### 3.293 $$\int \frac{\sqrt{b x+c x^2}}{(d+e x)^5} \, dx$$

Optimal. Leaf size=258 $\frac{\sqrt{b x+c x^2} \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right ) (x (2 c d-b e)+b d)}{64 d^3 (d+e x)^2 (c d-b e)^3}-\frac{b^2 \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{128 d^{7/2} (c d-b e)^{7/2}}-\frac{5 e \left (b x+c x^2\right )^{3/2} (2 c d-b e)}{24 d^2 (d+e x)^3 (c d-b e)^2}-\frac{e \left (b x+c x^2\right )^{3/2}}{4 d (d+e x)^4 (c d-b e)}$

[Out]

((16*c^2*d^2 - 16*b*c*d*e + 5*b^2*e^2)*(b*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(64*d^3*(c*d - b*e)^3*(d + e
*x)^2) - (e*(b*x + c*x^2)^(3/2))/(4*d*(c*d - b*e)*(d + e*x)^4) - (5*e*(2*c*d - b*e)*(b*x + c*x^2)^(3/2))/(24*d
^2*(c*d - b*e)^2*(d + e*x)^3) - (b^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])])/(128*d^(7/2)*(c*d - b*e)^(7/2))

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Rubi [A]  time = 0.319279, antiderivative size = 258, 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 = {744, 806, 720, 724, 206} $\frac{\sqrt{b x+c x^2} \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right ) (x (2 c d-b e)+b d)}{64 d^3 (d+e x)^2 (c d-b e)^3}-\frac{b^2 \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{128 d^{7/2} (c d-b e)^{7/2}}-\frac{5 e \left (b x+c x^2\right )^{3/2} (2 c d-b e)}{24 d^2 (d+e x)^3 (c d-b e)^2}-\frac{e \left (b x+c x^2\right )^{3/2}}{4 d (d+e x)^4 (c d-b e)}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[b*x + c*x^2]/(d + e*x)^5,x]

[Out]

((16*c^2*d^2 - 16*b*c*d*e + 5*b^2*e^2)*(b*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(64*d^3*(c*d - b*e)^3*(d + e
*x)^2) - (e*(b*x + c*x^2)^(3/2))/(4*d*(c*d - b*e)*(d + e*x)^4) - (5*e*(2*c*d - b*e)*(b*x + c*x^2)^(3/2))/(24*d
^2*(c*d - b*e)^2*(d + e*x)^3) - (b^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])])/(128*d^(7/2)*(c*d - b*e)^(7/2))

