3.2443 \(\int \frac {(5-x) (2+5 x+3 x^2)^{5/2}}{(3+2 x)^7} \, dx\)

Optimal. Leaf size=167 \[ \frac {(116 x+109) \left (3 x^2+5 x+2\right )^{5/2}}{120 (2 x+3)^6}+\frac {(328 x+437) \left (3 x^2+5 x+2\right )^{3/2}}{1920 (2 x+3)^4}+\frac {(10952 x+14083) \sqrt {3 x^2+5 x+2}}{25600 (2 x+3)^2}-\frac {9}{128} \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )+\frac {13931 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{51200 \sqrt {5}} \]

[Out]

1/1920*(437+328*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^4+1/120*(109+116*x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^6-9/128*arctanh
(1/6*(5+6*x)*3^(1/2)/(3*x^2+5*x+2)^(1/2))*3^(1/2)+13931/256000*arctanh(1/10*(7+8*x)*5^(1/2)/(3*x^2+5*x+2)^(1/2
))*5^(1/2)+1/25600*(14083+10952*x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^2

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Rubi [A]  time = 0.10, antiderivative size = 167, 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 = {810, 843, 621, 206, 724} \[ \frac {(116 x+109) \left (3 x^2+5 x+2\right )^{5/2}}{120 (2 x+3)^6}+\frac {(328 x+437) \left (3 x^2+5 x+2\right )^{3/2}}{1920 (2 x+3)^4}+\frac {(10952 x+14083) \sqrt {3 x^2+5 x+2}}{25600 (2 x+3)^2}-\frac {9}{128} \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )+\frac {13931 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{51200 \sqrt {5}} \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^7,x]

[Out]

((14083 + 10952*x)*Sqrt[2 + 5*x + 3*x^2])/(25600*(3 + 2*x)^2) + ((437 + 328*x)*(2 + 5*x + 3*x^2)^(3/2))/(1920*
(3 + 2*x)^4) + ((109 + 116*x)*(2 + 5*x + 3*x^2)^(5/2))/(120*(3 + 2*x)^6) - (9*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqr
t[3]*Sqrt[2 + 5*x + 3*x^2])])/128 + (13931*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(51200*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 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 810

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

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)^7} \, dx &=\frac {(109+116 x) \left (2+5 x+3 x^2\right )^{5/2}}{120 (3+2 x)^6}-\frac {1}{240} \int \frac {(215+180 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx\\ &=\frac {(437+328 x) \left (2+5 x+3 x^2\right )^{3/2}}{1920 (3+2 x)^4}+\frac {(109+116 x) \left (2+5 x+3 x^2\right )^{5/2}}{120 (3+2 x)^6}+\frac {\int \frac {(-18330-21600 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{38400}\\ &=\frac {(14083+10952 x) \sqrt {2+5 x+3 x^2}}{25600 (3+2 x)^2}+\frac {(437+328 x) \left (2+5 x+3 x^2\right )^{3/2}}{1920 (3+2 x)^4}+\frac {(109+116 x) \left (2+5 x+3 x^2\right )^{5/2}}{120 (3+2 x)^6}-\frac {\int \frac {1108140+1296000 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{3072000}\\ &=\frac {(14083+10952 x) \sqrt {2+5 x+3 x^2}}{25600 (3+2 x)^2}+\frac {(437+328 x) \left (2+5 x+3 x^2\right )^{3/2}}{1920 (3+2 x)^4}+\frac {(109+116 x) \left (2+5 x+3 x^2\right )^{5/2}}{120 (3+2 x)^6}-\frac {27}{128} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx+\frac {13931 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{51200}\\ &=\frac {(14083+10952 x) \sqrt {2+5 x+3 x^2}}{25600 (3+2 x)^2}+\frac {(437+328 x) \left (2+5 x+3 x^2\right )^{3/2}}{1920 (3+2 x)^4}+\frac {(109+116 x) \left (2+5 x+3 x^2\right )^{5/2}}{120 (3+2 x)^6}-\frac {27}{64} \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )-\frac {13931 \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{25600}\\ &=\frac {(14083+10952 x) \sqrt {2+5 x+3 x^2}}{25600 (3+2 x)^2}+\frac {(437+328 x) \left (2+5 x+3 x^2\right )^{3/2}}{1920 (3+2 x)^4}+\frac {(109+116 x) \left (2+5 x+3 x^2\right )^{5/2}}{120 (3+2 x)^6}-\frac {9}{128} \sqrt {3} \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )+\frac {13931 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{51200 \sqrt {5}}\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 120, normalized size = 0.72 \[ \frac {-41793 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )-54000 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )+\frac {10 \sqrt {3 x^2+5 x+2} \left (1351296 x^5+7629680 x^4+18217760 x^3+22854480 x^2+14921560 x+4015849\right )}{(2 x+3)^6}}{768000} \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^7,x]

