\left (x e + d\right )}^{\frac {5}{2}} A c^{7} d^{6} e + 36 \, {\left (x e + d\right )}^{\frac {3}{2}} B b c^{6} d^{7} e - 72 \, {\left (x e + d\right )}^{\frac {3}{2}} A c^{7} d^{7} e - 12 \, \sqrt {x e + d} B b c^{6} d^{8} e + 24 \, \sqrt {x e + d} A c^{7} d^{8} e - 35 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{2} c^{5} d^{4} e^{2} + 60 \, {\left (x e + d\right )}^{\frac {7}{2}} A b c^{6} d^{4} e^{2} + 123 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{2} c^{5} d^{5} e^{2} - 216 \, {\left (x e + d\right )}^{\frac {5}{2}} A b c^{6} d^{5} e^{2} - 141 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{2} c^{5} d^{6} e^{2} + 252 \, {\left (x e + d\right )}^{\frac {3}{2}} A b c^{6} d^{6} e^{2} + 53 \, \sqrt {x e + d} B b^{2} c^{5} d^{7} e^{2} - 96 \, \sqrt {x e + d} A b c^{6} d^{7} e^{2} + 24 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{3} c^{4} d^{3} e^{3} - 28 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{2} c^{5} d^{3} e^{3} - 125 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{3} c^{4} d^{4} e^{3} + 175 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{2} c^{5} d^{4} e^{3} + 182 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{3} c^{4} d^{5} e^{3} - 274 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{2} c^{5} d^{5} e^{3} - 81 \, \sqrt {x e + d} B b^{3} c^{4} d^{6} e^{3} + 127 \, \sqrt {x e + d} A b^{2} c^{5} d^{6} e^{3} - 16 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{4} c^{3} d^{2} e^{4} - 18 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{3} c^{4} d^{2} e^{4} + 96 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{4} c^{3} d^{3} e^{4} + 10 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{3} c^{4} d^{3} e^{4} - 160 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{4} c^{3} d^{4} e^{4} + 55 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{3} c^{4} d^{4} e^{4} + 80 \, \sqrt {x e + d} B b^{4} c^{3} d^{5} e^{4} - 45 \, \sqrt {x e + d} A b^{3} c^{4} d^{5} e^{4} + 4 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{5} c^{2} d e^{5} + 32 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{4} c^{3} d e^{5} - 44 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{5} c^{2} d^{2} e^{5} - 140 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{4} c^{3} d^{2} e^{5} + 100 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{5} c^{2} d^{3} e^{5} + 180 