3.294 $$\int \frac{\sqrt{b x+c x^2}}{(d+e x)^6} \, dx$$

Optimal. Leaf size=337 $-\frac{e \left (b x+c x^2\right )^{3/2} \left (35 b^2 e^2-108 b c d e+108 c^2 d^2\right )}{240 d^3 (d+e x)^3 (c d-b e)^3}+\frac{\sqrt{b x+c x^2} (2 c d-b e) \left (7 b^2 e^2-16 b c d e+16 c^2 d^2\right ) (x (2 c d-b e)+b d)}{128 d^4 (d+e x)^2 (c d-b e)^4}-\frac{b^2 (2 c d-b e) \left (7 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{256 d^{9/2} (c d-b e)^{9/2}}-\frac{7 e \left (b x+c x^2\right )^{3/2} (2 c d-b e)}{40 d^2 (d+e x)^4 (c d-b e)^2}-\frac{e \left (b x+c x^2\right )^{3/2}}{5 d (d+e x)^5 (c d-b e)}$

[Out]

((2*c*d - b*e)*(16*c^2*d^2 - 16*b*c*d*e + 7*b^2*e^2)*(b*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(128*d^4*(c*d
- b*e)^4*(d + e*x)^2) - (e*(b*x + c*x^2)^(3/2))/(5*d*(c*d - b*e)*(d + e*x)^5) - (7*e*(2*c*d - b*e)*(b*x + c*x^
2)^(3/2))/(40*d^2*(c*d - b*e)^2*(d + e*x)^4) - (e*(108*c^2*d^2 - 108*b*c*d*e + 35*b^2*e^2)*(b*x + c*x^2)^(3/2)
)/(240*d^3*(c*d - b*e)^3*(d + e*x)^3) - (b^2*(2*c*d - b*e)*(16*c^2*d^2 - 16*b*c*d*e + 7*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])])/(256*d^(9/2)*(c*d - b*e)^(9/2))

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Rubi [A]  time = 0.45576, antiderivative size = 337, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.286, Rules used = {744, 834, 806, 720, 724, 206} $-\frac{e \left (b x+c x^2\right )^{3/2} \left (35 b^2 e^2-108 b c d e+108 c^2 d^2\right )}{240 d^3 (d+e x)^3 (c d-b e)^3}+\frac{\sqrt{b x+c x^2} (2 c d-b e) \left (7 b^2 e^2-16 b c d e+16 c^2 d^2\right ) (x (2 c d-b e)+b d)}{128 d^4 (d+e x)^2 (c d-b e)^4}-\frac{b^2 (2 c d-b e) \left (7 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{256 d^{9/2} (c d-b e)^{9/2}}-\frac{7 e \left (b x+c x^2\right )^{3/2} (2 c d-b e)}{40 d^2 (d+e x)^4 (c d-b e)^2}-\frac{e \left (b x+c x^2\right )^{3/2}}{5 d (d+e x)^5 (c d-b e)}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[b*x + c*x^2]/(d + e*x)^6,x]

[Out]

((2*c*d - b*e)*(16*c^2*d^2 - 16*b*c*d*e + 7*b^2*e^2)*(b*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(128*d^4*(c*d
- b*e)^4*(d + e*x)^2) - (e*(b*x + c*x^2)^(3/2))/(5*d*(c*d - b*e)*(d + e*x)^5) - (7*e*(2*c*d - b*e)*(b*x + c*x^
2)^(3/2))/(40*d^2*(c*d - b*e)^2*(d + e*x)^4) - (e*(108*c^2*d^2 - 108*b*c*d*e + 35*b^2*e^2)*(b*x + c*x^2)^(3/2)
)/(240*d^3*(c*d - b*e)^3*(d + e*x)^3) - (b^2*(2*c*d - b*e)*(16*c^2*d^2 - 16*b*c*d*e + 7*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])])/(256*d^(9/2)*(c*d - b*e)^(9/2))

