3.1555 \(\int \frac{(b+2 c x) \sqrt{a+b x+c x^2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=307 \[ \frac{\left (a+b x+c x^2\right )^{3/2} \left (-4 c e (4 a e+b d)+5 b^2 e^2+4 c^2 d^2\right )}{24 (d+e x)^3 \left (a e^2-b d e+c d^2\right )^2}-\frac{5 e \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{64 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^3}+\frac{5 e \left (b^2-4 a c\right )^2 (2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{128 \left (a e^2-b d e+c d^2\right )^{7/2}}+\frac{\left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{4 (d+e x)^4 \left (a e^2-b d e+c d^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.458677, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {834, 806, 720, 724, 206} \[ \frac{\left (a+b x+c x^2\right )^{3/2} \left (-4 c e (4 a e+b d)+5 b^2 e^2+4 c^2 d^2\right )}{24 (d+e x)^3 \left (a e^2-b d e+c d^2\right )^2}-\frac{5 e \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{64 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^3}+\frac{5 e \left (b^2-4 a c\right )^2 (2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{128 \left (a e^2-b d e+c d^2\right )^{7/2}}+\frac{\left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{4 (d+e x)^4 \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*(b^2 - 4*a*c)*e*(2*c*d - b*e)*(b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(64*(c*d^2 - b*d*e +
a*e^2)^3*(d + e*x)^2) + ((2*c*d - b*e)*(a + b*x + c*x^2)^(3/2))/(4*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^4) + ((4*
c^2*d^2 + 5*b^2*e^2 - 4*c*e*(b*d + 4*a*e))*(a + b*x + c*x^2)^(3/2))/(24*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^3)
+ (5*(b^2 - 4*a*c)^2*e*(2*c*d - b*e)*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*S
qrt[a + b*x + c*x^2])])/(128*(c*d^2 - b*d*e + a*e^2)^(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{(b+2 c x) \sqrt{a+b x+c x^2}}{(d+e x)^5} \, dx &=\frac{(2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{4 \left (c d^2-b d e+a e^2\right ) (d+e x)^4}-\frac{\int \frac{\left (\frac{1}{2} \left (-2 b c d+5 b^2 e-16 a c e\right )-c (2 c d-b e) x\right ) \sqrt{a+b x+c x^2}}{(d+e x)^4} \, dx}{4 \left (c d^2-b d e+a e^2\right )}\\ &=\frac{(2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{4 \left (c d^2-b d e+a e^2\right ) (d+e x)^4}+\frac{\left (4 c^2 d^2+5 b^2 e^2-4 c e (b d+4 a e)\right ) \left (a+b x+c x^2\right )^{3/2}}{24 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}-\frac{\left (5 \left (b^2-4 a c\right ) e (2 c d-b e)\right ) \int \frac{\sqrt{a+b x+c x^2}}{(d+e x)^3} \, dx}{16 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{5 \left (b^2-4 a c\right ) e (2 c d-b e) (b d-2 a e+(2 c d-b e) x) \sqrt{a+b x+c x^2}}{64 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}+\frac{(2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{4 \left (c d^2-b d e+a e^2\right ) (d+e x)^4}+\frac{\left (4 c^2 d^2+5 b^2 e^2-4 c e (b d+4 a e)\right ) \left (a+b x+c x^2\right )^{3/2}}{24 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}+\frac{\left (5 \left (b^2-4 a c\right )^2 e (2 c d-b e)\right ) \int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{128 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{5 \left (b^2-4 a c\right ) e (2 c d-b e) (b d-2 a e+(2 c d-b e) x) \sqrt{a+b x+c x^2}}{64 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}+\frac{(2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{4 \left (c d^2-b d e+a e^2\right ) (d+e x)^4}+\frac{\left (4 c^2 d^2+5 b^2 e^2-4 c e (b d+4 a e)\right ) \left (a+b x+c x^2\right )^{3/2}}{24 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}-\frac{\left (5 \left (b^2-4 a c\right )^2 e (2 c d-b e)\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac{-b d+2 a e-(2 c d-b e) x}{\sqrt{a+b x+c x^2}}\right )}{64 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{5 \left (b^2-4 a c\right ) e (2 c d-b e) (b d-2 a e+(2 c d-b e) x) \sqrt{a+b x+c x^2}}{64 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}+\frac{(2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{4 \left (c d^2-b d e+a e^2\right ) (d+e x)^4}+\frac{\left (4 c^2 d^2+5 b^2 e^2-4 c e (b d+4 a e)\right ) \left (a+b x+c x^2\right )^{3/2}}{24 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}+\frac{5 \left (b^2-4 a c\right )^2 e (2 c d-b e) \tanh ^{-1}\left (\frac{b d-2 a e+(2 c d-b e) x}{2 \sqrt{c d^2-b d e+a e^2} \sqrt{a+b x+c x^2}}\right )}{128 \left (c d^2-b d e+a e^2\right )^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.829124, size = 290, normalized size = 0.94 \[ \frac{\frac{(a+x (b+c x))^{3/2} \left (-4 c e (4 a e+b