3.2380 \(\int \frac{1}{(d+e x)^3 \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=209 \[ \frac{\left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{5/2}}-\frac{3 e \sqrt{a+b x+c x^2} (2 c d-b e)}{4 (d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac{e \sqrt{a+b x+c x^2}}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.218934, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {744, 806, 724, 206} \[ \frac{\left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{5/2}}-\frac{3 e \sqrt{a+b x+c x^2} (2 c d-b e)}{4 (d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac{e \sqrt{a+b x+c x^2}}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^3*Sqrt[a + b*x + c*x^2]),x]

[Out]

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

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 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)^3 \sqrt{a+b x+c x^2}} \, dx &=-\frac{e \sqrt{a+b x+c x^2}}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac{\int \frac{\frac{1}{2} (-4 c d+3 b e)+c e x}{(d+e x)^2 \sqrt{a+b x+c x^2}} \, dx}{2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{e \sqrt{a+b x+c x^2}}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac{3 e (2 c d-b e) \sqrt{a+b x+c x^2}}{4 \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac{\left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) \int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{8 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{e \sqrt{a+b x+c x^2}}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac{3 e (2 c d-b e) \sqrt{a+b x+c x^2}}{4 \left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac{\left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a 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 )}{4 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{e \sqrt{a+b x+c x^2}}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac{3 e (2 c d-b e) \sqrt{a+b x+c x^2}}{4 \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac{\left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) \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 )}{8 \left (c d^2-b d e+a e^2\right )^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.304092, size = 205, normalized size = 0.98 \[ -\frac{\left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )}{8 \left (e (a e-b d)+c d^2\right )^{5/2}}-\frac{3 e \sqrt{a+x (b+c x)} (2 c d-b e)}{4 (d+e x) \left (e (a e-b d)+c d^2\right )^2}-\frac{e \sqrt{a+x (b+c x)}}{2 (d+e x)^2 \left (e (a e-b d)+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^3*Sqrt[a + b*x + c*x^2]),x]

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.237, size = 959, normalized size = 4.6 \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)^3/(c*x^2+b*x+a)^(1/2),x)

[Out]

-1/2/e/(a*e^2-b*d*e+c*d^2)/(d/e+x)^2*((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+3/4*e/(
a*e^2-b*d*e+c*d^2)^2/(d/e+x)*((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b-3/2/(a*e^2-b*
d*e+c*d^2)^2/(d/e+x)*((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*c*d-3/8*e/(a*e^2-b*d*e+
c*d^2)^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c
*d^2)/e^2)^(1/2)*((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*b^2+3/2/(a*e^2-b*
d*e+c*d^2)^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d
*e+c*d^2)/e^2)^(1/2)*((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*b*c*d-3/2/e/(
a*e^2-b*d*e+c*d^2)^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a
*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*c^2*d
^2+1/2/e*c/(a*e^2-b*d*e+c*d^2)/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/
e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((d/e+x)^2*c+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e
+x))

