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

Optimal. Leaf size=85 \[ -\frac {2 (47 x+37)}{5 \left (3 x^2+5 x+2\right )^{3/2}}+\frac {12 (836 x+701)}{25 \sqrt {3 x^2+5 x+2}}+\frac {104 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{25 \sqrt {5}} \]

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Rubi [A]  time = 0.05, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {822, 12, 724, 206} \begin {gather*} -\frac {2 (47 x+37)}{5 \left (3 x^2+5 x+2\right )^{3/2}}+\frac {12 (836 x+701)}{25 \sqrt {3 x^2+5 x+2}}+\frac {104 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{25 \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(37 + 47*x))/(5*(2 + 5*x + 3*x^2)^(3/2)) + (12*(701 + 836*x))/(25*Sqrt[2 + 5*x + 3*x^2]) + (104*ArcTanh[(7
+ 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(25*Sqrt[5])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*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] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rubi steps

\begin {align*} \int \frac {5-x}{(3+2 x) \left (2+5 x+3 x^2\right )^{5/2}} \, dx &=-\frac {2 (37+47 x)}{5 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {2}{15} \int \frac {807+564 x}{(3+2 x) \left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac {2 (37+47 x)}{5 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {12 (701+836 x)}{25 \sqrt {2+5 x+3 x^2}}+\frac {4}{75} \int \frac {78}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 (37+47 x)}{5 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {12 (701+836 x)}{25 \sqrt {2+5 x+3 x^2}}+\frac {104}{25} \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 (37+47 x)}{5 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {12 (701+836 x)}{25 \sqrt {2+5 x+3 x^2}}-\frac {208}{25} \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=-\frac {2 (37+47 x)}{5 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {12 (701+836 x)}{25 \sqrt {2+5 x+3 x^2}}+\frac {104 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{25 \sqrt {5}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 72, normalized size = 0.85 \begin {gather*} \frac {2}{125} \left (\frac {5 \left (15048 x^3+37698 x^2+30827 x+8227\right )}{\left (3 x^2+5 x+2\right )^{3/2}}-52 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((5*(8227 + 30827*x + 37698*x^2 + 15048*x^3))/(2 + 5*x + 3*x^2)^(3/2) - 52*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sq
rt[5]*Sqrt[2 + 5*x + 3*x^2])]))/125

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IntegrateAlgebraic [A]  time = 0.39, size = 81, normalized size = 0.95 \begin {gather*} \frac {208 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{25 \sqrt {5}}+\frac {2 \sqrt {3 x^2+5 x+2} \left (15048 x^3+37698 x^2+30827 x+8227\right )}{25 (x+1)^2 (3 x+2)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[2 + 5*x + 3*x^2]*(8227 + 30827*x + 37698*x^2 + 15048*x^3))/(25*(1 + x)^2*(2 + 3*x)^2) + (208*ArcTanh[S
qrt[2 + 5*x + 3*x^2]/(Sqrt[5]*(1 + x))])/(25*Sqrt[5])

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fricas [A]  time = 0.41, size = 125, normalized size = 1.47 \begin {gather*} \frac {2 \, {\left (26 \, \sqrt {5} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \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 ) + 5 \, {\left (15048 \, x^{3} + 37698 \, x^{2} + 30827 \, x + 8227\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}}{125 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/125*(26*sqrt(5)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log((4*sqrt(5)*sqrt(3*x^2 + 5*x + 2)*(8*x + 7) + 124*x^
2 + 212*x + 89)/(4*x^2 + 12*x + 9)) + 5*(15048*x^3 + 37698*x^2 + 30827*x + 8227)*sqrt(3*x^2 + 5*x + 2))/(9*x^4
+ 30*x^3 + 37*x^2 + 20*x + 4)

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giac [A]  time = 0.28, size = 102, normalized size = 1.20 \begin {gather*} \frac {104}{125} \, \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 {2 \, {\left ({\left (6 \, {\left (2508 \, x + 6283\right )} x + 30827\right )} x + 8227\right )}}{25 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

104/125*sqrt(5)*log(abs(-4*sqrt(3)*x - 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))/abs(-4*sqrt(3)*x + 2*s
qrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))) + 2/25*((6*(2508*x + 6283)*x + 30827)*x + 8227)/(3*x^2 + 5*x +
2)^(3/2)

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maple [B]  time = 0.01, size = 144, normalized size = 1.69 \begin {gather*} -\frac {104 \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 )}{125}+\frac {6 x +5}{3 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {8 \left (6 x +5\right )}{\sqrt {3 x^{2}+5 x +2}}+\frac {13}{15 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}-\frac {52 \left (6 x +5\right )}{15 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {\frac {11232 x}{25}+\frac {1872}{5}}{\sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}+\frac {52}{25 \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(6*x+5)/(3*x^2+5*x+2)^(3/2)-8*(6*x+5)/(3*x^2+5*x+2)^(1/2)+13/15/(-4*x+3*(x+3/2)^2-19/4)^(3/2)-52/15*(6*x+5
)/(-4*x+3*(x+3/2)^2-19/4)^(3/2)+1872/25*(6*x+5)/(-4*x+3*(x+3/2)^2-19/4)^(1/2)+52/25/(-4*x+3*(x+3/2)^2-19/4)^(1
/2)-104/125*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 = 1.45, size = 101, normalized size = 1.19 \begin {gather*} -\frac {104}{125} \, \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 {10032 \, x}{25 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} + \frac {8412}{25 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {94 \, x}{5 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {74}{5 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-104/125*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 10032/25*x/sqrt(3*x^
2 + 5*x + 2) + 8412/25/sqrt(3*x^2 + 5*x + 2) - 94/5*x/(3*x^2 + 5*x + 2)^(3/2) - 74/5/(3*x^2 + 5*x + 2)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x}{18 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 87 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 164 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 151 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 68 x \sqrt {3 x^{2} + 5 x + 2} + 12 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{18 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 87 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 164 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 151 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 68 x \sqrt {3 x^{2} + 5 x + 2} + 12 \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)/(3*x**2+5*x+2)**(5/2),x)

[Out]

-Integral(x/(18*x**5*sqrt(3*x**2 + 5*x + 2) + 87*x**4*sqrt(3*x**2 + 5*x + 2) + 164*x**3*sqrt(3*x**2 + 5*x + 2)
+ 151*x**2*sqrt(3*x**2 + 5*x + 2) + 68*x*sqrt(3*x**2 + 5*x + 2) + 12*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-
5/(18*x**5*sqrt(3*x**2 + 5*x + 2) + 87*x**4*sqrt(3*x**2 + 5*x + 2) + 164*x**3*sqrt(3*x**2 + 5*x + 2) + 151*x**
2*sqrt(3*x**2 + 5*x + 2) + 68*x*sqrt(3*x**2 + 5*x + 2) + 12*sqrt(3*x**2 + 5*x + 2)), x)

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