3.385 \(\int \frac{1}{(d+e x)^{5/2} (b x+c x^2)^3} \, dx\)

Optimal. Leaf size=470 \[ \frac{c x (2 c d-b e) \left (-7 b^2 e^2-12 b c d e+12 c^2 d^2\right )+b (c d-b e) \left (-7 b^2 e^2-3 b c d e+12 c^2 d^2\right )}{4 b^4 d^2 \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)^2}+\frac{e (2 c d-b e) \left (2 b^2 c^2 d^2 e^2+10 b^3 c d e^3-35 b^4 e^4-24 b c^3 d^3 e+12 c^4 d^4\right )}{4 b^4 d^4 \sqrt{d+e x} (c d-b e)^4}+\frac{e \left (27 b^2 c^2 d^2 e^2+45 b^3 c d e^3-35 b^4 e^4-144 b c^3 d^3 e+72 c^4 d^4\right )}{12 b^4 d^3 (d+e x)^{3/2} (c d-b e)^3}+\frac{c^{9/2} \left (143 b^2 e^2-156 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{9/2}}-\frac{\left (35 b^2 e^2+60 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5 d^{9/2}}-\frac{c x (2 c d-b e)+b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 (d+e x)^{3/2} (c d-b e)} \]

[Out]

(e*(72*c^4*d^4 - 144*b*c^3*d^3*e + 27*b^2*c^2*d^2*e^2 + 45*b^3*c*d*e^3 - 35*b^4*e^4))/(12*b^4*d^3*(c*d - b*e)^
3*(d + e*x)^(3/2)) + (e*(2*c*d - b*e)*(12*c^4*d^4 - 24*b*c^3*d^3*e + 2*b^2*c^2*d^2*e^2 + 10*b^3*c*d*e^3 - 35*b
^4*e^4))/(4*b^4*d^4*(c*d - b*e)^4*Sqrt[d + e*x]) - (b*(c*d - b*e) + c*(2*c*d - b*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*c^2*d^2 - 3*b*c*d*e - 7*b^2*e^2) + c*(2*c*d - b*e)*(12*c^2
*d^2 - 12*b*c*d*e - 7*b^2*e^2)*x)/(4*b^4*d^2*(c*d - b*e)^2*(d + e*x)^(3/2)*(b*x + c*x^2)) - ((48*c^2*d^2 + 60*
b*c*d*e + 35*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5*d^(9/2)) + (c^(9/2)*(48*c^2*d^2 - 156*b*c*d*e + 1
43*b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*(c*d - b*e)^(9/2))

________________________________________________________________________________________

Rubi [A]  time = 0.878529, antiderivative size = 470, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {740, 822, 828, 826, 1166, 208} \[ \frac{c x (2 c d-b e) \left (-7 b^2 e^2-12 b c d e+12 c^2 d^2\right )+b (c d-b e) \left (-7 b^2 e^2-3 b c d e+12 c^2 d^2\right )}{4 b^4 d^2 \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)^2}+\frac{e (2 c d-b e) \left (2 b^2 c^2 d^2 e^2+10 b^3 c d e^3-35 b^4 e^4-24 b c^3 d^3 e+12 c^4 d^4\right )}{4 b^4 d^4 \sqrt{d+e x} (c d-b e)^4}+\frac{e \left (27 b^2 c^2 d^2 e^2+45 b^3 c d e^3-35 b^4 e^4-144 b c^3 d^3 e+72 c^4 d^4\right )}{12 b^4 d^3 (d+e x)^{3/2} (c d-b e)^3}+\frac{c^{9/2} \left (143 b^2 e^2-156 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{9/2}}-\frac{\left (35 b^2 e^2+60 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5 d^{9/2}}-\frac{c x (2 c d-b e)+b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 (d+e x)^{3/2} (c d-b e)} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(5/2)*(b*x + c*x^2)^3),x]

[Out]

(e*(72*c^4*d^4 - 144*b*c^3*d^3*e + 27*b^2*c^2*d^2*e^2 + 45*b^3*c*d*e^3 - 35*b^4*e^4))/(12*b^4*d^3*(c*d - b*e)^
3*(d + e*x)^(3/2)) + (e*(2*c*d - b*e)*(12*c^4*d^4 - 24*b*c^3*d^3*e + 2*b^2*c^2*d^2*e^2 + 10*b^3*c*d*e^3 - 35*b
^4*e^4))/(4*b^4*d^4*(c*d - b*e)^4*Sqrt[d + e*x]) - (b*(c*d - b*e) + c*(2*c*d - b*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*c^2*d^2 - 3*b*c*d*e - 7*b^2*e^2) + c*(2*c*d - b*e)*(12*c^2
*d^2 - 12*b*c*d*e - 7*b^2*e^2)*x)/(4*b^4*d^2*(c*d - b*e)^2*(d + e*x)^(3/2)*(b*x + c*x^2)) - ((48*c^2*d^2 + 60*
b*c*d*e + 35*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5*d^(9/2)) + (c^(9/2)*(48*c^2*d^2 - 156*b*c*d*e + 1
43*b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*(c*d - b*e)^(9/2))

Rule 740

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*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*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b
*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 822