d)+5 b^2 e^2+4 c^2 d^2\right )}{(d+e x)^3}+\frac{15}{2} e \left (b^2-4 a c\right ) (b e-2 c d) \left (\frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )}{8 \left (e (a e-b d)+c d^2\right )^{3/2}}+\frac{\sqrt{a+x (b+c x)} (-2 a e+b (d-e x)+2 c d x)}{4 (d+e x)^2 \left (e (a e-b d)+c d^2\right )}\right )+\frac{6 (a+x (b+c x))^{3/2} (2 c d-b e) \left (e (a e-b d)+c d^2\right )}{(d+e x)^4}}{24 \left (e (a e-b d)+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((6*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))*(a + x*(b + c*x))^(3/2))/(d + e*x)^4 + ((4*c^2*d^2 + 5*b^2*e^2 -
4*c*e*(b*d + 4*a*e))*(a + x*(b + c*x))^(3/2))/(d + e*x)^3 + (15*(b^2 - 4*a*c)*e*(-2*c*d + b*e)*((Sqrt[a + x*(b
+ c*x)]*(-2*a*e + 2*c*d*x + b*(d - e*x)))/(4*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) + ((b^2 - 4*a*c)*ArcTanh
[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(8*(c*d^2 + e*(
-(b*d) + a*e))^(3/2))))/2)/(24*(c*d^2 + e*(-(b*d) + a*e))^2)

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Maple [B]  time = 0.021, size = 10723, normalized size = 34.9 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 176.948, size = 8436, normalized size = 27.48 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/768*(15*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^5*e - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^4*e^2 + (2*(b^4*c
- 8*a*b^2*c^2 + 16*a^2*c^3)*d*e^5 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*e^6)*x^4 + 4*(2*(b^4*c - 8*a*b^2*c^2 + 1
6*a^2*c^3)*d^2*e^4 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d*e^5)*x^3 + 6*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^3
*e^3 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^2*e^4)*x^2 + 4*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^4*e^2 - (b^5
- 8*a*b^3*c + 16*a^2*b*c^2)*d^3*e^3)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)
*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 + 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(
b*d - 2*a*e + (2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^2))
- 4*(128*a*c^4*d^7 - 48*a^4*b*e^7 - 6*(5*b^3*c^2 + 52*a*b*c^3)*d^6*e + (45*b^4*c + 448*a*b^2*c^2 - 16*a^2*c^3
)*d^5*e^2 - (15*b^5 + 412*a*b^3*c + 112*a^2*b*c^2)*d^4*e^3 + (133*a*b^4 + 472*a^2*b^2*c - 176*a^3*c^2)*d^3*e^4
