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

Optimal. Leaf size=112 \[ \frac {1}{60} (39-5 x) \left (3 x^2+2\right )^{5/2}+\frac {7}{96} (130-53 x) \left (3 x^2+2\right )^{3/2}+\frac {7}{64} (2275-691 x) \sqrt {3 x^2+2}-\frac {15925}{128} \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )-\frac {162673 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{128 \sqrt {3}} \]

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Rubi [A]  time = 0.08, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {815, 844, 215, 725, 206} \begin {gather*} \frac {1}{60} (39-5 x) \left (3 x^2+2\right )^{5/2}+\frac {7}{96} (130-53 x) \left (3 x^2+2\right )^{3/2}+\frac {7}{64} (2275-691 x) \sqrt {3 x^2+2}-\frac {15925}{128} \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )-\frac {162673 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{128 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(7*(2275 - 691*x)*Sqrt[2 + 3*x^2])/64 + (7*(130 - 53*x)*(2 + 3*x^2)^(3/2))/96 + ((39 - 5*x)*(2 + 3*x^2)^(5/2))
/60 - (162673*ArcSinh[Sqrt[3/2]*x])/(128*Sqrt[3]) - (15925*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2
])])/128

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 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

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 815

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*(a + c*x^2)^p)/(c*e^2*(m + 2*p + 1)*(m
+ 2*p + 2)), x] + Dist[(2*p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a + c*x^2)^(p - 1)*Simp[f*a
*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f*d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))
*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !R
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*
m, 2*p])

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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] &&  !IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{3+2 x} \, dx &=\frac {1}{60} (39-5 x) \left (2+3 x^2\right )^{5/2}+\frac {1}{72} \int \frac {(756-2226 x) \left (2+3 x^2\right )^{3/2}}{3+2 x} \, dx\\ &=\frac {7}{96} (130-53 x) \left (2+3 x^2\right )^{3/2}+\frac {1}{60} (39-5 x) \left (2+3 x^2\right )^{5/2}+\frac {\int \frac {(152712-1044792 x) \sqrt {2+3 x^2}}{3+2 x} \, dx}{3456}\\ &=\frac {7}{64} (2275-691 x) \sqrt {2+3 x^2}+\frac {7}{96} (130-53 x) \left (2+3 x^2\right )^{3/2}+\frac {1}{60} (39-5 x) \left (2+3 x^2\right )^{5/2}+\frac {\int \frac {44942688-210824208 x}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{82944}\\ &=\frac {7}{64} (2275-691 x) \sqrt {2+3 x^2}+\frac {7}{96} (130-53 x) \left (2+3 x^2\right )^{3/2}+\frac {1}{60} (39-5 x) \left (2+3 x^2\right )^{5/2}-\frac {162673}{128} \int \frac {1}{\sqrt {2+3 x^2}} \, dx+\frac {557375}{128} \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx\\ &=\frac {7}{64} (2275-691 x) \sqrt {2+3 x^2}+\frac {7}{96} (130-53 x) \left (2+3 x^2\right )^{3/2}+\frac {1}{60} (39-5 x) \left (2+3 x^2\right )^{5/2}-\frac {162673 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{128 \sqrt {3}}-\frac {557375}{128} \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )\\ &=\frac {7}{64} (2275-691 x) \sqrt {2+3 x^2}+\frac {7}{96} (130-53 x) \left (2+3 x^2\right )^{3/2}+\frac {1}{60} (39-5 x) \left (2+3 x^2\right )^{5/2}-\frac {162673 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{128 \sqrt {3}}-\frac {15925}{128} \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 90, normalized size = 0.80 \begin {gather*} \frac {-238875 \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )-2 \sqrt {3 x^2+2} \left (720 x^5-5616 x^4+12090 x^3-34788 x^2+80295 x-259571\right )-813365 \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{1920} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[2 + 3*x^2]*(-259571 + 80295*x - 34788*x^2 + 12090*x^3 - 5616*x^4 + 720*x^5) - 813365*Sqrt[3]*ArcSinh[
Sqrt[3/2]*x] - 238875*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/1920

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IntegrateAlgebraic [A]  time = 0.54, size = 119, normalized size = 1.06 \begin {gather*} \frac {162673 \log \left (\sqrt {3 x^2+2}-\sqrt {3} x\right )}{128 \sqrt {3}}+\frac {15925}{64} \sqrt {35} \tanh ^{-1}\left (-\frac {2 \sqrt {3 x^2+2}}{\sqrt {35}}+2 \sqrt {\frac {3}{35}} x+3 \sqrt {\frac {3}{35}}\right )+\frac {1}{960} \sqrt {3 x^2+2} \left (-720 x^5+5616 x^4-12090 x^3+34788 x^2-80295 x+259571\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + 3*x^2]*(259571 - 80295*x + 34788*x^2 - 12090*x^3 + 5616*x^4 - 720*x^5))/960 + (15925*Sqrt[35]*ArcTan
h[3*Sqrt[3/35] + 2*Sqrt[3/35]*x - (2*Sqrt[2 + 3*x^2])/Sqrt[35]])/64 + (162673*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^
2]])/(128*Sqrt[3])