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 828

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 826

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 1166

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]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{1}{(d+e x)^{5/2} \left (b x+c x^2\right )^3} \, dx &=-\frac{b (c d-b e)+c (2 c d-b 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 c^2 d^2-3 b c d e-7 b^2 e^2\right )+\frac{9}{2} c e (2 c d-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{b (c d-b e)+c (2 c d-b 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 c^2 d^2-3 b c d e-7 b^2 e^2\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-7 b^2 e^2\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 c^2 d^2+60 b c d e+35 b^2 e^2\right )+\frac{5}{4} c e (2 c d-b e) \left (12 c^2 d^2-12 b c d e-7 b^2 e^2\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 c^4 d^4-144 b c^3 d^3 e+27 b^2 c^2 d^2 e^2+45 b^3 c d e^3-35 b^4 e^4\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}-\frac{b (c d-b e)+c (2 c d-b 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 c^2 d^2-3 b c d e-7 b^2 e^2\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-7 b^2 e^2\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 c^2 d^2+60 b c d e+35 b^2 e^2\right )+\frac{1}{4} c e \left (72 c^4 d^4-144 b c^3 d^3 e+27 b^2 c^2 d^2 e^2+45 b^3 c d e^3-35 b^4 e^4\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 c^4 d^4-144 b c^3 d^3 e+27 b^2 c^2 d^2 e^2+45 b^3 c d e^3-35 b^4 e^4\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac{e (2 c d-b e) \left (12 c^4 d^4-24 b c^3 d^3 e+2 b^2 c^2 d^2 e^2+10 b^3 c d e^3-35 b^4 e^4\right )}{4 b^4 d^4 (c d-b e)^4 \sqrt{d+e x}}-\frac{b (c d-b e)+c (2 c d-b 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 c^2 d^2-3 b c d e-7 b^2 e^2\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-7 b^2 e^2\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 c^2 d^2+60 b c d e+35 b^2 e^2\right )+\frac{1}{4} c e (2 c d-b e) \left (12 c^4 d^4-24 b c^3 d^3 e+2 b^2 c^2 d^2 e^2+10 b^3 c d e^3-35 b^4 e^4\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 c^4 d^4-144 b c^3 d^3 e+27 b^2 c^2 d^2 e^2+45 b^3 c d e^3-35 b^4 e^4\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac{e (2 c d-b e) \left (12 c^4 d^4-24 b c^3 d^3 e+2 b^2 c^2 d^2 e^2+10 b^3 c d e^3-35 b^4 e^4\right )}{4 b^4 d^4 (c d-b e)^4 \sqrt{d+e x}}-\frac{b (c d-b e)+c (2 c d-b 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 c^2 d^2-3 b c d e-7 b^2 e^2\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-7 b^2 e^2\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{\operatorname{Subst}\left (\int \frac{\frac{1}{4} e (c d-b e)^4 \left (48 c^2 d^2+60 b c d e+35 b^2 e^2\right )-\frac{1}{4} c d e (2 c d-b e) \left (12 c^4 d^4-24 b c^3 d^3 e+2 b^2 c^2 d^2 e^2+10 b^3 c d e^3-35 b^4 e^4\right )+\frac{1}{4} c e (2 c d-b e) \left (12 c^4 d^4-24 b c^3 d^3 e+2 b^2 c^2 d^2 e^2+10 b^3 c d e^3-35 b^4 e^4\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 c^4 d^4-144 b c^3 d^3 e+27 b^2 c^2 d^2 e^2+45 b^3 c d e^3-35 b^4 e^4\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac{e (2 c d-b e) \left (12 c^4 d^4-24 b c^3 d^3 e+2 b^2 c^2 d^2 e^2+10 b^3 c d e^3-35 b^4 e^4\right )}{4 b^4 d^4 (c d-b e)^4 \sqrt{d+e x}}-\frac{b (c d-b e)+c (2 c d-b 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 c^2 d^2-3 b c d e-7 b^2 e^2\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-7 b^2 e^2\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 c^2 d^2+60 b c d e+35 b^2 e^2\right )\right ) \operatorname{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^5 \left (48 c^2 d^2-156 b c d e+143 b^2 e^2\right )\right ) \operatorname{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 c^4 d^4-144 b c^3 d^3 e+27 b^2 c^2 d^2 e^2+45 b^3 c d e^3-35 b^4 e^4\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac{e (2 c d-b e) \left (12 c^4 d^4-24 b c^3 d^3 e+2 b^2 c^2 d^2 e^2+10 b^3 c d e^3-35 b^4 e^4\right )}{4 b^4 d^4 (c d-b e)^4 \sqrt{d+e x}}-\frac{b (c d-b e)+c (2 c d-b 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 c^2 d^2-3 b c d e-7 b^2 e^2\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-7 b^2 e^2\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 c^2 d^2+60 b c d e+35 b^2 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5 d^{9/2}}+\frac{c^{9/2} \left (48 c^2 d^2-156 b c d e+143 b^2 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}}\\ \end{align*}

Mathematica [C]  time = 0.293585, size = 301, normalized size = 0.64 \[ \frac{-x^2 \left ((b+c x) \left (3 b c d (b e-c d) \left (-2 b^2 c d e^2+7 b^3 e^3-36 b c^2 d^2 e+24 c^3 d^3\right )-(b+c x) \left ((c d-b e)^3 \left (35 b^2 e^2+60 b c d e+48 c^2 d^2\right ) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{e x}{d}+1\right )-c^3 d^3 \left (143 b^2 e^2-156 b c d e+48 c^2 d^2\right ) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{c (d+e x)}{c d-b e}\right )\right )\right )+3 b^2 c d \left (7 b^2 e^2+3 b c d e-12 c^2 d^2\right ) (c d-b e)^2\right )+6 b^4 d^2 (b e-c d)^3+3 b^3 d x (c d-b e)^3 (7 b e+8 c d)}{12 b^5 d^3 x^2 (b+c x)^2 (d+e x)^{3/2} (c d-b e)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(5/2)*(b*x + c*x^2)^3),x]