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 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 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)^6} \, dx &=-\frac{e \left (b x+c x^2\right )^{3/2}}{5 d (c d-b e) (d+e x)^5}-\frac{\int \frac{\left (\frac{1}{2} (-10 c d+7 b e)+2 c e x\right ) \sqrt{b x+c x^2}}{(d+e x)^5} \, dx}{5 d (c d-b e)}\\ &=-\frac{e \left (b x+c x^2\right )^{3/2}}{5 d (c d-b e) (d+e x)^5}-\frac{7 e (2 c d-b e) \left (b x+c x^2\right )^{3/2}}{40 d^2 (c d-b e)^2 (d+e x)^4}+\frac{\int \frac{\left (\frac{1}{4} \left (80 c^2 d^2-94 b c d e+35 b^2 e^2\right )-\frac{7}{2} c e (2 c d-b e) x\right ) \sqrt{b x+c x^2}}{(d+e x)^4} \, dx}{20 d^2 (c d-b e)^2}\\ &=-\frac{e \left (b x+c x^2\right )^{3/2}}{5 d (c d-b e) (d+e x)^5}-\frac{7 e (2 c d-b e) \left (b x+c x^2\right )^{3/2}}{40 d^2 (c d-b e)^2 (d+e x)^4}-\frac{e \left (108 c^2 d^2-108 b c d e+35 b^2 e^2\right ) \left (b x+c x^2\right )^{3/2}}{240 d^3 (c d-b e)^3 (d+e x)^3}+\frac{\left ((2 c d-b e) \left (16 c^2 d^2-16 b c d e+7 b^2 e^2\right )\right ) \int \frac{\sqrt{b x+c x^2}}{(d+e x)^3} \, dx}{32 d^3 (c d-b e)^3}\\ &=\frac{(2 c d-b e) \left (16 c^2 d^2-16 b c d e+7 b^2 e^2\right ) (b d+(2 c d-b e) x) \sqrt{b x+c x^2}}{128 d^4 (c d-b e)^4 (d+e x)^2}-\frac{e \left (b x+c x^2\right )^{3/2}}{5 d (c d-b e) (d+e x)^5}-\frac{7 e (2 c d-b e) \left (b x+c x^2\right )^{3/2}}{40 d^2 (c d-b e)^2 (d+e x)^4}-\frac{e \left (108 c^2 d^2-108 b c d e+35 b^2 e^2\right ) \left (b x+c x^2\right )^{3/2}}{240 d^3 (c d-b e)^3 (d+e x)^3}-\frac{\left (b^2 (2 c d-b e) \left (16 c^2 d^2-16 b c d e+7 b^2 e^2\right )\right ) \int \frac{1}{(d+e x) \sqrt{b x+c x^2}} \, dx}{256 d^4 (c d-b e)^4}\\ &=\frac{(2 c d-b e) \left (16 c^2 d^2-16 b c d e+7 b^2 e^2\right ) (b d+(2 c d-b e) x) \sqrt{b x+c x^2}}{128 d^4 (c d-b e)^4 (d+e x)^2}-\frac{e \left (b x+c x^2\right )^{3/2}}{5 d (c d-b e) (d+e x)^5}-\frac{7 e (2 c d-b e) \left (b x+c x^2\right )^{3/2}}{40 d^2 (c d-b e)^2 (d+e x)^4}-\frac{e \left (108 c^2 d^2-108 b c d e+35 b^2 e^2\right ) \left (b x+c x^2\right )^{3/2}}{240 d^3 (c d-b e)^3 (d+e x)^3}+\frac{\left (b^2 (2 c d-b e) \left (16 c^2 d^2-16 b c d e+7 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 )}{128 d^4 (c d-b e)^4}\\ &=\frac{(2 c d-b e) \left (16 c^2 d^2-16 b c d e+7 b^2 e^2\right ) (b d+(2 c d-b e) x) \sqrt{b x+c x^2}}{128 d^4 (c d-b e)^4 (d+e x)^2}-\frac{e \left (b x+c x^2\right )^{3/2}}{5 d (c d-b e) (d+e x)^5}-\frac{7 e (2 c d-b e) \left (b x+c x^2\right )^{3/2}}{40 d^2 (c d-b e)^2 (d+e x)^4}-\frac{e \left (108 c^2 d^2-108 b c d e+35 b^2 e^2\right ) \left (b x+c x^2\right )^{3/2}}{240 d^3 (c d-b e)^3 (d+e x)^3}-\frac{b^2 (2 c d-b e) \left (16 c^2 d^2-16 b c d e+7 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 )}{256 d^{9/2} (c d-b e)^{9/2}}\\ \end{align*}

