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

Optimal. Leaf size=81 \[ -\frac {(139 x+121) (2 x+3)^{3/2}}{3 \left (3 x^2+5 x+2\right )}+30 \sqrt {2 x+3}-130 \tanh ^{-1}\left (\sqrt {2 x+3}\right )+100 \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \]

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Rubi [A]  time = 0.07, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {818, 824, 826, 1166, 207} \begin {gather*} -\frac {(139 x+121) (2 x+3)^{3/2}}{3 \left (3 x^2+5 x+2\right )}+30 \sqrt {2 x+3}-130 \tanh ^{-1}\left (\sqrt {2 x+3}\right )+100 \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

30*Sqrt[3 + 2*x] - ((3 + 2*x)^(3/2)*(121 + 139*x))/(3*(2 + 5*x + 3*x^2)) - 130*ArcTanh[Sqrt[3 + 2*x]] + 100*Sq
rt[5/3]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]]

Rule 207

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

Rule 818

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g
- c*(b*e*f + b*d*g + 2*a*e*g))*x))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d +
e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a
*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m +
2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &&
RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 824

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

Rule 826

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

Rule 1166

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

Rubi steps

\begin {align*} \int \frac {(5-x) (3+2 x)^{5/2}}{\left (2+5 x+3 x^2\right )^2} \, dx &=-\frac {(3+2 x)^{3/2} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}+\frac {1}{3} \int \frac {\sqrt {3+2 x} (-60+135 x)}{2+5 x+3 x^2} \, dx\\ &=30 \sqrt {3+2 x}-\frac {(3+2 x)^{3/2} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}+\frac {1}{9} \int \frac {-1080-495 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=30 \sqrt {3+2 x}-\frac {(3+2 x)^{3/2} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}+\frac {2}{9} \operatorname {Subst}\left (\int \frac {-675-495 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right )\\ &=30 \sqrt {3+2 x}-\frac {(3+2 x)^{3/2} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}+390 \operatorname {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right )-500 \operatorname {Subst}\left (\int \frac {1}{-5+3 x^2} \, dx,x,\sqrt {3+2 x}\right )\\ &=30 \sqrt {3+2 x}-\frac {(3+2 x)^{3/2} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}-130 \tanh ^{-1}\left (\sqrt {3+2 x}\right )+100 \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 92, normalized size = 1.14 \begin {gather*} -\frac {\sqrt {2 x+3} \left (8 x^2+209 x+183\right )+390 \left (3 x^2+5 x+2\right ) \tanh ^{-1}\left (\sqrt {2 x+3}\right )-100 \sqrt {15} \left (3 x^2+5 x+2\right ) \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right )}{9 x^2+15 x+6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[3 + 2*x]*(183 + 209*x + 8*x^2) + 390*(2 + 5*x + 3*x^2)*ArcTanh[Sqrt[3 + 2*x]] - 100*Sqrt[15]*(2 + 5*x
+ 3*x^2)*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/(6 + 15*x + 9*x^2))

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IntegrateAlgebraic [A]  time = 0.22, size = 98, normalized size = 1.21 \begin {gather*} -\frac {2 \left (4 (2 x+3)^{5/2}+185 (2 x+3)^{3/2}-225 \sqrt {2 x+3}\right )}{3 \left (3 (2 x+3)^2-8 (2 x+3)+5\right )}-130 \tanh ^{-1}\left (\sqrt {2 x+3}\right )+100 \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(-225*Sqrt[3 + 2*x] + 185*(3 + 2*x)^(3/2) + 4*(3 + 2*x)^(5/2)))/(3*(5 - 8*(3 + 2*x) + 3*(3 + 2*x)^2)) - 13
0*ArcTanh[Sqrt[3 + 2*x]] + 100*Sqrt[5/3]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]]

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fricas [A]  time = 0.40, size = 124, normalized size = 1.53 \begin {gather*} \frac {50 \, \sqrt {5} \sqrt {3} {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\frac {\sqrt {5} \sqrt {3} \sqrt {2 \, x + 3} + 3 \, x + 7}{3 \, x + 2}\right ) - 195 \, {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) + 195 \, {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) - {\left (8 \, x^{2} + 209 \, x + 183\right )} \sqrt {2 \, x + 3}}{3 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(50*sqrt(5)*sqrt(3)*(3*x^2 + 5*x + 2)*log((sqrt(5)*sqrt(3)*sqrt(2*x + 3) + 3*x + 7)/(3*x + 2)) - 195*(3*x^
2 + 5*x + 2)*log(sqrt(2*x + 3) + 1) + 195*(3*x^2 + 5*x + 2)*log(sqrt(2*x + 3) - 1) - (8*x^2 + 209*x + 183)*sqr
t(2*x + 3))/(3*x^2 + 5*x + 2)