Rule 744

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

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 720

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*
(d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^p)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(p*(b^2 -
4*a*c))/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[
{a, b, c, d, e}, 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] && EqQ[m +
2*p + 2, 0] && GtQ[p, 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{\sqrt{b x+c x^2}}{(d+e x)^5} \, dx &=-\frac{e \left (b x+c x^2\right )^{3/2}}{4 d (c d-b e) (d+e x)^4}-\frac{\int \frac{\left (\frac{1}{2} (-8 c d+5 b e)+c e x\right ) \sqrt{b x+c x^2}}{(d+e x)^4} \, dx}{4 d (c d-b e)}\\ &=-\frac{e \left (b x+c x^2\right )^{3/2}}{4 d (c d-b e) (d+e x)^4}-\frac{5 e (2 c d-b e) \left (b x+c x^2\right )^{3/2}}{24 d^2 (c d-b e)^2 (d+e x)^3}+\frac{\left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right ) \int \frac{\sqrt{b x+c x^2}}{(d+e x)^3} \, dx}{16 d^2 (c d-b e)^2}\\ &=\frac{\left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right ) (b d+(2 c d-b e) x) \sqrt{b x+c x^2}}{64 d^3 (c d-b e)^3 (d+e x)^2}-\frac{e \left (b x+c x^2\right )^{3/2}}{4 d (c d-b e) (d+e x)^4}-\frac{5 e (2 c d-b e) \left (b x+c x^2\right )^{3/2}}{24 d^2 (c d-b e)^2 (d+e x)^3}-\frac{\left (b^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}{128 d^3 (c d-b e)^3}\\ &=\frac{\left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right ) (b d+(2 c d-b e) x) \sqrt{b x+c x^2}}{64 d^3 (c d-b e)^3 (d+e x)^2}-\frac{e \left (b x+c x^2\right )^{3/2}}{4 d (c d-b e) (d+e x)^4}-\frac{5 e (2 c d-b e) \left (b x+c x^2\right )^{3/2}}{24 d^2 (c d-b e)^2 (d+e x)^3}+\frac{\left (b^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 )}{64 d^3 (c d-b e)^3}\\ &=\frac{\left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right ) (b d+(2 c d-b e) x) \sqrt{b x+c x^2}}{64 d^3 (c d-b e)^3 (d+e x)^2}-\frac{e \left (b x+c x^2\right )^{3/2}}{4 d (c d-b e) (d+e x)^4}-\frac{5 e (2 c d-b e) \left (b x+c x^2\right )^{3/2}}{24 d^2 (c d-b e)^2 (d+e x)^3}-\frac{b^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 )}{128 d^{7/2} (c d-b e)^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.760626, size = 243, normalized size = 0.94 $\frac{\sqrt{x (b+c x)} \left (\frac{3 (d+e x)^2 \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \left (b^2 (d+e x)^2 \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )+\sqrt{d} \sqrt{x} \sqrt{b+c x} \sqrt{b e-c d} (-b d+b e x-2 c d x)\right )}{d^{5/2} \sqrt{b+c x} (b e-c d)^{5/2}}+\frac{40 e x^{3/2} (b+c x) (d+e x) (2 c d-b e)}{d (c d-b e)}+48 e x^{3/2} (b+c x)\right )}{192 d \sqrt{x} (d+e x)^4 (b e-c d)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(Sqrt[x*(b + c*x)]*(48*e*x^(3/2)*(b + c*x) + (40*e*(2*c*d - b*e)*x^(3/2)*(b + c*x)*(d + e*x))/(d*(c*d - b*e))
+ (3*(16*c^2*d^2 - 16*b*c*d*e + 5*b^2*e^2)*(d + e*x)^2*(Sqrt[d]*Sqrt[-(c*d) + b*e]*Sqrt[x]*Sqrt[b + c*x]*(-(b*
d) - 2*c*d*x + b*e*x) + b^2*(d + e*x)^2*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])]))/(d^(5/2
)*(-(c*d) + b*e)^(5/2)*Sqrt[b + c*x])))/(192*d*(-(c*d) + b*e)*Sqrt[x]*(d + e*x)^4)