________________________________________________________________________________________

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)^3/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 19.7656, size = 2838, normalized size = 13.58 \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)^3/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[-1/16*((8*c^2*d^4 - 8*b*c*d^3*e + (3*b^2 - 4*a*c)*d^2*e^2 + (8*c^2*d^2*e^2 - 8*b*c*d*e^3 + (3*b^2 - 4*a*c)*e^
4)*x^2 + 2*(8*c^2*d^3*e - 8*b*c*d^2*e^2 + (3*b^2 - 4*a*c)*d*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*(8*c^2*d^4*e - 13*b*c*d^3*e^2 - 7*a*b*d*e^4 + 2*a^2*e^5 + 5*(b^2 + 2*a*c)*d^2
*e^3 + 3*(2*c^2*d^3*e^2 - 3*b*c*d^2*e^3 - a*b*e^5 + (b^2 + 2*a*c)*d*e^4)*x)*sqrt(c*x^2 + b*x + a))/(c^3*d^8 -
3*b*c^2*d^7*e - 3*a^2*b*d^3*e^5 + a^3*d^2*e^6 + 3*(b^2*c + a*c^2)*d^6*e^2 - (b^3 + 6*a*b*c)*d^5*e^3 + 3*(a*b^2
+ a^2*c)*d^4*e^4 + (c^3*d^6*e^2 - 3*b*c^2*d^5*e^3 - 3*a^2*b*d*e^7 + a^3*e^8 + 3*(b^2*c + a*c^2)*d^4*e^4 - (b^
3 + 6*a*b*c)*d^3*e^5 + 3*(a*b^2 + a^2*c)*d^2*e^6)*x^2 + 2*(c^3*d^7*e - 3*b*c^2*d^6*e^2 - 3*a^2*b*d^2*e^6 + a^3
*d*e^7 + 3*(b^2*c + a*c^2)*d^5*e^3 - (b^3 + 6*a*b*c)*d^4*e^4 + 3*(a*b^2 + a^2*c)*d^3*e^5)*x), 1/8*((8*c^2*d^4
- 8*b*c*d^3*e + (3*b^2 - 4*a*c)*d^2*e^2 + (8*c^2*d^2*e^2 - 8*b*c*d*e^3 + (3*b^2 - 4*a*c)*e^4)*x^2 + 2*(8*c^2*d
^3*e - 8*b*c*d^2*e^2 + (3*b^2 - 4*a*c)*d*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*(8*c^2*d^4*e - 13*b*c*d^3*e^2 - 7*a*b*d*e^4 + 2*a^2*
e^5 + 5*(b^2 + 2*a*c)*d^2*e^3 + 3*(2*c^2*d^3*e^2 - 3*b*c*d^2*e^3 - a*b*e^5 + (b^2 + 2*a*c)*d*e^4)*x)*sqrt(c*x^
2 + b*x + a))/(c^3*d^8 - 3*b*c^2*d^7*e - 3*a^2*b*d^3*e^5 + a^3*d^2*e^6 + 3*(b^2*c + a*c^2)*d^6*e^2 - (b^3 + 6*
a*b*c)*d^5*e^3 + 3*(a*b^2 + a^2*c)*d^4*e^4 + (c^3*d^6*e^2 - 3*b*c^2*d^5*e^3 - 3*a^2*b*d*e^7 + a^3*e^8 + 3*(b^2
*c + a*c^2)*d^4*e^4 - (b^3 + 6*a*b*c)*d^3*e^5 + 3*(a*b^2 + a^2*c)*d^2*e^6)*x^2 + 2*(c^3*d^7*e - 3*b*c^2*d^6*e^
2 - 3*a^2*b*d^2*e^6 + a^3*d*e^7 + 3*(b^2*c + a*c^2)*d^5*e^3 - (b^3 + 6*a*b*c)*d^4*e^4 + 3*(a*b^2 + a^2*c)*d^3*
e^5)*x)]

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d + e x\right )^{3} \sqrt{a + b x + c x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**3/(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral(1/((d + e*x)**3*sqrt(a + b*x + c*x**2)), x)

________________________________________________________________________________________

Giac [B]  time = 1.20098, size = 1060, normalized size = 5.07 \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)^3/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

1/4*(8*c^2*d^2 - 8*b*c*d*e + 3*b^2*e^2 - 4*a*c*e^2)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c)*d
)/sqrt(-c*d^2 + b*d*e - a*e^2))/((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)
*sqrt(-c*d^2 + b*d*e - a*e^2)) - 1/4*(8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*c^2*d^2*e + 24*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^2*c^(5/2)*d^3 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b*c^(3/2)*d^2*e + 24*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))*b*c^2*d^3 - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b*c*d*e^2 - 20*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))*b^2*c*d^2*e - 40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*c^2*d^2*e + 6*b^2*c^(3/2)*d^3 + 9*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*sqrt(c)*d*e^2 - 12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*c^(3/2)*d*e^2
- 3*b^3*sqrt(c)*d^2*e - 20*a*b*c^(3/2)*d^2*e + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^2*e^3 - 4*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^3*a*c*e^3 + 5*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^3*d*e^2 + 28*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))*a*b*c*d*e^2 + 11*a*b^2*sqrt(c)*d*e^2 + 12*a^2*c^(3/2)*d*e^2 - 5*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))*a*b^2*e^3 - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*c*e^3 - 8*a^2*b*sqrt(c)*e^3)/((c^2*d^4 - 2*b*c*d^3
*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*e + 2*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))*sqrt(c)*d + b*d - a*e)^2)