- 2*(127*a^2*b^3 + 44*a^3*b*c)*d^2*e^5 + 8*(23*a^3*b^2 - 4*a^4*c)*d*e^6 + (32*c^5*d^6*e - 96*b*c^4*d^5*e^2 +
4*(19*b^2*c^3 + 44*a*c^4)*d^4*e^3 + 8*(b^3*c^2 - 44*a*b*c^3)*d^3*e^4 - (35*b^4*c - 256*a*b^2*c^2 - 16*a^2*c^3)
*d^2*e^5 + (15*b^5 - 80*a*b^3*c - 16*a^2*b*c^2)*d*e^6 - (15*a*b^4 - 100*a^2*b^2*c + 128*a^3*c^2)*e^7)*x^3 + (1
28*c^5*d^7 - 400*b*c^4*d^6*e + 352*(b^2*c^3 + 2*a*c^4)*d^5*e^2 - 2*(3*b^3*c^2 + 748*a*b*c^3)*d^4*e^3 - (129*b^
4*c - 1080*a*b^2*c^2 - 304*a^2*c^3)*d^3*e^4 + (55*b^5 - 268*a*b^3*c - 432*a^2*b*c^2)*d^2*e^5 - (65*a*b^4 - 408
*a^2*b^2*c + 272*a^3*c^2)*d*e^6 + 2*(5*a^2*b^3 - 28*a^3*b*c)*e^7)*x^2 + (128*b*c^4*d^7 - 4*(123*b^2*c^3 - 68*a
*c^4)*d^6*e + 4*(157*b^3*c^2 - 28*a*b*c^3)*d^5*e^2 - (337*b^4*c + 456*a*b^2*c^2 + 304*a^2*c^3)*d^4*e^3 + (73*b
^5 + 360*a*b^3*c + 912*a^2*b*c^2)*d^3*e^4 - (109*a*b^4 + 372*a^2*b^2*c + 704*a^3*c^2)*d^2*e^5 + 4*(11*a^2*b^3
+ 108*a^3*b*c)*d*e^6 - 8*(a^3*b^2 + 16*a^4*c)*e^7)*x)*sqrt(c*x^2 + b*x + a))/(c^4*d^12 - 4*b*c^3*d^11*e - 4*a^
3*b*d^5*e^7 + a^4*d^4*e^8 + 2*(3*b^2*c^2 + 2*a*c^3)*d^10*e^2 - 4*(b^3*c + 3*a*b*c^2)*d^9*e^3 + (b^4 + 12*a*b^2
*c + 6*a^2*c^2)*d^8*e^4 - 4*(a*b^3 + 3*a^2*b*c)*d^7*e^5 + 2*(3*a^2*b^2 + 2*a^3*c)*d^6*e^6 + (c^4*d^8*e^4 - 4*b
*c^3*d^7*e^5 - 4*a^3*b*d*e^11 + a^4*e^12 + 2*(3*b^2*c^2 + 2*a*c^3)*d^6*e^6 - 4*(b^3*c + 3*a*b*c^2)*d^5*e^7 + (
b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^8 - 4*(a*b^3 + 3*a^2*b*c)*d^3*e^9 + 2*(3*a^2*b^2 + 2*a^3*c)*d^2*e^10)*x^4
+ 4*(c^4*d^9*e^3 - 4*b*c^3*d^8*e^4 - 4*a^3*b*d^2*e^10 + a^4*d*e^11 + 2*(3*b^2*c^2 + 2*a*c^3)*d^7*e^5 - 4*(b^3*
c + 3*a*b*c^2)*d^6*e^6 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^5*e^7 - 4*(a*b^3 + 3*a^2*b*c)*d^4*e^8 + 2*(3*a^2*b^2
+ 2*a^3*c)*d^3*e^9)*x^3 + 6*(c^4*d^10*e^2 - 4*b*c^3*d^9*e^3 - 4*a^3*b*d^3*e^9 + a^4*d^2*e^10 + 2*(3*b^2*c^2 +
2*a*c^3)*d^8*e^4 - 4*(b^3*c + 3*a*b*c^2)*d^7*e^5 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^6*e^6 - 4*(a*b^3 + 3*a^2*
b*c)*d^5*e^7 + 2*(3*a^2*b^2 + 2*a^3*c)*d^4*e^8)*x^2 + 4*(c^4*d^11*e - 4*b*c^3*d^10*e^2 - 4*a^3*b*d^4*e^8 + a^4
*d^3*e^9 + 2*(3*b^2*c^2 + 2*a*c^3)*d^9*e^3 - 4*(b^3*c + 3*a*b*c^2)*d^8*e^4 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^
7*e^5 - 4*(a*b^3 + 3*a^2*b*c)*d^6*e^6 + 2*(3*a^2*b^2 + 2*a^3*c)*d^5*e^7)*x), 1/384*(15*(2*(b^4*c - 8*a*b^2*c^2