Mathematica [A]  time = 1.27726, size = 309, normalized size = 0.92 $\frac{\sqrt{x (b+c x)} \left (\frac{(d+e x)^2 \left (8 e x^{3/2} (b+c x) \left (35 b^2 e^2-108 b c d e+108 c^2 d^2\right )+\frac{15 (d+e x) (2 c d-b e) \left (7 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \left (\sqrt{d} \sqrt{x} \sqrt{b+c x} \sqrt{b e-c d} (b (d-e x)+2 c d x)-b^2 (d+e x)^2 \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )\right )}{d^{3/2} \sqrt{b+c x} (b e-c d)^{3/2}}\right )}{d^2 (c d-b e)^2}+\frac{336 e x^{3/2} (b+c x) (d+e x) (2 c d-b e)}{d (c d-b e)}+384 e x^{3/2} (b+c x)\right )}{1920 d \sqrt{x} (d+e x)^5 (b e-c d)}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sqrt[b*x + c*x^2]/(d + e*x)^6,x]

[Out]

(Sqrt[x*(b + c*x)]*(384*e*x^(3/2)*(b + c*x) + (336*e*(2*c*d - b*e)*x^(3/2)*(b + c*x)*(d + e*x))/(d*(c*d - b*e)
) + ((d + e*x)^2*(8*e*(108*c^2*d^2 - 108*b*c*d*e + 35*b^2*e^2)*x^(3/2)*(b + c*x) + (15*(2*c*d - b*e)*(16*c^2*d
^2 - 16*b*c*d*e + 7*b^2*e^2)*(d + e*x)*(Sqrt[d]*Sqrt[-(c*d) + b*e]*Sqrt[x]*Sqrt[b + c*x]*(2*c*d*x + b*(d - e*x
)) - b^2*(d + e*x)^2*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])]))/(d^(3/2)*(-(c*d) + b*e)^(3
/2)*Sqrt[b + c*x])))/(d^2*(c*d - b*e)^2)))/(1920*d*(-(c*d) + b*e)*Sqrt[x]*(d + e*x)^5)

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Maple [B]  time = 0.241, size = 6533, normalized size = 19.4 \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)^6,x)

[Out]

result too large to display

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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)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.78922, size = 5138, normalized size = 15.25 \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)^6,x, algorithm="fricas")

[Out]