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Maple [B]  time = 0.222, size = 4819, normalized size = 18.7 \begin{align*} \text{output too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/e^3/d/(b*e-c*d)/(d/e+x)^4*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)-25/16/e/(b*e-c*d)^4*ln
((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(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))*c^(7/2)*b+1/
8/e^3*c^3/(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))-5/64*e^3/d^4/(b*e-c*d)^4*(c*(d/e+
x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b^4-3/4/e/d/(b*e-c*d)^3*c^3*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+
x)-d*(b*e-c*d)/e^2)^(1/2)-1/8*c^2/d^2/(b*e-c*d)^3/(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)+17/32/d^2/(b*e-c*d)^3*c^(3/2)*ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(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))*b^2+45/32/d/(b*e-c*d)^4*ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(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))*c^(5/2)*b^2-5/8/d/(b*e-c*d)^4/(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)*c^3+1/8*c^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
)*x+25/16/d/(b*e-c*d)^4*(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^3+1/8/e^2*c^(5/2)/d/(b*e
-c*d)^2*ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(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))+5/
8/e^2*d/(b*e-c*d)^4*ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(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))*c^(9/2)+3/4/e^2/(b*e-c*d)^3*c^(7/2)*ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(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))-5/8/e/(b*e-c*d)^4*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2
)*c^4+13/16/d^2/(b*e-c*d)^3*c^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b-1/8/e*c^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)^(1/2)+5/8/d/(b*e-c*d)^4*c^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+3/4/e^3*d/(b*e-c*d)^3*c^4/(-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))+45/128*e/d^2/(b*e-c*d)^4/(-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^4*c-15/16*e/d^2/(b*
e-c*d)^4*c^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*b-15/8/e^2*d/(b*e-c*d)^4/(-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^4*b+15/32*e^2/d^3/(b*e-c*d)^4*c^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*b^2-15/32*e^2/d^3/(b*e-c*d)^4/(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)*b^2*c+15/16/e/d/(b*e-c*d)^3*c^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))*b^2+15/1
6*e/d^2/(b*e-c*d)^4/(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)*b*c^2-1/16*e*c^2/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*b-1/8/e^2*c^2/d/(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))*b+1/16*e*c/d^3/(b*e-c*d)^3/(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)*b-5/64*e^3/d^4/(b*e-c*d)^4*c*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2
)*x*b^3-5/64*e/d^3/(b*e-c*d)^3*c^(1/2)*ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(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))*b^3-5/4/d/(b*e-c*d)^4/(-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^3*
c^2-5/8/d^2/(b*e-c*d)^3/(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)^(3/2)*b*c-3/16/d^2/(b*e-
c*d)^3*c/(-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^3-1/16/e*c^(3/2)/d^2/(b*e-c*d)^2*ln((1/2*(b*
e-2*c*d)/e+(d/e+x)*c)/c^(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))*b-45/32*e/d^2/(b*e-c*
d)^4*(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*c^2+1/8/e*c/d^2/(b*e-c*d)^2/(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)^(3/2)-3/2/e^2/(b*e-c*d)^3*c^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+35/16/e/(b*e-c*d)^4/(-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*c^3+5/6
4*e^2/d^3/(b*e-c*d)^4*ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(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))*c^(1/2)*b^4-35/64*e/d^2/(b*e-c*d)^4*ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(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))*c^(3/2)*b^3+5/8/e^3*d^2/(b*e-c*d)^4/(-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^5-9/8/e/d/(b*e-c*d)^3*c^(5/2)*ln((1/2*(b*e-2*c*d)/e+(d/e+x)*c)/c^(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))*b-5/128*e^2/d^3/(b*e-c*d)^4/(-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^5-7/32*e/d^3/(b*e-c*d)^3*c*(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+5/24/
e/d^2/(b*e-c*d)^2/(d/e+x)^3*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*b-5/12/e^2/d/(b*e-c*d)^2
/(d/e+x)^3*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c+5/32*e/d^3/(b*e-c*d)^3/(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)^(3/2)*b^2+5/8/e/d/(b*e-c*d)^3/(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)^(3/2)*c^2+5/64*e^3/d^4/(b*e-c*d)^4/(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)*b^3+35/64*e^2/d^3/(b*e-c*d)^4*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b^3
*c

