3.1578 \(\int \frac{b+2 c x}{(d+e x)^2 \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=141 \[ \frac{\sqrt{a+b x+c x^2} (2 c d-b e)}{(d+e x) \left (a e^2-b d e+c d^2\right )}-\frac{e \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.115026, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {806, 724, 206} \[ \frac{\sqrt{a+b x+c x^2} (2 c d-b e)}{(d+e x) \left (a e^2-b d e+c d^2\right )}-\frac{e \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.122869, size = 138, normalized size = 0.98 \[ \frac{e \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )}{2 \left (e (a e-b d)+c d^2\right )^{3/2}}+\frac{\sqrt{a+x (b+c x)} (2 c d-b e)}{(d+e x) \left (e (a e-b d)+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.013, size = 860, normalized size = 6.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*c/e^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*(x+d/e)+2*((a*e^2-b*d*e+c
*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))-1/(a*e^2-b*d*e+c*
d^2)/(x+d/e)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b+2/e/(a*e^2-b*d*e+c*d^2)/(x+d/
e)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*c*d+1/2/(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*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d
/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*b^2-2/e/(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*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x
+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*b*c*d+2/e^2/(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*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/
2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*c^2*d^2

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.41776, size = 1485, normalized size = 10.53 \begin{align*} \left [-\frac{{\left ({\left (b^{2} - 4 \, a c\right )} e^{2} x +{\left (b^{2} - 4 \, a c\right )} d e\right )} \sqrt{c d^{2} - b d e + a e^{2}} \log \left (\frac{8 \, a b d e - 8 \, a^{2} e^{2} -{\left (b^{2} + 4 \, a c\right )} d^{2} -{\left (8 \, c^{2} d^{2} - 8 \, b c d e +{\left (b^{2} + 4 \, a c\right )} e^{2}\right )} x^{2} - 4 \, \sqrt{c d^{2} - b d e + a e^{2}} \sqrt{c x^{2} + b x + a}{\left (b d - 2 \, a e +{\left (2 \, c d - b e\right )} x\right )} - 2 \,{\left (4 \, b c d^{2} + 4 \, a b e^{2} -{\left (3 \, b^{2} + 4 \, a c\right )} d e\right )} x}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 4 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} \sqrt{c x^{2} + b x + a}}{4 \,{\left (c^{2} d^{5} - 2 \, b c d^{4} e - 2 \, a b d^{2} e^{3} + a^{2} d e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{3} e^{2} +{\left (c^{2} d^{4} e - 2 \, b c d^{3} e^{2} - 2 \, a b d e^{4} + a^{2} e^{5} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{3}\right )} x\right )}}, -\frac{{\left ({\left (b^{2} - 4 \, a c\right )} e^{2} x +{\left (b^{2} - 4 \, a c\right )} d e\right )} \sqrt{-c d^{2} + b d e - a e^{2}} \arctan \left (-\frac{\sqrt{-c d^{2} + b d e - a e^{2}} \sqrt{c x^{2} + b x + a}{\left (b d - 2 \, a e +{\left (2 \, c d - b e\right )} x\right )}}{2 \,{\left (a c d^{2} - a b d e + a^{2} e^{2} +{\left (c^{2} d^{2} - b c d e + a c e^{2}\right )} x^{2} +{\left (b c d^{2} - b^{2} d e + a b e^{2}\right )} x\right )}}\right ) - 2 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} \sqrt{c x^{2} + b x + a}}{2 \,{\left (c^{2} d^{5} - 2 \, b c d^{4} e - 2 \, a b d^{2} e^{3} + a^{2} d e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{3} e^{2} +{\left (c^{2} d^{4} e - 2 \, b c d^{3} e^{2} - 2 \, a b d e^{4} + a^{2} e^{5} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{3}\right )} x\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(((b^2 - 4*a*c)*e^2*x + (b^2 - 4*a*c)*d*e)*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*(2*c^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*sqrt(c*x^2 + b*x + a))/(c^2*d^5 - 2*b*
c*d^4*e - 2*a*b*d^2*e^3 + a^2*d*e^4 + (b^2 + 2*a*c)*d^3*e^2 + (c^2*d^4*e - 2*b*c*d^3*e^2 - 2*a*b*d*e^4 + a^2*e
^5 + (b^2 + 2*a*c)*d^2*e^3)*x), -1/2*(((b^2 - 4*a*c)*e^2*x + (b^2 - 4*a*c)*d*e)*sqrt(-c*d^2 + b*d*e - a*e^2)*a
rctan(-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*(2*c^2*d^3 - 3*b*c*d^
2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*sqrt(c*x^2 + b*x + a))/(c^2*d^5 - 2*b*c*d^4*e - 2*a*b*d^2*e^3 + a^2*d*e^4
+ (b^2 + 2*a*c)*d^3*e^2 + (c^2*d^4*e - 2*b*c*d^3*e^2 - 2*a*b*d*e^4 + a^2*e^5 + (b^2 + 2*a*c)*d^2*e^3)*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [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((2*c*x+b)/(e*x+d)^2/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

Timed out