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

Optimal. Leaf size=109 \[ \frac {41 x+26}{210 (2 x+3) \left (3 x^2+2\right )^{3/2}}+\frac {277 \sqrt {3 x^2+2}}{5145 (2 x+3)}+\frac {507 x+34}{1470 (2 x+3) \sqrt {3 x^2+2}}-\frac {176 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{1715 \sqrt {35}} \]

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Rubi [A]  time = 0.06, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {823, 807, 725, 206} \begin {gather*} \frac {41 x+26}{210 (2 x+3) \left (3 x^2+2\right )^{3/2}}+\frac {277 \sqrt {3 x^2+2}}{5145 (2 x+3)}+\frac {507 x+34}{1470 (2 x+3) \sqrt {3 x^2+2}}-\frac {176 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{1715 \sqrt {35}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(26 + 41*x)/(210*(3 + 2*x)*(2 + 3*x^2)^(3/2)) + (34 + 507*x)/(1470*(3 + 2*x)*Sqrt[2 + 3*x^2]) + (277*Sqrt[2 +
3*x^2])/(5145*(3 + 2*x)) - (176*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(1715*Sqrt[35])

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 725

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

Rule 807

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

Rule 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + 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)^2 \left (2+3 x^2\right )^{5/2}} \, dx &=\frac {26+41 x}{210 (3+2 x) \left (2+3 x^2\right )^{3/2}}-\frac {1}{630} \int \frac {-1362-738 x}{(3+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx\\ &=\frac {26+41 x}{210 (3+2 x) \left (2+3 x^2\right )^{3/2}}+\frac {34+507 x}{1470 (3+2 x) \sqrt {2+3 x^2}}+\frac {\int \frac {12240+91260 x}{(3+2 x)^2 \sqrt {2+3 x^2}} \, dx}{132300}\\ &=\frac {26+41 x}{210 (3+2 x) \left (2+3 x^2\right )^{3/2}}+\frac {34+507 x}{1470 (3+2 x) \sqrt {2+3 x^2}}+\frac {277 \sqrt {2+3 x^2}}{5145 (3+2 x)}+\frac {176 \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{1715}\\ &=\frac {26+41 x}{210 (3+2 x) \left (2+3 x^2\right )^{3/2}}+\frac {34+507 x}{1470 (3+2 x) \sqrt {2+3 x^2}}+\frac {277 \sqrt {2+3 x^2}}{5145 (3+2 x)}-\frac {176 \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )}{1715}\\ &=\frac {26+41 x}{210 (3+2 x) \left (2+3 x^2\right )^{3/2}}+\frac {34+507 x}{1470 (3+2 x) \sqrt {2+3 x^2}}+\frac {277 \sqrt {2+3 x^2}}{5145 (3+2 x)}-\frac {176 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{1715 \sqrt {35}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 101, normalized size = 0.93 \begin {gather*} \frac {35 \left (4986 x^4+10647 x^3+7362 x^2+9107 x+3966\right )-1056 \sqrt {35} \sqrt {3 x^2+2} \left (6 x^3+9 x^2+4 x+6\right ) \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{360150 (2 x+3) \left (3 x^2+2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(35*(3966 + 9107*x + 7362*x^2 + 10647*x^3 + 4986*x^4) - 1056*Sqrt[35]*Sqrt[2 + 3*x^2]*(6 + 4*x + 9*x^2 + 6*x^3
)*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(360150*(3 + 2*x)*(2 + 3*x^2)^(3/2))

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IntegrateAlgebraic [F]  time = 0.94, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{5/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.43, size = 134, normalized size = 1.23 \begin {gather*} \frac {528 \, \sqrt {35} {\left (18 \, x^{5} + 27 \, x^{4} + 24 \, x^{3} + 36 \, x^{2} + 8 \, x + 12\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) + 35 \, {\left (4986 \, x^{4} + 10647 \, x^{3} + 7362 \, x^{2} + 9107 \, x + 3966\right )} \sqrt {3 \, x^{2} + 2}}{360150 \, {\left (18 \, x^{5} + 27 \, x^{4} + 24 \, x^{3} + 36 \, x^{2} + 8 \, x + 12\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/360150*(528*sqrt(35)*(18*x^5 + 27*x^4 + 24*x^3 + 36*x^2 + 8*x + 12)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4)
+ 93*x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) + 35*(4986*x^4 + 10647*x^3 + 7362*x^2 + 9107*x + 3966)*sqrt(3*x^2 +
2))/(18*x^5 + 27*x^4 + 24*x^3 + 36*x^2 + 8*x + 12)

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giac [B]  time = 0.31, size = 233, normalized size = 2.14 \begin {gather*} -\frac {1}{360150} \, \sqrt {35} {\left (277 \, \sqrt {35} \sqrt {3} - 1056 \, \log \left (\sqrt {35} \sqrt {3} - 9\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {176 \, \sqrt {35} \log \left (\sqrt {35} {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )} - 9\right )}{60025 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} + \frac {\frac {\frac {\frac {7 \, {\left (\frac {4813}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} + \frac {4368}{{\left (2 \, x + 3\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}\right )}}{2 \, x + 3} - \frac {53523}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{2 \, x + 3} + \frac {19269}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{2 \, x + 3} - \frac {2493}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{10290 \, {\left (\frac {18}{2 \, x + 3} - \frac {35}{{\left (2 \, x + 3\right )}^{2}} - 3\right )} \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/360150*sqrt(35)*(277*sqrt(35)*sqrt(3) - 1056*log(sqrt(35)*sqrt(3) - 9))*sgn(1/(2*x + 3)) - 176/60025*sqrt(3
5)*log(sqrt(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3)) - 9)/sgn(1/(2*x + 3)) + 1/1029
0*(((7*(4813/sgn(1/(2*x + 3)) + 4368/((2*x + 3)*sgn(1/(2*x + 3))))/(2*x + 3) - 53523/sgn(1/(2*x + 3)))/(2*x +
3) + 19269/sgn(1/(2*x + 3)))/(2*x + 3) - 2493/sgn(1/(2*x + 3)))/((18/(2*x + 3) - 35/(2*x + 3)^2 - 3)*sqrt(-18/
(2*x + 3) + 35/(2*x + 3)^2 + 3))