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.55067, size = 3313, normalized size = 12.84 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/384*(3*(16*b^2*c^2*d^6 - 16*b^3*c*d^5*e + 5*b^4*d^4*e^2 + (16*b^2*c^2*d^2*e^4 - 16*b^3*c*d*e^5 + 5*b^4*e^6
)*x^4 + 4*(16*b^2*c^2*d^3*e^3 - 16*b^3*c*d^2*e^4 + 5*b^4*d*e^5)*x^3 + 6*(16*b^2*c^2*d^4*e^2 - 16*b^3*c*d^3*e^3
+ 5*b^4*d^2*e^4)*x^2 + 4*(16*b^2*c^2*d^5*e - 16*b^3*c*d^4*e^2 + 5*b^4*d^3*e^3)*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*(48*b*c^3*d^7 - 96*b^2*c^2*d^6*e
+ 63*b^3*c*d^5*e^2 - 15*b^4*d^4*e^3 + (16*c^4*d^5*e^2 - 40*b*c^3*d^4*e^3 + 62*b^2*c^2*d^3*e^4 - 53*b^3*c*d^2*
e^5 + 15*b^4*d*e^6)*x^3 + (64*c^4*d^6*e - 168*b*c^3*d^5*e^2 + 244*b^2*c^2*d^4*e^3 - 195*b^3*c*d^3*e^4 + 55*b^4
*d^2*e^5)*x^2 + (96*c^4*d^7 - 272*b*c^3*d^6*e + 374*b^2*c^2*d^5*e^2 - 271*b^3*c*d^4*e^3 + 73*b^4*d^3*e^4)*x)*s
qrt(c*x^2 + b*x))/(c^4*d^12 - 4*b*c^3*d^11*e + 6*b^2*c^2*d^10*e^2 - 4*b^3*c*d^9*e^3 + b^4*d^8*e^4 + (c^4*d^8*e
^4 - 4*b*c^3*d^7*e^5 + 6*b^2*c^2*d^6*e^6 - 4*b^3*c*d^5*e^7 + b^4*d^4*e^8)*x^4 + 4*(c^4*d^9*e^3 - 4*b*c^3*d^8*e
^4 + 6*b^2*c^2*d^7*e^5 - 4*b^3*c*d^6*e^6 + b^4*d^5*e^7)*x^3 + 6*(c^4*d^10*e^2 - 4*b*c^3*d^9*e^3 + 6*b^2*c^2*d^
8*e^4 - 4*b^3*c*d^7*e^5 + b^4*d^6*e^6)*x^2 + 4*(c^4*d^11*e - 4*b*c^3*d^10*e^2 + 6*b^2*c^2*d^9*e^3 - 4*b^3*c*d^
8*e^4 + b^4*d^7*e^5)*x), -1/192*(3*(16*b^2*c^2*d^6 - 16*b^3*c*d^5*e + 5*b^4*d^4*e^2 + (16*b^2*c^2*d^2*e^4 - 16
*b^3*c*d*e^5 + 5*b^4*e^6)*x^4 + 4*(16*b^2*c^2*d^3*e^3 - 16*b^3*c*d^2*e^4 + 5*b^4*d*e^5)*x^3 + 6*(16*b^2*c^2*d^
4*e^2 - 16*b^3*c*d^3*e^3 + 5*b^4*d^2*e^4)*x^2 + 4*(16*b^2*c^2*d^5*e - 16*b^3*c*d^4*e^2 + 5*b^4*d^3*e^3)*x)*sqr
t(-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)) - (48*b*c^3*d^7 - 96*b^2*c^2
*d^6*e + 63*b^3*c*d^5*e^2 - 15*b^4*d^4*e^3 + (16*c^4*d^5*e^2 - 40*b*c^3*d^4*e^3 + 62*b^2*c^2*d^3*e^4 - 53*b^3*
c*d^2*e^5 + 15*b^4*d*e^6)*x^3 + (64*c^4*d^6*e - 168*b*c^3*d^5*e^2 + 244*b^2*c^2*d^4*e^3 - 195*b^3*c*d^3*e^4 +
55*b^4*d^2*e^5)*x^2 + (96*c^4*d^7 - 272*b*c^3*d^6*e + 374*b^2*c^2*d^5*e^2 - 271*b^3*c*d^4*e^3 + 73*b^4*d^3*e^4
)*x)*sqrt(c*x^2 + b*x))/(c^4*d^12 - 4*b*c^3*d^11*e + 6*b^2*c^2*d^10*e^2 - 4*b^3*c*d^9*e^3 + b^4*d^8*e^4 + (c^4
*d^8*e^4 - 4*b*c^3*d^7*e^5 + 6*b^2*c^2*d^6*e^6 - 4*b^3*c*d^5*e^7 + b^4*d^4*e^8)*x^4 + 4*(c^4*d^9*e^3 - 4*b*c^3
*d^8*e^4 + 6*b^2*c^2*d^7*e^5 - 4*b^3*c*d^6*e^6 + b^4*d^5*e^7)*x^3 + 6*(c^4*d^10*e^2 - 4*b*c^3*d^9*e^3 + 6*b^2*
c^2*d^8*e^4 - 4*b^3*c*d^7*e^5 + b^4*d^6*e^6)*x^2 + 4*(c^4*d^11*e - 4*b*c^3*d^10*e^2 + 6*b^2*c^2*d^9*e^3 - 4*b^
3*c*d^8*e^4 + b^4*d^7*e^5)*x)]

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

[Out]