[Out]

((10*Sqrt[2 + 5*x + 3*x^2]*(4015849 + 14921560*x + 22854480*x^2 + 18217760*x^3 + 7629680*x^4 + 1351296*x^5))/(
3 + 2*x)^6 - 41793*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])] - 54000*Sqrt[3]*ArcTanh[(5 +
6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/768000

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fricas [A]  time = 0.78, size = 223, normalized size = 1.34 \[ \frac {54000 \, \sqrt {3} {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 41793 \, \sqrt {5} {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 20 \, {\left (1351296 \, x^{5} + 7629680 \, x^{4} + 18217760 \, x^{3} + 22854480 \, x^{2} + 14921560 \, x + 4015849\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{1536000 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} \]

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)^7,x, algorithm="fricas")

[Out]

1/1536000*(54000*sqrt(3)*(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729)*log(-4*sqrt(3)*sqr
t(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) + 41793*sqrt(5)*(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 +
4860*x^2 + 2916*x + 729)*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*(1351296*x^5 + 7629680*x^4 + 18217760*x^3 + 22854480*x^2 + 14921560*x + 4015849)*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)

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giac [B]  time = 0.37, size = 444, normalized size = 2.66 \[ \frac {13931}{256000} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac {9}{128} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) + \frac {20435424 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 269619696 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 4893810640 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 17834042400 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 129909086880 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 219870810528 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 791797675536 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 672745449240 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 1187868124850 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 460902113505 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 318216938187 \, \sqrt {3} x + 32907940848 \, \sqrt {3} - 318216938187 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{76800 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{6}} \]

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)^7,x, algorithm="giac")

[Out]

13931/256000*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))) + 9/128*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2
+ 5*x + 2)) - 5)) + 1/76800*(20435424*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^11 + 269619696*sqrt(3)*(sqrt(3)*x -
sqrt(3*x^2 + 5*x + 2))^10 + 4893810640*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 + 17834042400*sqrt(3)*(sqrt(3)*x
- sqrt(3*x^2 + 5*x + 2))^8 + 129909086880*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 + 219870810528*sqrt(3)*(sqrt(
3)*x - sqrt(3*x^2 + 5*x + 2))^6 + 791797675536*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 672745449240*sqrt(3)*(s
qrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 1187868124850*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 460902113505*sqrt(
3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 318216938187*sqrt(3)*x + 32907940848*sqrt(3) - 318216938187*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)
^6

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maple [B]  time = 0.07, size = 300, normalized size = 1.80 \[ -\frac {13931 \sqrt {5}\, \arctanh \left (\frac {2 \left (-4 x -\frac {7}{2}\right ) \sqrt {5}}{5 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{256000}-\frac {9 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\right )}{128}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{1920 \left (x +\frac {3}{2}\right )^{6}}-\frac {709 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{30000 \left (x +\frac {3}{2}\right )^{3}}-\frac {23 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{2400 \left (x +\frac {3}{2}\right )^{5}}-\frac {22271 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{600000 \left (x +\frac {3}{2}\right )^{2}}+\frac {6089 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{250000}-\frac {1001 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{120000}-\frac {6089 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{125000 \left (x +\frac {3}{2}\right )}-\frac {431 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{32000}+\frac {13931 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{256000}-\frac {249 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{16000 \left (x +\frac {3}{2}\right )^{4}}+\frac {13931 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{480000}+\frac {13931 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{1000000} \]

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)^7,x)

[Out]

