3.303 \(\int \frac{A+B \log (x)}{\sqrt{a+b \log (x)}} \, dx\)

Optimal. Leaf size=69 \[ \frac{\sqrt{\pi } e^{-\frac{a}{b}} (2 A b-B (2 a+b)) \text{Erfi}\left (\frac{\sqrt{a+b \log (x)}}{\sqrt{b}}\right )}{2 b^{3/2}}+\frac{B x \sqrt{a+b \log (x)}}{b} \]

[Out]

((2*A*b - (2*a + b)*B)*Sqrt[Pi]*Erfi[Sqrt[a + b*Log[x]]/Sqrt[b]])/(2*b^(3/2)*E^(a/b)) + (B*x*Sqrt[a + b*Log[x]
])/b

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Rubi [A]  time = 0.0690086, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {2294, 2299, 2180, 2204} \[ \frac{\sqrt{\pi } e^{-\frac{a}{b}} (2 A b-B (2 a+b)) \text{Erfi}\left (\frac{\sqrt{a+b \log (x)}}{\sqrt{b}}\right )}{2 b^{3/2}}+\frac{B x \sqrt{a+b \log (x)}}{b} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[x])/Sqrt[a + b*Log[x]],x]

[Out]

((2*A*b - (2*a + b)*B)*Sqrt[Pi]*Erfi[Sqrt[a + b*Log[x]]/Sqrt[b]])/(2*b^(3/2)*E^(a/b)) + (B*x*Sqrt[a + b*Log[x]
])/b

Rule 2294

Int[((A_.) + Log[(c_.)*((d_.) + (e_.)*(x_))^(n_.)]*(B_.))/Sqrt[Log[(c_.)*((d_.) + (e_.)*(x_))^(n_.)]*(b_.) + (
a_)], x_Symbol] :> Simp[(B*(d + e*x)*Sqrt[a + b*Log[c*(d + e*x)^n]])/(b*e), x] + Dist[(2*A*b - B*(2*a + b*n))/
(2*b), Int[1/Sqrt[a + b*Log[c*(d + e*x)^n]], x], x] /; FreeQ[{a, b, c, d, e, A, B, n}, x]

Rule 2299

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[1/(n*c^(1/n)), Subst[Int[E^(x/n)*(a + b*x)^p
, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[1/n]

Rule 2180

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - (c*
f)/d) + (f*g*x^2)/d), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rubi steps

\begin{align*} \int \frac{A+B \log (x)}{\sqrt{a+b \log (x)}} \, dx &=\frac{B x \sqrt{a+b \log (x)}}{b}+\frac{(2 A b-(2 a+b) B) \int \frac{1}{\sqrt{a+b \log (x)}} \, dx}{2 b}\\ &=\frac{B x \sqrt{a+b \log (x)}}{b}+\frac{(2 A b-(2 a+b) B) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\log (x)\right )}{2 b}\\ &=\frac{B x \sqrt{a+b \log (x)}}{b}+\frac{(2 A b-(2 a+b) B) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \log (x)}\right )}{b^2}\\ &=\frac{(2 A b-(2 a+b) B) e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \log (x)}}{\sqrt{b}}\right )}{2 b^{3/2}}+\frac{B x \sqrt{a+b \log (x)}}{b}\\ \end{align*}

Mathematica [A]  time = 0.145826, size = 80, normalized size = 1.16 \[ \frac{e^{-\frac{a}{b}} (2 A b-B (2 a+b)) \sqrt{-\frac{a+b \log (x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \log (x)}{b}\right )+2 B x (a+b \log (x))}{2 b \sqrt{a+b \log (x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[x])/Sqrt[a + b*Log[x]],x]

[Out]

(2*B*x*(a + b*Log[x]) + ((2*A*b - (2*a + b)*B)*Gamma[1/2, -((a + b*Log[x])/b)]*Sqrt[-((a + b*Log[x])/b)])/E^(a
/b))/(2*b*Sqrt[a + b*Log[x]])

