3.1170 \(\int \frac{(A+B x) \sqrt{b x+c x^2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=310 \[ \frac{\sqrt{b x+c x^2} (x (2 c d-b e)+b d) \left (b^2 e (5 A e+3 B d)-8 b c d (2 A e+B d)+16 A c^2 d^2\right )}{64 d^3 (d+e x)^2 (c d-b e)^3}-\frac{b^2 \left (b^2 e (5 A e+3 B d)-8 b c d (2 A e+B d)+16 A c^2 d^2\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{128 d^{7/2} (c d-b e)^{7/2}}-\frac{\left (b x+c x^2\right )^{3/2} (5 A e (2 c d-b e)-B d (3 b e+2 c d))}{24 d^2 (d+e x)^3 (c d-b e)^2}+\frac{\left (b x+c x^2\right )^{3/2} (B d-A e)}{4 d (d+e x)^4 (c d-b e)} \]

[Out]

((16*A*c^2*d^2 - 8*b*c*d*(B*d + 2*A*e) + b^2*e*(3*B*d + 5*A*e))*(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) + ((B*d - A*e)*(b*x + c*x^2)^(3/2))/(4*d*(c*d - b*e)*(d + e*x)^4) - ((5*A*e*(2
*c*d - b*e) - B*d*(2*c*d + 3*b*e))*(b*x + c*x^2)^(3/2))/(24*d^2*(c*d - b*e)^2*(d + e*x)^3) - (b^2*(16*A*c^2*d^
2 - 8*b*c*d*(B*d + 2*A*e) + b^2*e*(3*B*d + 5*A*e))*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 = 0.405385, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {834, 806, 720, 724, 206} \[ \frac{\sqrt{b x+c x^2} (x (2 c d-b e)+b d) \left (b^2 e (5 A e+3 B d)-8 b c d (2 A e+B d)+16 A c^2 d^2\right )}{64 d^3 (d+e x)^2 (c d-b e)^3}-\frac{b^2 \left (b^2 e (5 A e+3 B d)-8 b c d (2 A e+B d)+16 A c^2 d^2\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{128 d^{7/2} (c d-b e)^{7/2}}-\frac{\left (b x+c x^2\right )^{3/2} (5 A e (2 c d-b e)-B d (3 b e+2 c d))}{24 d^2 (d+e x)^3 (c d-b e)^2}+\frac{\left (b x+c x^2\right )^{3/2} (B d-A e)}{4 d (d+e x)^4 (c d-b e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((16*A*c^2*d^2 - 8*b*c*d*(B*d + 2*A*e) + b^2*e*(3*B*d + 5*A*e))*(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) + ((B*d - A*e)*(b*x + c*x^2)^(3/2))/(4*d*(c*d - b*e)*(d + e*x)^4) - ((5*A*e*(2
*c*d - b*e) - B*d*(2*c*d + 3*b*e))*(b*x + c*x^2)^(3/2))/(24*d^2*(c*d - b*e)^2*(d + e*x)^3) - (b^2*(16*A*c^2*d^
2 - 8*b*c*d*(B*d + 2*A*e) + b^2*e*(3*B*d + 5*A*e))*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 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{(A+B x) \sqrt{b x+c x^2}}{(d+e x)^5} \, dx &=\frac{(B d-A 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} (3 b B d-8 A c d+5 A b e)-c (B d-A e) x\right ) \sqrt{b x+c x^2}}{(d+e x)^4} \, dx}{4 d (c d-b e)}\\ &=\frac{(B d-A e) \left (b x+c x^2\right )^{3/2}}{4 d (c d-b e) (d+e x)^4}-\frac{(5 A e (2 c d-b e)-B d (2 c d+3 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 A c^2 d^2-8 b c d (B d+2 A e)+b^2 e (3 B d+5 A e)\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 A c^2 d^2-8 b c d (B d+2 A e)+b^2 e (3 B d+5 A