### 3.335 $$\int \frac{1}{(d+e x)^2 (b x+c x^2)^{5/2}} \, dx$$

Optimal. Leaf size=348 $\frac{e \sqrt{b x+c x^2} \left (12 b^2 c^2 d^2 e^2+20 b^3 c d e^3-15 b^4 e^4-64 b c^3 d^3 e+32 c^4 d^4\right )}{3 b^4 d^3 (d+e x) (c d-b e)^3}+\frac{2 \left (c x (2 c d-b e) \left (-5 b^2 e^2-8 b c d e+8 c^2 d^2\right )+b (c d-b e) \left (-5 b^2 e^2-2 b c d e+8 c^2 d^2\right )\right )}{3 b^4 d^2 \sqrt{b x+c x^2} (d+e x) (c d-b e)^2}-\frac{2 (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (d+e x) (c d-b e)}+\frac{5 e^4 (2 c d-b e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 d^{7/2} (c d-b e)^{7/2}}$

[Out]

(-2*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(3*b^2*d*(c*d - b*e)*(d + e*x)*(b*x + c*x^2)^(3/2)) + (2*(b*(c*d - b*
e)*(8*c^2*d^2 - 2*b*c*d*e - 5*b^2*e^2) + c*(2*c*d - b*e)*(8*c^2*d^2 - 8*b*c*d*e - 5*b^2*e^2)*x))/(3*b^4*d^2*(c
*d - b*e)^2*(d + e*x)*Sqrt[b*x + c*x^2]) + (e*(32*c^4*d^4 - 64*b*c^3*d^3*e + 12*b^2*c^2*d^2*e^2 + 20*b^3*c*d*e
^3 - 15*b^4*e^4)*Sqrt[b*x + c*x^2])/(3*b^4*d^3*(c*d - b*e)^3*(d + e*x)) + (5*e^4*(2*c*d - b*e)*ArcTanh[(b*d +
(2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(2*d^(7/2)*(c*d - b*e)^(7/2))

