3.22.99 \(\int \frac {(5-x) (2+5 x+3 x^2)^{5/2}}{(3+2 x)^4} \, dx\)

Optimal. Leaf size=165 \[ -\frac {(x+8) \left (3 x^2+5 x+2\right )^{5/2}}{6 (2 x+3)^3}+\frac {5 (43 x+93) \left (3 x^2+5 x+2\right )^{3/2}}{48 (2 x+3)^2}-\frac {5 (343 x+736) \sqrt {3 x^2+5 x+2}}{64 (2 x+3)}+\frac {13505 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{256 \sqrt {3}}-\frac {3487}{256} \sqrt {5} \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right ) \]

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Rubi [A]  time = 0.11, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {812, 843, 621, 206, 724} \begin {gather*} -\frac {(x+8) \left (3 x^2+5 x+2\right )^{5/2}}{6 (2 x+3)^3}+\frac {5 (43 x+93) \left (3 x^2+5 x+2\right )^{3/2}}{48 (2 x+3)^2}-\frac {5 (343 x+736) \sqrt {3 x^2+5 x+2}}{64 (2 x+3)}+\frac {13505 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{256 \sqrt {3}}-\frac {3487}{256} \sqrt {5} \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^4,x]

[Out]

(-5*(736 + 343*x)*Sqrt[2 + 5*x + 3*x^2])/(64*(3 + 2*x)) + (5*(93 + 43*x)*(2 + 5*x + 3*x^2)^(3/2))/(48*(3 + 2*x
)^2) - ((8 + x)*(2 + 5*x + 3*x^2)^(5/2))/(6*(3 + 2*x)^3) + (13505*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x +
3*x^2])])/(256*Sqrt[3]) - (3487*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/256

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 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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 812

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

Rule 843

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

Rubi steps

\begin {align*} \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^4} \, dx &=-\frac {(8+x) \left (2+5 x+3 x^2\right )^{5/2}}{6 (3+2 x)^3}-\frac {5}{72} \int \frac {(-216-258 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^3} \, dx\\ &=\frac {5 (93+43 x) \left (2+5 x+3 x^2\right )^{3/2}}{48 (3+2 x)^2}-\frac {(8+x) \left (2+5 x+3 x^2\right )^{5/2}}{6 (3+2 x)^3}+\frac {5}{768} \int \frac {(-7032-8232 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^2} \, dx\\ &=-\frac {5 (736+343 x) \sqrt {2+5 x+3 x^2}}{64 (3+2 x)}+\frac {5 (93+43 x) \left (2+5 x+3 x^2\right )^{3/2}}{48 (3+2 x)^2}-\frac {(8+x) \left (2+5 x+3 x^2\right )^{5/2}}{6 (3+2 x)^3}-\frac {5 \int \frac {-110784-129648 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{6144}\\ &=-\frac {5 (736+343 x) \sqrt {2+5 x+3 x^2}}{64 (3+2 x)}+\frac {5 (93+43 x) \left (2+5 x+3 x^2\right )^{3/2}}{48 (3+2 x)^2}-\frac {(8+x) \left (2+5 x+3 x^2\right )^{5/2}}{6 (3+2 x)^3}+\frac {13505}{256} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx-\frac {17435}{256} \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {5 (736+343 x) \sqrt {2+5 x+3 x^2}}{64 (3+2 x)}+\frac {5 (93+43 x) \left (2+5 x+3 x^2\right )^{3/2}}{48 (3+2 x)^2}-\frac {(8+x) \left (2+5 x+3 x^2\right )^{5/2}}{6 (3+2 x)^3}+\frac {13505}{128} \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )+\frac {17435}{128} \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=-\frac {5 (736+343 x) \sqrt {2+5 x+3 x^2}}{64 (3+2 x)}+\frac {5 (93+43 x) \left (2+5 x+3 x^2\right )^{3/2}}{48 (3+2 x)^2}-\frac {(8+x) \left (2+5 x+3 x^2\right )^{5/2}}{6 (3+2 x)^3}+\frac {13505 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{256 \sqrt {3}}-\frac {3487}{256} \sqrt {5} \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 120, normalized size = 0.73 \begin {gather*} \frac {1}{768} \left (10461 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )+13505 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )-\frac {4 \sqrt {3 x^2+5 x+2} \left (288 x^5-1896 x^4+1944 x^3+64332 x^2+143533 x+89224\right )}{(2 x+3)^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^4,x]

[Out]

((-4*Sqrt[2 + 5*x + 3*x^2]*(89224 + 143533*x + 64332*x^2 + 1944*x^3 - 1896*x^4 + 288*x^5))/(3 + 2*x)^3 + 10461
*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])] + 13505*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 1
5*x + 9*x^2])])/768

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IntegrateAlgebraic [A]  time = 0.75, size = 121, normalized size = 0.73 \begin {gather*} \frac {13505 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{128 \sqrt {3}}-\frac {3487}{128} \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )+\frac {\sqrt {3 x^2+5 x+2} \left (-288 x^5+1896 x^4-1944 x^3-64332 x^2-143533 x-89224\right )}{192 (2 x+3)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^4,x]

[Out]

(Sqrt[2 + 5*x + 3*x^2]*(-89224 - 143533*x - 64332*x^2 - 1944*x^3 + 1896*x^4 - 288*x^5))/(192*(3 + 2*x)^3) + (1
3505*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[3]*(1 + x))])/(128*Sqrt[3]) - (3487*Sqrt[5]*ArcTanh[Sqrt[2 + 5*x + 3*
x^2]/(Sqrt[5]*(1 + x))])/128

