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

Optimal. Leaf size=296 \[ -\frac{e \sqrt{b x+c x^2} (2 c d-b e) \left (15 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{4 b^2 d^3 (d+e x) (c d-b e)^3}-\frac{e \sqrt{b x+c x^2} \left (5 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{2 b^2 d^2 (d+e x)^2 (c d-b e)^2}+\frac{3 e^2 \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 d^{7/2} (c d-b e)^{7/2}}-\frac{2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (d+e x)^2 (c d-b e)} \]

[Out]

(-2*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(b^2*d*(c*d - b*e)*(d + e*x)^2*Sqrt[b*x + c*x^2]) - (e*(8*c^2*d^2 - 8
*b*c*d*e + 5*b^2*e^2)*Sqrt[b*x + c*x^2])/(2*b^2*d^2*(c*d - b*e)^2*(d + e*x)^2) - (e*(2*c*d - b*e)*(8*c^2*d^2 -
8*b*c*d*e + 15*b^2*e^2)*Sqrt[b*x + c*x^2])/(4*b^2*d^3*(c*d - b*e)^3*(d + e*x)) + (3*e^2*(16*c^2*d^2 - 16*b*c*
d*e + 5*b^2*e^2)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(8*d^(7/2)*(c
*d - b*e)^(7/2))

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Rubi [A]  time = 0.343582, antiderivative size = 296, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {740, 834, 806, 724, 206} \[ -\frac{e \sqrt{b x+c x^2} (2 c d-b e) \left (15 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{4 b^2 d^3 (d+e x) (c d-b e)^3}-\frac{e \sqrt{b x+c x^2} \left (5 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{2 b^2 d^2 (d+e x)^2 (c d-b e)^2}+\frac{3 e^2 \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 d^{7/2} (c d-b e)^{7/2}}-\frac{2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (d+e x)^2 (c d-b e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(b^2*d*(c*d - b*e)*(d + e*x)^2*Sqrt[b*x + c*x^2]) - (e*(8*c^2*d^2 - 8
*b*c*d*e + 5*b^2*e^2)*Sqrt[b*x + c*x^2])/(2*b^2*d^2*(c*d - b*e)^2*(d + e*x)^2) - (e*(2*c*d - b*e)*(8*c^2*d^2 -
8*b*c*d*e + 15*b^2*e^2)*Sqrt[b*x + c*x^2])/(4*b^2*d^3*(c*d - b*e)^3*(d + e*x)) + (3*e^2*(16*c^2*d^2 - 16*b*c*
d*e + 5*b^2*e^2)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(8*d^(7/2)*(c
*d - b*e)^(7/2))

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 834

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

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(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] && EqQ[Sim
plify[m + 2*p + 3], 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]

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])