-13/1920/(x+3/2)^6*(-4*x+3*(x+3/2)^2-19/4)^(7/2)-709/30000/(x+3/2)^3*(-4*x+3*(x+3/2)^2-19/4)^(7/2)-23/2400/(x+
3/2)^5*(-4*x+3*(x+3/2)^2-19/4)^(7/2)-22271/600000/(x+3/2)^2*(-4*x+3*(x+3/2)^2-19/4)^(7/2)+6089/250000*(6*x+5)*
(-4*x+3*(x+3/2)^2-19/4)^(5/2)-1001/120000*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(3/2)-6089/125000/(x+3/2)*(-4*x+3*(x
+3/2)^2-19/4)^(7/2)-431/32000*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(1/2)-13931/256000*5^(1/2)*arctanh(2/5*(-4*x-7/2
)*5^(1/2)/(-16*x+12*(x+3/2)^2-19)^(1/2))-9/128*3^(1/2)*ln(1/3*(3*x+5/2)*3^(1/2)+(-4*x+3*(x+3/2)^2-19/4)^(1/2))
+13931/256000*(-16*x+12*(x+3/2)^2-19)^(1/2)-249/16000/(x+3/2)^4*(-4*x+3*(x+3/2)^2-19/4)^(7/2)+13931/480000*(-4
*x+3*(x+3/2)^2-19/4)^(3/2)+13931/1000000*(-4*x+3*(x+3/2)^2-19/4)^(5/2)

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maxima [B]  time = 1.32, size = 343, normalized size = 2.05 \[ \frac {22271}{200000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{30 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {23 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{75 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {249 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{1000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {709 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{3750 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {22271 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{150000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {1001}{20000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {6089}{480000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {6089 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{50000 \, {\left (2 \, x + 3\right )}} - \frac {1293}{16000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {9}{128} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) - \frac {13931}{256000} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) + \frac {5311}{128000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]

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)^7,x, algorithm="maxima")

[Out]

22271/200000*(3*x^2 + 5*x + 2)^(5/2) - 13/30*(3*x^2 + 5*x + 2)^(7/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 +
4860*x^2 + 2916*x + 729) - 23/75*(3*x^2 + 5*x + 2)^(7/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243
) - 249/1000*(3*x^2 + 5*x + 2)^(7/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 709/3750*(3*x^2 + 5*x + 2)^(7/
2)/(8*x^3 + 36*x^2 + 54*x + 27) - 22271/150000*(3*x^2 + 5*x + 2)^(7/2)/(4*x^2 + 12*x + 9) - 1001/20000*(3*x^2
+ 5*x + 2)^(3/2)*x - 6089/480000*(3*x^2 + 5*x + 2)^(3/2) - 6089/50000*(3*x^2 + 5*x + 2)^(5/2)/(2*x + 3) - 1293
/16000*sqrt(3*x^2 + 5*x + 2)*x - 9/128*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) - 13931/256000*s
qrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 5311/128000*sqrt(3*x^2 + 5*x +
2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2}}{{\left (2\,x+3\right )}^7} \,d x \]

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)^7,x)

[Out]

-int(((x - 5)*(5*x + 3*x^2 + 2)^(5/2))/(2*x + 3)^7, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- \frac {20 \sqrt {3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\right )\, dx - \int \left (- \frac {96 x \sqrt {3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\right )\, dx - \int \left (- \frac {165 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\right )\, dx - \int \left (- \frac {113 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\right )\, dx - \int \left (- \frac {15 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\right )\, dx - \int \frac {9 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\, dx \]

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)**7,x)

[Out]

-Integral(-20*sqrt(3*x**2 + 5*x + 2)/(128*x**7 + 1344*x**6 + 6048*x**5 + 15120*x**4 + 22680*x**3 + 20412*x**2
+ 10206*x + 2187), x) - Integral(-96*x*sqrt(3*x**2 + 5*x + 2)/(128*x**7 + 1344*x**6 + 6048*x**5 + 15120*x**4 +
22680*x**3 + 20412*x**2 + 10206*x + 2187), x) - Integral(-165*x**2*sqrt(3*x**2 + 5*x + 2)/(128*x**7 + 1344*x*
*6 + 6048*x**5 + 15120*x**4 + 22680*x**3 + 20412*x**2 + 10206*x + 2187), x) - Integral(-113*x**3*sqrt(3*x**2 +
5*x + 2)/(128*x**7 + 1344*x**6 + 6048*x**5 + 15120*x**4 + 22680*x**3 + 20412*x**2 + 10206*x + 2187), x) - Int
egral(-15*x**4*sqrt(3*x**2 + 5*x + 2)/(128*x**7 + 1344*x**6 + 6048*x**5 + 15120*x**4 + 22680*x**3 + 20412*x**2
+ 10206*x + 2187), x) - Integral(9*x**5*sqrt(3*x**2 + 5*x + 2)/(128*x**7 + 1344*x**6 + 6048*x**5 + 15120*x**4
+ 22680*x**3 + 20412*x**2 + 10206*x + 2187), x)

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