3.15.54 \(\int \frac {1}{(2+3 x) \sqrt {1+5 x}} \, dx\) [1454]

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

[Out]

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

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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 = {65, 209} \begin {gather*} \frac {2 \text {ArcTan}\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 65

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 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 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} \text {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 = 0.03, size = 25, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1+5 x}\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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Maple [A]
time = 0.16, size = 19, normalized size = 0.76

method	result	size
derivativedivides	\(\frac {2 \arctan \left (\frac {\sqrt {21}\, \sqrt {5 x +1}}{7}\right ) \sqrt {21}}{21}\)	\(19\)
default	\(\frac {2 \arctan \left (\frac {\sqrt {21}\, \sqrt {5 x +1}}{7}\right ) \sqrt {21}}{21}\)	\(19\)
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}+21\right ) \ln \left (-\frac {15 \RootOf \left (\textit {\_Z}^{2}+21\right ) x -4 \RootOf \left (\textit {\_Z}^{2}+21\right )-42 \sqrt {5 x +1}}{2+3 x}\right )}{21}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.53, 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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Fricas [A]
time = 0.73, 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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Sympy [C] Result contains complex when optimal does not.
time = 0.51, 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 {1}{\left |{x + \frac {2}{3}}\right |} > \frac {15}{7} \\- \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, 1/Abs(x + 2/3) > 15/7), (-2*sqrt(21)*asin(sqrt
(105)/(15*sqrt(x + 2/3)))/21, True))

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Giac [A]
time = 0.94, 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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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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