Rubi steps

\begin{align*} \int \frac{1}{(d+e x)^3 \left (b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x)^2 \sqrt{b x+c x^2}}-\frac{2 \int \frac{\frac{1}{2} b e (4 c d-5 b e)+2 c e (2 c d-b e) x}{(d+e x)^3 \sqrt{b x+c x^2}} \, dx}{b^2 d (c d-b e)}\\ &=-\frac{2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x)^2 \sqrt{b x+c x^2}}-\frac{e \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right ) \sqrt{b x+c x^2}}{2 b^2 d^2 (c d-b e)^2 (d+e x)^2}+\frac{\int \frac{-\frac{1}{4} b e \left (8 c^2 d^2-28 b c d e+15 b^2 e^2\right )-\frac{1}{2} c e \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right ) x}{(d+e x)^2 \sqrt{b x+c x^2}} \, dx}{b^2 d^2 (c d-b e)^2}\\ &=-\frac{2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x)^2 \sqrt{b x+c x^2}}-\frac{e \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right ) \sqrt{b x+c x^2}}{2 b^2 d^2 (c d-b e)^2 (d+e x)^2}-\frac{e (2 c d-b e) \left (8 c^2 d^2-8 b c d e+15 b^2 e^2\right ) \sqrt{b x+c x^2}}{4 b^2 d^3 (c d-b e)^3 (d+e x)}+\frac{\left (3 e^2 \left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right )\right ) \int \frac{1}{(d+e x) \sqrt{b x+c x^2}} \, dx}{8 d^3 (c d-b e)^3}\\ &=-\frac{2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x)^2 \sqrt{b x+c x^2}}-\frac{e \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right ) \sqrt{b x+c x^2}}{2 b^2 d^2 (c d-b e)^2 (d+e x)^2}-\frac{e (2 c d-b e) \left (8 c^2 d^2-8 b c d e+15 b^2 e^2\right ) \sqrt{b x+c x^2}}{4 b^2 d^3 (c d-b e)^3 (d+e x)}-\frac{\left (3 e^2 \left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right )\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 )}{4 d^3 (c d-b e)^3}\\ &=-\frac{2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x)^2 \sqrt{b x+c x^2}}-\frac{e \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right ) \sqrt{b x+c x^2}}{2 b^2 d^2 (c d-b e)^2 (d+e x)^2}-\frac{e (2 c d-b e) \left (8 c^2 d^2-8 b c d e+15 b^2 e^2\right ) \sqrt{b x+c x^2}}{4 b^2 d^3 (c d-b e)^3 (d+e x)}+\frac{3 e^2 \left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right ) \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 )}{8 d^{7/2} (c d-b e)^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.892341, size = 247, normalized size = 0.83 \[ \frac{\frac{2 b^2 c^2 d e (19 e x-18 d)+b^3 c e^2 (43 d-15 e x)-15 b^4 e^3+8 b c^3 d^2 (d-3 e x)+16 c^4 d^3 x}{b^2 d^2 (c d-b e)^2}-\frac{3 e^2 \sqrt{x} \sqrt{b+c x} \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{d^{5/2} (b e-c d)^{5/2}}+\frac{5 e (2 c d-b e)}{d (d+e x) (c d-b e)}+\frac{2 e}{(d+e x)^2}}{4 d \sqrt{x (b+c x)} (b e-c d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*e)/(d + e*x)^2 + (5*e*(2*c*d - b*e))/(d*(c*d - b*e)*(d + e*x)) + (-15*b^4*e^3 + 16*c^4*d^3*x + b^3*c*e^2*(
43*d - 15*e*x) + 8*b*c^3*d^2*(d - 3*e*x) + 2*b^2*c^2*d*e*(-18*d + 19*e*x))/(b^2*d^2*(c*d - b*e)^2) - (3*e^2*(1
6*c^2*d^2 - 16*b*c*d*e + 5*b^2*e^2)*Sqrt[x]*Sqrt[b + c*x]*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b
+ c*x])])/(d^(5/2)*(-(c*d) + b*e)^(5/2)))/(4*d*(-(c*d) + b*e)*Sqrt[x*(b + c*x)])

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Maple [B]  time = 0.244, size = 1612, normalized size = 5.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/e/d/(b*e-c*d)/(d/e+x)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)+5/4*e/d^2/(b*e-c*d)^2/(d
/e+x)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b-5/2/d/(b*e-c*d)^2/(d/e+x)/(c*(d/e+x)^2+(b*e-
2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*c-15/4*e^3/d^3/(b*e-c*d)^3/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c
*d)/e^2)^(1/2)*b^2+75/4*e^2/d^2/(b*e-c*d)^3/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b*c-30*e
/d/(b*e-c*d)^3/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*c^2-15/4*e^3/d^3/(b*e-c*d)^3/(c*(d/e+
x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*c*b+45/2*e^2/d^2/(b*e-c*d)^3/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d
/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*c^2-45*e/d/(b*e-c*d)^3/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1
/2)*x*c^3+30/(b*e-c*d)^3/b^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*c^4+15/(b*e-c*d)^3/b/
(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*c^3+15/8*e^3/d^3/(b*e-c*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*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d
*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b^2-15/2*e^2/d^2/(b*e-c*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*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/
e+x))*b*c+15/2*e/d/(b*e-c*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*(d/e+x)+2*(-d*(b*
e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*c^2-13*e/d^2/(b*e-c*d)^2
*c^2/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x+26/d/(b*e-c*d)^2*c^3/b^2/(c*(d/e+x)^2+(b*e-
2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x-8*e/d^2/(b*e-c*d)^2*c/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)
/e^2)^(1/2)+13/d/(b*e-c*d)^2*c^2/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)+3/2*e*c/d^2/(b*e-
c*d)^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/
e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.53763, size = 3347, normalized size = 11.31 \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)^3/(c*x^2+b*x)^(3/2),x, algorithm="fricas")

[Out]