[-1/3840*(15*(32*b^2*c^3*d^8 - 48*b^3*c^2*d^7*e + 30*b^4*c*d^6*e^2 - 7*b^5*d^5*e^3 + (32*b^2*c^3*d^3*e^5 - 48*
b^3*c^2*d^2*e^6 + 30*b^4*c*d*e^7 - 7*b^5*e^8)*x^5 + 5*(32*b^2*c^3*d^4*e^4 - 48*b^3*c^2*d^3*e^5 + 30*b^4*c*d^2*
e^6 - 7*b^5*d*e^7)*x^4 + 10*(32*b^2*c^3*d^5*e^3 - 48*b^3*c^2*d^4*e^4 + 30*b^4*c*d^3*e^5 - 7*b^5*d^2*e^6)*x^3 +
10*(32*b^2*c^3*d^6*e^2 - 48*b^3*c^2*d^5*e^3 + 30*b^4*c*d^4*e^4 - 7*b^5*d^3*e^5)*x^2 + 5*(32*b^2*c^3*d^7*e - 4
8*b^3*c^2*d^6*e^2 + 30*b^4*c*d^5*e^3 - 7*b^5*d^4*e^4)*x)*sqrt(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e)*x + 2*sq
rt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d)) - 2*(480*b*c^4*d^9 - 1200*b^2*c^3*d^8*e + 1170*b^3*c^2*d^7*e^2
- 555*b^4*c*d^6*e^3 + 105*b^5*d^5*e^4 + (96*c^5*d^6*e^3 - 288*b*c^4*d^5*e^4 + 668*b^2*c^3*d^4*e^5 - 856*b^3*c
^2*d^3*e^6 + 485*b^4*c*d^2*e^7 - 105*b^5*d*e^8)*x^4 + 2*(240*c^5*d^7*e^2 - 744*b*c^4*d^6*e^3 + 1622*b^2*c^3*d^
5*e^4 - 2007*b^3*c^2*d^4*e^5 + 1134*b^4*c*d^3*e^6 - 245*b^5*d^2*e^7)*x^3 + 2*(480*c^5*d^8*e - 1560*b*c^4*d^7*e
^2 + 3178*b^2*c^3*d^6*e^3 - 3729*b^3*c^2*d^5*e^4 + 2079*b^4*c*d^4*e^5 - 448*b^5*d^3*e^6)*x^2 + 10*(96*c^5*d^9
- 336*b*c^4*d^8*e + 642*b^2*c^3*d^7*e^2 - 697*b^3*c^2*d^6*e^3 + 374*b^4*c*d^5*e^4 - 79*b^5*d^4*e^5)*x)*sqrt(c*
x^2 + b*x))/(c^5*d^15 - 5*b*c^4*d^14*e + 10*b^2*c^3*d^13*e^2 - 10*b^3*c^2*d^12*e^3 + 5*b^4*c*d^11*e^4 - b^5*d^
10*e^5 + (c^5*d^10*e^5 - 5*b*c^4*d^9*e^6 + 10*b^2*c^3*d^8*e^7 - 10*b^3*c^2*d^7*e^8 + 5*b^4*c*d^6*e^9 - b^5*d^5
*e^10)*x^5 + 5*(c^5*d^11*e^4 - 5*b*c^4*d^10*e^5 + 10*b^2*c^3*d^9*e^6 - 10*b^3*c^2*d^8*e^7 + 5*b^4*c*d^7*e^8 -
b^5*d^6*e^9)*x^4 + 10*(c^5*d^12*e^3 - 5*b*c^4*d^11*e^4 + 10*b^2*c^3*d^10*e^5 - 10*b^3*c^2*d^9*e^6 + 5*b^4*c*d^
8*e^7 - b^5*d^7*e^8)*x^3 + 10*(c^5*d^13*e^2 - 5*b*c^4*d^12*e^3 + 10*b^2*c^3*d^11*e^4 - 10*b^3*c^2*d^10*e^5 + 5
*b^4*c*d^9*e^6 - b^5*d^8*e^7)*x^2 + 5*(c^5*d^14*e - 5*b*c^4*d^13*e^2 + 10*b^2*c^3*d^12*e^3 - 10*b^3*c^2*d^11*e