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Rubi [A]  time = 0.348716, antiderivative size = 348, 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, 822, 806, 724, 206} $\frac{e \sqrt{b x+c x^2} \left (12 b^2 c^2 d^2 e^2+20 b^3 c d e^3-15 b^4 e^4-64 b c^3 d^3 e+32 c^4 d^4\right )}{3 b^4 d^3 (d+e x) (c d-b e)^3}+\frac{2 \left (c x (2 c d-b e) \left (-5 b^2 e^2-8 b c d e+8 c^2 d^2\right )+b (c d-b e) \left (-5 b^2 e^2-2 b c d e+8 c^2 d^2\right )\right )}{3 b^4 d^2 \sqrt{b x+c x^2} (d+e x) (c d-b e)^2}-\frac{2 (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (d+e x) (c d-b e)}+\frac{5 e^4 (2 c d-b e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 d^{7/2} (c d-b e)^{7/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(3*b^2*d*(c*d - b*e)*(d + e*x)*(b*x + c*x^2)^(3/2)) + (2*(b*(c*d - b*
e)*(8*c^2*d^2 - 2*b*c*d*e - 5*b^2*e^2) + c*(2*c*d - b*e)*(8*c^2*d^2 - 8*b*c*d*e - 5*b^2*e^2)*x))/(3*b^4*d^2*(c
*d - b*e)^2*(d + e*x)*Sqrt[b*x + c*x^2]) + (e*(32*c^4*d^4 - 64*b*c^3*d^3*e + 12*b^2*c^2*d^2*e^2 + 20*b^3*c*d*e
^3 - 15*b^4*e^4)*Sqrt[b*x + c*x^2])/(3*b^4*d^3*(c*d - b*e)^3*(d + e*x)) + (5*e^4*(2*c*d - b*e)*ArcTanh[(b*d +
(2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(2*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 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 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)^2 \left (b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) (d+e x) \left (b x+c x^2\right )^{3/2}}-\frac{2 \int \frac{\frac{1}{2} \left (8 c^2 d^2-2 b c d e-5 b^2 e^2\right )+3 c e (2 c d-b e) x}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2 d (c d-b e)}\\ &=-\frac{2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) (d+e x) \left (b x+c x^2\right )^{3/2}}+\frac{2 \left (b (c d-b e) \left (8 c^2 d^2-2 b c d e-5 b^2 e^2\right )+c (2 c d-b e) \left (8 c^2 d^2-8 b c d e-5 b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 (d+e x) \sqrt{b x+c x^2}}+\frac{4 \int \frac{\frac{1}{4} b e \left (16 c^3 d^3-16 b c^2 d^2 e-10 b^2 c d e^2+15 b^3 e^3\right )+\frac{1}{2} c e (2 c d-b 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}{3 b^4 d^2 (c d-b e)^2}\\ &=-\frac{2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) (d+e x) \left (b x+c x^2\right )^{3/2}}+\frac{2 \left (b (c d-b e) \left (8 c^2 d^2-2 b c d e-5 b^2 e^2\right )+c (2 c d-b e) \left (8 c^2 d^2-8 b c d e-5 b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 (d+e x) \sqrt{b x+c x^2}}+\frac{e \left (32 c^4 d^4-64 b c^3 d^3 e+12 b^2 c^2 d^2 e^2+20 b^3 c d e^3-15 b^4 e^4\right ) \sqrt{b x+c x^2}}{3 b^4 d^3 (c d-b e)^3 (d+e x)}+\frac{\left (5 e^4 (2 c d-b e)\right ) \int \frac{1}{(d+e x) \sqrt{b x+c x^2}} \, dx}{2 d^3 (c d-b e)^3}\\ &=-\frac{2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) (d+e x) \left (b x+c x^2\right )^{3/2}}+\frac{2 \left (b (c d-b e) \left (8 c^2 d^2-2 b c d e-5 b^2 e^2\right )+c (2 c d-b e) \left (8 c^2 d^2-8 b c d e-5 b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 (d+e x) \sqrt{b x+c x^2}}+\frac{e \left (32 c^4 d^4-64 b c^3 d^3 e+12 b^2 c^2 d^2 e^2+20 b^3 c d e^3-15 b^4 e^4\right ) \sqrt{b x+c x^2}}{3 b^4 d^3 (c d-b e)^3 (d+e x)}-\frac{\left (5 e^4 (2 c d-b e)\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 )}{d^3 (c d-b e)^3}\\ &=-\frac{2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) (d+e x) \left (b x+c x^2\right )^{3/2}}+\frac{2 \left (b (c d-b e) \left (8 c^2 d^2-2 b c d e-5 b^2 e^2\right )+c (2 c d-b e) \left (8 c^2 d^2-8 b c d e-5 b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 (d+e x) \sqrt{b x+c x^2}}+\frac{e \left (32 c^4 d^4-64 b c^3 d^3 e+12 b^2 c^2 d^2 e^2+20 b^3 c d e^3-15 b^4 e^4\right ) \sqrt{b x+c x^2}}{3 b^4 d^3 (c d-b e)^3 (d+e x)}+\frac{5 e^4 (2 c d-b e) \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 )}{2 d^{7/2} (c d-b e)^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.988586, size = 316, normalized size = 0.91 $\frac{x \left (\frac{c x^2 (b+c x)^2 \left (-12 b^2 c^2 d^2 e^2-20 b^3 c d e^3+15 b^4 e^4+64 b c^3 d^3 e-32 c^4 d^4\right )}{b^4 d^2 (c d-b e)^2}+\frac{c x^2 (b+c x) \left (-10 b^2 c d e^2+15 b^3 e^3-16 b c^2 d^2 e+16 c^3 d^3\right )}{b^3 d^2 (b e-c d)}+\frac{3 x (b+c x) \left (5 b^2 e^2-4 c^2 d^2\right )}{b^2 d^2}+\frac{15 e^4 x^{3/2} (b+c x)^{5/2} (b e-2 c d) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{d^{5/2} (b e-c d)^{5/2}}+\frac{3 e (b+c x)}{d+e x}+\frac{(b+c x) (2 c d-5 b e)}{b d}\right )}{3 d (x (b+c x))^{5/2} (b e-c d)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(x*(((2*c*d - 5*b*e)*(b + c*x))/(b*d) + (3*(-4*c^2*d^2 + 5*b^2*e^2)*x*(b + c*x))/(b^2*d^2) + (c*(16*c^3*d^3 -
16*b*c^2*d^2*e - 10*b^2*c*d*e^2 + 15*b^3*e^3)*x^2*(b + c*x))/(b^3*d^2*(-(c*d) + b*e)) + (c*(-32*c^4*d^4 + 64*b
*c^3*d^3*e - 12*b^2*c^2*d^2*e^2 - 20*b^3*c*d*e^3 + 15*b^4*e^4)*x^2*(b + c*x)^2)/(b^4*d^2*(c*d - b*e)^2) + (3*e
*(b + c*x))/(d + e*x) + (15*e^4*(-2*c*d + b*e)*x^(3/2)*(b + c*x)^(5/2)*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sq
rt[d]*Sqrt[b + c*x])])/(d^(5/2)*(-(c*d) + b*e)^(5/2))))/(3*d*(-(c*d) + b*e)*(x*(b + c*x))^(5/2))