[-1/8*(3*((16*b^2*c^3*d^2*e^4 - 16*b^3*c^2*d*e^5 + 5*b^4*c*e^6)*x^4 + (32*b^2*c^3*d^3*e^3 - 16*b^3*c^2*d^2*e^4
- 6*b^4*c*d*e^5 + 5*b^5*e^6)*x^3 + (16*b^2*c^3*d^4*e^2 + 16*b^3*c^2*d^3*e^3 - 27*b^4*c*d^2*e^4 + 10*b^5*d*e^5
)*x^2 + (16*b^3*c^2*d^4*e^2 - 16*b^4*c*d^3*e^3 + 5*b^5*d^2*e^4)*x)*sqrt(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e
)*x - 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d)) + 2*(8*b*c^4*d^7 - 32*b^2*c^3*d^6*e + 48*b^3*c^2*d^5
*e^2 - 32*b^4*c*d^4*e^3 + 8*b^5*d^3*e^4 + (16*c^5*d^5*e^2 - 40*b*c^4*d^4*e^3 + 62*b^2*c^3*d^3*e^4 - 53*b^3*c^2
*d^2*e^5 + 15*b^4*c*d*e^6)*x^3 + (32*c^5*d^6*e - 72*b*c^4*d^5*e^2 + 80*b^2*c^3*d^4*e^3 - 27*b^3*c^2*d^3*e^4 -
28*b^4*c*d^2*e^5 + 15*b^5*d*e^6)*x^2 + (16*c^5*d^7 - 24*b*c^4*d^6*e - 16*b^2*c^3*d^5*e^2 + 80*b^3*c^2*d^4*e^3
- 81*b^4*c*d^3*e^4 + 25*b^5*d^2*e^5)*x)*sqrt(c*x^2 + b*x))/((b^2*c^5*d^8*e^2 - 4*b^3*c^4*d^7*e^3 + 6*b^4*c^3*d
^6*e^4 - 4*b^5*c^2*d^5*e^5 + b^6*c*d^4*e^6)*x^4 + (2*b^2*c^5*d^9*e - 7*b^3*c^4*d^8*e^2 + 8*b^4*c^3*d^7*e^3 - 2
*b^5*c^2*d^6*e^4 - 2*b^6*c*d^5*e^5 + b^7*d^4*e^6)*x^3 + (b^2*c^5*d^10 - 2*b^3*c^4*d^9*e - 2*b^4*c^3*d^8*e^2 +
8*b^5*c^2*d^7*e^3 - 7*b^6*c*d^6*e^4 + 2*b^7*d^5*e^5)*x^2 + (b^3*c^4*d^10 - 4*b^4*c^3*d^9*e + 6*b^5*c^2*d^8*e^2
- 4*b^6*c*d^7*e^3 + b^7*d^6*e^4)*x), 1/4*(3*((16*b^2*c^3*d^2*e^4 - 16*b^3*c^2*d*e^5 + 5*b^4*c*e^6)*x^4 + (32*
b^2*c^3*d^3*e^3 - 16*b^3*c^2*d^2*e^4 - 6*b^4*c*d*e^5 + 5*b^5*e^6)*x^3 + (16*b^2*c^3*d^4*e^2 + 16*b^3*c^2*d^3*e
^3 - 27*b^4*c*d^2*e^4 + 10*b^5*d*e^5)*x^2 + (16*b^3*c^2*d^4*e^2 - 16*b^4*c*d^3*e^3 + 5*b^5*d^2*e^4)*x)*sqrt(-c
*d^2 + b*d*e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) - (8*b*c^4*d^7 - 32*b^2*c^3*d^6*
e + 48*b^3*c^2*d^5*e^2 - 32*b^4*c*d^4*e^3 + 8*b^5*d^3*e^4 + (16*c^5*d^5*e^2 - 40*b*c^4*d^4*e^3 + 62*b^2*c^3*d^
3*e^4 - 53*b^3*c^2*d^2*e^5 + 15*b^4*c*d*e^6)*x^3 + (32*c^5*d^6*e - 72*b*c^4*d^5*e^2 + 80*b^2*c^3*d^4*e^3 - 27*
b^3*c^2*d^3*e^4 - 28*b^4*c*d^2*e^5 + 15*b^5*d*e^6)*x^2 + (16*c^5*d^7 - 24*b*c^4*d^6*e - 16*b^2*c^3*d^5*e^2 + 8
0*b^3*c^2*d^4*e^3 - 81*b^4*c*d^3*e^4 + 25*b^5*d^2*e^5)*x)*sqrt(c*x^2 + b*x))/((b^2*c^5*d^8*e^2 - 4*b^3*c^4*d^7
*e^3 + 6*b^4*c^3*d^6*e^4 - 4*b^5*c^2*d^5*e^5 + b^6*c*d^4*e^6)*x^4 + (2*b^2*c^5*d^9*e - 7*b^3*c^4*d^8*e^2 + 8*b
^4*c^3*d^7*e^3 - 2*b^5*c^2*d^6*e^4 - 2*b^6*c*d^5*e^5 + b^7*d^4*e^6)*x^3 + (b^2*c^5*d^10 - 2*b^3*c^4*d^9*e - 2*
b^4*c^3*d^8*e^2 + 8*b^5*c^2*d^7*e^3 - 7*b^6*c*d^6*e^4 + 2*b^7*d^5*e^5)*x^2 + (b^3*c^4*d^10 - 4*b^4*c^3*d^9*e +
