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

Optimal. Leaf size=197 \[ -\frac {(3 x+11) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^6}-\frac {7 (1046 x+1301) \left (3 x^2+5 x+2\right )^{5/2}}{1920 (2 x+3)^5}-\frac {7 (31174 x+40201) \left (3 x^2+5 x+2\right )^{3/2}}{25600 (2 x+3)^3}+\frac {63 (20678 x+44365) \sqrt {3 x^2+5 x+2}}{102400 (2 x+3)}-\frac {8547 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{1024}+\frac {6620481 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{204800 \sqrt {5}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.13, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {812, 810, 843, 621, 206, 724} \begin {gather*} -\frac {(3 x+11) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^6}-\frac {7 (1046 x+1301) \left (3 x^2+5 x+2\right )^{5/2}}{1920 (2 x+3)^5}-\frac {7 (31174 x+40201) \left (3 x^2+5 x+2\right )^{3/2}}{25600 (2 x+3)^3}+\frac {63 (20678 x+44365) \sqrt {3 x^2+5 x+2}}{102400 (2 x+3)}-\frac {8547 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{1024}+\frac {6620481 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{204800 \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(63*(44365 + 20678*x)*Sqrt[2 + 5*x + 3*x^2])/(102400*(3 + 2*x)) - (7*(40201 + 31174*x)*(2 + 5*x + 3*x^2)^(3/2)
)/(25600*(3 + 2*x)^3) - (7*(1301 + 1046*x)*(2 + 5*x + 3*x^2)^(5/2))/(1920*(3 + 2*x)^5) - ((11 + 3*x)*(2 + 5*x
+ 3*x^2)^(7/2))/(12*(3 + 2*x)^6) - (8547*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/1024 +
(6620481*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(204800*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 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 )^{7/2}}{(3+2 x)^7} \, dx &=-\frac {(11+3 x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^6}-\frac {7}{96} \int \frac {(-172-204 x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^6} \, dx\\ &=-\frac {7 (1301+1046 x) \left (2+5 x+3 x^2\right )^{5/2}}{1920 (3+2 x)^5}-\frac {(11+3 x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^6}+\frac {7 \int \frac {(31932+37032 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx}{15360}\\ &=-\frac {7 (40201+31174 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600 (3+2 x)^3}-\frac {7 (1301+1046 x) \left (2+5 x+3 x^2\right )^{5/2}}{1920 (3+2 x)^5}-\frac {(11+3 x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^6}-\frac {7 \int \frac {(-3816504-4466448 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^2} \, dx}{1228800}\\ &=\frac {63 (44365+20678 x) \sqrt {2+5 x+3 x^2}}{102400 (3+2 x)}-\frac {7 (40201+31174 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600 (3+2 x)^3}-\frac {7 (1301+1046 x) \left (2+5 x+3 x^2\right )^{5/2}}{1920 (3+2 x)^5}-\frac {(11+3 x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^6}+\frac {7 \int \frac {-60096816-70329600 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{9830400}\\ &=\frac {63 (44365+20678 x) \sqrt {2+5 x+3 x^2}}{102400 (3+2 x)}-\frac {7 (40201+31174 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600 (3+2 x)^3}-\frac {7 (1301+1046 x) \left (2+5 x+3 x^2\right )^{5/2}}{1920 (3+2 x)^5}-\frac {(11+3 x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^6}-\frac {25641 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{1024}+\frac {6620481 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{204800}\\ &=\frac {63 (44365+20678 x) \sqrt {2+5 x+3 x^2}}{102400 (3+2 x)}-\frac {7 (40201+31174 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600 (3+2 x)^3}-\frac {7 (1301+1046 x) \left (2+5 x+3 x^2\right )^{5/2}}{1920 (3+2 x)^5}-\frac {(11+3 x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^6}-\frac {25641}{512} \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )-\frac {6620481 \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{102400}\\ &=\frac {63 (44365+20678 x) \sqrt {2+5 x+3 x^2}}{102400 (3+2 x)}-\frac {7 (40201+31174 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600 (3+2 x)^3}-\frac {7 (1301+1046 x) \left (2+5 x+3 x^2\right )^{5/2}}{1920 (3+2 x)^5}-\frac {(11+3 x) \left (2+5 x+3 x^2\right )^{7/2}}{12 (3+2 x)^6}-\frac {8547 \sqrt {3} \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{1024}+\frac {6620481 \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]  time = 0.17, size = 130, normalized size = 0.66 \begin {gather*} \frac {-19861443 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )-25641000 \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 (2073600 x^7-23155200 x^6-550079616 x^5-2968126160 x^4-7425343520 x^3-9799959120 x^2-6648875480 x-1835461379\right )}{(2 x+3)^6}}{3072000} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-10*Sqrt[2 + 5*x + 3*x^2]*(-1835461379 - 6648875480*x - 9799959120*x^2 - 7425343520*x^3 - 2968126160*x^4 - 5
50079616*x^5 - 23155200*x^6 + 2073600*x^7))/(3 + 2*x)^6 - 19861443*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[
2 + 5*x + 3*x^2])] - 25641000*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/3072000

