3.22.87 \(\int \frac {(5-x) (2+5 x+3 x^2)^{3/2}}{(3+2 x)^6} \, dx\)

Optimal. Leaf size=124 \[ -\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{25 (2 x+3)^5}+\frac {47 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{400 (2 x+3)^4}-\frac {141 (8 x+7) \sqrt {3 x^2+5 x+2}}{16000 (2 x+3)^2}+\frac {141 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{32000 \sqrt {5}} \]

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Rubi [A]  time = 0.06, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {806, 720, 724, 206} \begin {gather*} -\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{25 (2 x+3)^5}+\frac {47 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{400 (2 x+3)^4}-\frac {141 (8 x+7) \sqrt {3 x^2+5 x+2}}{16000 (2 x+3)^2}+\frac {141 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{32000 \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^6,x]

[Out]

(-141*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(16000*(3 + 2*x)^2) + (47*(7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2))/(400*(3 +
2*x)^4) - (13*(2 + 5*x + 3*x^2)^(5/2))/(25*(3 + 2*x)^5) + (141*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x
^2])])/(32000*Sqrt[5])

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 720

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

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rubi steps

\begin {align*} \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx &=-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{25 (3+2 x)^5}+\frac {47}{10} \int \frac {\left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx\\ &=\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{400 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{25 (3+2 x)^5}-\frac {141}{800} \int \frac {\sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx\\ &=-\frac {141 (7+8 x) \sqrt {2+5 x+3 x^2}}{16000 (3+2 x)^2}+\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{400 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{25 (3+2 x)^5}+\frac {141 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{32000}\\ &=-\frac {141 (7+8 x) \sqrt {2+5 x+3 x^2}}{16000 (3+2 x)^2}+\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{400 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{25 (3+2 x)^5}-\frac {141 \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{16000}\\ &=-\frac {141 (7+8 x) \sqrt {2+5 x+3 x^2}}{16000 (3+2 x)^2}+\frac {47 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{400 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{25 (3+2 x)^5}+\frac {141 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{32000 \sqrt {5}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 128, normalized size = 1.03 \begin {gather*} -\frac {83200 \left (3 x^2+5 x+2\right )^{5/2}-47 (2 x+3) \left (-30 (8 x+7) \sqrt {3 x^2+5 x+2} (2 x+3)^2+400 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}-3 \sqrt {5} (2 x+3)^4 \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )\right )}{160000 (2 x+3)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^6,x]

[Out]

-1/160000*(83200*(2 + 5*x + 3*x^2)^(5/2) - 47*(3 + 2*x)*(-30*(3 + 2*x)^2*(7 + 8*x)*Sqrt[2 + 5*x + 3*x^2] + 400
*(7 + 8*x)*(2 + 5*x + 3*x^2)^(3/2) - 3*Sqrt[5]*(3 + 2*x)^4*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2]
)]))/(3 + 2*x)^5

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IntegrateAlgebraic [A]  time = 0.56, size = 81, normalized size = 0.65 \begin {gather*} \frac {141 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{16000 \sqrt {5}}+\frac {\sqrt {3 x^2+5 x+2} \left (6336 x^4+66616 x^3+131516 x^2+90126 x+19031\right )}{16000 (2 x+3)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((5 - x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^6,x]

[Out]

(Sqrt[2 + 5*x + 3*x^2]*(19031 + 90126*x + 131516*x^2 + 66616*x^3 + 6336*x^4))/(16000*(3 + 2*x)^5) + (141*ArcTa
nh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[5]*(1 + x))])/(16000*Sqrt[5])

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fricas [A]  time = 0.42, size = 140, normalized size = 1.13 \begin {gather*} \frac {141 \, \sqrt {5} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 20 \, {\left (6336 \, x^{4} + 66616 \, x^{3} + 131516 \, x^{2} + 90126 \, x + 19031\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{320000 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^6,x, algorithm="fricas")

[Out]

1/320000*(141*sqrt(5)*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)*log((4*sqrt(5)*sqrt(3*x^2 + 5*x +
2)*(8*x + 7) + 124*x^2 + 212*x + 89)/(4*x^2 + 12*x + 9)) + 20*(6336*x^4 + 66616*x^3 + 131516*x^2 + 90126*x + 1
9031)*sqrt(3*x^2 + 5*x + 2))/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)