[Out]

(6*b^4*d^2*(-(c*d) + b*e)^3 + 3*b^3*d*(c*d - b*e)^3*(8*c*d + 7*b*e)*x - x^2*(3*b^2*c*d*(c*d - b*e)^2*(-12*c^2*
d^2 + 3*b*c*d*e + 7*b^2*e^2) + (b + c*x)*(3*b*c*d*(-(c*d) + b*e)*(24*c^3*d^3 - 36*b*c^2*d^2*e - 2*b^2*c*d*e^2
+ 7*b^3*e^3) - (b + c*x)*(-(c^3*d^3*(48*c^2*d^2 - 156*b*c*d*e + 143*b^2*e^2)*Hypergeometric2F1[-3/2, 1, -1/2,
(c*(d + e*x))/(c*d - b*e)]) + (c*d - b*e)^3*(48*c^2*d^2 + 60*b*c*d*e + 35*b^2*e^2)*Hypergeometric2F1[-3/2, 1,
-1/2, 1 + (e*x)/d]))))/(12*b^5*d^3*(c*d - b*e)^3*x^2*(b + c*x)^2*(d + e*x)^(3/2))

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Maple [A]  time = 0.253, size = 582, normalized size = 1.2 \begin{align*} 6\,{\frac{{e}^{6}b}{{d}^{4} \left ( be-cd \right ) ^{4}\sqrt{ex+d}}}-12\,{\frac{{e}^{5}c}{{d}^{3} \left ( be-cd \right ) ^{4}\sqrt{ex+d}}}+{\frac{2\,{e}^{5}}{3\,{d}^{3} \left ( be-cd \right ) ^{3}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}-{\frac{23\,{e}^{2}{c}^{6}}{4\,{b}^{3} \left ( be-cd \right ) ^{4} \left ( cex+be \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+3\,{\frac{e{c}^{7} \left ( ex+d \right ) ^{3/2}d}{{b}^{4} \left ( be-cd \right ) ^{4} \left ( cex+be \right ) ^{2}}}-{\frac{25\,{e}^{3}{c}^{5}}{4\,{b}^{2} \left ( be-cd \right ) ^{4} \left ( cex+be \right ) ^{2}}\sqrt{ex+d}}+{\frac{37\,{e}^{2}{c}^{6}d}{4\,{b}^{3} \left ( be-cd \right ) ^{4} \left ( cex+be \right ) ^{2}}\sqrt{ex+d}}-3\,{\frac{e{c}^{7}\sqrt{ex+d}{d}^{2}}{{b}^{4} \left ( be-cd \right ) ^{4} \left ( cex+be \right ) ^{2}}}-{\frac{143\,{e}^{2}{c}^{5}}{4\,{b}^{3} \left ( be-cd \right ) ^{4}}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}}+39\,{\frac{e{c}^{6}d}{{b}^{4} \left ( be-cd \right ) ^{4}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-12\,{\frac{{c}^{7}{d}^{2}}{{b}^{5} \left ( be-cd \right ) ^{4}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+{\frac{11}{4\,{b}^{3}{d}^{4}{x}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+3\,{\frac{ \left ( ex+d \right ) ^{3/2}c}{e{b}^{4}{d}^{3}{x}^{2}}}-{\frac{13}{4\,{b}^{3}{d}^{3}{x}^{2}}\sqrt{ex+d}}-3\,{\frac{\sqrt{ex+d}c}{e{b}^{4}{d}^{2}{x}^{2}}}-{\frac{35\,{e}^{2}}{4\,{b}^{3}}{\it Artanh} \left ({\sqrt{ex+d}{\frac{1}{\sqrt{d}}}} \right ){d}^{-{\frac{9}{2}}}}-15\,{\frac{ce}{{b}^{4}{d}^{7/2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }-12\,{\frac{{c}^{2}}{{b}^{5}{d}^{5/2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(5/2)/(c*x^2+b*x)^3,x)

[Out]

6*e^6/d^4/(b*e-c*d)^4/(e*x+d)^(1/2)*b-12*e^5/d^3/(b*e-c*d)^4/(e*x+d)^(1/2)*c+2/3*e^5/d^3/(b*e-c*d)^3/(e*x+d)^(
3/2)-23/4*e^2*c^6/b^3/(b*e-c*d)^4/(c*e*x+b*e)^2*(e*x+d)^(3/2)+3*e*c^7/b^4/(b*e-c*d)^4/(c*e*x+b*e)^2*(e*x+d)^(3
/2)*d-25/4*e^3*c^5/b^2/(b*e-c*d)^4/(c*e*x+b*e)^2*(e*x+d)^(1/2)+37/4*e^2*c^6/b^3/(b*e-c*d)^4/(c*e*x+b*e)^2*(e*x
+d)^(1/2)*d-3*e*c^7/b^4/(b*e-c*d)^4/(c*e*x+b*e)^2*(e*x+d)^(1/2)*d^2-143/4*e^2*c^5/b^3/(b*e-c*d)^4/((b*e-c*d)*c
)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))+39*e*c^6/b^4/(b*e-c*d)^4/((b*e-c*d)*c)^(1/2)*arctan((e*x+d
)^(1/2)*c/((b*e-c*d)*c)^(1/2))*d-12*c^7/b^5/(b*e-c*d)^4/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*
c)^(1/2))*d^2+11/4/b^3/d^4/x^2*(e*x+d)^(3/2)+3/e/b^4/d^3/x^2*(e*x+d)^(3/2)*c-13/4/b^3/d^3/x^2*(e*x+d)^(1/2)-3/
e/b^4/d^2/x^2*(e*x+d)^(1/2)*c-35/4*e^2/b^3/d^(9/2)*arctanh((e*x+d)^(1/2)/d^(1/2))-15*e/b^4/d^(7/2)*arctanh((e*
x+d)^(1/2)/d^(1/2))*c-12/b^5/d^(5/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*c^2

