3.15.25 \(\int \frac {5-x}{(3+2 x) (2+3 x^2)^{5/2}} \, dx\) [1425]

Optimal. Leaf size=73 \[ \frac {26+41 x}{210 \left (2+3 x^2\right )^{3/2}}+\frac {312+2137 x}{7350 \sqrt {2+3 x^2}}-\frac {104 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{1225 \sqrt {35}} \]

[Out]

1/210*(26+41*x)/(3*x^2+2)^(3/2)-104/42875*arctanh(1/35*(4-9*x)*35^(1/2)/(3*x^2+2)^(1/2))*35^(1/2)+1/7350*(312+
2137*x)/(3*x^2+2)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 73, 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 = {837, 12, 739, 212} \begin {gather*} \frac {41 x+26}{210 \left (3 x^2+2\right )^{3/2}}+\frac {2137 x+312}{7350 \sqrt {3 x^2+2}}-\frac {104 \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 + 3*x^2)^(5/2)),x]

[Out]

(26 + 41*x)/(210*(2 + 3*x^2)^(3/2)) + (312 + 2137*x)/(7350*Sqrt[2 + 3*x^2]) - (104*ArcTanh[(4 - 9*x)/(Sqrt[35]
*Sqrt[2 + 3*x^2])])/(1225*Sqrt[35])

Rule 12

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

Rule 212

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

Rule 739

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 837

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] +
Dist[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) \left (2+3 x^2\right )^{5/2}} \, dx &=\frac {26+41 x}{210 \left (2+3 x^2\right )^{3/2}}-\frac {1}{630} \int \frac {-1206-492 x}{(3+2 x) \left (2+3 x^2\right )^{3/2}} \, dx\\ &=\frac {26+41 x}{210 \left (2+3 x^2\right )^{3/2}}+\frac {312+2137 x}{7350 \sqrt {2+3 x^2}}+\frac {\int \frac {11232}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{132300}\\ &=\frac {26+41 x}{210 \left (2+3 x^2\right )^{3/2}}+\frac {312+2137 x}{7350 \sqrt {2+3 x^2}}+\frac {104 \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{1225}\\ &=\frac {26+41 x}{210 \left (2+3 x^2\right )^{3/2}}+\frac {312+2137 x}{7350 \sqrt {2+3 x^2}}-\frac {104 \text {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}{210 \left (2+3 x^2\right )^{3/2}}+\frac {312+2137 x}{7350 \sqrt {2+3 x^2}}-\frac {104 \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 = 0.66, size = 63, normalized size = 0.86 \begin {gather*} \frac {\frac {35 \left (1534+5709 x+936 x^2+6411 x^3\right )}{\left (2+3 x^2\right )^{3/2}}-624 \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{257250} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((35*(1534 + 5709*x + 936*x^2 + 6411*x^3))/(2 + 3*x^2)^(3/2) - 624*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2
+ 3*x^2])])/257250

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(121\) vs. \(2(58)=116\).
time = 0.61, size = 122, normalized size = 1.67

method	result	size
trager	\(\frac {6411 x^{3}+936 x^{2}+5709 x +1534}{7350 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {104 \RootOf \left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {-9 \RootOf \left (\textit {\_Z}^{2}-35\right ) x +35 \sqrt {3 x^{2}+2}+4 \RootOf \left (\textit {\_Z}^{2}-35\right )}{2 x +3}\right )}{42875}\)	\(74\)
default	\(-\frac {x}{12 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {x}{12 \sqrt {3 x^{2}+2}}+\frac {13}{105 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {39 x}{140 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {1833 x}{4900 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}+\frac {52}{1225 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}-\frac {104 \sqrt {35}\, \arctanh \left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{42875}\)	\(122\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12/(3*x^2+2)^(3/2)*x-1/12*x/(3*x^2+2)^(1/2)+13/105/(3*(x+3/2)^2-9*x-19/4)^(3/2)+39/140*x/(3*(x+3/2)^2-9*x-1
9/4)^(3/2)+1833/4900*x/(3*(x+3/2)^2-9*x-19/4)^(1/2)+52/1225/(3*(x+3/2)^2-9*x-19/4)^(1/2)-104/42875*35^(1/2)*ar
ctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))

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Maxima [A]
time = 0.49, size = 81, normalized size = 1.11 \begin {gather*} \frac {104}{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 {2137 \, x}{7350 \, \sqrt {3 \, x^{2} + 2}} + \frac {52}{1225 \, \sqrt {3 \, x^{2} + 2}} + \frac {41 \, x}{210 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} + \frac {13}{105 \, {\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)/(3*x^2+2)^(5/2),x, algorithm="maxima")

[Out]

104/42875*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 2137/7350*x/sqrt(3*x^2 + 2
) + 52/1225/sqrt(3*x^2 + 2) + 41/210*x/(3*x^2 + 2)^(3/2) + 13/105/(3*x^2 + 2)^(3/2)

