3.15.6 \(\int \frac {5-x}{(3+2 x)^5 \sqrt {2+3 x^2}} \, dx\) [1406]

Optimal. Leaf size=121 \[ -\frac {13 \sqrt {2+3 x^2}}{140 (3+2 x)^4}-\frac {97 \sqrt {2+3 x^2}}{2100 (3+2 x)^3}-\frac {87 \sqrt {2+3 x^2}}{4900 (3+2 x)^2}-\frac {991 \sqrt {2+3 x^2}}{171500 (3+2 x)}+\frac {27 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{42875 \sqrt {35}} \]

[Out]

27/1500625*arctanh(1/35*(4-9*x)*35^(1/2)/(3*x^2+2)^(1/2))*35^(1/2)-13/140*(3*x^2+2)^(1/2)/(3+2*x)^4-97/2100*(3
*x^2+2)^(1/2)/(3+2*x)^3-87/4900*(3*x^2+2)^(1/2)/(3+2*x)^2-991/171500*(3*x^2+2)^(1/2)/(3+2*x)

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Rubi [A]
time = 0.05, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {849, 821, 739, 212} \begin {gather*} -\frac {991 \sqrt {3 x^2+2}}{171500 (2 x+3)}-\frac {87 \sqrt {3 x^2+2}}{4900 (2 x+3)^2}-\frac {97 \sqrt {3 x^2+2}}{2100 (2 x+3)^3}-\frac {13 \sqrt {3 x^2+2}}{140 (2 x+3)^4}+\frac {27 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{42875 \sqrt {35}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-13*Sqrt[2 + 3*x^2])/(140*(3 + 2*x)^4) - (97*Sqrt[2 + 3*x^2])/(2100*(3 + 2*x)^3) - (87*Sqrt[2 + 3*x^2])/(4900
*(3 + 2*x)^2) - (991*Sqrt[2 + 3*x^2])/(171500*(3 + 2*x)) + (27*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/
(42875*Sqrt[35])

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 821

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 849

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)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

Rubi steps

\begin {align*} \int \frac {5-x}{(3+2 x)^5 \sqrt {2+3 x^2}} \, dx &=-\frac {13 \sqrt {2+3 x^2}}{140 (3+2 x)^4}-\frac {1}{140} \int \frac {-164+117 x}{(3+2 x)^4 \sqrt {2+3 x^2}} \, dx\\ &=-\frac {13 \sqrt {2+3 x^2}}{140 (3+2 x)^4}-\frac {97 \sqrt {2+3 x^2}}{2100 (3+2 x)^3}+\frac {\int \frac {3024-4074 x}{(3+2 x)^3 \sqrt {2+3 x^2}} \, dx}{14700}\\ &=-\frac {13 \sqrt {2+3 x^2}}{140 (3+2 x)^4}-\frac {97 \sqrt {2+3 x^2}}{2100 (3+2 x)^3}-\frac {87 \sqrt {2+3 x^2}}{4900 (3+2 x)^2}-\frac {\int \frac {-21840+54810 x}{(3+2 x)^2 \sqrt {2+3 x^2}} \, dx}{1029000}\\ &=-\frac {13 \sqrt {2+3 x^2}}{140 (3+2 x)^4}-\frac {97 \sqrt {2+3 x^2}}{2100 (3+2 x)^3}-\frac {87 \sqrt {2+3 x^2}}{4900 (3+2 x)^2}-\frac {991 \sqrt {2+3 x^2}}{171500 (3+2 x)}-\frac {27 \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{42875}\\ &=-\frac {13 \sqrt {2+3 x^2}}{140 (3+2 x)^4}-\frac {97 \sqrt {2+3 x^2}}{2100 (3+2 x)^3}-\frac {87 \sqrt {2+3 x^2}}{4900 (3+2 x)^2}-\frac {991 \sqrt {2+3 x^2}}{171500 (3+2 x)}+\frac {27 \text {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )}{42875}\\ &=-\frac {13 \sqrt {2+3 x^2}}{140 (3+2 x)^4}-\frac {97 \sqrt {2+3 x^2}}{2100 (3+2 x)^3}-\frac {87 \sqrt {2+3 x^2}}{4900 (3+2 x)^2}-\frac {991 \sqrt {2+3 x^2}}{171500 (3+2 x)}+\frac {27 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{42875 \sqrt {35}}\\ \end {align*}