6*b^5*c^2*d^8*e^2 - 4*b^6*c*d^7*e^3 + b^7*d^6*e^4)*x)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (x \left (b + c x\right )\right )^{\frac{3}{2}} \left (d + e x\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.97243, size = 975, normalized size = 3.29 \begin{align*} -\frac{2 \,{\left (\frac{{\left (2 \, c^{4} d^{6} - 3 \, b c^{3} d^{5} e + 3 \, b^{2} c^{2} d^{4} e^{2} - b^{3} c d^{3} e^{3}\right )} x}{b^{2} c^{3} d^{9} - 3 \, b^{3} c^{2} d^{8} e + 3 \, b^{4} c d^{7} e^{2} - b^{5} d^{6} e^{3}} + \frac{b c^{3} d^{6} - 3 \, b^{2} c^{2} d^{5} e + 3 \, b^{3} c d^{4} e^{2} - b^{4} d^{3} e^{3}}{b^{2} c^{3} d^{9} - 3 \, b^{3} c^{2} d^{8} e + 3 \, b^{4} c d^{7} e^{2} - b^{5} d^{6} e^{3}}\right )}}{\sqrt{c x^{2} + b x}} + \frac{3 \,{\left (16 \, c^{2} d^{2} e^{2} - 16 \, b c d e^{3} + 5 \, b^{2} e^{4}\right )} \arctan \left (-\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} + b d e}}\right )}{4 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} \sqrt{-c d^{2} + b d e}} - \frac{56 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} c^{\frac{5}{2}} d^{3} e^{2} + 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} c^{2} d^{2} e^{3} + 56 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b c^{2} d^{3} e^{2} - 48 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} b c^{\frac{3}{2}} d^{2} e^{3} + 14 \, b^{2} c^{\frac{3}{2}} d^{3} e^{2} - 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} b c d e^{4} - 44 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b^{2} c d^{2} e^{3} + 13 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} b^{2} \sqrt{c} d e^{4} - 7 \, b^{3} \sqrt{c} d^{2} e^{3} + 7 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} b^{2} e^{5} + 9 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b^{3} d e^{4}}{4 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )}{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} e + 2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} d + b d\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*((2*c^4*d^6 - 3*b*c^3*d^5*e + 3*b^2*c^2*d^4*e^2 - b^3*c*d^3*e^3)*x/(b^2*c^3*d^9 - 3*b^3*c^2*d^8*e + 3*b^4*c
*d^7*e^2 - b^5*d^6*e^3) + (b*c^3*d^6 - 3*b^2*c^2*d^5*e + 3*b^3*c*d^4*e^2 - b^4*d^3*e^3)/(b^2*c^3*d^9 - 3*b^3*c
^2*d^8*e + 3*b^4*c*d^7*e^2 - b^5*d^6*e^3))/sqrt(c*x^2 + b*x) + 3/4*(16*c^2*d^2*e^2 - 16*b*c*d*e^3 + 5*b^2*e^4)
*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))/((c^3*d^6 - 3*b*c^2*d^5*e + 3*b
^2*c*d^4*e^2 - b^3*d^3*e^3)*sqrt(-c*d^2 + b*d*e)) - 1/4*(56*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*c^(5/2)*d^3*e^2
+ 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*c^2*d^2*e^3 + 56*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b*c^2*d^3*e^2 - 48*(sq
rt(c)*x - sqrt(c*x^2 + b*x))^2*b*c^(3/2)*d^2*e^3 + 14*b^2*c^(3/2)*d^3*e^2 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))
^3*b*c*d*e^4 - 44*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^2*c*d^2*e^3 + 13*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^2*sqr
t(c)*d*e^4 - 7*b^3*sqrt(c)*d^2*e^3 + 7*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b^2*e^5 + 9*(sqrt(c)*x - sqrt(c*x^2 +
b*x))*b^3*d*e^4)/((c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3)*((sqrt(c)*x - sqrt(c*x^2 + b*x))^
2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c)*d + b*d)^2)