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

Optimal. Leaf size=104 \[ -\frac {89 \left (3 x^2+2\right )^{3/2}}{2940 (2 x+3)^3}-\frac {13 \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}-\frac {33 (4-9 x) \sqrt {3 x^2+2}}{8575 (2 x+3)^2}-\frac {198 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{8575 \sqrt {35}} \]

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Rubi [A]  time = 0.05, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {835, 807, 721, 725, 206} \begin {gather*} -\frac {89 \left (3 x^2+2\right )^{3/2}}{2940 (2 x+3)^3}-\frac {13 \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}-\frac {33 (4-9 x) \sqrt {3 x^2+2}}{8575 (2 x+3)^2}-\frac {198 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{8575 \sqrt {35}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-33*(4 - 9*x)*Sqrt[2 + 3*x^2])/(8575*(3 + 2*x)^2) - (13*(2 + 3*x^2)^(3/2))/(140*(3 + 2*x)^4) - (89*(2 + 3*x^2
)^(3/2))/(2940*(3 + 2*x)^3) - (198*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(8575*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 721

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*(-2*a*e + (2*c*
d)*x)*(a + c*x^2)^p)/(2*(m + 1)*(c*d^2 + a*e^2)), x] - Dist[(4*a*c*p)/(2*(m + 1)*(c*d^2 + a*e^2)), Int[(d + e*
x)^(m + 2)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2,
0] && GtQ[p, 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 835

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) \sqrt {2+3 x^2}}{(3+2 x)^5} \, dx &=-\frac {13 \left (2+3 x^2\right )^{3/2}}{140 (3+2 x)^4}-\frac {1}{140} \int \frac {(-164+39 x) \sqrt {2+3 x^2}}{(3+2 x)^4} \, dx\\ &=-\frac {13 \left (2+3 x^2\right )^{3/2}}{140 (3+2 x)^4}-\frac {89 \left (2+3 x^2\right )^{3/2}}{2940 (3+2 x)^3}+\frac {66}{245} \int \frac {\sqrt {2+3 x^2}}{(3+2 x)^3} \, dx\\ &=-\frac {33 (4-9 x) \sqrt {2+3 x^2}}{8575 (3+2 x)^2}-\frac {13 \left (2+3 x^2\right )^{3/2}}{140 (3+2 x)^4}-\frac {89 \left (2+3 x^2\right )^{3/2}}{2940 (3+2 x)^3}+\frac {198 \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{8575}\\ &=-\frac {33 (4-9 x) \sqrt {2+3 x^2}}{8575 (3+2 x)^2}-\frac {13 \left (2+3 x^2\right )^{3/2}}{140 (3+2 x)^4}-\frac {89 \left (2+3 x^2\right )^{3/2}}{2940 (3+2 x)^3}-\frac {198 \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )}{8575}\\ &=-\frac {33 (4-9 x) \sqrt {2+3 x^2}}{8575 (3+2 x)^2}-\frac {13 \left (2+3 x^2\right )^{3/2}}{140 (3+2 x)^4}-\frac {89 \left (2+3 x^2\right )^{3/2}}{2940 (3+2 x)^3}-\frac {198 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{8575 \sqrt {35}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 70, normalized size = 0.67 \begin {gather*} -\frac {198 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{8575 \sqrt {35}}-\frac {\sqrt {3 x^2+2} \left (2217 x^3+10134 x^2-304 x+26028\right )}{51450 (2 x+3)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/51450*(Sqrt[2 + 3*x^2]*(26028 - 304*x + 10134*x^2 + 2217*x^3))/(3 + 2*x)^4 - (198*ArcTanh[(4 - 9*x)/(Sqrt[3
5]*Sqrt[2 + 3*x^2])])/(8575*Sqrt[35])

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IntegrateAlgebraic [A]  time = 1.05, size = 86, normalized size = 0.83 \begin {gather*} \frac {396 \tanh ^{-1}\left (-\frac {2 \sqrt {3 x^2+2}}{\sqrt {35}}+2 \sqrt {\frac {3}{35}} x+3 \sqrt {\frac {3}{35}}\right )}{8575 \sqrt {35}}+\frac {\sqrt {3 x^2+2} \left (-2217 x^3-10134 x^2+304 x-26028\right )}{51450 (2 x+3)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + 3*x^2]*(-26028 + 304*x - 10134*x^2 - 2217*x^3))/(51450*(3 + 2*x)^4) + (396*ArcTanh[3*Sqrt[3/35] + 2*
Sqrt[3/35]*x - (2*Sqrt[2 + 3*x^2])/Sqrt[35]])/(8575*Sqrt[35])