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Mathematica [A]
time = 0.68, size = 83, normalized size = 0.69 \begin {gather*} \frac {-\frac {35 \sqrt {2+3 x^2} \left (70389+79423 x+35892 x^2+5946 x^3\right )}{(3+2 x)^4}-162 \sqrt {35} \tanh ^{-1}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{4501875} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-35*Sqrt[2 + 3*x^2]*(70389 + 79423*x + 35892*x^2 + 5946*x^3))/(3 + 2*x)^4 - 162*Sqrt[35]*ArcTanh[(3*Sqrt[3]
+ 2*Sqrt[3]*x - 2*Sqrt[2 + 3*x^2])/Sqrt[35]])/4501875

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Maple [A]
time = 0.71, size = 116, normalized size = 0.96

method	result	size
risch	\(-\frac {17838 x^{5}+107676 x^{4}+250161 x^{3}+282951 x^{2}+158846 x +140778}{128625 \left (2 x +3\right )^{4} \sqrt {3 x^{2}+2}}+\frac {27 \sqrt {35}\, \arctanh \left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{1500625}\)	\(75\)
trager	\(-\frac {\left (5946 x^{3}+35892 x^{2}+79423 x +70389\right ) \sqrt {3 x^{2}+2}}{128625 \left (2 x +3\right )^{4}}-\frac {27 \RootOf \left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {9 \RootOf \left (\textit {\_Z}^{2}-35\right ) x -4 \RootOf \left (\textit {\_Z}^{2}-35\right )+35 \sqrt {3 x^{2}+2}}{2 x +3}\right )}{1500625}\)	\(81\)
default	\(-\frac {97 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{16800 \left (x +\frac {3}{2}\right )^{3}}-\frac {87 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{19600 \left (x +\frac {3}{2}\right )^{2}}-\frac {991 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{343000 \left (x +\frac {3}{2}\right )}+\frac {27 \sqrt {35}\, \arctanh \left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{1500625}-\frac {13 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{2240 \left (x +\frac {3}{2}\right )^{4}}\)	\(116\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)/(2*x+3)^5/(3*x^2+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-97/16800/(x+3/2)^3*(3*(x+3/2)^2-9*x-19/4)^(1/2)-87/19600/(x+3/2)^2*(3*(x+3/2)^2-9*x-19/4)^(1/2)-991/343000/(x
+3/2)*(3*(x+3/2)^2-9*x-19/4)^(1/2)+27/1500625*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1
/2))-13/2240/(x+3/2)^4*(3*(x+3/2)^2-9*x-19/4)^(1/2)

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Maxima [A]
time = 0.49, size = 137, normalized size = 1.13 \begin {gather*} -\frac {27}{1500625} \, \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 {13 \, \sqrt {3 \, x^{2} + 2}}{140 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {97 \, \sqrt {3 \, x^{2} + 2}}{2100 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {87 \, \sqrt {3 \, x^{2} + 2}}{4900 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {991 \, \sqrt {3 \, x^{2} + 2}}{171500 \, {\left (2 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^5/(3*x^2+2)^(1/2),x, algorithm="maxima")

[Out]

-27/1500625*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) - 13/140*sqrt(3*x^2 + 2)/(
16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 97/2100*sqrt(3*x^2 + 2)/(8*x^3 + 36*x^2 + 54*x + 27) - 87/4900*sqrt(
3*x^2 + 2)/(4*x^2 + 12*x + 9) - 991/171500*sqrt(3*x^2 + 2)/(2*x + 3)

