3.2.12 \(\int \sqrt {-2+5 x+3 x^2} \, dx\) [112]

Optimal. Leaf size=62 \[ \frac {1}{12} (5+6 x) \sqrt {-2+5 x+3 x^2}-\frac {49 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {-2+5 x+3 x^2}}\right )}{24 \sqrt {3}} \]

[Out]

-49/72*arctanh(1/6*(5+6*x)*3^(1/2)/(3*x^2+5*x-2)^(1/2))*3^(1/2)+1/12*(5+6*x)*(3*x^2+5*x-2)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {626, 635, 212} \begin {gather*} \frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x-2}-\frac {49 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x-2}}\right )}{24 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[-2 + 5*x + 3*x^2],x]

[Out]

((5 + 6*x)*Sqrt[-2 + 5*x + 3*x^2])/12 - (49*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[-2 + 5*x + 3*x^2])])/(24*Sqrt[3]
)

Rule 212

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

Rule 626

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^p/(2*c*(2*p + 1
))), x] - Dist[p*((b^2 - 4*a*c)/(2*c*(2*p + 1))), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \sqrt {-2+5 x+3 x^2} \, dx &=\frac {1}{12} (5+6 x) \sqrt {-2+5 x+3 x^2}-\frac {49}{24} \int \frac {1}{\sqrt {-2+5 x+3 x^2}} \, dx\\ &=\frac {1}{12} (5+6 x) \sqrt {-2+5 x+3 x^2}-\frac {49}{12} \text {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {-2+5 x+3 x^2}}\right )\\ &=\frac {1}{12} (5+6 x) \sqrt {-2+5 x+3 x^2}-\frac {49 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {-2+5 x+3 x^2}}\right )}{24 \sqrt {3}}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 56, normalized size = 0.90 \begin {gather*} \frac {1}{36} \left (3 (5+6 x) \sqrt {-2+5 x+3 x^2}-49 \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {-\frac {2}{3}+\frac {5 x}{3}+x^2}}{2+x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-2 + 5*x + 3*x^2],x]

[Out]

(3*(5 + 6*x)*Sqrt[-2 + 5*x + 3*x^2] - 49*Sqrt[3]*ArcTanh[Sqrt[-2/3 + (5*x)/3 + x^2]/(2 + x)])/36

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Maple [A]
time = 0.50, size = 50, normalized size = 0.81

method	result	size
default	\(\frac {\left (5+6 x \right ) \sqrt {3 x^{2}+5 x -2}}{12}-\frac {49 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x -2}\right ) \sqrt {3}}{72}\)	\(50\)
risch	\(\frac {\left (5+6 x \right ) \sqrt {3 x^{2}+5 x -2}}{12}-\frac {49 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x -2}\right ) \sqrt {3}}{72}\)	\(50\)
trager	\(\left (\frac {5}{12}+\frac {x}{2}\right ) \sqrt {3 x^{2}+5 x -2}+\frac {49 \RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \RootOf \left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}+5 x -2}-5 \RootOf \left (\textit {\_Z}^{2}-3\right )\right )}{72}\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(5+6*x)*(3*x^2+5*x-2)^(1/2)-49/72*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 = 0.53, size = 58, normalized size = 0.94 \begin {gather*} \frac {1}{2} \, \sqrt {3 \, x^{2} + 5 \, x - 2} x - \frac {49}{72} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x - 2} + 6 \, x + 5\right ) + \frac {5}{12} \, \sqrt {3 \, x^{2} + 5 \, x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 2.65, size = 58, normalized size = 0.94 \begin {gather*} \frac {1}{12} \, \sqrt {3 \, x^{2} + 5 \, x - 2} {\left (6 \, x + 5\right )} + \frac {49}{144} \, \sqrt {3} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x - 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*sqrt(3*x^2 + 5*x - 2)*(6*x + 5) + 49/144*sqrt(3)*log(-4*sqrt(3)*sqrt(3*x^2 + 5*x - 2)*(6*x + 5) + 72*x^2
+ 120*x + 1)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {3 x^{2} + 5 x - 2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(3*x**2 + 5*x - 2), x)

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Giac [A]
time = 0.73, size = 54, normalized size = 0.87 \begin {gather*} \frac {1}{12} \, \sqrt {3 \, x^{2} + 5 \, x - 2} {\left (6 \, x + 5\right )} + \frac {49}{72} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x - 2}\right )} - 5 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.22, size = 48, normalized size = 0.77 \begin {gather*} \left (\frac {x}{2}+\frac {5}{12}\right )\,\sqrt {3\,x^2+5\,x-2}-\frac {49\,\sqrt {3}\,\ln \left (\sqrt {3\,x^2+5\,x-2}+\frac {\sqrt {3}\,\left (3\,x+\frac {5}{2}\right )}{3}\right )}{72} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + 3*x^2 - 2)^(1/2),x)

[Out]

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

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