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Maple [B]  time = 0.219, size = 1857, normalized size = 5.3 \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)^2/(c*x^2+b*x)^(5/2),x)

[Out]

80/3/(b*e-c*d)^2*c^3/b^3/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)+1/d/(b*e-c*d)/(d/e+x)/(c*(d
/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)-10/3/(b*e-c*d)^2/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(
b*e-c*d)/e^2)^(3/2)*c^2+64/3*c^2/d/(b*e-c*d)/b^3/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)-5/3
*e^2/d^2/(b*e-c*d)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*b+5*e/d/(b*e-c*d)^2/(c*(d/e+x)^
2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c+5*e^4/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-15*e^3/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)*c-20/3/(
b*e-c*d)^2/b^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*x*c^3+160/3/(b*e-c*d)^2*c^4/b^4/(c*(d
/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x+10*e^2/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)*c^2-5/2*e^4/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
-8/3*c/d/(b*e-c*d)/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)-160/3*e/d/(b*e-c*d)^2*c^3/b^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-20*e^3/d^2/(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^2+20*e^2/d/(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^3+20/3*e/d/(b*e-c*d)^2/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*x*c^2+40/3*e
^2/d^2/(b*e-c*d)^2*c^2/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+5*e^3/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))*c-80/3*e/d/(b*e-c*d)^2*c^2/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)-5/3*e^2/d^2/(b*e-c*d)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(
3/2)*x*c+5*e^4/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+20/3*e^2/d^2/(b*e
-c*d)^2*c/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)-16/3*c^2/d/(b*e-c*d)/b^2/(c*(d/e+x)^2+(b
*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*x+128/3*c^3/d/(b*e-c*d)/b^4/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b
*e-c*d)/e^2)^(1/2)*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)^2/(c*x^2+b*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.74634, size = 3553, normalized size = 10.21 \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)^2/(c*x^2+b*x)^(5/2),x, algorithm="fricas")

[Out]