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{4} c^{3} d^{3} e^{5} - 60 \, \sqrt {x e + d} B b^{5} c^{2} d^{4} e^{5} - 80 \, \sqrt {x e + d} A b^{4} c^{3} d^{4} e^{5} - 11 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{5} c^{2} e^{6} + 8 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{6} c d e^{6} + 99 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{5} c^{2} d e^{6} - 32 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{6} c d^{2} e^{6} - 199 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{5} c^{2} d^{2} e^{6} + 24 \, \sqrt {x e + d} B b^{6} c d^{3} e^{6} + 123 \, \sqrt {x e + d} A b^{5} c^{2} d^{3} e^{6} - 22 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{6} c e^{7} + 4 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{7} d e^{7} + 80 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{6} c d e^{7} - 4 \, \sqrt {x e + d} B b^{7} d^{2} e^{7} - 66 \, \sqrt {x e + d} A b^{6} c d^{2} e^{7} - 11 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{7} e^{8} + 13 \, \sqrt {x e + d} A b^{7} d e^{8}}{4 \, {\left (b^{4} c^{4} d^{8} - 4 \, b^{5} c^{3} d^{7} e + 6 \, b^{6} c^{2} d^{6} e^{2} - 4 \, b^{7} c d^{5} e^{3} + b^{8} d^{4} e^{4}\right )} {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + {\left (x e + d\right )} b e - b d e\right )}^{2}} - \frac {{\left (24 \, B b c d^{2} - 48 \, A c^{2} d^{2} + 20 \, B b^{2} d e - 60 \, A b c d e - 35 \, A b^{2} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{4 \, b^{5} \sqrt {-d} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(5/2)/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

1/4*(24*B*b*c^6*d^2 - 48*A*c^7*d^2 - 88*B*b^2*c^5*d*e + 156*A*b*c^6*d*e + 99*B*b^3*c^4*e^2 - 143*A*b^2*c^5*e^2
)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b^5*c^4*d^4 - 4*b^6*c^3*d^3*e + 6*b^7*c^2*d^2*e^2 - 4*b^8*c*d
*e^3 + b^9*e^4)*sqrt(-c^2*d + b*c*e)) + 2/3*(15*(x*e + d)*B*c*d^2*e^4 + B*c*d^3*e^4 - 6*(x*e + d)*B*b*d*e^5 -
18*(x*e + d)*A*c*d*e^5 - B*b*d^2*e^5 - A*c*d^2*e^5 + 9*(x*e + d)*A*b*e^6 + A*b*d*e^6)/((c^4*d^8 - 4*b*c^3*d^7*
e + 6*b^2*c^2*d^6*e^2 - 4*b^3*c*d^5*e^3 + b^4*d^4*e^4)*(x*e + d)^(3/2)) - 1/4*(12*(x*e + d)^(7/2)*B*b*c^6*d^5*
e - 24*(x*e + d)^(7/2)*A*c^7*d^5*e - 36*(x*e + d)^(5/2)*B*b*c^6*d^6*e + 72*(x*e + d)^(5/2)*A*c^7*d^6*e + 36*(x
*e + d)^(3/2)*B*b*c^6*d^7*e - 72*(x*e + d)^(3/2)*A*c^7*d^7*e - 12*sqrt(x*e + d)*B*b*c^6*d^8*e + 24*sqrt(x*e +