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Fricas [A]
time = 2.29, size = 103, normalized size = 1.41 \begin {gather*} \frac {312 \, \sqrt {35} {\left (9 \, x^{4} + 12 \, x^{2} + 4\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 (6411 \, x^{3} + 936 \, x^{2} + 5709 \, x + 1534\right )} \sqrt {3 \, x^{2} + 2}}{257250 \, {\left (9 \, x^{4} + 12 \, x^{2} + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/257250*(312*sqrt(35)*(9*x^4 + 12*x^2 + 4)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*
x^2 + 12*x + 9)) + 35*(6411*x^3 + 936*x^2 + 5709*x + 1534)*sqrt(3*x^2 + 2))/(9*x^4 + 12*x^2 + 4)

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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} + 2} + 27 x^{4} \sqrt {3 x^{2} + 2} + 24 x^{3} \sqrt {3 x^{2} + 2} + 36 x^{2} \sqrt {3 x^{2} + 2} + 8 x \sqrt {3 x^{2} + 2} + 12 \sqrt {3 x^{2} + 2}}\, dx - \int \left (- \frac {5}{18 x^{5} \sqrt {3 x^{2} + 2} + 27 x^{4} \sqrt {3 x^{2} + 2} + 24 x^{3} \sqrt {3 x^{2} + 2} + 36 x^{2} \sqrt {3 x^{2} + 2} + 8 x \sqrt {3 x^{2} + 2} + 12 \sqrt {3 x^{2} + 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+2)**(5/2),x)

[Out]

-Integral(x/(18*x**5*sqrt(3*x**2 + 2) + 27*x**4*sqrt(3*x**2 + 2) + 24*x**3*sqrt(3*x**2 + 2) + 36*x**2*sqrt(3*x
**2 + 2) + 8*x*sqrt(3*x**2 + 2) + 12*sqrt(3*x**2 + 2)), x) - Integral(-5/(18*x**5*sqrt(3*x**2 + 2) + 27*x**4*s
qrt(3*x**2 + 2) + 24*x**3*sqrt(3*x**2 + 2) + 36*x**2*sqrt(3*x**2 + 2) + 8*x*sqrt(3*x**2 + 2) + 12*sqrt(3*x**2
+ 2)), x)

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Giac [A]
time = 1.50, size = 93, normalized size = 1.27 \begin {gather*} \frac {104}{42875} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) + \frac {3 \, {\left ({\left (2137 \, x + 312\right )} x + 1903\right )} x + 1534}{7350 \, {\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)/(3*x^2+2)^(5/2),x, algorithm="giac")

[Out]

104/42875*sqrt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sqrt(3*x^2 + 2))/(2*sqrt(3)*x - sqrt(35) +
3*sqrt(3) - 2*sqrt(3*x^2 + 2))) + 1/7350*(3*((2137*x + 312)*x + 1903)*x + 1534)/(3*x^2 + 2)^(3/2)

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Mupad [B]
time = 0.14, size = 218, normalized size = 2.99 \begin {gather*} \frac {\sqrt {35}\,\left (104\,\ln \left (x+\frac {3}{2}\right )-104\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )\right )}{42875}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {-\frac {123}{560}+\frac {\sqrt {6}\,39{}\mathrm {i}}{560}}{x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}-\frac {\sqrt {6}\,\left (-\frac {41}{280}+\frac {\sqrt {6}\,13{}\mathrm {i}}{280}\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 {123}{560}+\frac {\sqrt {6}\,39{}\mathrm {i}}{560}}{x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}+\frac {\sqrt {6}\,\left (\frac {41}{280}+\frac {\sqrt {6}\,13{}\mathrm {i}}{280}\right )\,1{}\mathrm {i}}{2\,{\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {6}\,\left (-3744+\sqrt {6}\,7113{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{1058400\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\left (3744+\sqrt {6}\,7113{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{1058400\,\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)*(3*x^2 + 2)^(5/2)),x)

[Out]

(35^(1/2)*(104*log(x + 3/2) - 104*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9)))/42875 - (3^(1/2)*(x^
2 + 2/3)^(1/2)*(((6^(1/2)*39i)/560 - 123/560)/(x - (6^(1/2)*1i)/3) - (6^(1/2)*((6^(1/2)*13i)/280 - 41/280)*1i)
/(2*(x - (6^(1/2)*1i)/3)^2)))/27 + (3^(1/2)*(x^2 + 2/3)^(1/2)*(((6^(1/2)*39i)/560 + 123/560)/(x + (6^(1/2)*1i)
/3) + (6^(1/2)*((6^(1/2)*13i)/280 + 41/280)*1i)/(2*(x + (6^(1/2)*1i)/3)^2)))/27 - (3^(1/2)*6^(1/2)*(6^(1/2)*71
13i - 3744)*(x^2 + 2/3)^(1/2)*1i)/(1058400*(x + (6^(1/2)*1i)/3)) - (3^(1/2)*6^(1/2)*(6^(1/2)*7113i + 3744)*(x^
2 + 2/3)^(1/2)*1i)/(1058400*(x - (6^(1/2)*1i)/3))

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