3.22.82 \(\int \frac {(5-x) (2+5 x+3 x^2)^{3/2}}{3+2 x} \, dx\)

Optimal. Leaf size=123 \[ \frac {1}{48} (47-6 x) \left (3 x^2+5 x+2\right )^{3/2}+\frac {1}{128} (175-414 x) \sqrt {3 x^2+5 x+2}-\frac {2011 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{256 \sqrt {3}}+\frac {65}{32} \sqrt {5} \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right ) \]

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Rubi [A]  time = 0.08, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {814, 843, 621, 206, 724} \begin {gather*} \frac {1}{48} (47-6 x) \left (3 x^2+5 x+2\right )^{3/2}+\frac {1}{128} (175-414 x) \sqrt {3 x^2+5 x+2}-\frac {2011 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{256 \sqrt {3}}+\frac {65}{32} \sqrt {5} \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((175 - 414*x)*Sqrt[2 + 5*x + 3*x^2])/128 + ((47 - 6*x)*(2 + 5*x + 3*x^2)^(3/2))/48 - (2011*ArcTanh[(5 + 6*x)/
(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(256*Sqrt[3]) + (65*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*
x^2])])/32

Rule 206

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

Rule 621

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 724

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

Rule 814

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

Rule 843

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

Rubi steps

\begin {align*} \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{3+2 x} \, dx &=\frac {1}{48} (47-6 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{96} \int \frac {(1083+1242 x) \sqrt {2+5 x+3 x^2}}{3+2 x} \, dx\\ &=\frac {1}{128} (175-414 x) \sqrt {2+5 x+3 x^2}+\frac {1}{48} (47-6 x) \left (2+5 x+3 x^2\right )^{3/2}+\frac {\int \frac {-61794-72396 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{4608}\\ &=\frac {1}{128} (175-414 x) \sqrt {2+5 x+3 x^2}+\frac {1}{48} (47-6 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {2011}{256} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx+\frac {325}{32} \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {1}{128} (175-414 x) \sqrt {2+5 x+3 x^2}+\frac {1}{48} (47-6 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {2011}{128} \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )-\frac {325}{16} \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=\frac {1}{128} (175-414 x) \sqrt {2+5 x+3 x^2}+\frac {1}{48} (47-6 x) \left (2+5 x+3 x^2\right )^{3/2}-\frac {2011 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{256 \sqrt {3}}+\frac {65}{32} \sqrt {5} \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 103, normalized size = 0.84 \begin {gather*} \frac {1}{768} \left (-1560 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )-2011 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )-2 \sqrt {3 x^2+5 x+2} \left (144 x^3-888 x^2-542 x-1277\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[2 + 5*x + 3*x^2]*(-1277 - 542*x - 888*x^2 + 144*x^3) - 1560*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqr
t[2 + 5*x + 3*x^2])] - 2011*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/768

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IntegrateAlgebraic [A]  time = 0.61, size = 104, normalized size = 0.85 \begin {gather*} -\frac {2011 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{128 \sqrt {3}}+\frac {65}{16} \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )+\frac {1}{384} \sqrt {3 x^2+5 x+2} \left (-144 x^3+888 x^2+542 x+1277\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + 5*x + 3*x^2]*(1277 + 542*x + 888*x^2 - 144*x^3))/384 - (2011*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[3]*
(1 + x))])/(128*Sqrt[3]) + (65*Sqrt[5]*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[5]*(1 + x))])/16

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fricas [A]  time = 0.42, size = 119, normalized size = 0.97 \begin {gather*} -\frac {1}{384} \, {\left (144 \, x^{3} - 888 \, x^{2} - 542 \, x - 1277\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {2011}{1536} \, \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 ) + \frac {65}{64} \, \sqrt {5} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/384*(144*x^3 - 888*x^2 - 542*x - 1277)*sqrt(3*x^2 + 5*x + 2) + 2011/1536*sqrt(3)*log(-4*sqrt(3)*sqrt(3*x^2
+ 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) + 65/64*sqrt(5)*log((4*sqrt(5)*sqrt(3*x^2 + 5*x + 2)*(8*x + 7) + 1
24*x^2 + 212*x + 89)/(4*x^2 + 12*x + 9))

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giac [A]  time = 0.30, size = 136, normalized size = 1.11 \begin {gather*} -\frac {1}{384} \, {\left (2 \, {\left (12 \, {\left (6 \, x - 37\right )} x - 271\right )} x - 1277\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {65}{32} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac {2011}{768} \, \sqrt {3} \log \left ({\left | -6 \, \sqrt {3} x - 5 \, \sqrt {3} + 6 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/384*(2*(12*(6*x - 37)*x - 271)*x - 1277)*sqrt(3*x^2 + 5*x + 2) + 65/32*sqrt(5)*log(abs(-4*sqrt(3)*x - 2*sqr
t(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))/abs(-4*sqrt(3)*x + 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2)
)) + 2011/768*sqrt(3)*log(abs(-6*sqrt(3)*x - 5*sqrt(3) + 6*sqrt(3*x^2 + 5*x + 2)))

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maple [A]  time = 0.05, size = 183, normalized size = 1.49 \begin {gather*} -\frac {65 \sqrt {5}\, \arctanh \left (\frac {2 \left (-4 x -\frac {7}{2}\right ) \sqrt {5}}{5 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{32}-\frac {377 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\right )}{144}-\frac {\sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{2304}-\frac {\left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{48}+\frac {\left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{384}+\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{12}-\frac {13 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{24}+\frac {65 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{32} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*(6*x+5)*(3*x^2+5*x+2)^(3/2)+1/384*(6*x+5)*(3*x^2+5*x+2)^(1/2)-1/2304*3^(1/2)*ln(1/3*(3*x+5/2)*3^(1/2)+(3
*x^2+5*x+2)^(1/2))+13/12*(-4*x+3*(x+3/2)^2-19/4)^(3/2)-13/24*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(1/2)-377/144*3^(
1/2)*ln(1/3*(3*x+5/2)*3^(1/2)+(-4*x+3*(x+3/2)^2-19/4)^(1/2))+65/32*(-16*x+12*(x+3/2)^2-19)^(1/2)-65/32*5^(1/2)
*arctanh(2/5*(-4*x-7/2)*5^(1/2)/(-16*x+12*(x+3/2)^2-19)^(1/2))

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maxima [A]  time = 1.51, size = 128, normalized size = 1.04 \begin {gather*} -\frac {1}{8} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {47}{48} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {207}{64} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {2011}{768} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) - \frac {65}{32} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) + \frac {175}{128} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(3*x^2 + 5*x + 2)^(3/2)*x + 47/48*(3*x^2 + 5*x + 2)^(3/2) - 207/64*sqrt(3*x^2 + 5*x + 2)*x - 2011/768*sqr
t(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) - 65/32*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x
+ 3) + 5/2/abs(2*x + 3) - 2) + 175/128*sqrt(3*x^2 + 5*x + 2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {10 \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \left (- \frac {23 x \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \left (- \frac {10 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{2 x + 3}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-10*sqrt(3*x**2 + 5*x + 2)/(2*x + 3), x) - Integral(-23*x*sqrt(3*x**2 + 5*x + 2)/(2*x + 3), x) - Int
egral(-10*x**2*sqrt(3*x**2 + 5*x + 2)/(2*x + 3), x) - Integral(3*x**3*sqrt(3*x**2 + 5*x + 2)/(2*x + 3), x)

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