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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(1/(e*x+d)^(5/2)/(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 105.111, size = 14445, normalized size = 30.73 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

[1/24*(3*((48*c^8*d^7*e^2 - 156*b*c^7*d^6*e^3 + 143*b^2*c^6*d^5*e^4)*x^6 + 2*(48*c^8*d^8*e - 108*b*c^7*d^7*e^2
- 13*b^2*c^6*d^6*e^3 + 143*b^3*c^5*d^5*e^4)*x^5 + (48*c^8*d^9 + 36*b*c^7*d^8*e - 433*b^2*c^6*d^7*e^2 + 416*b^
3*c^5*d^6*e^3 + 143*b^4*c^4*d^5*e^4)*x^4 + 2*(48*b*c^7*d^9 - 108*b^2*c^6*d^8*e - 13*b^3*c^5*d^7*e^2 + 143*b^4*
c^4*d^6*e^3)*x^3 + (48*b^2*c^6*d^9 - 156*b^3*c^5*d^8*e + 143*b^4*c^4*d^7*e^2)*x^2)*sqrt(c/(c*d - b*e))*log((c*
e*x + 2*c*d - b*e + 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x + b)) + 3*((48*c^8*d^6*e^2 - 132*b*c
^7*d^5*e^3 + 83*b^2*c^6*d^4*e^4 + 28*b^3*c^5*d^3*e^5 + 18*b^4*c^4*d^2*e^6 - 80*b^5*c^3*d*e^7 + 35*b^6*c^2*e^8)
*x^6 + 2*(48*c^8*d^7*e - 84*b*c^7*d^6*e^2 - 49*b^2*c^6*d^5*e^3 + 111*b^3*c^5*d^4*e^4 + 46*b^4*c^4*d^3*e^5 - 62
*b^5*c^3*d^2*e^6 - 45*b^6*c^2*d*e^7 + 35*b^7*c*e^8)*x^5 + (48*c^8*d^8 + 60*b*c^7*d^7*e - 397*b^2*c^6*d^6*e^2 +
228*b^3*c^5*d^5*e^3 + 213*b^4*c^4*d^4*e^4 + 20*b^5*c^3*d^3*e^5 - 267*b^6*c^2*d^2*e^6 + 60*b^7*c*d*e^7 + 35*b^
8*e^8)*x^4 + 2*(48*b*c^7*d^8 - 84*b^2*c^6*d^7*e - 49*b^3*c^5*d^6*e^2 + 111*b^4*c^4*d^5*e^3 + 46*b^5*c^3*d^4*e^
4 - 62*b^6*c^2*d^3*e^5 - 45*b^7*c*d^2*e^6 + 35*b^8*d*e^7)*x^3 + (48*b^2*c^6*d^8 - 132*b^3*c^5*d^7*e + 83*b^4*c
^4*d^6*e^2 + 28*b^5*c^3*d^5*e^3 + 18*b^6*c^2*d^4*e^4 - 80*b^7*c*d^3*e^5 + 35*b^8*d^2*e^6)*x^2)*sqrt(d)*log((e*
x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(6*b^4*c^4*d^8 - 24*b^5*c^3*d^7*e + 36*b^6*c^2*d^6*e^2 - 24*b^7*c*d^
5*e^3 + 6*b^8*d^4*e^4 - 3*(24*b*c^7*d^6*e^2 - 60*b^2*c^6*d^5*e^3 + 28*b^3*c^5*d^4*e^4 + 18*b^4*c^4*d^3*e^5 - 8
0*b^5*c^3*d^2*e^6 + 35*b^6*c^2*d*e^7)*x^5 - (144*b*c^7*d^7*e - 252*b^2*c^6*d^6*e^2 - 105*b^3*c^5*d^5*e^3 + 240
*b^4*c^4*d^4*e^4 - 212*b^5*c^3*d^3*e^5 - 340*b^6*c^2*d^2*e^6 + 210*b^7*c*d*e^7)*x^4 - (72*b*c^7*d^8 + 36*b^2*c
^6*d^7*e - 438*b^3*c^5*d^6*e^2 + 255*b^4*c^4*d^5*e^3 + 180*b^5*c^3*d^4*e^4 - 565*b^6*c^2*d^3*e^5 + 40*b^7*c*d^
2*e^6 + 105*b^8*d*e^7)*x^3 - (108*b^2*c^6*d^8 - 225*b^3*c^5*d^7*e + 180*b^5*c^3*d^5*e^3 - 30*b^6*c^2*d^4*e^4 -
278*b^7*c*d^3*e^5 + 140*b^8*d^2*e^6)*x^2 - 3*(8*b^3*c^5*d^8 - 25*b^4*c^4*d^7*e + 20*b^5*c^3*d^6*e^2 + 10*b^6*
c^2*d^5*e^3 - 20*b^7*c*d^4*e^4 + 7*b^8*d^3*e^5)*x)*sqrt(e*x + d))/((b^5*c^6*d^9*e^2 - 4*b^6*c^5*d^8*e^3 + 6*b^
7*c^4*d^7*e^4 - 4*b^8*c^3*d^6*e^5 + b^9*c^2*d^5*e^6)*x^6 + 2*(b^5*c^6*d^10*e - 3*b^6*c^5*d^9*e^2 + 2*b^7*c^4*d
^8*e^3 + 2*b^8*c^3*d^7*e^4 - 3*b^9*c^2*d^6*e^5 + b^10*c*d^5*e^6)*x^5 + (b^5*c^6*d^11 - 9*b^7*c^4*d^9*e^2 + 16*