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fricas [A]  time = 0.41, size = 119, normalized size = 1.14 \begin {gather*} \frac {594 \, \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 ) - 35 \, {\left (2217 \, x^{3} + 10134 \, x^{2} - 304 \, x + 26028\right )} \sqrt {3 \, x^{2} + 2}}{1800750 \, {\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*x^2+2)^(1/2)/(3+2*x)^5,x, algorithm="fricas")

[Out]

1/1800750*(594*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)) - 35*(2217*x^3 + 10134*x^2 - 304*x + 26028)*sqrt(3*x^2 + 2))/(16*x^4 + 9
6*x^3 + 216*x^2 + 216*x + 81)

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giac [B]  time = 0.22, size = 181, normalized size = 1.74 \begin {gather*} \frac {1}{9604000} \, \sqrt {35} {\left (739 \, \sqrt {35} \sqrt {3} + 6336 \, \log \left (\sqrt {35} \sqrt {3} - 9\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {198}{300125} \, \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 ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {1}{823200} \, {\left (\frac {35 \, {\left (\frac {7 \, {\left (\frac {1365 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}{2 \, x + 3} - 257 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} + 9 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} + 2217 \, \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9604000*sqrt(35)*(739*sqrt(35)*sqrt(3) + 6336*log(sqrt(35)*sqrt(3) - 9))*sgn(1/(2*x + 3)) - 198/300125*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)) - 1/823
200*(35*(7*(1365*sgn(1/(2*x + 3))/(2*x + 3) - 257*sgn(1/(2*x + 3)))/(2*x + 3) + 9*sgn(1/(2*x + 3)))/(2*x + 3)
+ 2217*sgn(1/(2*x + 3)))*sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3)

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maple [A]  time = 0.06, size = 149, normalized size = 1.43 \begin {gather*} \frac {891 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\, x}{300125}-\frac {198 \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 )}{300125}-\frac {89 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{23520 \left (x +\frac {3}{2}\right )^{3}}-\frac {33 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{17150 \left (x +\frac {3}{2}\right )^{2}}-\frac {297 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{300125 \left (x +\frac {3}{2}\right )}+\frac {198 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{300125}-\frac {13 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{2240 \left (x +\frac {3}{2}\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-89/23520/(x+3/2)^3*(-9*x+3*(x+3/2)^2-19/4)^(3/2)-33/17150/(x+3/2)^2*(-9*x+3*(x+3/2)^2-19/4)^(3/2)-297/300125/
(x+3/2)*(-9*x+3*(x+3/2)^2-19/4)^(3/2)+198/300125*(-36*x+12*(x+3/2)^2-19)^(1/2)-198/300125*35^(1/2)*arctanh(2/3
5*(-9*x+4)*35^(1/2)/(-36*x+12*(x+3/2)^2-19)^(1/2))+891/300125*(-9*x+3*(x+3/2)^2-19/4)^(1/2)*x-13/2240/(x+3/2)^
4*(-9*x+3*(x+3/2)^2-19/4)^(3/2)

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maxima [A]  time = 1.17, size = 148, normalized size = 1.42 \begin {gather*} \frac {198}{300125} \, \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 {99}{17150} \, \sqrt {3 \, x^{2} + 2} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{140 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {89 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{2940 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {66 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{8575 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {297 \, \sqrt {3 \, x^{2} + 2}}{17150 \, {\left (2 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

198/300125*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 99/17150*sqrt(3*x^2 + 2)
- 13/140*(3*x^2 + 2)^(3/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 89/2940*(3*x^2 + 2)^(3/2)/(8*x^3 + 36*x^
2 + 54*x + 27) - 66/8575*(3*x^2 + 2)^(3/2)/(4*x^2 + 12*x + 9) - 297/17150*sqrt(3*x^2 + 2)/(2*x + 3)

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mupad [B]  time = 1.86, size = 140, normalized size = 1.35 \begin {gather*} \frac {198\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{300125}-\frac {198\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{300125}-\frac {13\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{256\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}-\frac {739\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{274400\,\left (x+\frac {3}{2}\right )}-\frac {3\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{15680\,\left (x^2+3\,x+\frac {9}{4}\right )}+\frac {257\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{13440\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(198*35^(1/2)*log(x + 3/2))/300125 - (198*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/3001
25 - (13*3^(1/2)*(x^2 + 2/3)^(1/2))/(256*((27*x)/2 + (27*x^2)/2 + 6*x^3 + x^4 + 81/16)) - (739*3^(1/2)*(x^2 +
2/3)^(1/2))/(274400*(x + 3/2)) - (3*3^(1/2)*(x^2 + 2/3)^(1/2))/(15680*(3*x + x^2 + 9/4)) + (257*3^(1/2)*(x^2 +
2/3)^(1/2))/(13440*((27*x)/4 + (9*x^2)/2 + x^3 + 27/8))

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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*x**2+2)**(1/2)/(3+2*x)**5,x)

[Out]

Timed out

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