\(\int \frac {(5-x) (2+5 x+3 x^2)^{7/2}}{(3+2 x)^8} \, dx\) [2459]

Optimal result

Integrand size = 27, antiderivative size = 197 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^8} \, dx=-\frac {3 (131465+61278 x) \sqrt {2+5 x+3 x^2}}{102400 (3+2 x)}+\frac {(39767+30858 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600 (3+2 x)^3}+\frac {3 (135+106 x) \left (2+5 x+3 x^2\right )^{5/2}}{640 (3+2 x)^5}+\frac {(269+266 x) \left (2+5 x+3 x^2\right )^{7/2}}{280 (3+2 x)^7}+\frac {603}{512} \sqrt {3} \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )-\frac {934161 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{204800 \sqrt {5}} \]

[Out]

1/25600*(39767+30858*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^3+3/640*(135+106*x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^5+1/280*(2
69+266*x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^7+603/512*arctanh(1/6*(5+6*x)*3^(1/2)/(3*x^2+5*x+2)^(1/2))*3^(1/2)-93416
1/1024000*arctanh(1/10*(7+8*x)*5^(1/2)/(3*x^2+5*x+2)^(1/2))*5^(1/2)-3/102400*(131465+61278*x)*(3*x^2+5*x+2)^(1
/2)/(3+2*x)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {824, 826, 857, 635, 212, 738} \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^8} \, dx=\frac {603}{512} \sqrt {3} \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )-\frac {934161 \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{204800 \sqrt {5}}+\frac {(266 x+269) \left (3 x^2+5 x+2\right )^{7/2}}{280 (2 x+3)^7}+\frac {3 (106 x+135) \left (3 x^2+5 x+2\right )^{5/2}}{640 (2 x+3)^5}+\frac {(30858 x+39767) \left (3 x^2+5 x+2\right )^{3/2}}{25600 (2 x+3)^3}-\frac {3 (61278 x+131465) \sqrt {3 x^2+5 x+2}}{102400 (2 x+3)} \]

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^8,x]

[Out]

(-3*(131465 + 61278*x)*Sqrt[2 + 5*x + 3*x^2])/(102400*(3 + 2*x)) + ((39767 + 30858*x)*(2 + 5*x + 3*x^2)^(3/2))
/(25600*(3 + 2*x)^3) + (3*(135 + 106*x)*(2 + 5*x + 3*x^2)^(5/2))/(640*(3 + 2*x)^5) + ((269 + 266*x)*(2 + 5*x +
3*x^2)^(7/2))/(280*(3 + 2*x)^7) + (603*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/512 - (9
34161*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(204800*Sqrt[5])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 635

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 738

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 824

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((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), 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 826