^4 + 5*b^4*c*d^10*e^5 - b^5*d^9*e^6)*x), -1/1920*(15*(32*b^2*c^3*d^8 - 48*b^3*c^2*d^7*e + 30*b^4*c*d^6*e^2 - 7
*b^5*d^5*e^3 + (32*b^2*c^3*d^3*e^5 - 48*b^3*c^2*d^2*e^6 + 30*b^4*c*d*e^7 - 7*b^5*e^8)*x^5 + 5*(32*b^2*c^3*d^4*
e^4 - 48*b^3*c^2*d^3*e^5 + 30*b^4*c*d^2*e^6 - 7*b^5*d*e^7)*x^4 + 10*(32*b^2*c^3*d^5*e^3 - 48*b^3*c^2*d^4*e^4 +
30*b^4*c*d^3*e^5 - 7*b^5*d^2*e^6)*x^3 + 10*(32*b^2*c^3*d^6*e^2 - 48*b^3*c^2*d^5*e^3 + 30*b^4*c*d^4*e^4 - 7*b^
5*d^3*e^5)*x^2 + 5*(32*b^2*c^3*d^7*e - 48*b^3*c^2*d^6*e^2 + 30*b^4*c*d^5*e^3 - 7*b^5*d^4*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)) - (480*b*c^4*d^9 - 1200*b^2*c^3*d^8*e
+ 1170*b^3*c^2*d^7*e^2 - 555*b^4*c*d^6*e^3 + 105*b^5*d^5*e^4 + (96*c^5*d^6*e^3 - 288*b*c^4*d^5*e^4 + 668*b^2*c
^3*d^4*e^5 - 856*b^3*c^2*d^3*e^6 + 485*b^4*c*d^2*e^7 - 105*b^5*d*e^8)*x^4 + 2*(240*c^5*d^7*e^2 - 744*b*c^4*d^6
*e^3 + 1622*b^2*c^3*d^5*e^4 - 2007*b^3*c^2*d^4*e^5 + 1134*b^4*c*d^3*e^6 - 245*b^5*d^2*e^7)*x^3 + 2*(480*c^5*d^
8*e - 1560*b*c^4*d^7*e^2 + 3178*b^2*c^3*d^6*e^3 - 3729*b^3*c^2*d^5*e^4 + 2079*b^4*c*d^4*e^5 - 448*b^5*d^3*e^6)
*x^2 + 10*(96*c^5*d^9 - 336*b*c^4*d^8*e + 642*b^2*c^3*d^7*e^2 - 697*b^3*c^2*d^6*e^3 + 374*b^4*c*d^5*e^4 - 79*b
^5*d^4*e^5)*x)*sqrt(c*x^2 + b*x))/(c^5*d^15 - 5*b*c^4*d^14*e + 10*b^2*c^3*d^13*e^2 - 10*b^3*c^2*d^12*e^3 + 5*b
^4*c*d^11*e^4 - b^5*d^10*e^5 + (c^5*d^10*e^5 - 5*b*c^4*d^9*e^6 + 10*b^2*c^3*d^8*e^7 - 10*b^3*c^2*d^7*e^8 + 5*b
^4*c*d^6*e^9 - b^5*d^5*e^10)*x^5 + 5*(c^5*d^11*e^4 - 5*b*c^4*d^10*e^5 + 10*b^2*c^3*d^9*e^6 - 10*b^3*c^2*d^8*e^
7 + 5*b^4*c*d^7*e^8 - b^5*d^6*e^9)*x^4 + 10*(c^5*d^12*e^3 - 5*b*c^4*d^11*e^4 + 10*b^2*c^3*d^10*e^5 - 10*b^3*c^
2*d^9*e^6 + 5*b^4*c*d^8*e^7 - b^5*d^7*e^8)*x^3 + 10*(c^5*d^13*e^2 - 5*b*c^4*d^12*e^3 + 10*b^2*c^3*d^11*e^4 - 1
0*b^3*c^2*d^10*e^5 + 5*b^4*c*d^9*e^6 - b^5*d^8*e^7)*x^2 + 5*(c^5*d^14*e - 5*b*c^4*d^13*e^2 + 10*b^2*c^3*d^12*e
^3 - 10*b^3*c^2*d^11*e^4 + 5*b^4*c*d^10*e^5 - b^5*d^9*e^6)*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)**6,x)