d)*A*c^7*d^8*e - 35*(x*e + d)^(7/2)*B*b^2*c^5*d^4*e^2 + 60*(x*e + d)^(7/2)*A*b*c^6*d^4*e^2 + 123*(x*e + d)^(5/
2)*B*b^2*c^5*d^5*e^2 - 216*(x*e + d)^(5/2)*A*b*c^6*d^5*e^2 - 141*(x*e + d)^(3/2)*B*b^2*c^5*d^6*e^2 + 252*(x*e
+ d)^(3/2)*A*b*c^6*d^6*e^2 + 53*sqrt(x*e + d)*B*b^2*c^5*d^7*e^2 - 96*sqrt(x*e + d)*A*b*c^6*d^7*e^2 + 24*(x*e +
d)^(7/2)*B*b^3*c^4*d^3*e^3 - 28*(x*e + d)^(7/2)*A*b^2*c^5*d^3*e^3 - 125*(x*e + d)^(5/2)*B*b^3*c^4*d^4*e^3 + 1
75*(x*e + d)^(5/2)*A*b^2*c^5*d^4*e^3 + 182*(x*e + d)^(3/2)*B*b^3*c^4*d^5*e^3 - 274*(x*e + d)^(3/2)*A*b^2*c^5*d
^5*e^3 - 81*sqrt(x*e + d)*B*b^3*c^4*d^6*e^3 + 127*sqrt(x*e + d)*A*b^2*c^5*d^6*e^3 - 16*(x*e + d)^(7/2)*B*b^4*c
^3*d^2*e^4 - 18*(x*e + d)^(7/2)*A*b^3*c^4*d^2*e^4 + 96*(x*e + d)^(5/2)*B*b^4*c^3*d^3*e^4 + 10*(x*e + d)^(5/2)*
A*b^3*c^4*d^3*e^4 - 160*(x*e + d)^(3/2)*B*b^4*c^3*d^4*e^4 + 55*(x*e + d)^(3/2)*A*b^3*c^4*d^4*e^4 + 80*sqrt(x*e
+ d)*B*b^4*c^3*d^5*e^4 - 45*sqrt(x*e + d)*A*b^3*c^4*d^5*e^4 + 4*(x*e + d)^(7/2)*B*b^5*c^2*d*e^5 + 32*(x*e + d
)^(7/2)*A*b^4*c^3*d*e^5 - 44*(x*e + d)^(5/2)*B*b^5*c^2*d^2*e^5 - 140*(x*e + d)^(5/2)*A*b^4*c^3*d^2*e^5 + 100*(
x*e + d)^(3/2)*B*b^5*c^2*d^3*e^5 + 180*(x*e + d)^(3/2)*A*b^4*c^3*d^3*e^5 - 60*sqrt(x*e + d)*B*b^5*c^2*d^4*e^5
- 80*sqrt(x*e + d)*A*b^4*c^3*d^4*e^5 - 11*(x*e + d)^(7/2)*A*b^5*c^2*e^6 + 8*(x*e + d)^(5/2)*B*b^6*c*d*e^6 + 99
*(x*e + d)^(5/2)*A*b^5*c^2*d*e^6 - 32*(x*e + d)^(3/2)*B*b^6*c*d^2*e^6 - 199*(x*e + d)^(3/2)*A*b^5*c^2*d^2*e^6
+ 24*sqrt(x*e + d)*B*b^6*c*d^3*e^6 + 123*sqrt(x*e + d)*A*b^5*c^2*d^3*e^6 - 22*(x*e + d)^(5/2)*A*b^6*c*e^7 + 4*
(x*e + d)^(3/2)*B*b^7*d*e^7 + 80*(x*e + d)^(3/2)*A*b^6*c*d*e^7 - 4*sqrt(x*e + d)*B*b^7*d^2*e^7 - 66*sqrt(x*e +
d)*A*b^6*c*d^2*e^7 - 11*(x*e + d)^(3/2)*A*b^7*e^8 + 13*sqrt(x*e + d)*A*b^7*d*e^8)/((b^4*c^4*d^8 - 4*b^5*c^3*d
^7*e + 6*b^6*c^2*d^6*e^2 - 4*b^7*c*d^5*e^3 + b^8*d^4*e^4)*((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)
*b*e - b*d*e)^2) - 1/4*(24*B*b*c*d^2 - 48*A*c^2*d^2 + 20*B*b^2*d*e - 60*A*b*c*d*e - 35*A*b^2*e^2)*arctan(sqrt(
x*e + d)/sqrt(-d))/(b^5*sqrt(-d)*d^4)

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Mupad [B]
time = 6.93, size = 2500, normalized size = 3.88 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(14598144*A^3*b^9*c^27*d^32*e^4 - 884736*A^3*b^8*c^28*d^33*e^3 - ((d + e*x)^(1/2)*(589824*A^2*b^12*c^27*d^3
6*e^2 - 10616832*A^2*b^13*c^26*d^35*e^3 + 89518080*A^2*b^14*c^25*d^34*e^4 - 468971520*A^2*b^15*c^24*d^33*e^5 +