+ 16*a^2*c^3)*d^5*e - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^4*e^2 + (2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*e^5
- (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*e^6)*x^4 + 4*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^2*e^4 - (b^5 - 8*a*b^3
*c + 16*a^2*b*c^2)*d*e^5)*x^3 + 6*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^3*e^3 - (b^5 - 8*a*b^3*c + 16*a^2*b*
c^2)*d^2*e^4)*x^2 + 4*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^4*e^2 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^3*e^3
)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e
+ (2*c*d - b*e)*x)/(a*c*d^2 - a*b*d*e + a^2*e^2 + (c^2*d^2 - b*c*d*e + a*c*e^2)*x^2 + (b*c*d^2 - b^2*d*e + a*b
*e^2)*x)) + 2*(128*a*c^4*d^7 - 48*a^4*b*e^7 - 6*(5*b^3*c^2 + 52*a*b*c^3)*d^6*e + (45*b^4*c + 448*a*b^2*c^2 - 1
6*a^2*c^3)*d^5*e^2 - (15*b^5 + 412*a*b^3*c + 112*a^2*b*c^2)*d^4*e^3 + (133*a*b^4 + 472*a^2*b^2*c - 176*a^3*c^2
)*d^3*e^4 - 2*(127*a^2*b^3 + 44*a^3*b*c)*d^2*e^5 + 8*(23*a^3*b^2 - 4*a^4*c)*d*e^6 + (32*c^5*d^6*e - 96*b*c^4*d
^5*e^2 + 4*(19*b^2*c^3 + 44*a*c^4)*d^4*e^3 + 8*(b^3*c^2 - 44*a*b*c^3)*d^3*e^4 - (35*b^4*c - 256*a*b^2*c^2 - 16
*a^2*c^3)*d^2*e^5 + (15*b^5 - 80*a*b^3*c - 16*a^2*b*c^2)*d*e^6 - (15*a*b^4 - 100*a^2*b^2*c + 128*a^3*c^2)*e^7)
*x^3 + (128*c^5*d^7 - 400*b*c^4*d^6*e + 352*(b^2*c^3 + 2*a*c^4)*d^5*e^2 - 2*(3*b^3*c^2 + 748*a*b*c^3)*d^4*e^3
- (129*b^4*c - 1080*a*b^2*c^2 - 304*a^2*c^3)*d^3*e^4 + (55*b^5 - 268*a*b^3*c - 432*a^2*b*c^2)*d^2*e^5 - (65*a*
b^4 - 408*a^2*b^2*c + 272*a^3*c^2)*d*e^6 + 2*(5*a^2*b^3 - 28*a^3*b*c)*e^7)*x^2 + (128*b*c^4*d^7 - 4*(123*b^2*c
^3 - 68*a*c^4)*d^6*e + 4*(157*b^3*c^2 - 28*a*b*c^3)*d^5*e^2 - (337*b^4*c + 456*a*b^2*c^2 + 304*a^2*c^3)*d^4*e^
3 + (73*b^5 + 360*a*b^3*c + 912*a^2*b*c^2)*d^3*e^4 - (109*a*b^4 + 372*a^2*b^2*c + 704*a^3*c^2)*d^2*e^5 + 4*(11
*a^2*b^3 + 108*a^3*b*c)*d*e^6 - 8*(a^3*b^2 + 16*a^4*c)*e^7)*x)*sqrt(c*x^2 + b*x + a))/(c^4*d^12 - 4*b*c^3*d^11
*e - 4*a^3*b*d^5*e^7 + a^4*d^4*e^8 + 2*(3*b^2*c^2 + 2*a*c^3)*d^10*e^2 - 4*(b^3*c + 3*a*b*c^2)*d^9*e^3 + (b^4 +
12*a*b^2*c + 6*a^2*c^2)*d^8*e^4 - 4*(a*b^3 + 3*a^2*b*c)*d^7*e^5 + 2*(3*a^2*b^2 + 2*a^3*c)*d^6*e^6 + (c^4*d^8*
e^4 - 4*b*c^3*d^7*e^5 - 4*a^3*b*d*e^11 + a^4*e^12 + 2*(3*b^2*c^2 + 2*a*c^3)*d^6*e^6 - 4*(b^3*c + 3*a*b*c^2)*d^
