3.25.97 \(\int \frac {(5-x) (3+2 x)^2}{\sqrt {2+5 x+3 x^2}} \, dx\) [2497]

Optimal. Leaf size=87 \[ -\frac {1}{9} (3+2 x)^2 \sqrt {2+5 x+3 x^2}+\frac {1}{54} (699+194 x) \sqrt {2+5 x+3 x^2}+\frac {1147 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{108 \sqrt {3}} \]

[Out]

1147/324*arctanh(1/6*(5+6*x)*3^(1/2)/(3*x^2+5*x+2)^(1/2))*3^(1/2)-1/9*(3+2*x)^2*(3*x^2+5*x+2)^(1/2)+1/54*(699+
194*x)*(3*x^2+5*x+2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {846, 793, 635, 212} \begin {gather*} -\frac {1}{9} \sqrt {3 x^2+5 x+2} (2 x+3)^2+\frac {1}{54} (194 x+699) \sqrt {3 x^2+5 x+2}+\frac {1147 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{108 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(3 + 2*x)^2)/Sqrt[2 + 5*x + 3*x^2],x]

[Out]

-1/9*((3 + 2*x)^2*Sqrt[2 + 5*x + 3*x^2]) + ((699 + 194*x)*Sqrt[2 + 5*x + 3*x^2])/54 + (1147*ArcTanh[(5 + 6*x)/
(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(108*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 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]

Rule 793

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p +
3))), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(
a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 846

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m
- 1)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m*(c*e*f + c*d*g - b*e*g) + e*(p
+ 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 0])

Rubi steps

\begin {align*} \int \frac {(5-x) (3+2 x)^2}{\sqrt {2+5 x+3 x^2}} \, dx &=-\frac {1}{9} (3+2 x)^2 \sqrt {2+5 x+3 x^2}+\frac {1}{9} \int \frac {(3+2 x) \left (\frac {301}{2}+97 x\right )}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {1}{9} (3+2 x)^2 \sqrt {2+5 x+3 x^2}+\frac {1}{54} (699+194 x) \sqrt {2+5 x+3 x^2}+\frac {1147}{108} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {1}{9} (3+2 x)^2 \sqrt {2+5 x+3 x^2}+\frac {1}{54} (699+194 x) \sqrt {2+5 x+3 x^2}+\frac {1147}{54} \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}{9} (3+2 x)^2 \sqrt {2+5 x+3 x^2}+\frac {1}{54} (699+194 x) \sqrt {2+5 x+3 x^2}+\frac {1147 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{108 \sqrt {3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.23, size = 61, normalized size = 0.70 \begin {gather*} \frac {1}{162} \left (-3 \sqrt {2+5 x+3 x^2} \left (-645-122 x+24 x^2\right )+1147 \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*Sqrt[2 + 5*x + 3*x^2]*(-645 - 122*x + 24*x^2) + 1147*Sqrt[3]*ArcTanh[Sqrt[2/3 + (5*x)/3 + x^2]/(1 + x)])/1
62

________________________________________________________________________________________

Maple [A]
time = 0.08, size = 77, normalized size = 0.89

method	result	size
risch	\(-\frac {\left (24 x^{2}-122 x -645\right ) \sqrt {3 x^{2}+5 x +2}}{54}+\frac {1147 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{324}\)	\(55\)
trager	\(\left (-\frac {4}{9} x^{2}+\frac {61}{27} x +\frac {215}{18}\right ) \sqrt {3 x^{2}+5 x +2}-\frac {1147 \RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \RootOf \left (\textit {\_Z}^{2}-3\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{324}\)	\(66\)
default	\(-\frac {4 x^{2} \sqrt {3 x^{2}+5 x +2}}{9}+\frac {61 x \sqrt {3 x^{2}+5 x +2}}{27}+\frac {215 \sqrt {3 x^{2}+5 x +2}}{18}+\frac {1147 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{324}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/9*x^2*(3*x^2+5*x+2)^(1/2)+61/27*x*(3*x^2+5*x+2)^(1/2)+215/18*(3*x^2+5*x+2)^(1/2)+1147/324*ln(1/3*(5/2+3*x)*
3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.58, size = 75, normalized size = 0.86 \begin {gather*} -\frac {4}{9} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x^{2} + \frac {61}{27} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {1147}{324} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {215}{18} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/9*sqrt(3*x^2 + 5*x + 2)*x^2 + 61/27*sqrt(3*x^2 + 5*x + 2)*x + 1147/324*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5
*x + 2) + 6*x + 5) + 215/18*sqrt(3*x^2 + 5*x + 2)

________________________________________________________________________________________

Fricas [A]
time = 3.23, size = 63, normalized size = 0.72 \begin {gather*} -\frac {1}{54} \, {\left (24 \, x^{2} - 122 \, x - 645\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {1147}{648} \, \sqrt {3} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/54*(24*x^2 - 122*x - 645)*sqrt(3*x^2 + 5*x + 2) + 1147/648*sqrt(3)*log(4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x
+ 5) + 72*x^2 + 120*x + 49)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {51 x}{\sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac {8 x^{2}}{\sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac {4 x^{3}}{\sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {45}{\sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-51*x/sqrt(3*x**2 + 5*x + 2), x) - Integral(-8*x**2/sqrt(3*x**2 + 5*x + 2), x) - Integral(4*x**3/sqr
t(3*x**2 + 5*x + 2), x) - Integral(-45/sqrt(3*x**2 + 5*x + 2), x)

________________________________________________________________________________________

Giac [A]
time = 2.04, size = 59, normalized size = 0.68 \begin {gather*} -\frac {1}{54} \, {\left (2 \, {\left (12 \, x - 61\right )} x - 645\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {1147}{324} \, \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((5-x)*(3+2*x)^2/(3*x^2+5*x+2)^(1/2),x, algorithm="giac")

[Out]

-1/54*(2*(12*x - 61)*x - 645)*sqrt(3*x^2 + 5*x + 2) - 1147/324*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*
x^2 + 5*x + 2)) - 5))

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {{\left (2\,x+3\right )}^2\,\left (x-5\right )}{\sqrt {3\,x^2+5\,x+2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________