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 857

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*} \text {integral}& = \frac {(269+266 x) \left (2+5 x+3 x^2\right )^{7/2}}{280 (3+2 x)^7}-\frac {1}{240} \int \frac {(513+522 x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^6} \, dx \\ & = \frac {3 (135+106 x) \left (2+5 x+3 x^2\right )^{5/2}}{640 (3+2 x)^5}+\frac {(269+266 x) \left (2+5 x+3 x^2\right )^{7/2}}{280 (3+2 x)^7}+\frac {\int \frac {(-78210-91260 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx}{38400} \\ & = \frac {(39767+30858 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600 (3+2 x)^3}+\frac {3 (135+106 x) \left (2+5 x+3 x^2\right )^{5/2}}{640 (3+2 x)^5}+\frac {(269+266 x) \left (2+5 x+3 x^2\right )^{7/2}}{280 (3+2 x)^7}-\frac {\int \frac {(9426420+11030040 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^2} \, dx}{3072000} \\ & = -\frac {3 (131465+61278 x) \sqrt {2+5 x+3 x^2}}{102400 (3+2 x)}+\frac {(39767+30858 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600 (3+2 x)^3}+\frac {3 (135+106 x) \left (2+5 x+3 x^2\right )^{5/2}}{640 (3+2 x)^5}+\frac {(269+266 x) \left (2+5 x+3 x^2\right )^{7/2}}{280 (3+2 x)^7}+\frac {\int \frac {148396680+173664000 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{24576000} \\ & = -\frac {3 (131465+61278 x) \sqrt {2+5 x+3 x^2}}{102400 (3+2 x)}+\frac {(39767+30858 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600 (3+2 x)^3}+\frac {3 (135+106 x) \left (2+5 x+3 x^2\right )^{5/2}}{640 (3+2 x)^5}+\frac {(269+266 x) \left (2+5 x+3 x^2\right )^{7/2}}{280 (3+2 x)^7}+\frac {1809}{512} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx-\frac {934161 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{204800} \\ & = -\frac {3 (131465+61278 x) \sqrt {2+5 x+3 x^2}}{102400 (3+2 x)}+\frac {(39767+30858 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600 (3+2 x)^3}+\frac {3 (135+106 x) \left (2+5 x+3 x^2\right )^{5/2}}{640 (3+2 x)^5}+\frac {(269+266 x) \left (2+5 x+3 x^2\right )^{7/2}}{280 (3+2 x)^7}+\frac {1809}{256} \text {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )+\frac {934161 \text {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{102400} \\ & = -\frac {3 (131465+61278 x) \sqrt {2+5 x+3 x^2}}{102400 (3+2 x)}+\frac {(39767+30858 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600 (3+2 x)^3}+\frac {3 (135+106 x) \left (2+5 x+3 x^2\right )^{5/2}}{640 (3+2 x)^5}+\frac {(269+266 x) \left (2+5 x+3 x^2\right )^{7/2}}{280 (3+2 x)^7}+\frac {603}{512} \sqrt {3} \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )-\frac {934161 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{204800 \sqrt {5}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.62 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^8} \, dx=\frac {-\frac {5 \sqrt {2+5 x+3 x^2} \left (1810375853+7753535702 x+13975079520 x^2+13619671040 x^3+7622049520 x^4+2361590432 x^5+338443008 x^6+9676800 x^7\right )}{(3+2 x)^7}-6539127 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )+8442000 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{3584000} \]

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^8,x]

[Out]

((-5*Sqrt[2 + 5*x + 3*x^2]*(1810375853 + 7753535702*x + 13975079520*x^2 + 13619671040*x^3 + 7622049520*x^4 + 2
361590432*x^5 + 338443008*x^6 + 9676800*x^7))/(3 + 2*x)^7 - 6539127*Sqrt[5]*ArcTanh[Sqrt[2/5 + x + (3*x^2)/5]/
(1 + x)] + 8442000*Sqrt[3]*ArcTanh[Sqrt[2/3 + (5*x)/3 + x^2]/(1 + x)])/3584000

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.64

method	result	size
risch	\(-\frac {29030400 x^{9}[phone] x^{8}[phone] x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]}{716800 \left (3+2 x \right )^{7} \sqrt {3 x^{2}+5 x +2}}+\frac {603 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{512}+\frac {934161 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{1024000}\)	\(127\)
trager	\(-\frac {\left (9676800 x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {3 x^{2}+5 x +2}}{716800 \left (3+2 x \right )^{7}}+\frac {603 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{512}-\frac {934161 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{1024000}\)	\(148\)
default	\(-\frac {3 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{896 \left (x +\frac {3}{2}\right )^{6}}-\frac {3 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{500 \left (x +\frac {3}{2}\right )^{5}}-\frac {4719 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{560000 \left (x +\frac {3}{2}\right )^{4}}-\frac {5147 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{350000 \left (x +\frac {3}{2}\right )^{3}}-\frac {15267 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{1000000 \left (x +\frac {3}{2}\right )^{2}}+\frac {78423 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{1750000}-\frac {78423 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{875000 \left (x +\frac {3}{2}\right )}+\frac {47541 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{1000000}+\frac {14777 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{160000}+\frac {29661 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{128000}+\frac {603 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}\right ) \sqrt {3}}{512}+\frac {934161 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{1024000}-\frac {934161 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{4000000}-\frac {311387 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{640000}-\frac {934161 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{7000000}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{4480 \left (x +\frac {3}{2}\right )^{7}}-\frac {934161 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{1024000}\)	\(358\)