5*e^7 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^8 - 4*(a*b^3 + 3*a^2*b*c)*d^3*e^9 + 2*(3*a^2*b^2 + 2*a^3*c)*d^2*e
^10)*x^4 + 4*(c^4*d^9*e^3 - 4*b*c^3*d^8*e^4 - 4*a^3*b*d^2*e^10 + a^4*d*e^11 + 2*(3*b^2*c^2 + 2*a*c^3)*d^7*e^5
- 4*(b^3*c + 3*a*b*c^2)*d^6*e^6 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^5*e^7 - 4*(a*b^3 + 3*a^2*b*c)*d^4*e^8 + 2*(
3*a^2*b^2 + 2*a^3*c)*d^3*e^9)*x^3 + 6*(c^4*d^10*e^2 - 4*b*c^3*d^9*e^3 - 4*a^3*b*d^3*e^9 + a^4*d^2*e^10 + 2*(3*
b^2*c^2 + 2*a*c^3)*d^8*e^4 - 4*(b^3*c + 3*a*b*c^2)*d^7*e^5 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^6*e^6 - 4*(a*b^3
+ 3*a^2*b*c)*d^5*e^7 + 2*(3*a^2*b^2 + 2*a^3*c)*d^4*e^8)*x^2 + 4*(c^4*d^11*e - 4*b*c^3*d^10*e^2 - 4*a^3*b*d^4*
e^8 + a^4*d^3*e^9 + 2*(3*b^2*c^2 + 2*a*c^3)*d^9*e^3 - 4*(b^3*c + 3*a*b*c^2)*d^8*e^4 + (b^4 + 12*a*b^2*c + 6*a^
2*c^2)*d^7*e^5 - 4*(a*b^3 + 3*a^2*b*c)*d^6*e^6 + 2*(3*a^2*b^2 + 2*a^3*c)*d^5*e^7)*x)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (b + 2 c x\right ) \sqrt{a + b x + c x^{2}}}{\left (d + e x\right )^{5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((b + 2*c*x)*sqrt(a + b*x + c*x**2)/(d + e*x)**5, x)

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Giac [B]  time = 19.2871, size = 4416, normalized size = 14.38 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x+a)^(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 + a*e^2/(x*e + d)^2
)*(2*(4*((20*c^4*d^6*e^11*sgn(1/(x*e + d)) - 60*b*c^3*d^5*e^12*sgn(1/(x*e + d)) + 61*b^2*c^2*d^4*e^13*sgn(1/(x
*e + d)) + 56*a*c^3*d^4*e^13*sgn(1/(x*e + d)) - 22*b^3*c*d^3*e^14*sgn(1/(x*e + d)) - 112*a*b*c^2*d^3*e^14*sgn(
1/(x*e + d)) + b^4*d^2*e^15*sgn(1/(x*e + d)) + 58*a*b^2*c*d^2*e^15*sgn(1/(x*e + d)) + 52*a^2*c^2*d^2*e^15*sgn(
1/(x*e + d)) - 2*a*b^3*d*e^16*sgn(1/(x*e + d)) - 52*a^2*b*c*d*e^16*sgn(1/(x*e + d)) + a^2*b^2*e^17*sgn(1/(x*e
+ d)) + 16*a^3*c*e^17*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*a*c^3*d^6*e^10
- 4*b^3*c*d^5*e^11 - 12*a*b*c^2*d^5*e^11 + b^4*d^4*e^12 + 12*a*b^2*c*d^4*e^12 + 6*a^2*c^2*d^4*e^12 - 4*a*b^3*
d^3*e^13 - 12*a^2*b*c*d^3*e^13 + 6*a^2*b^2*d^2*e^14 + 4*a^3*c*d^2*e^14 - 4*a^3*b*d*e^15 + a^4*e^16) - 6*(2*c^4
*d^7*e^12*sgn(1/(x*e + d)) - 7*b*c^3*d^6*e^13*sgn(1/(x*e + d)) + 9*b^2*c^2*d^5*e^14*sgn(1/(x*e + d)) + 6*a*c^3
*d^5*e^14*sgn(1/(x*e + d)) - 5*b^3*c*d^4*e^15*sgn(1/(x*e + d)) - 15*a*b*c^2*d^4*e^15*sgn(1/(x*e + d)) + b^4*d^