[Out]

Timed out

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Giac [B]  time = 1.66808, size = 2836, normalized size = 8.42 \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)^6,x, algorithm="giac")

[Out]

1/128*(32*b^2*c^3*d^3 - 48*b^3*c^2*d^2*e + 30*b^4*c*d*e^2 - 7*b^5*e^3)*arctan(((sqrt(c)*x - sqrt(c*x^2 + b*x))
*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))/((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)*sqrt(-c*d^2 + b*d*e)) + 1/1920*(7680*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*c^(13/2)*d^8*e + 3072*(sqrt(c)*x
- sqrt(c*x^2 + b*x))^5*c^7*d^9 + 9216*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*b*c^6*d^8*e + 7680*(sqrt(c)*x - sqrt(
c*x^2 + b*x))^4*b*c^(13/2)*d^9 - 30720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*b*c^(11/2)*d^7*e^2 - 3840*(sqrt(c)*x
- sqrt(c*x^2 + b*x))^4*b^2*c^(11/2)*d^8*e + 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b^2*c^6*d^9 - 50048*(sqrt(c
)*x - sqrt(c*x^2 + b*x))^5*b^2*c^5*d^7*e^2 - 11520*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b^3*c^5*d^8*e + 3840*(sqr
t(c)*x - sqrt(c*x^2 + b*x))^2*b^3*c^(11/2)*d^9 + 70720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*b^2*c^(9/2)*d^6*e^3 -
17600*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*b^3*c^(9/2)*d^7*e^2 - 7200*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^4*c^(9
/2)*d^8*e + 960*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^4*c^5*d^9 + 15040*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*b^2*c^4*
d^5*e^4 + 129280*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*b^3*c^4*d^6*e^3 + 14080*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b
^4*c^4*d^7*e^2 - 1920*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^5*c^4*d^8*e + 96*b^5*c^(9/2)*d^9 + 4320*(sqrt(c)*x - s
qrt(c*x^2 + b*x))^8*b^2*c^(7/2)*d^4*e^5 - 52000*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*b^3*c^(7/2)*d^5*e^4 + 81920*
(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*b^4*c^(7/2)*d^6*e^3 + 13760*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^5*c^(7/2)*d^
7*e^2 - 192*b^6*c^(7/2)*d^8*e + 480*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*b^2*c^3*d^3*e^6 - 20320*(sqrt(c)*x - sqr
t(c*x^2 + b*x))^7*b^3*c^3*d^4*e^5 - 120680*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*b^4*c^3*d^5*e^4 + 14080*(sqrt(c)*
x - sqrt(c*x^2 + b*x))^3*b^5*c^3*d^6*e^3 + 4280*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^6*c^3*d^7*e^2 - 6480*(sqrt(c
)*x - sqrt(c*x^2 + b*x))^8*b^3*c^(5/2)*d^3*e^6 + 7260*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*b^4*c^(5/2)*d^4*e^5 -
85780*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*b^5*c^(5/2)*d^5*e^4 - 6340*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^6*c^(5/
2)*d^6*e^3 + 476*b^7*c^(5/2)*d^7*e^2 - 720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*b^3*c^2*d^2*e^7 + 10740*(sqrt(c)*
x - sqrt(c*x^2 + b*x))^7*b^4*c^2*d^3*e^6 + 47944*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*b^5*c^2*d^4*e^5 - 25220*(sq
rt(c)*x - sqrt(c*x^2 + b*x))^3*b^6*c^2*d^5*e^4 - 3080*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^7*c^2*d^6*e^3 + 4050*(
sqrt(c)*x - sqrt(c*x^2 + b*x))^8*b^4*c^(3/2)*d^2*e^7 + 9310*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*b^5*c^(3/2)*d^3*
e^6 + 35330*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*b^6*c^(3/2)*d^4*e^5 - 1750*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^7
*c^(3/2)*d^5*e^4 - 380*b^8*c^(3/2)*d^6*e^3 + 450*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*b^4*c*d*e^8 - 1190*(sqrt(c)
*x - sqrt(c*x^2 + b*x))^7*b^5*c*d^2*e^7 - 4658*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*b^6*c*d^3*e^6 + 10510*(sqrt(c
)*x - sqrt(c*x^2 + b*x))^3*b^7*c*d^4*e^5 + 600*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^8*c*d^5*e^4 - 945*(sqrt(c)*x
- sqrt(c*x^2 + b*x))^8*b^5*sqrt(c)*d*e^8 - 3430*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*b^6*sqrt(c)*d^2*e^7 - 4480*(
sqrt(c)*x - sqrt(c*x^2 + b*x))^4*b^7*sqrt(c)*d^3*e^6 + 1470*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^8*sqrt(c)*d^4*
e^5 + 105*b^9*sqrt(c)*d^5*e^4 - 105*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*b^5*e^9 - 490*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^7*b^6*d*e^8 - 896*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*b^7*d^2*e^7 - 790*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*
b^8*d^3*e^6 + 105*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^9*d^4*e^5)/((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)*x - sqrt(c*x^2 + b*x))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x))
*sqrt(c)*d + b*d)^5)