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giac [B]  time = 0.31, size = 359, normalized size = 2.90 \begin {gather*} \frac {141}{160000} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac {146256 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 654456 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 415048 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} - 15455452 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} - 140042336 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} - 207568854 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} - 544555762 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} - 286352757 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} - 252454821 \, \sqrt {3} x - 31985676 \, \sqrt {3} + 252454821 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{16000 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^6,x, algorithm="giac")

[Out]

141/160000*sqrt(5)*log(abs(-4*sqrt(3)*x - 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))/abs(-4*sqrt(3)*x +
2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))) - 1/16000*(146256*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 + 65
4456*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 + 415048*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 - 15455452*s
qrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 - 140042336*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 - 207568854*sqr
t(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 - 544555762*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 - 286352757*sqrt(
3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 - 252454821*sqrt(3)*x - 31985676*sqrt(3) + 252454821*sqrt(3*x^2 + 5*x
+ 2))/(2*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 6*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) + 11)^5

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maple [B]  time = 0.06, size = 211, normalized size = 1.70 \begin {gather*} -\frac {141 \sqrt {5}\, \arctanh \left (\frac {2 \left (-4 x -\frac {7}{2}\right ) \sqrt {5}}{5 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{160000}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{800 \left (x +\frac {3}{2}\right )^{5}}-\frac {47 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{1600 \left (x +\frac {3}{2}\right )^{4}}-\frac {47 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{1000 \left (x +\frac {3}{2}\right )^{3}}-\frac {1457 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{20000 \left (x +\frac {3}{2}\right )^{2}}-\frac {1363 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{12500 \left (x +\frac {3}{2}\right )}+\frac {47 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{100000}-\frac {141 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{20000}+\frac {141 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{160000}+\frac {1363 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{25000} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+5*x+2)^(3/2)/(2*x+3)^6,x)

[Out]

-13/800/(x+3/2)^5*(-4*x+3*(x+3/2)^2-19/4)^(5/2)-47/1600/(x+3/2)^4*(-4*x+3*(x+3/2)^2-19/4)^(5/2)-47/1000/(x+3/2
)^3*(-4*x+3*(x+3/2)^2-19/4)^(5/2)-1457/20000/(x+3/2)^2*(-4*x+3*(x+3/2)^2-19/4)^(5/2)-1363/12500/(x+3/2)*(-4*x+
3*(x+3/2)^2-19/4)^(5/2)+47/100000*(-4*x+3*(x+3/2)^2-19/4)^(3/2)-141/20000*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(1/2
)+141/160000*(-16*x+12*(x+3/2)^2-19)^(1/2)-141/160000*5^(1/2)*arctanh(2/5*(-4*x-7/2)*5^(1/2)/(-16*x+12*(x+3/2)
^2-19)^(1/2))+1363/25000*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(3/2)

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maxima [B]  time = 1.29, size = 241, normalized size = 1.94 \begin {gather*} \frac {4371}{20000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{25 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {47 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{100 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {47 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{125 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {1457 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{5000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {423}{10000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {141}{160000} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) - \frac {2679}{80000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {1363 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{5000 \, {\left (2 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^6,x, algorithm="maxima")

[Out]

4371/20000*(3*x^2 + 5*x + 2)^(3/2) - 13/25*(3*x^2 + 5*x + 2)^(5/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 81
0*x + 243) - 47/100*(3*x^2 + 5*x + 2)^(5/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 47/125*(3*x^2 + 5*x + 2
)^(5/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 1457/5000*(3*x^2 + 5*x + 2)^(5/2)/(4*x^2 + 12*x + 9) - 423/10000*sqrt(3
*x^2 + 5*x + 2)*x - 141/160000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2)
- 2679/80000*sqrt(3*x^2 + 5*x + 2) - 1363/5000*(3*x^2 + 5*x + 2)^(3/2)/(2*x + 3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{{\left (2\,x+3\right )}^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x - 5)*(5*x + 3*x^2 + 2)^(3/2))/(2*x + 3)^6,x)

[Out]

-int(((x - 5)*(5*x + 3*x^2 + 2)^(3/2))/(2*x + 3)^6, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {10 \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac {23 x \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac {10 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x**2+5*x+2)**(3/2)/(3+2*x)**6,x)

[Out]

-Integral(-10*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4 + 4320*x**3 + 4860*x**2 + 2916*x + 729),
x) - Integral(-23*x*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4 + 4320*x**3 + 4860*x**2 + 2916*x +
729), x) - Integral(-10*x**2*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4 + 4320*x**3 + 4860*x**2 +
2916*x + 729), x) - Integral(3*x**3*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4 + 4320*x**3 + 4860*
x**2 + 2916*x + 729), x)

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