3*e^16*sgn(1/(x*e + d)) + 12*a*b^2*c*d^3*e^16*sgn(1/(x*e + d)) + 6*a^2*c^2*d^3*e^16*sgn(1/(x*e + d)) - 3*a*b^3
*d^2*e^17*sgn(1/(x*e + d)) - 9*a^2*b*c*d^2*e^17*sgn(1/(x*e + d)) + 3*a^2*b^2*d*e^18*sgn(1/(x*e + d)) + 2*a^3*c
*d*e^18*sgn(1/(x*e + d)) - a^3*b*e^19*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*a*c^3*d^6*e^10 - 4*b^3*c*d^5*e^11 - 12*a*b*c^2*d^5*e^11 + b^4*d^4*e^12 + 12*a*b^2*c*d^4*e^12 + 6*a^2
*c^2*d^4*e^12 - 4*a*b^3*d^3*e^13 - 12*a^2*b*c*d^3*e^13 + 6*a^2*b^2*d^2*e^14 + 4*a^3*c*d^2*e^14 - 4*a^3*b*d*e^1
5 + a^4*e^16)*(x*e + d)))*e^(-1)/(x*e + d) - (16*c^4*d^5*e^10*sgn(1/(x*e + d)) - 40*b*c^3*d^4*e^11*sgn(1/(x*e
+ d)) + 22*b^2*c^2*d^3*e^12*sgn(1/(x*e + d)) + 72*a*c^3*d^3*e^12*sgn(1/(x*e + d)) + 7*b^3*c*d^2*e^13*sgn(1/(x*
e + d)) - 108*a*b*c^2*d^2*e^13*sgn(1/(x*e + d)) - 5*b^4*d*e^14*sgn(1/(x*e + d)) + 26*a*b^2*c*d*e^14*sgn(1/(x*e
+ d)) + 56*a^2*c^2*d*e^14*sgn(1/(x*e + d)) + 5*a*b^3*e^15*sgn(1/(x*e + d)) - 28*a^2*b*c*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*a*c^3*d^6*e^10 - 4*b^3*c*d^5*e^11 - 12*a*b*c^2*d^5*e
^11 + b^4*d^4*e^12 + 12*a*b^2*c*d^4*e^12 + 6*a^2*c^2*d^4*e^12 - 4*a*b^3*d^3*e^13 - 12*a^2*b*c*d^3*e^13 + 6*a^2
*b^2*d^2*e^14 + 4*a^3*c*d^2*e^14 - 4*a^3*b*d*e^15 + a^4*e^16))*e^(-1)/(x*e + d) - (32*c^4*d^4*e^9*sgn(1/(x*e +
d)) - 64*b*c^3*d^3*e^10*sgn(1/(x*e + d)) + 12*b^2*c^2*d^2*e^11*sgn(1/(x*e + d)) + 144*a*c^3*d^2*e^11*sgn(1/(x
*e + d)) + 20*b^3*c*d*e^12*sgn(1/(x*e + d)) - 144*a*b*c^2*d*e^12*sgn(1/(x*e + d)) - 15*b^4*e^13*sgn(1/(x*e + d
)) + 100*a*b^2*c*e^13*sgn(1/(x*e + d)) - 128*a^2*c^2*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^10 + 4*a*c^3*d^6*e^10 - 4*b^3*c*d^5*e^11 - 12*a*b*c^2*d^5*e^11 + b^4*d^4*e^12 + 12*a*b^2*c*d^4*
e^12 + 6*a^2*c^2*d^4*e^12 - 4*a*b^3*d^3*e^13 - 12*a^2*b*c*d^3*e^13 + 6*a^2*b^2*d^2*e^14 + 4*a^3*c*d^2*e^14 - 4
*a^3*b*d*e^15 + a^4*e^16)) + 15*(2*b^4*c*d*e^6*sgn(1/(x*e + d)) - 16*a*b^2*c^2*d*e^6*sgn(1/(x*e + d)) + 32*a^2
*c^3*d*e^6*sgn(1/(x*e + d)) - b^5*e^7*sgn(1/(x*e + d)) + 8*a*b^3*c*e^7*sgn(1/(x*e + d)) - 16*a^2*b*c^2*e^7*sgn
(1/(x*e + d)))*sqrt(c*d^2 - b*d*e + a*e^2)*log(abs(2*(c*d^2 - b*d*e + a*e^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 + a*e^2/(x*e + d)^2) + sqrt(c*d^2*e^2 - b*d*e^3 + a*e^4)*e^(-
