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

Optimal. Leaf size=49 \[ -\frac{2 \sqrt{x} (b c-a d)}{d^2}+\frac{2 c (b c-a d) \log \left (c+d \sqrt{x}\right )}{d^3}+\frac{b x}{d} \]

[Out]

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

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Rubi [A]  time = 0.0487507, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {376, 77} \[ -\frac{2 \sqrt{x} (b c-a d)}{d^2}+\frac{2 c (b c-a d) \log \left (c+d \sqrt{x}\right )}{d^3}+\frac{b x}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 376

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{g = Denominator[n]}, Dis
t[g, Subst[Int[x^(g - 1)*(a + b*x^(g*n))^p*(c + d*x^(g*n))^q, x], x, x^(1/g)], x]] /; FreeQ[{a, b, c, d, p, q}
, x] && NeQ[b*c - a*d, 0] && FractionQ[n]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.0406529, size = 41, normalized size = 0.84 \[ \frac{2 (a d-b c) \left (d \sqrt{x}-c \log \left (c+d \sqrt{x}\right )\right )}{d^3}+\frac{b x}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 59, normalized size = 1.2 \begin{align*}{\frac{bx}{d}}+2\,{\frac{a\sqrt{x}}{d}}-2\,{\frac{b\sqrt{x}c}{{d}^{2}}}-2\,{\frac{c\ln \left ( c+d\sqrt{x} \right ) a}{{d}^{2}}}+2\,{\frac{{c}^{2}\ln \left ( c+d\sqrt{x} \right ) b}{{d}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.966631, size = 63, normalized size = 1.29 \begin{align*} \frac{b d x - 2 \,{\left (b c - a d\right )} \sqrt{x}}{d^{2}} + \frac{2 \,{\left (b c^{2} - a c d\right )} \log \left (d \sqrt{x} + c\right )}{d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.30843, size = 111, normalized size = 2.27 \begin{align*} \frac{b d^{2} x + 2 \,{\left (b c^{2} - a c d\right )} \log \left (d \sqrt{x} + c\right ) - 2 \,{\left (b c d - a d^{2}\right )} \sqrt{x}}{d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.271116, size = 82, normalized size = 1.67 \begin{align*} \begin{cases} - \frac{2 a c \log{\left (\frac{c}{d} + \sqrt{x} \right )}}{d^{2}} + \frac{2 a \sqrt{x}}{d} + \frac{2 b c^{2} \log{\left (\frac{c}{d} + \sqrt{x} \right )}}{d^{3}} - \frac{2 b c \sqrt{x}}{d^{2}} + \frac{b x}{d} & \text{for}\: d \neq 0 \\\frac{a x + \frac{2 b x^{\frac{3}{2}}}{3}}{c} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*a*c*log(c/d + sqrt(x))/d**2 + 2*a*sqrt(x)/d + 2*b*c**2*log(c/d + sqrt(x))/d**3 - 2*b*c*sqrt(x)/d
**2 + b*x/d, Ne(d, 0)), ((a*x + 2*b*x**(3/2)/3)/c, True))

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Giac [A]  time = 1.0981, size = 66, normalized size = 1.35 \begin{align*} \frac{b d x - 2 \, b c \sqrt{x} + 2 \, a d \sqrt{x}}{d^{2}} + \frac{2 \,{\left (b c^{2} - a c d\right )} \log \left ({\left | d \sqrt{x} + c \right |}\right )}{d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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