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giac [A]  time = 0.30, size = 111, normalized size = 1.37 \begin {gather*} -\frac {50}{3} \, \sqrt {15} \log \left (\frac {{\left | -2 \, \sqrt {15} + 6 \, \sqrt {2 \, x + 3} \right |}}{2 \, {\left (\sqrt {15} + 3 \, \sqrt {2 \, x + 3}\right )}}\right ) - \frac {8}{9} \, \sqrt {2 \, x + 3} - \frac {2 \, {\left (587 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 695 \, \sqrt {2 \, x + 3}\right )}}{9 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}} - 65 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) + 65 \, \log \left ({\left | \sqrt {2 \, x + 3} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-50/3*sqrt(15)*log(1/2*abs(-2*sqrt(15) + 6*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3))) - 8/9*sqrt(2*x + 3) -
2/9*(587*(2*x + 3)^(3/2) - 695*sqrt(2*x + 3))/(3*(2*x + 3)^2 - 16*x - 19) - 65*log(sqrt(2*x + 3) + 1) + 65*log
(abs(sqrt(2*x + 3) - 1))

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maple [A]  time = 0.02, size = 95, normalized size = 1.17 \begin {gather*} \frac {100 \sqrt {15}\, \arctanh \left (\frac {\sqrt {15}\, \sqrt {2 x +3}}{5}\right )}{3}+65 \ln \left (-1+\sqrt {2 x +3}\right )-65 \ln \left (\sqrt {2 x +3}+1\right )-\frac {8 \sqrt {2 x +3}}{9}-\frac {850 \sqrt {2 x +3}}{27 \left (2 x +\frac {4}{3}\right )}-\frac {6}{\sqrt {2 x +3}+1}-\frac {6}{-1+\sqrt {2 x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/9*(2*x+3)^(1/2)-850/27*(2*x+3)^(1/2)/(2*x+4/3)+100/3*arctanh(1/5*15^(1/2)*(2*x+3)^(1/2))*15^(1/2)-6/((2*x+3
)^(1/2)+1)-65*ln((2*x+3)^(1/2)+1)-6/(-1+(2*x+3)^(1/2))+65*ln(-1+(2*x+3)^(1/2))

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maxima [A]  time = 1.39, size = 107, normalized size = 1.32 \begin {gather*} -\frac {50}{3} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) - \frac {8}{9} \, \sqrt {2 \, x + 3} - \frac {2 \, {\left (587 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 695 \, \sqrt {2 \, x + 3}\right )}}{9 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}} - 65 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) + 65 \, \log \left (\sqrt {2 \, x + 3} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-50/3*sqrt(15)*log(-(sqrt(15) - 3*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3))) - 8/9*sqrt(2*x + 3) - 2/9*(587*
(2*x + 3)^(3/2) - 695*sqrt(2*x + 3))/(3*(2*x + 3)^2 - 16*x - 19) - 65*log(sqrt(2*x + 3) + 1) + 65*log(sqrt(2*x
+ 3) - 1)

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mupad [B]  time = 0.08, size = 81, normalized size = 1.00 \begin {gather*} -\frac {\frac {1390\,\sqrt {2\,x+3}}{27}-\frac {1174\,{\left (2\,x+3\right )}^{3/2}}{27}}{\frac {16\,x}{3}-{\left (2\,x+3\right )}^2+\frac {19}{3}}-\frac {8\,\sqrt {2\,x+3}}{9}+\mathrm {atan}\left (\sqrt {2\,x+3}\,1{}\mathrm {i}\right )\,130{}\mathrm {i}-\frac {\sqrt {15}\,\mathrm {atan}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}\,1{}\mathrm {i}}{5}\right )\,100{}\mathrm {i}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan((2*x + 3)^(1/2)*1i)*130i - ((1390*(2*x + 3)^(1/2))/27 - (1174*(2*x + 3)^(3/2))/27)/((16*x)/3 - (2*x + 3)^
2 + 19/3) - (15^(1/2)*atan((15^(1/2)*(2*x + 3)^(1/2)*1i)/5)*100i)/3 - (8*(2*x + 3)^(1/2))/9

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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