1707439360*A^2*b^16*c^23*d^32*e^6 - 4579446784*A^2*b^17*c^22*d^31*e^7 + 9364822016*A^2*b^18*c^21*d^30*e^8 - 1
4937190400*A^2*b^19*c^20*d^29*e^9 + 18936107520*A^2*b^20*c^19*d^28*e^10 - 19535324160*A^2*b^21*c^18*d^27*e^11
+ 17074641408*A^2*b^22*c^17*d^26*e^12 - 13484230656*A^2*b^23*c^16*d^25*e^13 + 10265639040*A^2*b^24*c^15*d^24*e
^14 - 7643066880*A^2*b^25*c^14*d^23*e^15 + 5421597440*A^2*b^26*c^13*d^22*e^16 - 3708136960*A^2*b^27*c^12*d^21*
e^17 + 2608529792*A^2*b^28*c^11*d^20*e^18 - 1894041600*A^2*b^29*c^10*d^19*e^19 + 1274465280*A^2*b^30*c^9*d^18*
e^20 - 707773440*A^2*b^31*c^8*d^17*e^21 + 301648512*A^2*b^32*c^7*d^16*e^22 - 93688320*A^2*b^33*c^6*d^15*e^23 +
19930880*A^2*b^34*c^5*d^14*e^24 - 2598400*A^2*b^35*c^4*d^13*e^25 + 156800*A^2*b^36*c^3*d^12*e^26 + 147456*B^2
*b^14*c^25*d^36*e^2 - 2777088*B^2*b^15*c^24*d^35*e^3 + 24555520*B^2*b^16*c^23*d^34*e^4 - 135055360*B^2*b^17*c^
22*d^33*e^5 + 515884160*B^2*b^18*c^21*d^32*e^6 - 1446258176*B^2*b^19*c^20*d^31*e^7 + 3062171904*B^2*b^20*c^19*
d^30*e^8 - 4951119360*B^2*b^21*c^18*d^29*e^9 + 6076371840*B^2*b^22*c^17*d^28*e^10 - 5478190080*B^2*b^23*c^16*d
^27*e^11 + 3273549312*B^2*b^24*c^15*d^26*e^12 - 766116864*B^2*b^25*c^14*d^25*e^13 - 668122240*B^2*b^26*c^13*d^
24*e^14 + 721318400*B^2*b^27*c^12*d^23*e^15 - 107134720*B^2*b^28*c^11*d^22*e^16 - 366558720*B^2*b^29*c^10*d^21
*e^17 + 437847168*B^2*b^30*c^9*d^20*e^18 - 282501120*B^2*b^31*c^8*d^19*e^19 + 121989120*B^2*b^32*c^7*d^18*e^20
- 36495360*B^2*b^33*c^6*d^17*e^21 + 7344128*B^2*b^34*c^5*d^16*e^22 - 901120*B^2*b^35*c^4*d^15*e^23 + 51200*B^
2*b^36*c^3*d^14*e^24 - 589824*A*B*b^13*c^26*d^36*e^2 + 10862592*A*B*b^14*c^25*d^35*e^3 - 93818880*A*B*b^15*c^2
4*d^34*e^4 + 503726080*A*B*b^16*c^23*d^33*e^5 - 1878764800*A*B*b^17*c^22*d^32*e^6 + 5151263744*A*B*b^18*c^21*d
^31*e^7 - 10713545216*A*B*b^19*c^20*d^30*e^8 + 17186104320*A*B*b^20*c^19*d^29*e^9 - 21406851840*A*B*b^21*c^18*
d^28*e^10 + 20693207040*A*B*b^22*c^17*d^27*e^11 - 15463523328*A*B*b^23*c^16*d^26*e^12 + 8955257856*A*B*b^24*c^
15*d^25*e^13 - 4111491840*A*B*b^25*c^14*d^24*e^14 + 1413002240*A*B*b^26*c^13*d^23*e^15 + 178449920*A*B*b^27*c^
12*d^22*e^16 - 1280942080*A*B*b^28*c^11*d^21*e^17 + 1742746368*A*B*b^29*c^10*d^20*e^18 - 1489551360*A*B*b^30*c
^9*d^19*e^19 + 892446720*A*B*b^31*c^8*d^18*e^20 - 383708160*A*B*b^32*c^7*d^17*e^21 + 117055488*A*B*b^33*c^6*d^
16*e^22 - 24217600*A*B*b^34*c^5*d^15*e^23 + 3061760*A*B*b^35*c^4*d^14*e^24 - 179200*A*B*b^36*c^3*d^13*e^25) -