b^8*c^3*d^8*e^3 - 9*b^9*c^2*d^7*e^4 + b^11*d^5*e^6)*x^4 + 2*(b^6*c^5*d^11 - 3*b^7*c^4*d^10*e + 2*b^8*c^3*d^9*e
^2 + 2*b^9*c^2*d^8*e^3 - 3*b^10*c*d^7*e^4 + b^11*d^6*e^5)*x^3 + (b^7*c^4*d^11 - 4*b^8*c^3*d^10*e + 6*b^9*c^2*d
^9*e^2 - 4*b^10*c*d^8*e^3 + b^11*d^7*e^4)*x^2), 1/24*(6*((48*c^8*d^7*e^2 - 156*b*c^7*d^6*e^3 + 143*b^2*c^6*d^5
*e^4)*x^6 + 2*(48*c^8*d^8*e - 108*b*c^7*d^7*e^2 - 13*b^2*c^6*d^6*e^3 + 143*b^3*c^5*d^5*e^4)*x^5 + (48*c^8*d^9
+ 36*b*c^7*d^8*e - 433*b^2*c^6*d^7*e^2 + 416*b^3*c^5*d^6*e^3 + 143*b^4*c^4*d^5*e^4)*x^4 + 2*(48*b*c^7*d^9 - 10
8*b^2*c^6*d^8*e - 13*b^3*c^5*d^7*e^2 + 143*b^4*c^4*d^6*e^3)*x^3 + (48*b^2*c^6*d^9 - 156*b^3*c^5*d^8*e + 143*b^
4*c^4*d^7*e^2)*x^2)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(e*x + d)*sqrt(-c/(c*d - b*e))/(c*e*x + c*d))
+ 3*((48*c^8*d^6*e^2 - 132*b*c^7*d^5*e^3 + 83*b^2*c^6*d^4*e^4 + 28*b^3*c^5*d^3*e^5 + 18*b^4*c^4*d^2*e^6 - 80*
b^5*c^3*d*e^7 + 35*b^6*c^2*e^8)*x^6 + 2*(48*c^8*d^7*e - 84*b*c^7*d^6*e^2 - 49*b^2*c^6*d^5*e^3 + 111*b^3*c^5*d^
4*e^4 + 46*b^4*c^4*d^3*e^5 - 62*b^5*c^3*d^2*e^6 - 45*b^6*c^2*d*e^7 + 35*b^7*c*e^8)*x^5 + (48*c^8*d^8 + 60*b*c^
7*d^7*e - 397*b^2*c^6*d^6*e^2 + 228*b^3*c^5*d^5*e^3 + 213*b^4*c^4*d^4*e^4 + 20*b^5*c^3*d^3*e^5 - 267*b^6*c^2*d
^2*e^6 + 60*b^7*c*d*e^7 + 35*b^8*e^8)*x^4 + 2*(48*b*c^7*d^8 - 84*b^2*c^6*d^7*e - 49*b^3*c^5*d^6*e^2 + 111*b^4*
c^4*d^5*e^3 + 46*b^5*c^3*d^4*e^4 - 62*b^6*c^2*d^3*e^5 - 45*b^7*c*d^2*e^6 + 35*b^8*d*e^7)*x^3 + (48*b^2*c^6*d^8
- 132*b^3*c^5*d^7*e + 83*b^4*c^4*d^6*e^2 + 28*b^5*c^3*d^5*e^3 + 18*b^6*c^2*d^4*e^4 - 80*b^7*c*d^3*e^5 + 35*b^
8*d^2*e^6)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(6*b^4*c^4*d^8 - 24*b^5*c^3*d^7*e + 3
6*b^6*c^2*d^6*e^2 - 24*b^7*c*d^5*e^3 + 6*b^8*d^4*e^4 - 3*(24*b*c^7*d^6*e^2 - 60*b^2*c^6*d^5*e^3 + 28*b^3*c^5*d
^4*e^4 + 18*b^4*c^4*d^3*e^5 - 80*b^5*c^3*d^2*e^6 + 35*b^6*c^2*d*e^7)*x^5 - (144*b*c^7*d^7*e - 252*b^2*c^6*d^6*
e^2 - 105*b^3*c^5*d^5*e^3 + 240*b^4*c^4*d^4*e^4 - 212*b^5*c^3*d^3*e^5 - 340*b^6*c^2*d^2*e^6 + 210*b^7*c*d*e^7)
*x^4 - (72*b*c^7*d^8 + 36*b^2*c^6*d^7*e - 438*b^3*c^5*d^6*e^2 + 255*b^4*c^4*d^5*e^3 + 180*b^5*c^3*d^4*e^4 - 56
5*b^6*c^2*d^3*e^5 + 40*b^7*c*d^2*e^6 + 105*b^8*d*e^7)*x^3 - (108*b^2*c^6*d^8 - 225*b^3*c^5*d^7*e + 180*b^5*c^3
*d^5*e^3 - 30*b^6*c^2*d^4*e^4 - 278*b^7*c*d^3*e^5 + 140*b^8*d^2*e^6)*x^2 - 3*(8*b^3*c^5*d^8 - 25*b^4*c^4*d^7*e
+ 20*b^5*c^3*d^6*e^2 + 10*b^6*c^2*d^5*e^3 - 20*b^7*c*d^4*e^4 + 7*b^8*d^3*e^5)*x)*sqrt(e*x + d))/((b^5*c^6*d^9
*e^2 - 4*b^6*c^5*d^8*e^3 + 6*b^7*c^4*d^7*e^4 - 4*b^8*c^3*d^6*e^5 + b^9*c^2*d^5*e^6)*x^6 + 2*(b^5*c^6*d^10*e -
3*b^6*c^5*d^9*e^2 + 2*b^7*c^4*d^8*e^3 + 2*b^8*c^3*d^7*e^4 - 3*b^9*c^2*d^6*e^5 + b^10*c*d^5*e^6)*x^5 + (b^5*c^6
*d^11 - 9*b^7*c^4*d^9*e^2 + 16*b^8*c^3*d^8*e^3 - 9*b^9*c^2*d^7*e^4 + b^11*d^5*e^6)*x^4 + 2*(b^6*c^5*d^11 - 3*b
