3.53 \(\int \frac{1}{\sqrt{e x} (a+b x) (a c-b c x)} \, dx\)

Optimal. Leaf size=87 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e x}}{\sqrt{a} \sqrt{e}}\right )}{a^{3/2} \sqrt{b} c \sqrt{e}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e x}}{\sqrt{a} \sqrt{e}}\right )}{a^{3/2} \sqrt{b} c \sqrt{e}} \]

[Out]

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

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Rubi [A]  time = 0.0489624, antiderivative size = 87, 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, 212, 208, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e x}}{\sqrt{a} \sqrt{e}}\right )}{a^{3/2} \sqrt{b} c \sqrt{e}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e x}}{\sqrt{a} \sqrt{e}}\right )}{a^{3/2} \sqrt{b} c \sqrt{e}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[e*x]*(a + b*x)*(a*c - b*c*x)),x]

[Out]

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

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 212

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

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]

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]

Rubi steps

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

Mathematica [A]  time = 0.0246309, size = 61, normalized size = 0.7 \[ \frac{\sqrt{x} \left (\tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )+\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )\right )}{a^{3/2} \sqrt{b} c \sqrt{e x}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[e*x]*(a + b*x)*(a*c - b*c*x)),x]

[Out]

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

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Maple [A]  time = 0.008, size = 56, normalized size = 0.6 \begin{align*}{\frac{1}{ac}\arctan \left ({b\sqrt{ex}{\frac{1}{\sqrt{aeb}}}} \right ){\frac{1}{\sqrt{aeb}}}}+{\frac{1}{ac}{\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(1/(e*x)^(1/2)/(b*x+a)/(-b*c*x+a*c),x)

[Out]

1/c/a/(a*e*b)^(1/2)*arctan(b*(e*x)^(1/2)/(a*e*b)^(1/2))+1/c/a/(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(1/(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 = 2.1947, size = 392, normalized size = 4.51 \begin{align*} \left [-\frac{2 \, \sqrt{a b e} \arctan \left (\frac{\sqrt{a b e} \sqrt{e x}}{b e x}\right ) - \sqrt{a b e} \log \left (\frac{b e x + a e + 2 \, \sqrt{a b e} \sqrt{e x}}{b x - a}\right )}{2 \, a^{2} b c e}, -\frac{2 \, \sqrt{-a b e} \arctan \left (\frac{\sqrt{-a b e} \sqrt{e x}}{b e x}\right ) + \sqrt{-a b e} \log \left (\frac{b e x - a e - 2 \, \sqrt{-a b e} \sqrt{e x}}{b x + a}\right )}{2 \, a^{2} b c e}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.75659, size = 175, normalized size = 2.01 \begin{align*} \begin{cases} \frac{1}{a b c \sqrt{e} \sqrt{x}} + \frac{\operatorname{acoth}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}} \sqrt{b} c \sqrt{e}} + \frac{\operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}} \sqrt{b} c \sqrt{e}} & \text{for}\: \frac{\left |{b x}\right |}{\left |{a}\right |} > 1 \\\frac{1}{a b c \sqrt{e} \sqrt{x}} + \frac{\operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}} \sqrt{b} c \sqrt{e}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}} \sqrt{b} c \sqrt{e}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18141, size = 78, normalized size = 0.9 \begin{align*} \frac{\arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right ) e^{\left (-\frac{1}{2}\right )}}{\sqrt{a b} a c} - \frac{\arctan \left (\frac{b \sqrt{x} e^{\frac{1}{2}}}{\sqrt{-a b e}}\right )}{\sqrt{-a b e} a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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