((1225*A^2*b^4*e^4 + 2304*A^2*c^4*d^4 + 576*B^2*b^2*c^2*d^4 + 400*B^2*b^4*d^2*e^2 + 6960*A^2*b^2*c^2*d^2*e^2 +
5760*A^2*b*c^3*d^3*e + 4200*A^2*b^3*c*d*e^3 + 960*B^2*b^3*c*d^3*e - 2304*A*B*b*c^3*d^4 - 1400*A*B*b^4*d*e^3 -
4800*A*B*b^2*c^2*d^3*e - 4080*A*B*b^3*c*d^2*e^2)/(64*b^10*d^9))^(1/2)*((d + e*x)^(1/2)*((1225*A^2*b^4*e^4 + 2
304*A^2*c^4*d^4 + 576*B^2*b^2*c^2*d^4 + 400*B^2*b^4*d^2*e^2 + 6960*A^2*b^2*c^2*d^2*e^2 + 5760*A^2*b*c^3*d^3*e
+ 4200*A^2*b^3*c*d*e^3 + 960*B^2*b^3*c*d^3*e - 2304*A*B*b*c^3*d^4 - 1400*A*B*b^4*d*e^3 - 4800*A*B*b^2*c^2*d^3*
e - 4080*A*B*b^3*c*d^2*e^2)/(64*b^10*d^9))^(1/2)*(16384*b^22*c^23*d^41*e^2 - 335872*b^23*c^22*d^40*e^3 + 32768
00*b^24*c^21*d^39*e^4 - 20234240*b^25*c^20*d^38*e^5 + 88719360*b^26*c^19*d^37*e^6 - 293707776*b^27*c^18*d^36*e
^7 + 762052608*b^28*c^17*d^35*e^8 - 1587609600*b^29*c^16*d^34*e^9 + 2698936320*b^30*c^15*d^33*e^10 - 378380288
0*b^31*c^14*d^32*e^11 + 4402970624*b^32*c^13*d^31*e^12 - 4265377792*b^33*c^12*d^30*e^13 + 3439820800*b^34*c^11
*d^29*e^14 - 2302033920*b^35*c^10*d^28*e^15 + 1270087680*b^36*c^9*d^27*e^16 - 571539456*b^37*c^8*d^26*e^17 + 2
06389248*b^38*c^7*d^25*e^18 - 58368000*b^39*c^6*d^24*e^19 + 12451840*b^40*c^5*d^23*e^20 - 1884160*b^41*c^4*d^2
2*e^21 + 180224*b^42*c^3*d^21*e^22 - 8192*b^43*c^2*d^20*e^23) + 24576*A*b^18*c^24*d^38*e^3 - 466944*A*b^19*c^2
3*d^37*e^4 + 4185088*A*b^20*c^22*d^36*e^5 - 23500800*A*b^21*c^21*d^35*e^6 + 92710912*A*b^22*c^20*d^34*e^7 - 27
3566720*A*b^23*c^19*d^33*e^8 + 629578752*A*b^24*c^18*d^32*e^9 - 1169833984*A*b^25*c^17*d^31*e^10 + 1818910720*
A*b^26*c^16*d^30*e^11 - 2465058816*A*b^27*c^15*d^29*e^12 + 3031169024*A*b^28*c^14*d^28*e^13 - 3457871872*A*b^2
9*c^13*d^27*e^14 + 3626348544*A*b^30*c^12*d^26*e^15 - 3385559040*A*b^31*c^11*d^25*e^16 + 2714064896*A*b^32*c^1
0*d^24*e^17 - 1813512192*A*b^33*c^9*d^23*e^18 + 986251264*A*b^34*c^8*d^22*e^19 - 426815488*A*b^35*c^7*d^21*e^2
0 + 143109120*A*b^36*c^6*d^20*e^21 - 35796992*A*b^37*c^5*d^19*e^22 + 6285312*A*b^38*c^4*d^18*e^23 - 691200*A*b
^39*c^3*d^17*e^24 + 35840*A*b^40*c^2*d^16*e^25 - 12288*B*b^19*c^23*d^38*e^3 + 238592*B*b^20*c^22*d^37*e^4 - 21
87264*B*b^21*c^21*d^36*e^5 + 12492800*B*b^22*c^20*d^35*e^6 - 49401856*B*b^23*c^19*d^34*e^7 + 141926400*B*b^24*
c^18*d^33*e^8 - 300793856*B*b^25*c^17*d^32*e^9 + 460562432*B*b^26*c^16*d^31*e^10 - 455516160*B*b^27*c^15*d^30*
e^11 + 116267008*B*b^28*c^14*d^29*e^12 + 543981...

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