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fricas [A]  time = 0.42, size = 179, normalized size = 1.08 \begin {gather*} \frac {13505 \, \sqrt {3} {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 10461 \, \sqrt {5} {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\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 ) - 8 \, {\left (288 \, x^{5} - 1896 \, x^{4} + 1944 \, x^{3} + 64332 \, x^{2} + 143533 \, x + 89224\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{1536 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^4,x, algorithm="fricas")

[Out]

1/1536*(13505*sqrt(3)*(8*x^3 + 36*x^2 + 54*x + 27)*log(4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 12
0*x + 49) + 10461*sqrt(5)*(8*x^3 + 36*x^2 + 54*x + 27)*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)) - 8*(288*x^5 - 1896*x^4 + 1944*x^3 + 64332*x^2 + 143533*x + 89224)*sqrt(3
*x^2 + 5*x + 2))/(8*x^3 + 36*x^2 + 54*x + 27)

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giac [B]  time = 0.34, size = 315, normalized size = 1.91 \begin {gather*} -\frac {1}{128} \, {\left (2 \, {\left (12 \, x - 133\right )} x + 1197\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {3487}{256} \, \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 {13505}{768} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac {203604 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 1334970 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 10053790 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 12051375 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 20819415 \, \sqrt {3} x + 4639299 \, \sqrt {3} - 20819415 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{384 \, {\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 )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^4,x, algorithm="giac")

[Out]

-1/128*(2*(12*x - 133)*x + 1197)*sqrt(3*x^2 + 5*x + 2) - 3487/256*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))) - 135
05/768*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) - 1/384*(203604*(sqrt(3)*x - sqrt(
3*x^2 + 5*x + 2))^5 + 1334970*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 10053790*(sqrt(3)*x - sqrt(3*x^2
+ 5*x + 2))^3 + 12051375*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 20819415*sqrt(3)*x + 4639299*sqrt(3)
- 20819415*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)^3

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maple [A]  time = 0.06, size = 237, normalized size = 1.44 \begin {gather*} \frac {3487 \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 )}{256}+\frac {13505 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\right )}{768}+\frac {67 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{600 \left (x +\frac {3}{2}\right )^{2}}-\frac {197 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{125 \left (x +\frac {3}{2}\right )}-\frac {3487 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{1000}+\frac {329 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{240}+\frac {443 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{128}-\frac {3487 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{480}-\frac {3487 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{256}+\frac {197 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{250}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{120 \left (x +\frac {3}{2}\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+5*x+2)^(5/2)/(2*x+3)^4,x)

[Out]

67/600/(x+3/2)^2*(-4*x+3*(x+3/2)^2-19/4)^(7/2)-197/125/(x+3/2)*(-4*x+3*(x+3/2)^2-19/4)^(7/2)-3487/1000*(-4*x+3
*(x+3/2)^2-19/4)^(5/2)+329/240*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(3/2)+443/128*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(
1/2)+13505/768*3^(1/2)*ln(1/3*(3*x+5/2)*3^(1/2)+(-4*x+3*(x+3/2)^2-19/4)^(1/2))-3487/480*(-4*x+3*(x+3/2)^2-19/4
)^(3/2)-3487/256*(-16*x+12*(x+3/2)^2-19)^(1/2)+3487/256*5^(1/2)*arctanh(2/5*(-4*x-7/2)*5^(1/2)/(-16*x+12*(x+3/
2)^2-19)^(1/2))+197/250*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(5/2)-13/120/(x+3/2)^3*(-4*x+3*(x+3/2)^2-19/4)^(7/2)

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maxima [A]  time = 1.36, size = 220, normalized size = 1.33 \begin {gather*} -\frac {67}{200} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{15 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} + \frac {67 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{150 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {329}{40} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {197}{480} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {197 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{50 \, {\left (2 \, x + 3\right )}} + \frac {1329}{64} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {13505}{768} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) + \frac {3487}{256} \, \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 {159}{16} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^4,x, algorithm="maxima")

[Out]

-67/200*(3*x^2 + 5*x + 2)^(5/2) - 13/15*(3*x^2 + 5*x + 2)^(7/2)/(8*x^3 + 36*x^2 + 54*x + 27) + 67/150*(3*x^2 +
5*x + 2)^(7/2)/(4*x^2 + 12*x + 9) + 329/40*(3*x^2 + 5*x + 2)^(3/2)*x - 197/480*(3*x^2 + 5*x + 2)^(3/2) - 197/
50*(3*x^2 + 5*x + 2)^(5/2)/(2*x + 3) + 1329/64*sqrt(3*x^2 + 5*x + 2)*x + 13505/768*sqrt(3)*log(sqrt(3)*sqrt(3*
x^2 + 5*x + 2) + 3*x + 5/2) + 3487/256*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x +
3) - 2) - 159/16*sqrt(3*x^2 + 5*x + 2)

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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 )}^{5/2}}{{\left (2\,x+3\right )}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x - 5)*(5*x + 3*x^2 + 2)^(5/2))/(2*x + 3)^4,x)

[Out]

-int(((x - 5)*(5*x + 3*x^2 + 2)^(5/2))/(2*x + 3)^4, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {20 \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac {96 x \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac {165 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac {113 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac {15 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \frac {9 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x**2+5*x+2)**(5/2)/(3+2*x)**4,x)

[Out]

-Integral(-20*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x) - Integral(-96*x*sqrt(3*x
**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x) - Integral(-165*x**2*sqrt(3*x**2 + 5*x + 2)/(16
*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x) - Integral(-113*x**3*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 +
216*x**2 + 216*x + 81), x) - Integral(-15*x**4*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x +
81), x) - Integral(9*x**5*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x)

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