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Maple [F]  time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{(A+B\ln \left ( x \right ) ){\frac{1}{\sqrt{a+b\ln \left ( x \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*ln(x))/(a+b*ln(x))^(1/2),x)

[Out]

int((A+B*ln(x))/(a+b*ln(x))^(1/2),x)

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Maxima [B]  time = 1.27067, size = 211, normalized size = 3.06 \begin{align*} \frac{\frac{2 \, \sqrt{\pi } A \operatorname{erf}\left (\sqrt{b \log \left (x\right ) + a} \sqrt{-\frac{1}{b}}\right ) e^{\left (-\frac{a}{b}\right )}}{\sqrt{-\frac{1}{b}}} - \frac{2 \, \sqrt{\pi } B a \operatorname{erf}\left (\sqrt{b \log \left (x\right ) + a} \sqrt{-\frac{1}{b}}\right ) e^{\left (-\frac{a}{b}\right )}}{b \sqrt{-\frac{1}{b}}} - \frac{{\left (\frac{\sqrt{\pi } b \operatorname{erf}\left (\sqrt{b \log \left (x\right ) + a} \sqrt{-\frac{1}{b}}\right ) e^{\left (-\frac{a}{b}\right )}}{\sqrt{-\frac{1}{b}}} - 2 \, \sqrt{b \log \left (x\right ) + a} b e^{\left (\frac{b \log \left (x\right ) + a}{b} - \frac{a}{b}\right )}\right )} B}{b}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(x))/(a+b*log(x))^(1/2),x, algorithm="maxima")

[Out]

1/2*(2*sqrt(pi)*A*erf(sqrt(b*log(x) + a)*sqrt(-1/b))*e^(-a/b)/sqrt(-1/b) - 2*sqrt(pi)*B*a*erf(sqrt(b*log(x) +
a)*sqrt(-1/b))*e^(-a/b)/(b*sqrt(-1/b)) - (sqrt(pi)*b*erf(sqrt(b*log(x) + a)*sqrt(-1/b))*e^(-a/b)/sqrt(-1/b) -
2*sqrt(b*log(x) + a)*b*e^((b*log(x) + a)/b - a/b))*B/b)/b

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(x))/(a+b*log(x))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{A + B \log{\left (x \right )}}{\sqrt{a + b \log{\left (x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*ln(x))/(a+b*ln(x))**(1/2),x)

[Out]

Integral((A + B*log(x))/sqrt(a + b*log(x)), x)

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Giac [B]  time = 1.36718, size = 174, normalized size = 2.52 \begin{align*} -\frac{\sqrt{\pi } A \operatorname{erf}\left (-\frac{\sqrt{b \log \left (x\right ) + a} \sqrt{-b}}{b}\right ) e^{\left (-\frac{a}{b}\right )}}{\sqrt{-b}} + \frac{\sqrt{\pi } B \operatorname{erf}\left (-\frac{\sqrt{b \log \left (x\right ) + a} \sqrt{-b}}{b}\right ) e^{\left (-\frac{a}{b}\right )}}{2 \, \sqrt{-b}} + \frac{\sqrt{\pi } B a \operatorname{erf}\left (-\frac{\sqrt{b \log \left (x\right ) + a} \sqrt{-b}}{b}\right ) e^{\left (-\frac{a}{b}\right )}}{\sqrt{-b} b} + \frac{\sqrt{b \log \left (x\right ) + a} B x}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(x))/(a+b*log(x))^(1/2),x, algorithm="giac")

[Out]

-sqrt(pi)*A*erf(-sqrt(b*log(x) + a)*sqrt(-b)/b)*e^(-a/b)/sqrt(-b) + 1/2*sqrt(pi)*B*erf(-sqrt(b*log(x) + a)*sqr
t(-b)/b)*e^(-a/b)/sqrt(-b) + sqrt(pi)*B*a*erf(-sqrt(b*log(x) + a)*sqrt(-b)/b)*e^(-a/b)/(sqrt(-b)*b) + sqrt(b*l
og(x) + a)*B*x/b