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Fricas [A]
time = 2.87, size = 118, normalized size = 0.98 \begin {gather*} \frac {81 \, \sqrt {35} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\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 ) - 70 \, {\left (5946 \, x^{3} + 35892 \, x^{2} + 79423 \, x + 70389\right )} \sqrt {3 \, x^{2} + 2}}{9003750 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^5/(3*x^2+2)^(1/2),x, algorithm="fricas")

[Out]

1/9003750*(81*sqrt(35)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log((sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) - 93*x
^2 + 36*x - 43)/(4*x^2 + 12*x + 9)) - 70*(5946*x^3 + 35892*x^2 + 79423*x + 70389)*sqrt(3*x^2 + 2))/(16*x^4 + 9
6*x^3 + 216*x^2 + 216*x + 81)

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Sympy [F(-1)] Timed out
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/(3*x**2+2)**(1/2),x)

[Out]

Timed out

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Giac [A]
time = 1.05, size = 191, normalized size = 1.58 \begin {gather*} \frac {1}{12005000} \, \sqrt {35} {\left (991 \, \sqrt {35} \sqrt {3} - 216 \, \log \left (\sqrt {35} \sqrt {3} - 9\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {1}{1029000} \, {\left (\frac {35 \, {\left (\frac {7 \, {\left (\frac {97}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} + \frac {195}{{\left (2 \, x + 3\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}\right )}}{2 \, x + 3} + \frac {261}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}\right )}}{2 \, x + 3} + \frac {2973}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}\right )} \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {27 \, \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 )}{1500625 \, \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)^5/(3*x^2+2)^(1/2),x, algorithm="giac")

[Out]

1/12005000*sqrt(35)*(991*sqrt(35)*sqrt(3) - 216*log(sqrt(35)*sqrt(3) - 9))*sgn(1/(2*x + 3)) - 1/1029000*(35*(7
*(97/sgn(1/(2*x + 3)) + 195/((2*x + 3)*sgn(1/(2*x + 3))))/(2*x + 3) + 261/sgn(1/(2*x + 3)))/(2*x + 3) + 2973/s
gn(1/(2*x + 3)))*sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + 27/1500625*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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Mupad [B]
time = 0.21, size = 146, normalized size = 1.21 \begin {gather*} \frac {\sqrt {35}\,\left (\frac {2808\,\ln \left (x+\frac {3}{2}\right )}{42875}-\frac {2808\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{42875}\right )}{560}-\frac {\sqrt {35}\,\left (\frac {324\,\ln \left (x+\frac {3}{2}\right )}{8575}-\frac {324\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{8575}\right )}{280}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {18252}{42875\,\left (x+\frac {3}{2}\right )}+\frac {702}{1225\,{\left (x+\frac {3}{2}\right )}^2}+\frac {117}{175\,{\left (x+\frac {3}{2}\right )}^3}+\frac {39}{70\,{\left (x+\frac {3}{2}\right )}^4}\right )}{96}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {636}{8575\,\left (x+\frac {3}{2}\right )}+\frac {18}{245\,{\left (x+\frac {3}{2}\right )}^2}+\frac {2}{35\,{\left (x+\frac {3}{2}\right )}^3}\right )}{48} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x - 5)/((2*x + 3)^5*(3*x^2 + 2)^(1/2)),x)

[Out]

(35^(1/2)*((2808*log(x + 3/2))/42875 - (2808*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/42875))/56
0 - (35^(1/2)*((324*log(x + 3/2))/8575 - (324*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/8575))/28
0 - (3^(1/2)*(x^2 + 2/3)^(1/2)*(18252/(42875*(x + 3/2)) + 702/(1225*(x + 3/2)^2) + 117/(175*(x + 3/2)^3) + 39/
(70*(x + 3/2)^4)))/96 + (3^(1/2)*(x^2 + 2/3)^(1/2)*(636/(8575*(x + 3/2)) + 18/(245*(x + 3/2)^2) + 2/(35*(x + 3
/2)^3)))/48

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