[In]

int((5-x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^8,x,method=_RETURNVERBOSE)

[Out]

-1/716800*(29030400*x^9+1063713024*x^8+8796339936*x^7+35350986736*x^6+83692441584*x^5+125267692800*x^4+1203753
46786*x^3+72148965109*x^2+24558950669*x+3620751706)/(3+2*x)^7/(3*x^2+5*x+2)^(1/2)+603/512*ln(1/3*(5/2+3*x)*3^(
1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)+934161/1024000*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)
^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.26 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^8} \, dx=\frac {8442000 \, \sqrt {3} {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 6539127 \, \sqrt {5} {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )} \log \left (-\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} - 124 \, x^{2} - 212 \, x - 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 20 \, {\left (9676800 \, x^{7} + 338443008 \, x^{6} + 2361590432 \, x^{5} + 7622049520 \, x^{4} + 13619671040 \, x^{3} + 13975079520 \, x^{2} + 7753535702 \, x + 1810375853\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{14336000 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} \]

[In]

integrate((5-x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^8,x, algorithm="fricas")

[Out]

1/14336000*(8442000*sqrt(3)*(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 218
7)*log(4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) + 6539127*sqrt(5)*(128*x^7 + 1344*x^6
+ 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187)*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*(9676800*x^7 + 338443008*x^6 + 2361590432*x^5 + 7622049520
*x^4 + 13619671040*x^3 + 13975079520*x^2 + 7753535702*x + 1810375853)*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)

Sympy [F]

\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^8} \, dx=- \int \left (- \frac {40 \sqrt {3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\right )\, dx - \int \left (- \frac {292 x \sqrt {3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\right )\, dx - \int \left (- \frac {870 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\right )\, dx - \int \left (- \frac {1339 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\right )\, dx - \int \left (- \frac {1090 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\right )\, dx - \int \left (- \frac {396 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\right )\, dx - \int \frac {27 x^{7} \sqrt {3 x^{2} + 5 x + 2}}{256 x^{8} + 3072 x^{7} + 16128 x^{6} + 48384 x^{5} + 90720 x^{4} + 108864 x^{3} + 81648 x^{2} + 34992 x + 6561}\, dx \]

[In]

integrate((5-x)*(3*x**2+5*x+2)**(7/2)/(3+2*x)**8,x)

[Out]

-Integral(-40*sqrt(3*x**2 + 5*x + 2)/(256*x**8 + 3072*x**7 + 16128*x**6 + 48384*x**5 + 90720*x**4 + 108864*x**
3 + 81648*x**2 + 34992*x + 6561), x) - Integral(-292*x*sqrt(3*x**2 + 5*x + 2)/(256*x**8 + 3072*x**7 + 16128*x*
*6 + 48384*x**5 + 90720*x**4 + 108864*x**3 + 81648*x**2 + 34992*x + 6561), x) - Integral(-870*x**2*sqrt(3*x**2
+ 5*x + 2)/(256*x**8 + 3072*x**7 + 16128*x**6 + 48384*x**5 + 90720*x**4 + 108864*x**3 + 81648*x**2 + 34992*x
+ 6561), x) - Integral(-1339*x**3*sqrt(3*x**2 + 5*x + 2)/(256*x**8 + 3072*x**7 + 16128*x**6 + 48384*x**5 + 907
20*x**4 + 108864*x**3 + 81648*x**2 + 34992*x + 6561), x) - Integral(-1090*x**4*sqrt(3*x**2 + 5*x + 2)/(256*x**
8 + 3072*x**7 + 16128*x**6 + 48384*x**5 + 90720*x**4 + 108864*x**3 + 81648*x**2 + 34992*x + 6561), x) - Integr
al(-396*x**5*sqrt(3*x**2 + 5*x + 2)/(256*x**8 + 3072*x**7 + 16128*x**6 + 48384*x**5 + 90720*x**4 + 108864*x**3
+ 81648*x**2 + 34992*x + 6561), x) - Integral(27*x**7*sqrt(3*x**2 + 5*x + 2)/(256*x**8 + 3072*x**7 + 16128*x*
*6 + 48384*x**5 + 90720*x**4 + 108864*x**3 + 81648*x**2 + 34992*x + 6561), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 423 vs. \(2 (161) = 322\).