^7*c^4*d^10*e + 2*b^8*c^3*d^9*e^2 + 2*b^9*c^2*d^8*e^3 - 3*b^10*c*d^7*e^4 + b^11*d^6*e^5)*x^3 + (b^7*c^4*d^11 -
4*b^8*c^3*d^10*e + 6*b^9*c^2*d^9*e^2 - 4*b^10*c*d^8*e^3 + b^11*d^7*e^4)*x^2), 1/24*(6*((48*c^8*d^6*e^2 - 132*
b*c^7*d^5*e^3 + 83*b^2*c^6*d^4*e^4 + 28*b^3*c^5*d^3*e^5 + 18*b^4*c^4*d^2*e^6 - 80*b^5*c^3*d*e^7 + 35*b^6*c^2*e
^8)*x^6 + 2*(48*c^8*d^7*e - 84*b*c^7*d^6*e^2 - 49*b^2*c^6*d^5*e^3 + 111*b^3*c^5*d^4*e^4 + 46*b^4*c^4*d^3*e^5 -
62*b^5*c^3*d^2*e^6 - 45*b^6*c^2*d*e^7 + 35*b^7*c*e^8)*x^5 + (48*c^8*d^8 + 60*b*c^7*d^7*e - 397*b^2*c^6*d^6*e^
2 + 228*b^3*c^5*d^5*e^3 + 213*b^4*c^4*d^4*e^4 + 20*b^5*c^3*d^3*e^5 - 267*b^6*c^2*d^2*e^6 + 60*b^7*c*d*e^7 + 35
*b^8*e^8)*x^4 + 2*(48*b*c^7*d^8 - 84*b^2*c^6*d^7*e - 49*b^3*c^5*d^6*e^2 + 111*b^4*c^4*d^5*e^3 + 46*b^5*c^3*d^4
*e^4 - 62*b^6*c^2*d^3*e^5 - 45*b^7*c*d^2*e^6 + 35*b^8*d*e^7)*x^3 + (48*b^2*c^6*d^8 - 132*b^3*c^5*d^7*e + 83*b^
4*c^4*d^6*e^2 + 28*b^5*c^3*d^5*e^3 + 18*b^6*c^2*d^4*e^4 - 80*b^7*c*d^3*e^5 + 35*b^8*d^2*e^6)*x^2)*sqrt(-d)*arc
tan(sqrt(e*x + d)*sqrt(-d)/d) + 3*((48*c^8*d^7*e^2 - 156*b*c^7*d^6*e^3 + 143*b^2*c^6*d^5*e^4)*x^6 + 2*(48*c^8*
d^8*e - 108*b*c^7*d^7*e^2 - 13*b^2*c^6*d^6*e^3 + 143*b^3*c^5*d^5*e^4)*x^5 + (48*c^8*d^9 + 36*b*c^7*d^8*e - 433
*b^2*c^6*d^7*e^2 + 416*b^3*c^5*d^6*e^3 + 143*b^4*c^4*d^5*e^4)*x^4 + 2*(48*b*c^7*d^9 - 108*b^2*c^6*d^8*e - 13*b
^3*c^5*d^7*e^2 + 143*b^4*c^4*d^6*e^3)*x^3 + (48*b^2*c^6*d^9 - 156*b^3*c^5*d^8*e + 143*b^4*c^4*d^7*e^2)*x^2)*sq
rt(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e + 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x + b)) - 2*(
6*b^4*c^4*d^8 - 24*b^5*c^3*d^7*e + 36*b^6*c^2*d^6*e^2 - 24*b^7*c*d^5*e^3 + 6*b^8*d^4*e^4 - 3*(24*b*c^7*d^6*e^2
- 60*b^2*c^6*d^5*e^3 + 28*b^3*c^5*d^4*e^4 + 18*b^4*c^4*d^3*e^5 - 80*b^5*c^3*d^2*e^6 + 35*b^6*c^2*d*e^7)*x^5 -
(144*b*c^7*d^7*e - 252*b^2*c^6*d^6*e^2 - 105*b^3*c^5*d^5*e^3 + 240*b^4*c^4*d^4*e^4 - 212*b^5*c^3*d^3*e^5 - 34
0*b^6*c^2*d^2*e^6 + 210*b^7*c*d*e^7)*x^4 - (72*b*c^7*d^8 + 36*b^2*c^6*d^7*e - 438*b^3*c^5*d^6*e^2 + 255*b^4*c^
4*d^5*e^3 + 180*b^5*c^3*d^4*e^4 - 565*b^6*c^2*d^3*e^5 + 40*b^7*c*d^2*e^6 + 105*b^8*d*e^7)*x^3 - (108*b^2*c^6*d
^8 - 225*b^3*c^5*d^7*e + 180*b^5*c^3*d^5*e^3 - 30*b^6*c^2*d^4*e^4 - 278*b^7*c*d^3*e^5 + 140*b^8*d^2*e^6)*x^2 -
3*(8*b^3*c^5*d^8 - 25*b^4*c^4*d^7*e + 20*b^5*c^3*d^6*e^2 + 10*b^6*c^2*d^5*e^3 - 20*b^7*c*d^4*e^4 + 7*b^8*d^3*
e^5)*x)*sqrt(e*x + d))/((b^5*c^6*d^9*e^2 - 4*b^6*c^5*d^8*e^3 + 6*b^7*c^4*d^7*e^4 - 4*b^8*c^3*d^6*e^5 + b^9*c^2
*d^5*e^6)*x^6 + 2*(b^5*c^6*d^10*e - 3*b^6*c^5*d^9*e^2 + 2*b^7*c^4*d^8*e^3 + 2*b^8*c^3*d^7*e^4 - 3*b^9*c^2*d^6*
e^5 + b^10*c*d^5*e^6)*x^5 + (b^5*c^6*d^11 - 9*b^7*c^4*d^9*e^2 + 16*b^8*c^3*d^8*e^3 - 9*b^9*c^2*d^7*e^4 + b^11*
d^5*e^6)*x^4 + 2*(b^6*c^5*d^11 - 3*b^7*c^4*d^10*e + 2*b^8*c^3*d^9*e^2 + 2*b^9*c^2*d^8*e^3 - 3*b^10*c*d^7*e^4 +
