3.14.48 \(\int \frac {1}{(2+3 x) \sqrt {1+5 x}} \, dx\)

Optimal. Leaf size=25 \[ \frac {2 \tan ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {5 x+1}\right )}{\sqrt {21}} \]

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Rubi [A]  time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {63, 203} \begin {gather*} \frac {2 \tan ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {5 x+1}\right )}{\sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((2 + 3*x)*Sqrt[1 + 5*x]),x]

[Out]

(2*ArcTan[Sqrt[3/7]*Sqrt[1 + 5*x]])/Sqrt[21]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 203

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

Rubi steps

\begin {align*} \int \frac {1}{(2+3 x) \sqrt {1+5 x}} \, dx &=\frac {2}{5} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{5}+\frac {3 x^2}{5}} \, dx,x,\sqrt {1+5 x}\right )\\ &=\frac {2 \tan ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1+5 x}\right )}{\sqrt {21}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {5 x+1}\right )}{\sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((2 + 3*x)*Sqrt[1 + 5*x]),x]

[Out]

(2*ArcTan[Sqrt[3/7]*Sqrt[1 + 5*x]])/Sqrt[21]

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IntegrateAlgebraic [A]  time = 0.03, size = 25, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {5 x+1}\right )}{\sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((2 + 3*x)*Sqrt[1 + 5*x]),x]

[Out]

(2*ArcTan[Sqrt[3/7]*Sqrt[1 + 5*x]])/Sqrt[21]

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fricas [A]  time = 1.05, size = 18, normalized size = 0.72 \begin {gather*} \frac {2}{21} \, \sqrt {21} \arctan \left (\frac {1}{7} \, \sqrt {21} \sqrt {5 \, x + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*x)/(1+5*x)^(1/2),x, algorithm="fricas")

[Out]

2/21*sqrt(21)*arctan(1/7*sqrt(21)*sqrt(5*x + 1))

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giac [A]  time = 0.98, size = 18, normalized size = 0.72 \begin {gather*} \frac {2}{21} \, \sqrt {21} \arctan \left (\frac {1}{7} \, \sqrt {21} \sqrt {5 \, x + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*x)/(1+5*x)^(1/2),x, algorithm="giac")

[Out]

2/21*sqrt(21)*arctan(1/7*sqrt(21)*sqrt(5*x + 1))

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maple [A]  time = 0.01, size = 19, normalized size = 0.76 \begin {gather*} \frac {2 \sqrt {21}\, \arctan \left (\frac {\sqrt {21}\, \sqrt {5 x +1}}{7}\right )}{21} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x+2)/(1+5*x)^(1/2),x)

[Out]

2/21*arctan(1/7*21^(1/2)*(1+5*x)^(1/2))*21^(1/2)

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maxima [A]  time = 3.00, size = 18, normalized size = 0.72 \begin {gather*} \frac {2}{21} \, \sqrt {21} \arctan \left (\frac {1}{7} \, \sqrt {21} \sqrt {5 \, x + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*x)/(1+5*x)^(1/2),x, algorithm="maxima")

[Out]

2/21*sqrt(21)*arctan(1/7*sqrt(21)*sqrt(5*x + 1))

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mupad [B]  time = 0.06, size = 15, normalized size = 0.60 \begin {gather*} \frac {2\,\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {105\,x+21}}{7}\right )}{21} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((3*x + 2)*(5*x + 1)^(1/2)),x)

[Out]

(2*21^(1/2)*atan((105*x + 21)^(1/2)/7))/21

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sympy [A]  time = 1.12, size = 61, normalized size = 2.44 \begin {gather*} \begin {cases} \frac {2 \sqrt {21} i \operatorname {acosh}{\left (\frac {\sqrt {105}}{15 \sqrt {x + \frac {2}{3}}} \right )}}{21} & \text {for}\: \frac {7}{15 \left |{x + \frac {2}{3}}\right |} > 1 \\- \frac {2 \sqrt {21} \operatorname {asin}{\left (\frac {\sqrt {105}}{15 \sqrt {x + \frac {2}{3}}} \right )}}{21} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*x)/(1+5*x)**(1/2),x)

[Out]

Piecewise((2*sqrt(21)*I*acosh(sqrt(105)/(15*sqrt(x + 2/3)))/21, 7/(15*Abs(x + 2/3)) > 1), (-2*sqrt(21)*asin(sq
rt(105)/(15*sqrt(x + 2/3)))/21, True))

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