1)/(x*e + d)) - sqrt(c*d^2 - b*d*e + a*e^2)*(2*c*d - b*e)))/(c^5*d^10*e - 5*b*c^4*d^9*e^2 + 10*b^2*c^3*d^8*e^3
+ 5*a*c^4*d^8*e^3 - 10*b^3*c^2*d^7*e^4 - 20*a*b*c^3*d^7*e^4 + 5*b^4*c*d^6*e^5 + 30*a*b^2*c^2*d^6*e^5 + 10*a^2
*c^3*d^6*e^5 - b^5*d^5*e^6 - 20*a*b^3*c*d^5*e^6 - 30*a^2*b*c^2*d^5*e^6 + 5*a*b^4*d^4*e^7 + 30*a^2*b^2*c*d^4*e^
7 + 10*a^3*c^2*d^4*e^7 - 10*a^2*b^3*d^3*e^8 - 20*a^3*b*c*d^3*e^8 + 10*a^3*b^2*d^2*e^9 + 5*a^4*c*d^2*e^9 - 5*a^
4*b*d*e^10 + a^5*e^11) + (64*c^(11/2)*d^6*e - 192*b*c^(9/2)*d^5*e^2 + 152*b^2*c^(7/2)*d^4*e^3 + 352*a*c^(9/2)*
d^4*e^3 + 16*b^3*c^(5/2)*d^3*e^4 - 704*a*b*c^(7/2)*d^3*e^4 - 70*b^4*c^(3/2)*d^2*e^5 + 512*a*b^2*c^(5/2)*d^2*e^
5 + 32*a^2*c^(7/2)*d^2*e^5 - 30*sqrt(c*d^2 - b*d*e + a*e^2)*b^4*c*d*e^5*log(abs(2*c^(3/2)*d^2 - 2*b*sqrt(c)*d*
e - 2*sqrt(c*d^2 - b*d*e + a*e^2)*c*d + 2*a*sqrt(c)*e^2 + sqrt(c*d^2 - b*d*e + a*e^2)*b*e)) + 240*sqrt(c*d^2 -
b*d*e + a*e^2)*a*b^2*c^2*d*e^5*log(abs(2*c^(3/2)*d^2 - 2*b*sqrt(c)*d*e - 2*sqrt(c*d^2 - b*d*e + a*e^2)*c*d +
2*a*sqrt(c)*e^2 + sqrt(c*d^2 - b*d*e + a*e^2)*b*e)) - 480*sqrt(c*d^2 - b*d*e + a*e^2)*a^2*c^3*d*e^5*log(abs(2*
c^(3/2)*d^2 - 2*b*sqrt(c)*d*e - 2*sqrt(c*d^2 - b*d*e + a*e^2)*c*d + 2*a*sqrt(c)*e^2 + sqrt(c*d^2 - b*d*e + a*e
^2)*b*e)) + 30*b^5*sqrt(c)*d*e^6 - 160*a*b^3*c^(3/2)*d*e^6 - 32*a^2*b*c^(5/2)*d*e^6 + 15*sqrt(c*d^2 - b*d*e +
a*e^2)*b^5*e^6*log(abs(2*c^(3/2)*d^2 - 2*b*sqrt(c)*d*e - 2*sqrt(c*d^2 - b*d*e + a*e^2)*c*d + 2*a*sqrt(c)*e^2 +
sqrt(c*d^2 - b*d*e + a*e^2)*b*e)) - 120*sqrt(c*d^2 - b*d*e + a*e^2)*a*b^3*c*e^6*log(abs(2*c^(3/2)*d^2 - 2*b*s
qrt(c)*d*e - 2*sqrt(c*d^2 - b*d*e + a*e^2)*c*d + 2*a*sqrt(c)*e^2 + sqrt(c*d^2 - b*d*e + a*e^2)*b*e)) + 240*sqr
t(c*d^2 - b*d*e + a*e^2)*a^2*b*c^2*e^6*log(abs(2*c^(3/2)*d^2 - 2*b*sqrt(c)*d*e - 2*sqrt(c*d^2 - b*d*e + a*e^2)
*c*d + 2*a*sqrt(c)*e^2 + sqrt(c*d^2 - b*d*e + a*e^2)*b*e)) - 30*a*b^4*sqrt(c)*e^7 + 200*a^2*b^2*c^(3/2)*e^7 -
256*a^3*c^(5/2)*e^7)*sgn(1/(x*e + d))/(c^5*d^10 - 5*b*c^4*d^9*e + 10*b^2*c^3*d^8*e^2 + 5*a*c^4*d^8*e^2 - 10*b^
3*c^2*d^7*e^3 - 20*a*b*c^3*d^7*e^3 + 5*b^4*c*d^6*e^4 + 30*a*b^2*c^2*d^6*e^4 + 10*a^2*c^3*d^6*e^4 - b^5*d^5*e^5
- 20*a*b^3*c*d^5*e^5 - 30*a^2*b*c^2*d^5*e^5 + 5*a*b^4*d^4*e^6 + 30*a^2*b^2*c*d^4*e^6 + 10*a^3*c^2*d^4*e^6 - 1
0*a^2*b^3*d^3*e^7 - 20*a^3*b*c*d^3*e^7 + 10*a^3*b^2*d^2*e^8 + 5*a^4*c*d^2*e^8 - 5*a^4*b*d*e^9 + a^5*e^10))*e^2