e)\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{(B d-A e) \left (b x+c x^2\right )^{3/2}}{4 d (c d-b e) (d+e x)^4}-\frac{(5 A e (2 c d-b e)-B d (2 c d+3 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 A c^2 d^2-8 b c d (B d+2 A e)+b^2 e (3 B d+5 A e)\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 A c^2 d^2-8 b c d (B d+2 A e)+b^2 e (3 B d+5 A e)\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{(B d-A e) \left (b x+c x^2\right )^{3/2}}{4 d (c d-b e) (d+e x)^4}-\frac{(5 A e (2 c d-b e)-B d (2 c d+3 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 A c^2 d^2-8 b c d (B d+2 A e)+b^2 e (3 B d+5 A e)\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 A c^2 d^2-8 b c d (B d+2 A e)+b^2 e (3 B d+5 A e)\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{(B d-A e) \left (b x+c x^2\right )^{3/2}}{4 d (c d-b e) (d+e x)^4}-\frac{(5 A e (2 c d-b e)-B d (2 c d+3 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 A c^2 d^2-8 b c d (B d+2 A e)+b^2 e (3 B d+5 A e)\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 = 0.925234, size = 279, normalized size = 0.9 \[ \frac{\sqrt{x (b+c x)} \left (\frac{3 (d+e x)^2 \left (b^2 e (5 A e+3 B d)-8 b c d (2 A e+B d)+16 A c^2 d^2\right ) \left (b^2 (d+e x)^2 \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )+\sqrt{d} \sqrt{x} \sqrt{b+c x} \sqrt{b e-c d} (-b d+b e x-2 c d x)\right )}{d^{5/2} \sqrt{b+c x} (b e-c d)^{5/2}}-\frac{8 x^{3/2} (b+c x) (d+e x) (5 A e (b e-2 c d)+B d (3 b e+2 c d))}{d (c d-b e)}+48 x^{3/2} (b+c x) (A e-B d)\right )}{192 d \sqrt{x} (d+e x)^4 (b e-c d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x*(b + c*x)]*(48*(-(B*d) + A*e)*x^(3/2)*(b + c*x) - (8*(5*A*e*(-2*c*d + b*e) + B*d*(2*c*d + 3*b*e))*x^(3
/2)*(b + c*x)*(d + e*x))/(d*(c*d - b*e)) + (3*(16*A*c^2*d^2 - 8*b*c*d*(B*d + 2*A*e) + b^2*e*(3*B*d + 5*A*e))*(
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*ArcT
an[(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 = 0.017, size = 10550, normalized size = 34. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^(1/2)/(e*x+d)^5,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((B*x+A)*(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 = 1.24428, size = 4544, normalized size = 14.66 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/384*(3*(5*A*b^4*d^4*e^2 - 8*(B*b^3*c - 2*A*b^2*c^2)*d^6 + (3*B*b^4 - 16*A*b^3*c)*d^5*e + (5*A*b^4*e^6 - 8*
(B*b^3*c - 2*A*b^2*c^2)*d^2*e^4 + (3*B*b^4 - 16*A*b^3*c)*d*e^5)*x^4 + 4*(5*A*b^4*d*e^5 - 8*(B*b^3*c - 2*A*b^2*
c^2)*d^3*e^3 + (3*B*b^4 - 16*A*b^3*c)*d^2*e^4)*x^3 + 6*(5*A*b^4*d^2*e^4 - 8*(B*b^3*c - 2*A*b^2*c^2)*d^4*e^2 +
