3.1769 \(\int \frac{a+b x}{(c+d x) \sqrt{e+f x}} \, dx\)

Optimal. Leaf size=74 \[ \frac{2 (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{3/2} \sqrt{d e-c f}}+\frac{2 b \sqrt{e+f x}}{d f} \]

[Out]

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

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Rubi [A]  time = 0.0653447, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {80, 63, 208} \[ \frac{2 (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{3/2} \sqrt{d e-c f}}+\frac{2 b \sqrt{e+f x}}{d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 80

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.133969, size = 74, normalized size = 1. \[ \frac{2 (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{3/2} \sqrt{d e-c f}}+\frac{2 b \sqrt{e+f x}}{d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.008, size = 96, normalized size = 1.3 \begin{align*} 2\,{\frac{b\sqrt{fx+e}}{df}}+2\,{\frac{a}{\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-2\,{\frac{bc}{d\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.35149, size = 450, normalized size = 6.08 \begin{align*} \left [-\frac{\sqrt{d^{2} e - c d f}{\left (b c - a d\right )} f \log \left (\frac{d f x + 2 \, d e - c f - 2 \, \sqrt{d^{2} e - c d f} \sqrt{f x + e}}{d x + c}\right ) - 2 \,{\left (b d^{2} e - b c d f\right )} \sqrt{f x + e}}{d^{3} e f - c d^{2} f^{2}}, -\frac{2 \,{\left (\sqrt{-d^{2} e + c d f}{\left (b c - a d\right )} f \arctan \left (\frac{\sqrt{-d^{2} e + c d f} \sqrt{f x + e}}{d f x + d e}\right ) -{\left (b d^{2} e - b c d f\right )} \sqrt{f x + e}\right )}}{d^{3} e f - c d^{2} f^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-(sqrt(d^2*e - c*d*f)*(b*c - a*d)*f*log((d*f*x + 2*d*e - c*f - 2*sqrt(d^2*e - c*d*f)*sqrt(f*x + e))/(d*x + c)
) - 2*(b*d^2*e - b*c*d*f)*sqrt(f*x + e))/(d^3*e*f - c*d^2*f^2), -2*(sqrt(-d^2*e + c*d*f)*(b*c - a*d)*f*arctan(
sqrt(-d^2*e + c*d*f)*sqrt(f*x + e)/(d*f*x + d*e)) - (b*d^2*e - b*c*d*f)*sqrt(f*x + e))/(d^3*e*f - c*d^2*f^2)]

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Sympy [A]  time = 9.95572, size = 66, normalized size = 0.89 \begin{align*} \frac{2 b \sqrt{e + f x}}{d f} - \frac{2 \left (a d - b c\right ) \operatorname{atan}{\left (\frac{1}{\sqrt{\frac{d}{c f - d e}} \sqrt{e + f x}} \right )}}{d \sqrt{\frac{d}{c f - d e}} \left (c f - d e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b*sqrt(e + f*x)/(d*f) - 2*(a*d - b*c)*atan(1/(sqrt(d/(c*f - d*e))*sqrt(e + f*x)))/(d*sqrt(d/(c*f - d*e))*(c*
f - d*e))

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Giac [A]  time = 2.30807, size = 95, normalized size = 1.28 \begin{align*} -\frac{2 \,{\left (b c - a d\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{\sqrt{c d f - d^{2} e} d} + \frac{2 \, \sqrt{f x + e} b}{d f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(b*c - a*d)*arctan(sqrt(f*x + e)*d/sqrt(c*d*f - d^2*e))/(sqrt(c*d*f - d^2*e)*d) + 2*sqrt(f*x + e)*b/(d*f)