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

Optimal. Leaf size=82 \[ \frac {41 x+26}{70 (2 x+3) \sqrt {3 x^2+2}}+\frac {19 \sqrt {3 x^2+2}}{1225 (2 x+3)}-\frac {632 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{1225 \sqrt {35}} \]

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Rubi [A]  time = 0.04, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, 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}{70 (2 x+3) \sqrt {3 x^2+2}}+\frac {19 \sqrt {3 x^2+2}}{1225 (2 x+3)}-\frac {632 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{1225 \sqrt {35}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(26 + 41*x)/(70*(3 + 2*x)*Sqrt[2 + 3*x^2]) + (19*Sqrt[2 + 3*x^2])/(1225*(3 + 2*x)) - (632*ArcTanh[(4 - 9*x)/(S
qrt[35]*Sqrt[2 + 3*x^2])])/(1225*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 )^{3/2}} \, dx &=\frac {26+41 x}{70 (3+2 x) \sqrt {2+3 x^2}}-\frac {1}{210} \int \frac {-312-246 x}{(3+2 x)^2 \sqrt {2+3 x^2}} \, dx\\ &=\frac {26+41 x}{70 (3+2 x) \sqrt {2+3 x^2}}+\frac {19 \sqrt {2+3 x^2}}{1225 (3+2 x)}+\frac {632 \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{1225}\\ &=\frac {26+41 x}{70 (3+2 x) \sqrt {2+3 x^2}}+\frac {19 \sqrt {2+3 x^2}}{1225 (3+2 x)}-\frac {632 \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )}{1225}\\ &=\frac {26+41 x}{70 (3+2 x) \sqrt {2+3 x^2}}+\frac {19 \sqrt {2+3 x^2}}{1225 (3+2 x)}-\frac {632 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{1225 \sqrt {35}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 65, normalized size = 0.79 \begin {gather*} \frac {\frac {35 \left (114 x^2+1435 x+986\right )}{(2 x+3) \sqrt {3 x^2+2}}-1264 \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{85750} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((35*(986 + 1435*x + 114*x^2))/((3 + 2*x)*Sqrt[2 + 3*x^2]) - 1264*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2
+ 3*x^2])])/85750

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IntegrateAlgebraic [F]  time = 0.69, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {5-x}{(3+2 x)^2 \left (2+3 x^2\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.47, size = 104, normalized size = 1.27 \begin {gather*} \frac {632 \, \sqrt {35} {\left (6 \, x^{3} + 9 \, x^{2} + 4 \, x + 6\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 (114 \, x^{2} + 1435 \, x + 986\right )} \sqrt {3 \, x^{2} + 2}}{85750 \, {\left (6 \, x^{3} + 9 \, x^{2} + 4 \, x + 6\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)^(3/2),x, algorithm="fricas")

[Out]

1/85750*(632*sqrt(35)*(6*x^3 + 9*x^2 + 4*x + 6)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)
/(4*x^2 + 12*x + 9)) + 35*(114*x^2 + 1435*x + 986)*sqrt(3*x^2 + 2))/(6*x^3 + 9*x^2 + 4*x + 6)

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giac [B]  time = 0.27, size = 168, normalized size = 2.05 \begin {gather*} -\frac {1}{85750} \, \sqrt {35} {\left (19 \, \sqrt {35} \sqrt {3} - 1264 \, \log \left (\sqrt {35} \sqrt {3} - 9\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + \frac {\frac {\frac {1093}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} - \frac {1820}{{\left (2 \, x + 3\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{2 \, x + 3} + \frac {57}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{2450 \, \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3}} - \frac {632 \, \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 )}{42875 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\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)^(3/2),x, algorithm="giac")

[Out]

-1/85750*sqrt(35)*(19*sqrt(35)*sqrt(3) - 1264*log(sqrt(35)*sqrt(3) - 9))*sgn(1/(2*x + 3)) + 1/2450*((1093/sgn(
1/(2*x + 3)) - 1820/((2*x + 3)*sgn(1/(2*x + 3))))/(2*x + 3) + 57/sgn(1/(2*x + 3)))/sqrt(-18/(2*x + 3) + 35/(2*
x + 3)^2 + 3) - 632/42875*sqrt(35)*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))

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maple [A]  time = 0.05, size = 86, normalized size = 1.05 \begin {gather*} \frac {57 x}{2450 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}-\frac {632 \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 )}{42875}-\frac {13}{70 \left (x +\frac {3}{2}\right ) \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}+\frac {316}{1225 \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)^(3/2),x)

[Out]

-13/70/(x+3/2)/(-9*x+3*(x+3/2)^2-19/4)^(1/2)+316/1225/(-9*x+3*(x+3/2)^2-19/4)^(1/2)+57/2450/(-9*x+3*(x+3/2)^2-
19/4)^(1/2)*x-632/42875*35^(1/2)*arctanh(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 = 1.43, size = 86, normalized size = 1.05 \begin {gather*} \frac {632}{42875} \, \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 {57 \, x}{2450 \, \sqrt {3 \, x^{2} + 2}} + \frac {316}{1225 \, \sqrt {3 \, x^{2} + 2}} - \frac {13}{35 \, {\left (2 \, \sqrt {3 \, x^{2} + 2} x + 3 \, \sqrt {3 \, x^{2} + 2}\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)^(3/2),x, algorithm="maxima")

[Out]

632/42875*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 57/2450*x/sqrt(3*x^2 + 2)
+ 316/1225/sqrt(3*x^2 + 2) - 13/35/(2*sqrt(3*x^2 + 2)*x + 3*sqrt(3*x^2 + 2))

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mupad [B]  time = 0.12, size = 157, normalized size = 1.91 \begin {gather*} \frac {632\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{42875}-\frac {632\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{42875}+\frac {71\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{4900\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {71\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{4900\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {26\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1225\,\left (x+\frac {3}{2}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,199{}\mathrm {i}}{14700\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {\sqrt {3}\,\sqrt {6}\,\sqrt {x^2+\frac {2}{3}}\,199{}\mathrm {i}}{14700\,\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)^(3/2)),x)

[Out]

(632*35^(1/2)*log(x + 3/2))/42875 - (632*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/42875
+ (71*3^(1/2)*(x^2 + 2/3)^(1/2))/(4900*(x - (6^(1/2)*1i)/3)) + (71*3^(1/2)*(x^2 + 2/3)^(1/2))/(4900*(x + (6^(
1/2)*1i)/3)) - (26*3^(1/2)*(x^2 + 2/3)^(1/2))/(1225*(x + 3/2)) - (3^(1/2)*6^(1/2)*(x^2 + 2/3)^(1/2)*199i)/(147
00*(x - (6^(1/2)*1i)/3)) + (3^(1/2)*6^(1/2)*(x^2 + 2/3)^(1/2)*199i)/(14700*(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)**(3/2),x)

[Out]

Timed out

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