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maple [A]  time = 0.06, size = 119, normalized size = 1.09 \begin {gather*} -\frac {17 x}{490 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {277 x}{3430 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}-\frac {176 \sqrt {35}\, \arctanh \left (\frac {2 \left (-9 x +4\right ) \sqrt {35}}{35 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{60025}-\frac {13}{70 \left (x +\frac {3}{2}\right ) \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {22}{147 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {88}{1715 \sqrt {-9 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)/(2*x+3)^2/(3*x^2+2)^(5/2),x)

[Out]

-13/70/(x+3/2)/(-9*x+3*(x+3/2)^2-19/4)^(3/2)+22/147/(-9*x+3*(x+3/2)^2-19/4)^(3/2)-17/490/(-9*x+3*(x+3/2)^2-19/
4)^(3/2)*x+277/3430/(-9*x+3*(x+3/2)^2-19/4)^(1/2)*x+88/1715/(-9*x+3*(x+3/2)^2-19/4)^(1/2)-176/60025*35^(1/2)*a
rctanh(2/35*(-9*x+4)*35^(1/2)/(-36*x+12*(x+3/2)^2-19)^(1/2))

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maxima [A]  time = 1.22, size = 109, normalized size = 1.00 \begin {gather*} \frac {176}{60025} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {277 \, x}{3430 \, \sqrt {3 \, x^{2} + 2}} + \frac {88}{1715 \, \sqrt {3 \, x^{2} + 2}} - \frac {17 \, x}{490 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {13}{35 \, {\left (2 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + 3 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}\right )}} + \frac {22}{147 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

176/60025*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 277/3430*x/sqrt(3*x^2 + 2)
+ 88/1715/sqrt(3*x^2 + 2) - 17/490*x/(3*x^2 + 2)^(3/2) - 13/35/(2*(3*x^2 + 2)^(3/2)*x + 3*(3*x^2 + 2)^(3/2))
+ 22/147/(3*x^2 + 2)^(3/2)

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mupad [B]  time = 1.88, size = 270, normalized size = 2.48 \begin {gather*} \frac {\sqrt {35}\,\left (3464\,\ln \left (x+\frac {3}{2}\right )-3464\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )\right )}{1500625}+\frac {\sqrt {35}\,\left (\frac {1872\,\ln \left (x+\frac {3}{2}\right )}{42875}-\frac {1872\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{42875}\right )}{70}-\frac {104\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{42875\,\left (x+\frac {3}{2}\right )}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {-\frac {639}{19600}+\frac {\sqrt {6}\,597{}\mathrm {i}}{19600}}{x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}-\frac {\sqrt {6}\,\left (-\frac {213}{9800}+\frac {\sqrt {6}\,199{}\mathrm {i}}{9800}\right )\,1{}\mathrm {i}}{2\,{\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {\frac {639}{19600}+\frac {\sqrt {6}\,597{}\mathrm {i}}{19600}}{x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}+\frac {\sqrt {6}\,\left (\frac {213}{9800}+\frac {\sqrt {6}\,199{}\mathrm {i}}{9800}\right )\,1{}\mathrm {i}}{2\,{\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {6}\,\left (-41568+\sqrt {6}\,27711{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{12348000\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\left (41568+\sqrt {6}\,27711{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{12348000\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(35^(1/2)*(3464*log(x + 3/2) - 3464*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9)))/1500625 + (35^(1/2
)*((1872*log(x + 3/2))/42875 - (1872*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/42875))/70 - (104*
3^(1/2)*(x^2 + 2/3)^(1/2))/(42875*(x + 3/2)) - (3^(1/2)*(x^2 + 2/3)^(1/2)*(((6^(1/2)*597i)/19600 - 639/19600)/
(x - (6^(1/2)*1i)/3) - (6^(1/2)*((6^(1/2)*199i)/9800 - 213/9800)*1i)/(2*(x - (6^(1/2)*1i)/3)^2)))/27 + (3^(1/2
)*(x^2 + 2/3)^(1/2)*(((6^(1/2)*597i)/19600 + 639/19600)/(x + (6^(1/2)*1i)/3) + (6^(1/2)*((6^(1/2)*199i)/9800 +
213/9800)*1i)/(2*(x + (6^(1/2)*1i)/3)^2)))/27 - (3^(1/2)*6^(1/2)*(6^(1/2)*27711i - 41568)*(x^2 + 2/3)^(1/2)*1
i)/(12348000*(x + (6^(1/2)*1i)/3)) - (3^(1/2)*6^(1/2)*(6^(1/2)*27711i + 41568)*(x^2 + 2/3)^(1/2)*1i)/(12348000
*(x - (6^(1/2)*1i)/3))

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

[Out]

Timed out

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