3.9.62 \(\int \frac {\sqrt {2+3 x}}{\sqrt {1+x}} \, dx\) [862]

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

[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 = {52, 56, 221} \begin {gather*} \sqrt {x+1} \sqrt {3 x+2}-\frac {\sinh ^{-1}\left (\sqrt {3 x+2}\right )}{\sqrt {3}} \end {gather*}

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 52

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 56

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 221

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 {\text {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 = 0.08, size = 43, normalized size = 1.23 \begin {gather*} \sqrt {1+x} \sqrt {2+3 x}-\frac {\tanh ^{-1}\left (\sqrt {\frac {2+3 x}{3+3 x}}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1 + x]*Sqrt[2 + 3*x] - ArcTanh[Sqrt[(2 + 3*x)/(3 + 3*x)]]/Sqrt[3]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(66\) vs. \(2(27)=54\).
time = 0.56, size = 67, normalized size = 1.91

method	result	size
default	\(\sqrt {1+x}\, \sqrt {2+3 x}-\frac {\sqrt {\left (1+x \right ) \left (2+3 x \right )}\, \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{6 \sqrt {2+3 x}\, \sqrt {1+x}}\)	\(67\)
risch	\(\sqrt {1+x}\, \sqrt {2+3 x}-\frac {\sqrt {\left (1+x \right ) \left (2+3 x \right )}\, \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{6 \sqrt {2+3 x}\, \sqrt {1+x}}\)	\(67\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1+x)^(1/2)*(2+3*x)^(1/2)-1/6*((1+x)*(2+3*x))^(1/2)/(2+3*x)^(1/2)/(1+x)^(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 = 0.52, size = 41, normalized size = 1.17 \begin {gather*} -\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} \end {gather*}

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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Fricas [A]
time = 0.35, size = 52, normalized size = 1.49 \begin {gather*} \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} \end {gather*}

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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Sympy [C] Result contains complex when optimal does not.
time = 0.92, size = 100, normalized size = 2.86 \begin {gather*} \begin {cases} \sqrt {x + 1} \sqrt {3 x + 2} - \frac {\sqrt {3} \operatorname {acosh}{\left (\sqrt {3} \sqrt {x + 1} \right )}}{3} & \text {for}\: \left |{x + 1}\right | > \frac {1}{3} \\\frac {\sqrt {3} i \operatorname {asin}{\left (\sqrt {3} \sqrt {x + 1} \right )}}{3} - \frac {3 i \left (x + 1\right )^{\frac {3}{2}}}{\sqrt {- 3 x - 2}} + \frac {i \sqrt {x + 1}}{\sqrt {- 3 x - 2}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 3.40, size = 39, normalized size = 1.11 \begin {gather*} \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 )} \end {gather*}

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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Mupad [B]
time = 6.14, size = 172, normalized size = 4.91 \begin {gather*} \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} \end {gather*}

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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