Timed out

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Giac [B]  time = 1.92823, size = 1661, normalized size = 6.44 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(2*sqrt(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + b*e/(x*e + d) - b*d*e/(x*e + d)^2)*(2*(4*((2*c^3*d^5*e
^6*sgn(1/(x*e + d)) - 5*b*c^2*d^4*e^7*sgn(1/(x*e + d)) + 4*b^2*c*d^3*e^8*sgn(1/(x*e + d)) - b^3*d^2*e^9*sgn(1/
(x*e + d)))/(c^4*d^8*e^8 - 4*b*c^3*d^7*e^9 + 6*b^2*c^2*d^6*e^10 - 4*b^3*c*d^5*e^11 + b^4*d^4*e^12) - 6*(c^3*d^
6*e^7*sgn(1/(x*e + d)) - 3*b*c^2*d^5*e^8*sgn(1/(x*e + d)) + 3*b^2*c*d^4*e^9*sgn(1/(x*e + d)) - b^3*d^3*e^10*sg
n(1/(x*e + d)))*e^(-1)/((c^4*d^8*e^8 - 4*b*c^3*d^7*e^9 + 6*b^2*c^2*d^6*e^10 - 4*b^3*c*d^5*e^11 + b^4*d^4*e^12)
*(x*e + d)))*e^(-1)/(x*e + d) + (8*c^3*d^4*e^5*sgn(1/(x*e + d)) - 16*b*c^2*d^3*e^6*sgn(1/(x*e + d)) + 13*b^2*c
*d^2*e^7*sgn(1/(x*e + d)) - 5*b^3*d*e^8*sgn(1/(x*e + d)))/(c^4*d^8*e^8 - 4*b*c^3*d^7*e^9 + 6*b^2*c^2*d^6*e^10
- 4*b^3*c*d^5*e^11 + b^4*d^4*e^12))*e^(-1)/(x*e + d) + (16*c^3*d^3*e^4*sgn(1/(x*e + d)) - 24*b*c^2*d^2*e^5*sgn
(1/(x*e + d)) + 38*b^2*c*d*e^6*sgn(1/(x*e + d)) - 15*b^3*e^7*sgn(1/(x*e + d)))/(c^4*d^8*e^8 - 4*b*c^3*d^7*e^9
+ 6*b^2*c^2*d^6*e^10 - 4*b^3*c*d^5*e^11 + b^4*d^4*e^12)) - (48*b^2*c^2*d^2*e^2*log(abs(2*c*d - b*e - 2*sqrt(c*
d^2 - b*d*e)*sqrt(c))) + 32*sqrt(c*d^2 - b*d*e)*c^(7/2)*d^3 - 48*sqrt(c*d^2 - b*d*e)*b*c^(5/2)*d^2*e - 48*b^3*
c*d*e^3*log(abs(2*c*d - b*e - 2*sqrt(c*d^2 - b*d*e)*sqrt(c))) + 76*sqrt(c*d^2 - b*d*e)*b^2*c^(3/2)*d*e^2 + 15*
b^4*e^4*log(abs(2*c*d - b*e - 2*sqrt(c*d^2 - b*d*e)*sqrt(c))) - 30*sqrt(c*d^2 - b*d*e)*b^3*sqrt(c)*e^3)*sgn(1/
(x*e + d))/(sqrt(c*d^2 - b*d*e)*c^4*d^8*e^4 - 4*sqrt(c*d^2 - b*d*e)*b*c^3*d^7*e^5 + 6*sqrt(c*d^2 - b*d*e)*b^2*
c^2*d^6*e^6 - 4*sqrt(c*d^2 - b*d*e)*b^3*c*d^5*e^7 + sqrt(c*d^2 - b*d*e)*b^4*d^4*e^8) + 3*(16*b^2*c^2*d^2*sgn(1
/(x*e + d)) - 16*b^3*c*d*e*sgn(1/(x*e + d)) + 5*b^4*e^2*sgn(1/(x*e + d)))*log(abs(2*c*d - b*e - 2*sqrt(c*d^2 -
b*d*e)*(sqrt(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + b*e/(x*e + d) - b*d*e/(x*e + d)^2) + sqrt(c*d^2*e^2 -
b*d*e^3)*e^(-1)/(x*e + d))))/((c^4*d^8*e^2 - 4*b*c^3*d^7*e^3 + 6*b^2*c^2*d^6*e^4 - 4*b^3*c*d^5*e^5 + b^4*d^4*e
^6)*sqrt(c*d^2 - b*d*e)))*e^2