[1/6*(15*((2*b^4*c^3*d*e^5 - b^5*c^2*e^6)*x^5 + (2*b^4*c^3*d^2*e^4 + 3*b^5*c^2*d*e^5 - 2*b^6*c*e^6)*x^4 + (4*b
^5*c^2*d^2*e^4 - b^7*e^6)*x^3 + (2*b^6*c*d^2*e^4 - b^7*d*e^5)*x^2)*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*(2*b^3*c^4*d^7 - 8*b^4*c^3*d^6*e + 12*b^5*c^2*d^
5*e^2 - 8*b^6*c*d^4*e^3 + 2*b^7*d^3*e^4 - (32*c^7*d^6*e - 96*b*c^6*d^5*e^2 + 76*b^2*c^5*d^4*e^3 + 8*b^3*c^4*d^
3*e^4 - 35*b^4*c^3*d^2*e^5 + 15*b^5*c^2*d*e^6)*x^4 - 2*(16*c^7*d^7 - 24*b*c^6*d^6*e - 34*b^2*c^5*d^5*e^2 + 61*
b^3*c^4*d^4*e^3 - 4*b^4*c^3*d^3*e^4 - 30*b^5*c^2*d^2*e^5 + 15*b^6*c*d*e^6)*x^3 - 3*(16*b*c^6*d^7 - 44*b^2*c^5*
d^6*e + 26*b^3*c^4*d^5*e^2 + 16*b^4*c^3*d^4*e^3 - 14*b^5*c^2*d^3*e^4 - 5*b^6*c*d^2*e^5 + 5*b^7*d*e^6)*x^2 - 2*
(6*b^2*c^5*d^7 - 19*b^3*c^4*d^6*e + 16*b^4*c^3*d^5*e^2 + 6*b^5*c^2*d^4*e^3 - 14*b^6*c*d^3*e^4 + 5*b^7*d^2*e^5)
*x)*sqrt(c*x^2 + b*x))/((b^4*c^6*d^8*e - 4*b^5*c^5*d^7*e^2 + 6*b^6*c^4*d^6*e^3 - 4*b^7*c^3*d^5*e^4 + b^8*c^2*d
^4*e^5)*x^5 + (b^4*c^6*d^9 - 2*b^5*c^5*d^8*e - 2*b^6*c^4*d^7*e^2 + 8*b^7*c^3*d^6*e^3 - 7*b^8*c^2*d^5*e^4 + 2*b
^9*c*d^4*e^5)*x^4 + (2*b^5*c^5*d^9 - 7*b^6*c^4*d^8*e + 8*b^7*c^3*d^7*e^2 - 2*b^8*c^2*d^6*e^3 - 2*b^9*c*d^5*e^4
+ b^10*d^4*e^5)*x^3 + (b^6*c^4*d^9 - 4*b^7*c^3*d^8*e + 6*b^8*c^2*d^7*e^2 - 4*b^9*c*d^6*e^3 + b^10*d^5*e^4)*x^
2), 1/3*(15*((2*b^4*c^3*d*e^5 - b^5*c^2*e^6)*x^5 + (2*b^4*c^3*d^2*e^4 + 3*b^5*c^2*d*e^5 - 2*b^6*c*e^6)*x^4 + (
4*b^5*c^2*d^2*e^4 - b^7*e^6)*x^3 + (2*b^6*c*d^2*e^4 - b^7*d*e^5)*x^2)*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)) - (2*b^3*c^4*d^7 - 8*b^4*c^3*d^6*e + 12*b^5*c^2*d^5*e^2 - 8*b^6*c
*d^4*e^3 + 2*b^7*d^3*e^4 - (32*c^7*d^6*e - 96*b*c^6*d^5*e^2 + 76*b^2*c^5*d^4*e^3 + 8*b^3*c^4*d^3*e^4 - 35*b^4*
c^3*d^2*e^5 + 15*b^5*c^2*d*e^6)*x^4 - 2*(16*c^7*d^7 - 24*b*c^6*d^6*e - 34*b^2*c^5*d^5*e^2 + 61*b^3*c^4*d^4*e^3
- 4*b^4*c^3*d^3*e^4 - 30*b^5*c^2*d^2*e^5 + 15*b^6*c*d*e^6)*x^3 - 3*(16*b*c^6*d^7 - 44*b^2*c^5*d^6*e + 26*b^3*
c^4*d^5*e^2 + 16*b^4*c^3*d^4*e^3 - 14*b^5*c^2*d^3*e^4 - 5*b^6*c*d^2*e^5 + 5*b^7*d*e^6)*x^2 - 2*(6*b^2*c^5*d^7
- 19*b^3*c^4*d^6*e + 16*b^4*c^3*d^5*e^2 + 6*b^5*c^2*d^4*e^3 - 14*b^6*c*d^3*e^4 + 5*b^7*d^2*e^5)*x)*sqrt(c*x^2
+ b*x))/((b^4*c^6*d^8*e - 4*b^5*c^5*d^7*e^2 + 6*b^6*c^4*d^6*e^3 - 4*b^7*c^3*d^5*e^4 + b^8*c^2*d^4*e^5)*x^5 + (
b^4*c^6*d^9 - 2*b^5*c^5*d^8*e - 2*b^6*c^4*d^7*e^2 + 8*b^7*c^3*d^6*e^3 - 7*b^8*c^2*d^5*e^4 + 2*b^9*c*d^4*e^5)*x
^4 + (2*b^5*c^5*d^9 - 7*b^6*c^4*d^8*e + 8*b^7*c^3*d^7*e^2 - 2*b^8*c^2*d^6*e^3 - 2*b^9*c*d^5*e^4 + b^10*d^4*e^5
)*x^3 + (b^6*c^4*d^9 - 4*b^7*c^3*d^8*e + 6*b^8*c^2*d^7*e^2 - 4*b^9*c*d^6*e^3 + b^10*d^5*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)**2/(c*x**2+b*x)**(5/2),x)

[Out]

Timed out

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Giac [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)^2/(c*x^2+b*x)^(5/2),x, algorithm="giac")

[Out]

Timed out