3.1575 \(\int \frac{(b+2 c x) (d+e x)}{\sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=84 \[ \frac{e \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{4 c^{3/2}}+\frac{\sqrt{a+b x+c x^2} (-b e+4 c d+2 c e x)}{2 c} \]

[Out]

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

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Rubi [A]  time = 0.0527219, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {779, 621, 206} \[ \frac{e \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{4 c^{3/2}}+\frac{\sqrt{a+b x+c x^2} (-b e+4 c d+2 c e x)}{2 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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)}{\sqrt{a+b x+c x^2}} \, dx &=\frac{(4 c d-b e+2 c e x) \sqrt{a+b x+c x^2}}{2 c}+\frac{\left (\left (b^2-4 a c\right ) e\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{4 c}\\ &=\frac{(4 c d-b e+2 c e x) \sqrt{a+b x+c x^2}}{2 c}+\frac{\left (\left (b^2-4 a c\right ) e\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{2 c}\\ &=\frac{(4 c d-b e+2 c e x) \sqrt{a+b x+c x^2}}{2 c}+\frac{\left (b^2-4 a c\right ) e \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{4 c^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.122603, size = 82, normalized size = 0.98 \[ \frac{e \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{4 c^{3/2}}+\frac{\sqrt{a+x (b+c x)} (-b e+4 c d+2 c e x)}{2 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 117, normalized size = 1.4 \begin{align*} ex\sqrt{c{x}^{2}+bx+a}-{\frac{be}{2\,c}\sqrt{c{x}^{2}+bx+a}}+{\frac{{b}^{2}e}{4}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}-{ae\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}+2\,\sqrt{c{x}^{2}+bx+a}d \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.9328, size = 478, normalized size = 5.69 \begin{align*} \left [-\frac{{\left (b^{2} - 4 \, a c\right )} \sqrt{c} e \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{c} - 4 \, a c\right ) - 4 \,{\left (2 \, c^{2} e x + 4 \, c^{2} d - b c e\right )} \sqrt{c x^{2} + b x + a}}{8 \, c^{2}}, -\frac{{\left (b^{2} - 4 \, a c\right )} \sqrt{-c} e \arctan \left (\frac{\sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \,{\left (2 \, c^{2} e x + 4 \, c^{2} d - b c e\right )} \sqrt{c x^{2} + b x + a}}{4 \, c^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/8*((b^2 - 4*a*c)*sqrt(c)*e*log(-8*c^2*x^2 - 8*b*c*x - b^2 + 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) -
4*a*c) - 4*(2*c^2*e*x + 4*c^2*d - b*c*e)*sqrt(c*x^2 + b*x + a))/c^2, -1/4*((b^2 - 4*a*c)*sqrt(-c)*e*arctan(1/2
*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) - 2*(2*c^2*e*x + 4*c^2*d - b*c*e)*sqrt(c*
x^2 + b*x + a))/c^2]

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

[Out]

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

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Giac [A]  time = 1.45934, size = 113, normalized size = 1.35 \begin{align*} \frac{1}{2} \, \sqrt{c x^{2} + b x + a}{\left (2 \, x e + \frac{4 \, c d - b e}{c}\right )} - \frac{{\left (b^{2} e - 4 \, a c e\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{4 \, c^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(c*x^2 + b*x + a)*(2*x*e + (4*c*d - b*e)/c) - 1/4*(b^2*e - 4*a*c*e)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))*sqrt(c) - b))/c^(3/2)