b^11*d^6*e^5)*x^3 + (b^7*c^4*d^11 - 4*b^8*c^3*d^10*e + 6*b^9*c^2*d^9*e^2 - 4*b^10*c*d^8*e^3 + b^11*d^7*e^4)*x
^2), 1/12*(3*((48*c^8*d^7*e^2 - 156*b*c^7*d^6*e^3 + 143*b^2*c^6*d^5*e^4)*x^6 + 2*(48*c^8*d^8*e - 108*b*c^7*d^7
*e^2 - 13*b^2*c^6*d^6*e^3 + 143*b^3*c^5*d^5*e^4)*x^5 + (48*c^8*d^9 + 36*b*c^7*d^8*e - 433*b^2*c^6*d^7*e^2 + 41
6*b^3*c^5*d^6*e^3 + 143*b^4*c^4*d^5*e^4)*x^4 + 2*(48*b*c^7*d^9 - 108*b^2*c^6*d^8*e - 13*b^3*c^5*d^7*e^2 + 143*
b^4*c^4*d^6*e^3)*x^3 + (48*b^2*c^6*d^9 - 156*b^3*c^5*d^8*e + 143*b^4*c^4*d^7*e^2)*x^2)*sqrt(-c/(c*d - b*e))*ar
ctan(-(c*d - b*e)*sqrt(e*x + d)*sqrt(-c/(c*d - b*e))/(c*e*x + c*d)) + 3*((48*c^8*d^6*e^2 - 132*b*c^7*d^5*e^3 +
83*b^2*c^6*d^4*e^4 + 28*b^3*c^5*d^3*e^5 + 18*b^4*c^4*d^2*e^6 - 80*b^5*c^3*d*e^7 + 35*b^6*c^2*e^8)*x^6 + 2*(48
*c^8*d^7*e - 84*b*c^7*d^6*e^2 - 49*b^2*c^6*d^5*e^3 + 111*b^3*c^5*d^4*e^4 + 46*b^4*c^4*d^3*e^5 - 62*b^5*c^3*d^2
*e^6 - 45*b^6*c^2*d*e^7 + 35*b^7*c*e^8)*x^5 + (48*c^8*d^8 + 60*b*c^7*d^7*e - 397*b^2*c^6*d^6*e^2 + 228*b^3*c^5
*d^5*e^3 + 213*b^4*c^4*d^4*e^4 + 20*b^5*c^3*d^3*e^5 - 267*b^6*c^2*d^2*e^6 + 60*b^7*c*d*e^7 + 35*b^8*e^8)*x^4 +
2*(48*b*c^7*d^8 - 84*b^2*c^6*d^7*e - 49*b^3*c^5*d^6*e^2 + 111*b^4*c^4*d^5*e^3 + 46*b^5*c^3*d^4*e^4 - 62*b^6*c
^2*d^3*e^5 - 45*b^7*c*d^2*e^6 + 35*b^8*d*e^7)*x^3 + (48*b^2*c^6*d^8 - 132*b^3*c^5*d^7*e + 83*b^4*c^4*d^6*e^2 +
28*b^5*c^3*d^5*e^3 + 18*b^6*c^2*d^4*e^4 - 80*b^7*c*d^3*e^5 + 35*b^8*d^2*e^6)*x^2)*sqrt(-d)*arctan(sqrt(e*x +
d)*sqrt(-d)/d) - (6*b^4*c^4*d^8 - 24*b^5*c^3*d^7*e + 36*b^6*c^2*d^6*e^2 - 24*b^7*c*d^5*e^3 + 6*b^8*d^4*e^4 - 3
*(24*b*c^7*d^6*e^2 - 60*b^2*c^6*d^5*e^3 + 28*b^3*c^5*d^4*e^4 + 18*b^4*c^4*d^3*e^5 - 80*b^5*c^3*d^2*e^6 + 35*b^
6*c^2*d*e^7)*x^5 - (144*b*c^7*d^7*e - 252*b^2*c^6*d^6*e^2 - 105*b^3*c^5*d^5*e^3 + 240*b^4*c^4*d^4*e^4 - 212*b^
5*c^3*d^3*e^5 - 340*b^6*c^2*d^2*e^6 + 210*b^7*c*d*e^7)*x^4 - (72*b*c^7*d^8 + 36*b^2*c^6*d^7*e - 438*b^3*c^5*d^
6*e^2 + 255*b^4*c^4*d^5*e^3 + 180*b^5*c^3*d^4*e^4 - 565*b^6*c^2*d^3*e^5 + 40*b^7*c*d^2*e^6 + 105*b^8*d*e^7)*x^
3 - (108*b^2*c^6*d^8 - 225*b^3*c^5*d^7*e + 180*b^5*c^3*d^5*e^3 - 30*b^6*c^2*d^4*e^4 - 278*b^7*c*d^3*e^5 + 140*
b^8*d^2*e^6)*x^2 - 3*(8*b^3*c^5*d^8 - 25*b^4*c^4*d^7*e + 20*b^5*c^3*d^6*e^2 + 10*b^6*c^2*d^5*e^3 - 20*b^7*c*d^
4*e^4 + 7*b^8*d^3*e^5)*x)*sqrt(e*x + d))/((b^5*c^6*d^9*e^2 - 4*b^6*c^5*d^8*e^3 + 6*b^7*c^4*d^7*e^4 - 4*b^8*c^3
*d^6*e^5 + b^9*c^2*d^5*e^6)*x^6 + 2*(b^5*c^6*d^10*e - 3*b^6*c^5*d^9*e^2 + 2*b^7*c^4*d^8*e^3 + 2*b^8*c^3*d^7*e^
4 - 3*b^9*c^2*d^6*e^5 + b^10*c*d^5*e^6)*x^5 + (b^5*c^6*d^11 - 9*b^7*c^4*d^9*e^2 + 16*b^8*c^3*d^8*e^3 - 9*b^9*c
^2*d^7*e^4 + b^11*d^5*e^6)*x^4 + 2*(b^6*c^5*d^11 - 3*b^7*c^4*d^10*e + 2*b^8*c^3*d^9*e^2 + 2*b^9*c^2*d^8*e^3 -
3*b^10*c*d^7*e^4 + b^11*d^6*e^5)*x^3 + (b^7*c^4*d^11 - 4*b^8*c^3*d^10*e + 6*b^9*c^2*d^9*e^2 - 4*b^10*c*d^8*e^3
+ b^11*d^7*e^4)*x^2)]

