3.862 \(\int \frac {\sqrt {2+3 x}}{\sqrt {1+x}} \, dx\)

Optimal. Leaf size=35 \[ \sqrt {x+1} \sqrt {3 x+2}-\frac {\sinh ^{-1}\left (\sqrt {3 x+2}\right )}{\sqrt {3}} \]

[Out]

-1/3*arcsinh((2+3*x)^(1/2))*3^(1/2)+(1+x)^(1/2)*(2+3*x)^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {50, 54, 215} \[ \sqrt {x+1} \sqrt {3 x+2}-\frac {\sinh ^{-1}\left (\sqrt {3 x+2}\right )}{\sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[2 + 3*x]/Sqrt[1 + x],x]

[Out]

Sqrt[1 + x]*Sqrt[2 + 3*x] - ArcSinh[Sqrt[2 + 3*x]]/Sqrt[3]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin {align*} \int \frac {\sqrt {2+3 x}}{\sqrt {1+x}} \, dx &=\sqrt {1+x} \sqrt {2+3 x}-\frac {1}{2} \int \frac {1}{\sqrt {1+x} \sqrt {2+3 x}} \, dx\\ &=\sqrt {1+x} \sqrt {2+3 x}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\sqrt {2+3 x}\right )}{\sqrt {3}}\\ &=\sqrt {1+x} \sqrt {2+3 x}-\frac {\sinh ^{-1}\left (\sqrt {2+3 x}\right )}{\sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 49, normalized size = 1.40 \[ \frac {3 \sqrt {x+1} (3 x+2)-\sqrt {9 x+6} \sinh ^{-1}\left (\sqrt {3 x+2}\right )}{3 \sqrt {3 x+2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[2 + 3*x]/Sqrt[1 + x],x]

[Out]

(3*Sqrt[1 + x]*(2 + 3*x) - Sqrt[6 + 9*x]*ArcSinh[Sqrt[2 + 3*x]])/(3*Sqrt[2 + 3*x])

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fricas [A]  time = 0.42, size = 52, normalized size = 1.49 \[ \frac {1}{12} \, \sqrt {3} \log \left (-4 \, \sqrt {3} {\left (6 \, x + 5\right )} \sqrt {3 \, x + 2} \sqrt {x + 1} + 72 \, x^{2} + 120 \, x + 49\right ) + \sqrt {3 \, x + 2} \sqrt {x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^(1/2)/(1+x)^(1/2),x, algorithm="fricas")

[Out]

1/12*sqrt(3)*log(-4*sqrt(3)*(6*x + 5)*sqrt(3*x + 2)*sqrt(x + 1) + 72*x^2 + 120*x + 49) + sqrt(3*x + 2)*sqrt(x
+ 1)

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giac [A]  time = 0.35, size = 39, normalized size = 1.11 \[ \frac {1}{3} \, \sqrt {3} {\left (\sqrt {3 \, x + 3} \sqrt {3 \, x + 2} + \log \left (\sqrt {3 \, x + 3} - \sqrt {3 \, x + 2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^(1/2)/(1+x)^(1/2),x, algorithm="giac")

[Out]

1/3*sqrt(3)*(sqrt(3*x + 3)*sqrt(3*x + 2) + log(sqrt(3*x + 3) - sqrt(3*x + 2)))

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maple [B]  time = 0.01, size = 67, normalized size = 1.91 \[ -\frac {\sqrt {\left (x +1\right ) \left (3 x +2\right )}\, \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{6 \sqrt {3 x +2}\, \sqrt {x +1}}+\sqrt {x +1}\, \sqrt {3 x +2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x+2)^(1/2)/(x+1)^(1/2),x)

[Out]

(x+1)^(1/2)*(3*x+2)^(1/2)-1/6*((x+1)*(3*x+2))^(1/2)/(3*x+2)^(1/2)/(x+1)^(1/2)*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+
5*x+2)^(1/2))*3^(1/2)

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maxima [A]  time = 1.97, size = 41, normalized size = 1.17 \[ -\frac {1}{6} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \sqrt {3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^(1/2)/(1+x)^(1/2),x, algorithm="maxima")

[Out]

-1/6*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) + sqrt(3*x^2 + 5*x + 2)

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mupad [B]  time = 6.14, size = 172, normalized size = 4.91 \[ \frac {2\,\sqrt {3}\,\mathrm {atanh}\left (\frac {\sqrt {3}\,\left (\sqrt {2}-\sqrt {3\,x+2}\right )}{3\,\left (\sqrt {x+1}-1\right )}\right )}{3}-\frac {\frac {30\,\left (\sqrt {2}-\sqrt {3\,x+2}\right )}{\sqrt {x+1}-1}+\frac {10\,{\left (\sqrt {2}-\sqrt {3\,x+2}\right )}^3}{{\left (\sqrt {x+1}-1\right )}^3}+\frac {24\,\sqrt {2}\,{\left (\sqrt {2}-\sqrt {3\,x+2}\right )}^2}{{\left (\sqrt {x+1}-1\right )}^2}}{\frac {{\left (\sqrt {2}-\sqrt {3\,x+2}\right )}^4}{{\left (\sqrt {x+1}-1\right )}^4}-\frac {6\,{\left (\sqrt {2}-\sqrt {3\,x+2}\right )}^2}{{\left (\sqrt {x+1}-1\right )}^2}+9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x + 2)^(1/2)/(x + 1)^(1/2),x)

[Out]

(2*3^(1/2)*atanh((3^(1/2)*(2^(1/2) - (3*x + 2)^(1/2)))/(3*((x + 1)^(1/2) - 1))))/3 - ((30*(2^(1/2) - (3*x + 2)
^(1/2)))/((x + 1)^(1/2) - 1) + (10*(2^(1/2) - (3*x + 2)^(1/2))^3)/((x + 1)^(1/2) - 1)^3 + (24*2^(1/2)*(2^(1/2)
- (3*x + 2)^(1/2))^2)/((x + 1)^(1/2) - 1)^2)/((2^(1/2) - (3*x + 2)^(1/2))^4/((x + 1)^(1/2) - 1)^4 - (6*(2^(1/
2) - (3*x + 2)^(1/2))^2)/((x + 1)^(1/2) - 1)^2 + 9)

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sympy [A]  time = 1.62, size = 97, normalized size = 2.77 \[ \begin {cases} \frac {3 \left (x + 1\right )^{\frac {3}{2}}}{\sqrt {3 x + 2}} - \frac {\sqrt {x + 1}}{\sqrt {3 x + 2}} - \frac {\sqrt {3} \operatorname {acosh}{\left (\sqrt {3} \sqrt {x + 1} \right )}}{3} & \text {for}\: 3 \left |{x + 1}\right | > 1 \\i \sqrt {- 3 x - 2} \sqrt {x + 1} + \frac {\sqrt {3} i \operatorname {asin}{\left (\sqrt {3} \sqrt {x + 1} \right )}}{3} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**(1/2)/(1+x)**(1/2),x)

[Out]

Piecewise((3*(x + 1)**(3/2)/sqrt(3*x + 2) - sqrt(x + 1)/sqrt(3*x + 2) - sqrt(3)*acosh(sqrt(3)*sqrt(x + 1))/3,
3*Abs(x + 1) > 1), (I*sqrt(-3*x - 2)*sqrt(x + 1) + sqrt(3)*I*asin(sqrt(3)*sqrt(x + 1))/3, True))

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