(3*B*b^4 - 16*A*b^3*c)*d^3*e^3)*x^2 + 4*(5*A*b^4*d^3*e^3 - 8*(B*b^3*c - 2*A*b^2*c^2)*d^5*e + (3*B*b^4 - 16*A*b
^3*c)*d^4*e^2)*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*(15*A*b^4*d^4*e^3 + 24*(B*b^2*c^2 - 2*A*b*c^3)*d^7 - 3*(11*B*b^3*c - 32*A*b^2*c^2)*d^6*e + 9*(B*b
^4 - 7*A*b^3*c)*d^5*e^2 - (16*B*c^4*d^6*e + 15*A*b^4*d*e^6 - 8*(7*B*b*c^3 - 2*A*c^4)*d^5*e^2 + 2*(29*B*b^2*c^2
- 20*A*b*c^3)*d^4*e^3 - (27*B*b^3*c - 62*A*b^2*c^2)*d^3*e^4 + (9*B*b^4 - 53*A*b^3*c)*d^2*e^5)*x^3 - (64*B*c^4
*d^7 + 55*A*b^4*d^2*e^5 - 8*(29*B*b*c^3 - 8*A*c^4)*d^6*e + 4*(65*B*b^2*c^2 - 42*A*b*c^3)*d^5*e^2 - (125*B*b^3*
c - 244*A*b^2*c^2)*d^4*e^3 + 3*(11*B*b^4 - 65*A*b^3*c)*d^3*e^4)*x^2 - (73*A*b^4*d^3*e^4 + 16*(B*b*c^3 + 6*A*c^
4)*d^7 - 2*(55*B*b^2*c^2 + 136*A*b*c^3)*d^6*e + (127*B*b^3*c + 374*A*b^2*c^2)*d^5*e^2 - (33*B*b^4 + 271*A*b^3*
c)*d^4*e^3)*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), -1/192*(3*(5*A*b^4*d^4*e^2 - 8*(B*b^3*c - 2*A*b^2*c^2)*d^6 + (3*B*b^4
- 16*A*b^3*c)*d^5*e + (5*A*b^4*e^6 - 8*(B*b^3*c - 2*A*b^2*c^2)*d^2*e^4 + (3*B*b^4 - 16*A*b^3*c)*d*e^5)*x^4 +
4*(5*A*b^4*d*e^5 - 8*(B*b^3*c - 2*A*b^2*c^2)*d^3*e^3 + (3*B*b^4 - 16*A*b^3*c)*d^2*e^4)*x^3 + 6*(5*A*b^4*d^2*e^
4 - 8*(B*b^3*c - 2*A*b^2*c^2)*d^4*e^2 + (3*B*b^4 - 16*A*b^3*c)*d^3*e^3)*x^2 + 4*(5*A*b^4*d^3*e^3 - 8*(B*b^3*c
- 2*A*b^2*c^2)*d^5*e + (3*B*b^4 - 16*A*b^3*c)*d^4*e^2)*x)*sqrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^2 + b*d*e)*sq
rt(c*x^2 + b*x)/((c*d - b*e)*x)) + (15*A*b^4*d^4*e^3 + 24*(B*b^2*c^2 - 2*A*b*c^3)*d^7 - 3*(11*B*b^3*c - 32*A*b
^2*c^2)*d^6*e + 9*(B*b^4 - 7*A*b^3*c)*d^5*e^2 - (16*B*c^4*d^6*e + 15*A*b^4*d*e^6 - 8*(7*B*b*c^3 - 2*A*c^4)*d^5
*e^2 + 2*(29*B*b^2*c^2 - 20*A*b*c^3)*d^4*e^3 - (27*B*b^3*c - 62*A*b^2*c^2)*d^3*e^4 + (9*B*b^4 - 53*A*b^3*c)*d^
2*e^5)*x^3 - (64*B*c^4*d^7 + 55*A*b^4*d^2*e^5 - 8*(29*B*b*c^3 - 8*A*c^4)*d^6*e + 4*(65*B*b^2*c^2 - 42*A*b*c^3)
*d^5*e^2 - (125*B*b^3*c - 244*A*b^2*c^2)*d^4*e^3 + 3*(11*B*b^4 - 65*A*b^3*c)*d^3*e^4)*x^2 - (73*A*b^4*d^3*e^4
+ 16*(B*b*c^3 + 6*A*c^4)*d^7 - 2*(55*B*b^2*c^2 + 136*A*b*c^3)*d^6*e + (127*B*b^3*c + 374*A*b^2*c^2)*d^5*e^2 -
(33*B*b^4 + 271*A*b^3*c)*d^4*e^3)*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^1
0*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^1
0*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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x \left (b + c x\right )} \left (A + B x\right )}{\left (d + e x\right )^{5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x*(b + c*x))*(A + B*x)/(d + e*x)**5, x)

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Giac [B]  time = 2.10851, size = 2444, normalized size = 7.88 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(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*((10*B*c^3*d
