3.52 \(\int \frac{\sqrt{e x}}{(a+b x) (a c-b c x)} \, dx\)

Optimal. Leaf size=88 \[ \frac{\sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e x}}{\sqrt{a} \sqrt{e}}\right )}{\sqrt{a} b^{3/2} c}-\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e x}}{\sqrt{a} \sqrt{e}}\right )}{\sqrt{a} b^{3/2} c} \]

[Out]

-((Sqrt[e]*ArcTan[(Sqrt[b]*Sqrt[e*x])/(Sqrt[a]*Sqrt[e])])/(Sqrt[a]*b^(3/2)*c)) + (Sqrt[e]*ArcTanh[(Sqrt[b]*Sqr
t[e*x])/(Sqrt[a]*Sqrt[e])])/(Sqrt[a]*b^(3/2)*c)

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Rubi [A]  time = 0.0470194, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {73, 329, 298, 205, 208} \[ \frac{\sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e x}}{\sqrt{a} \sqrt{e}}\right )}{\sqrt{a} b^{3/2} c}-\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e x}}{\sqrt{a} \sqrt{e}}\right )}{\sqrt{a} b^{3/2} c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[e]*ArcTan[(Sqrt[b]*Sqrt[e*x])/(Sqrt[a]*Sqrt[e])])/(Sqrt[a]*b^(3/2)*c)) + (Sqrt[e]*ArcTanh[(Sqrt[b]*Sqr
t[e*x])/(Sqrt[a]*Sqrt[e])])/(Sqrt[a]*b^(3/2)*c)

Rule 73

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[(a*c + b*
d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && Integer
Q[m]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 0]

Rule 205

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

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

Mathematica [A]  time = 0.0191914, size = 63, normalized size = 0.72 \[ \frac{\sqrt{e x} \left (\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )-\tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )\right )}{\sqrt{a} b^{3/2} c \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.01, size = 59, normalized size = 0.7 \begin{align*} -{\frac{e}{bc}\arctan \left ({b\sqrt{ex}{\frac{1}{\sqrt{aeb}}}} \right ){\frac{1}{\sqrt{aeb}}}}+{\frac{e}{bc}{\it Artanh} \left ({b\sqrt{ex}{\frac{1}{\sqrt{aeb}}}} \right ){\frac{1}{\sqrt{aeb}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.17487, size = 406, normalized size = 4.61 \begin{align*} \left [\frac{2 \, \sqrt{\frac{e}{a b}} \arctan \left (\frac{\sqrt{e x} a \sqrt{\frac{e}{a b}}}{e x}\right ) + \sqrt{\frac{e}{a b}} \log \left (\frac{b e x + 2 \, \sqrt{e x} a b \sqrt{\frac{e}{a b}} + a e}{b x - a}\right )}{2 \, b c}, -\frac{2 \, \sqrt{-\frac{e}{a b}} \arctan \left (\frac{\sqrt{e x} a \sqrt{-\frac{e}{a b}}}{e x}\right ) - \sqrt{-\frac{e}{a b}} \log \left (\frac{b e x - 2 \, \sqrt{e x} a b \sqrt{-\frac{e}{a b}} - a e}{b x + a}\right )}{2 \, b c}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(2*sqrt(e/(a*b))*arctan(sqrt(e*x)*a*sqrt(e/(a*b))/(e*x)) + sqrt(e/(a*b))*log((b*e*x + 2*sqrt(e*x)*a*b*sqr
t(e/(a*b)) + a*e)/(b*x - a)))/(b*c), -1/2*(2*sqrt(-e/(a*b))*arctan(sqrt(e*x)*a*sqrt(-e/(a*b))/(e*x)) - sqrt(-e
/(a*b))*log((b*e*x - 2*sqrt(e*x)*a*b*sqrt(-e/(a*b)) - a*e)/(b*x + a)))/(b*c)]

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Sympy [A]  time = 1.80953, size = 173, normalized size = 1.97 \begin{align*} \begin{cases} - \frac{\sqrt{e} \sqrt{x}}{a b c} + \frac{\sqrt{e} \operatorname{acoth}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{\sqrt{a} b^{\frac{3}{2}} c} + \frac{\sqrt{e} \operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{\sqrt{a} b^{\frac{3}{2}} c} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x}\right |} > 1 \\- \frac{\sqrt{e} \sqrt{x}}{a b c} + \frac{\sqrt{e} \operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{\sqrt{a} b^{\frac{3}{2}} c} + \frac{\sqrt{e} \operatorname{atanh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{\sqrt{a} b^{\frac{3}{2}} c} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-sqrt(e)*sqrt(x)/(a*b*c) + sqrt(e)*acoth(sqrt(a)/(sqrt(b)*sqrt(x)))/(sqrt(a)*b**(3/2)*c) + sqrt(e)*
atan(sqrt(a)/(sqrt(b)*sqrt(x)))/(sqrt(a)*b**(3/2)*c), Abs(a)/(Abs(b)*Abs(x)) > 1), (-sqrt(e)*sqrt(x)/(a*b*c) +
sqrt(e)*atan(sqrt(a)/(sqrt(b)*sqrt(x)))/(sqrt(a)*b**(3/2)*c) + sqrt(e)*atanh(sqrt(a)/(sqrt(b)*sqrt(x)))/(sqrt
(a)*b**(3/2)*c), True))

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Giac [A]  time = 1.26176, size = 82, normalized size = 0.93 \begin{align*} -\frac{\arctan \left (\frac{b \sqrt{x} e^{\frac{1}{2}}}{\sqrt{-a b e}}\right ) e}{\sqrt{-a b e} b c} - \frac{\arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right ) e^{\frac{1}{2}}}{\sqrt{a b} b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctan(b*sqrt(x)*e^(1/2)/sqrt(-a*b*e))*e/(sqrt(-a*b*e)*b*c) - arctan(b*sqrt(x)/sqrt(a*b))*e^(1/2)/(sqrt(a*b)*
b*c)