Time = 0.28 (sec) , antiderivative size = 423, normalized size of antiderivative = 2.15 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^8} \, dx=\frac {45801}{1000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{35 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {3 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{14 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {24 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{125 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {4719 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{35000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {5147 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{43750 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {15267 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{250000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {142623}{500000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {16659}{4000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {78423 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{350000 \, {\left (2 \, x + 3\right )}} + \frac {44331}{80000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {15847}{640000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {88983}{64000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {603}{512} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) + \frac {934161}{1024000} \, \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 {340941}{512000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]

[In]

integrate((5-x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^8,x, algorithm="maxima")

[Out]

45801/1000000*(3*x^2 + 5*x + 2)^(7/2) - 13/35*(3*x^2 + 5*x + 2)^(9/2)/(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x
^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187) - 3/14*(3*x^2 + 5*x + 2)^(9/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 432
0*x^3 + 4860*x^2 + 2916*x + 729) - 24/125*(3*x^2 + 5*x + 2)^(9/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810
*x + 243) - 4719/35000*(3*x^2 + 5*x + 2)^(9/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 5147/43750*(3*x^2 +
5*x + 2)^(9/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 15267/250000*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) + 142623
/500000*(3*x^2 + 5*x + 2)^(5/2)*x + 16659/4000000*(3*x^2 + 5*x + 2)^(5/2) - 78423/350000*(3*x^2 + 5*x + 2)^(7/
2)/(2*x + 3) + 44331/80000*(3*x^2 + 5*x + 2)^(3/2)*x - 15847/640000*(3*x^2 + 5*x + 2)^(3/2) + 88983/64000*sqrt
(3*x^2 + 5*x + 2)*x + 603/512*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) + 934161/1024000*sqrt(5)*
log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 340941/512000*sqrt(3*x^2 + 5*x + 2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 509 vs. \(2 (161) = 322\).

Time = 0.36 (sec) , antiderivative size = 509, normalized size of antiderivative = 2.58 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^8} \, dx=-\frac {934161}{1024000} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac {603}{512} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac {27}{256} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {2310353472 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{13} + 39459777504 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{12} + 930047331808 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 4439192854544 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 42996771835920 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 98991221694624 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 500967391220544 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 626374342937616 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 1740466332835804 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 1179088946690970 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 1703610278292706 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 552456024942507 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 324453464706399 \, \sqrt {3} x + 28970271150072 \, \sqrt {3} - 324453464706399 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{716800 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{7}} \]

[In]

integrate((5-x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^8,x, algorithm="giac")

[Out]

-934161/1024000*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))) - 603/512*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(
3*x^2 + 5*x + 2)) - 5)) - 27/256*sqrt(3*x^2 + 5*x + 2) - 1/716800*(2310353472*(sqrt(3)*x - sqrt(3*x^2 + 5*x +
2))^13 + 39459777504*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^12 + 930047331808*(sqrt(3)*x - sqrt(3*x^2 + 5
*x + 2))^11 + 4439192854544*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 + 42996771835920*(sqrt(3)*x - sqrt(
3*x^2 + 5*x + 2))^9 + 98991221694624*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 + 500967391220544*(sqrt(3)*
x - sqrt(3*x^2 + 5*x + 2))^7 + 626374342937616*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 + 174046633283580
4*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 1179088946690970*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 170
3610278292706*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 552456024942507*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x +
2))^2 + 324453464706399*sqrt(3)*x + 28970271150072*sqrt(3) - 324453464706399*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)^7

Mupad [F(-1)]

Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^8} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{7/2}}{{\left (2\,x+3\right )}^8} \,d x \]

[In]

int(-((x - 5)*(5*x + 3*x^2 + 2)^(7/2))/(2*x + 3)^8,x)

[Out]

-int(((x - 5)*(5*x + 3*x^2 + 2)^(7/2))/(2*x + 3)^8, x)