^6*e^11*sgn(1/(x*e + d)) - 29*B*b*c^2*d^5*e^12*sgn(1/(x*e + d)) - 2*A*c^3*d^5*e^12*sgn(1/(x*e + d)) + 28*B*b^2
*c*d^4*e^13*sgn(1/(x*e + d)) + 5*A*b*c^2*d^4*e^13*sgn(1/(x*e + d)) - 9*B*b^3*d^3*e^14*sgn(1/(x*e + d)) - 4*A*b
^2*c*d^3*e^14*sgn(1/(x*e + d)) + A*b^3*d^2*e^15*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*(B*c^3*d^7*e^12*sgn(1/(x*e + d)) - 3*B*b*c^2*d^6*e^13*sgn(1/(x*
e + d)) - A*c^3*d^6*e^13*sgn(1/(x*e + d)) + 3*B*b^2*c*d^5*e^14*sgn(1/(x*e + d)) + 3*A*b*c^2*d^5*e^14*sgn(1/(x*
e + d)) - B*b^3*d^4*e^15*sgn(1/(x*e + d)) - 3*A*b^2*c*d^4*e^15*sgn(1/(x*e + d)) + A*b^3*d^3*e^16*sgn(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*B*c^3*d^5*e^10*sgn(1/(x*e + d)) - 24*B*b*c^2*d^4*e^11*sgn(1/(x*e + d)) + 8*A*c^3*d^4*e
^11*sgn(1/(x*e + d)) + 19*B*b^2*c*d^3*e^12*sgn(1/(x*e + d)) - 16*A*b*c^2*d^3*e^12*sgn(1/(x*e + d)) - 3*B*b^3*d
^2*e^13*sgn(1/(x*e + d)) + 13*A*b^2*c*d^2*e^13*sgn(1/(x*e + d)) - 5*A*b^3*d*e^14*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*B*c^3*d^4*
e^9*sgn(1/(x*e + d)) - 40*B*b*c^2*d^3*e^10*sgn(1/(x*e + d)) + 16*A*c^3*d^3*e^10*sgn(1/(x*e + d)) + 18*B*b^2*c*
d^2*e^11*sgn(1/(x*e + d)) - 24*A*b*c^2*d^2*e^11*sgn(1/(x*e + d)) - 9*B*b^3*d*e^12*sgn(1/(x*e + d)) + 38*A*b^2*
c*d*e^12*sgn(1/(x*e + d)) - 15*A*b^3*e^13*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^1
0 - 4*b^3*c*d^5*e^11 + b^4*d^4*e^12)) + (32*sqrt(c*d^2 - b*d*e)*B*c^(7/2)*d^4*e - 80*sqrt(c*d^2 - b*d*e)*B*b*c
^(5/2)*d^3*e^2 + 32*sqrt(c*d^2 - b*d*e)*A*c^(7/2)*d^3*e^2 - 24*B*b^3*c*d^2*e^4*log(abs(2*c*d - b*e - 2*sqrt(c*
d^2 - b*d*e)*sqrt(c))) + 48*A*b^2*c^2*d^2*e^4*log(abs(2*c*d - b*e - 2*sqrt(c*d^2 - b*d*e)*sqrt(c))) + 36*sqrt(
c*d^2 - b*d*e)*B*b^2*c^(3/2)*d^2*e^3 - 48*sqrt(c*d^2 - b*d*e)*A*b*c^(5/2)*d^2*e^3 + 9*B*b^4*d*e^5*log(abs(2*c*
d - b*e - 2*sqrt(c*d^2 - b*d*e)*sqrt(c))) - 48*A*b^3*c*d*e^5*log(abs(2*c*d - b*e - 2*sqrt(c*d^2 - b*d*e)*sqrt(
c))) - 18*sqrt(c*d^2 - b*d*e)*B*b^3*sqrt(c)*d*e^4 + 76*sqrt(c*d^2 - b*d*e)*A*b^2*c^(3/2)*d*e^4 + 15*A*b^4*e^6*
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)*A*b^3*sqrt(c)*e^5)*sgn(1/(x*e +
d))/(sqrt(c*d^2 - b*d*e)*c^4*d^8 - 4*sqrt(c*d^2 - b*d*e)*b*c^3*d^7*e + 6*sqrt(c*d^2 - b*d*e)*b^2*c^2*d^6*e^2
- 4*sqrt(c*d^2 - b*d*e)*b^3*c*d^5*e^3 + sqrt(c*d^2 - b*d*e)*b^4*d^4*e^4) + 3*(8*B*b^3*c*d^2*e^5*sgn(1/(x*e + d
)) - 16*A*b^2*c^2*d^2*e^5*sgn(1/(x*e + d)) - 3*B*b^4*d*e^6*sgn(1/(x*e + d)) + 16*A*b^3*c*d*e^6*sgn(1/(x*e + d)
) - 5*A*b^4*e^7*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
- 4*b*c^3*d^7*e^2 + 6*b^2*c^2*d^6*e^3 - 4*b^3*c*d^5*e^4 + b^4*d^4*e^5)*sqrt(c*d^2 - b*d*e)))*e^2