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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(1/(e*x+d)**(5/2)/(c*x**2+b*x)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.84377, size = 1272, normalized size = 2.71 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

-1/4*(48*c^7*d^2 - 156*b*c^6*d*e + 143*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*(18*(x*e + d)*c*
d*e^5 + c*d^2*e^5 - 9*(x*e + d)*b*e^6 - 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*(24*(x*e + d)^(7/2)*c^7*d^5*e - 72*(x*e + d)^(5/2)*c^7*d^6*e + 72*(x*
e + d)^(3/2)*c^7*d^7*e - 24*sqrt(x*e + d)*c^7*d^8*e - 60*(x*e + d)^(7/2)*b*c^6*d^4*e^2 + 216*(x*e + d)^(5/2)*b
*c^6*d^5*e^2 - 252*(x*e + d)^(3/2)*b*c^6*d^6*e^2 + 96*sqrt(x*e + d)*b*c^6*d^7*e^2 + 28*(x*e + d)^(7/2)*b^2*c^5
*d^3*e^3 - 175*(x*e + d)^(5/2)*b^2*c^5*d^4*e^3 + 274*(x*e + d)^(3/2)*b^2*c^5*d^5*e^3 - 127*sqrt(x*e + d)*b^2*c
^5*d^6*e^3 + 18*(x*e + d)^(7/2)*b^3*c^4*d^2*e^4 - 10*(x*e + d)^(5/2)*b^3*c^4*d^3*e^4 - 55*(x*e + d)^(3/2)*b^3*
c^4*d^4*e^4 + 45*sqrt(x*e + d)*b^3*c^4*d^5*e^4 - 32*(x*e + d)^(7/2)*b^4*c^3*d*e^5 + 140*(x*e + d)^(5/2)*b^4*c^
3*d^2*e^5 - 180*(x*e + d)^(3/2)*b^4*c^3*d^3*e^5 + 80*sqrt(x*e + d)*b^4*c^3*d^4*e^5 + 11*(x*e + d)^(7/2)*b^5*c^
2*e^6 - 99*(x*e + d)^(5/2)*b^5*c^2*d*e^6 + 199*(x*e + d)^(3/2)*b^5*c^2*d^2*e^6 - 123*sqrt(x*e + d)*b^5*c^2*d^3
*e^6 + 22*(x*e + d)^(5/2)*b^6*c*e^7 - 80*(x*e + d)^(3/2)*b^6*c*d*e^7 + 66*sqrt(x*e + d)*b^6*c*d^2*e^7 + 11*(x*
e + d)^(3/2)*b^7*e^8 - 13*sqrt(x*e + d)*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*(48*c^2*d
^2 + 60*b*c*d*e + 35*b^2*e^2)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^5*sqrt(-d)*d^4)