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fricas [A]  time = 0.44, size = 112, normalized size = 1.00 \begin {gather*} -\frac {1}{960} \, {\left (720 \, x^{5} - 5616 \, x^{4} + 12090 \, x^{3} - 34788 \, x^{2} + 80295 \, x - 259571\right )} \sqrt {3 \, x^{2} + 2} + \frac {162673}{768} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + \frac {15925}{256} \, \sqrt {35} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/960*(720*x^5 - 5616*x^4 + 12090*x^3 - 34788*x^2 + 80295*x - 259571)*sqrt(3*x^2 + 2) + 162673/768*sqrt(3)*lo
g(sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + 15925/256*sqrt(35)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^
2 - 36*x + 43)/(4*x^2 + 12*x + 9))

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giac [A]  time = 0.28, size = 125, normalized size = 1.12 \begin {gather*} -\frac {1}{960} \, {\left (3 \, {\left (2 \, {\left ({\left (24 \, {\left (5 \, x - 39\right )} x + 2015\right )} x - 5798\right )} x + 26765\right )} x - 259571\right )} \sqrt {3 \, x^{2} + 2} + \frac {162673}{384} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) + \frac {15925}{128} \, \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/960*(3*(2*((24*(5*x - 39)*x + 2015)*x - 5798)*x + 26765)*x - 259571)*sqrt(3*x^2 + 2) + 162673/384*sqrt(3)*l
og(-sqrt(3)*x + sqrt(3*x^2 + 2)) + 15925/128*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)))

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maple [A]  time = 0.05, size = 162, normalized size = 1.45 \begin {gather*} -\frac {\left (3 x^{2}+2\right )^{\frac {5}{2}} x}{12}-\frac {5 \left (3 x^{2}+2\right )^{\frac {3}{2}} x}{24}-\frac {5 \sqrt {3 x^{2}+2}\, x}{8}-\frac {117 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}} x}{32}-\frac {4797 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\, x}{64}-\frac {162673 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{384}-\frac {15925 \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 )}{128}+\frac {13 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{20}+\frac {455 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{48}+\frac {15925 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{128} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(3*x^2+2)^(5/2)*x-5/24*(3*x^2+2)^(3/2)*x-5/8*(3*x^2+2)^(1/2)*x-162673/384*arcsinh(1/2*6^(1/2)*x)*3^(1/2)
+13/20*(-9*x+3*(x+3/2)^2-19/4)^(5/2)-117/32*(-9*x+3*(x+3/2)^2-19/4)^(3/2)*x-4797/64*(-9*x+3*(x+3/2)^2-19/4)^(1
/2)*x+455/48*(-9*x+3*(x+3/2)^2-19/4)^(3/2)+15925/128*(-36*x+12*(x+3/2)^2-19)^(1/2)-15925/128*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 = 1.31, size = 116, normalized size = 1.04 \begin {gather*} -\frac {1}{12} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} x + \frac {13}{20} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} - \frac {371}{96} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {455}{48} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {4837}{64} \, \sqrt {3 \, x^{2} + 2} x - \frac {162673}{384} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {15925}{128} \, \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 {15925}{64} \, \sqrt {3 \, x^{2} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(3*x^2 + 2)^(5/2)*x + 13/20*(3*x^2 + 2)^(5/2) - 371/96*(3*x^2 + 2)^(3/2)*x + 455/48*(3*x^2 + 2)^(3/2) -
4837/64*sqrt(3*x^2 + 2)*x - 162673/384*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 15925/128*sqrt(35)*arcsinh(3/2*sqrt(6)
*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 15925/64*sqrt(3*x^2 + 2)

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mupad [B]  time = 1.81, size = 86, normalized size = 0.77 \begin {gather*} \frac {\sqrt {35}\,\left (1114750\,\ln \left (x+\frac {3}{2}\right )-1114750\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )\right )}{8960}-\frac {162673\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{384}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {9\,x^5}{4}-\frac {351\,x^4}{20}+\frac {1209\,x^3}{32}-\frac {8697\,x^2}{80}+\frac {16059\,x}{64}-\frac {259571}{320}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(35^(1/2)*(1114750*log(x + 3/2) - 1114750*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9)))/8960 - (1626
73*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/384 - (3^(1/2)*(x^2 + 2/3)^(1/2)*((16059*x)/64 - (8697*x^2)/80 + (120
9*x^3)/32 - (351*x^4)/20 + (9*x^5)/4 - 259571/320))/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*x**2+2)**(5/2)/(3+2*x),x)

[Out]

Timed out

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