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 1.01, size = 131, normalized size = 0.66 \begin {gather*} -\frac {8547}{512} \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )+\frac {6620481 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{102400 \sqrt {5}}+\frac {\sqrt {3 x^2+5 x+2} \left (-2073600 x^7+23155200 x^6+550079616 x^5+2968126160 x^4+7425343520 x^3+9799959120 x^2+6648875480 x+1835461379\right )}{307200 (2 x+3)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + 5*x + 3*x^2]*(1835461379 + 6648875480*x + 9799959120*x^2 + 7425343520*x^3 + 2968126160*x^4 + 5500796
16*x^5 + 23155200*x^6 - 2073600*x^7))/(307200*(3 + 2*x)^6) - (8547*Sqrt[3]*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt
[3]*(1 + x))])/512 + (6620481*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[5]*(1 + x))])/(102400*Sqrt[5])

________________________________________________________________________________________

fricas [A]  time = 0.45, size = 233, normalized size = 1.18 \begin {gather*} \frac {25641000 \, \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 ) + 19861443 \, \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 (2073600 \, x^{7} - 23155200 \, x^{6} - 550079616 \, x^{5} - 2968126160 \, x^{4} - 7425343520 \, x^{3} - 9799959120 \, x^{2} - 6648875480 \, x - 1835461379\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{6144000 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6144000*(25641000*sqrt(3)*(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729)*log(-4*sqrt(3)*
sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) + 19861443*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*(2073600*x^7 - 23155200*x^6 - 550079616*x^5 - 2968126160*x^4 - 7425343520*x^3 - 9799959120*x^
2 - 6648875480*x - 1835461379)*sqrt(3*x^2 + 5*x + 2))/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 291
6*x + 729)

________________________________________________________________________________________

giac [B]  time = 0.39, size = 467, normalized size = 2.37 \begin {gather*} -\frac {9}{512} \, \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x - 121\right )} + \frac {6620481}{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 {8547}{1024} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) + \frac {\sqrt {3} {\left (1761054624 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 78359519088 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 522182992240 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 6180007168800 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 16013156565600 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 85756996584864 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 107556795368496 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 284279833881720 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 172447244925750 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 205883289380025 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 48408731804817 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 15295619190024\right )}}{921600 \, {\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/512*sqrt(3*x^2 + 5*x + 2)*(6*x - 121) + 6620481/1024000*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))) + 8547/1024*
sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) + 1/921600*sqrt(3)*(1761054624*sqrt(3)*(s
qrt(3)*x - sqrt(3*x^2 + 5*x + 2))^11 + 78359519088*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 + 522182992240*sqrt(
3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 + 6180007168800*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 + 1601315656560
0*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 + 85756996584864*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 + 10755
6795368496*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 284279833881720*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))
^4 + 172447244925750*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 205883289380025*(sqrt(3)*x - sqrt(3*x^2 +
5*x + 2))^2 + 48408731804817*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) + 15295619190024)/(2*(sqrt(3)*x - sq
rt(3*x^2 + 5*x + 2))^2 + 6*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) + 11)^6

________________________________________________________________________________________

maple [B]  time = 0.07, size = 337, normalized size = 1.71 \begin {gather*} -\frac {6620481 \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 )}{1024000}-\frac {8547 \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 )}{1024}-\frac {1143 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{80000 \left (x +\frac {3}{2}\right )^{4}}-\frac {63693 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{1000000 \left (x +\frac {3}{2}\right )^{2}}-\frac {459 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{50000 \left (x +\frac {3}{2}\right )^{3}}-\frac {47169 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{250000}-\frac {349461 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{1000000}+\frac {47169 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{125000 \left (x +\frac {3}{2}\right )}-\frac {104517 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{160000}-\frac {210231 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{128000}+\frac {6620481 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{1024000}+\frac {2206827 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{640000}+\frac {6620481 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{4000000}+\frac {945783 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{1000000}-\frac {21 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{4000 \left (x +\frac {3}{2}\right )^{5}}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{1920 \left (x +\frac {3}{2}\right )^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1143/80000/(x+3/2)^4*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-63693/1000000/(x+3/2)^2*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-459/
50000/(x+3/2)^3*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-47169/250000*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(7/2)-349461/100000
0*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(5/2)+47169/125000/(x+3/2)*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-104517/160000*(6*x+
5)*(-4*x+3*(x+3/2)^2-19/4)^(3/2)-210231/128000*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(1/2)-6620481/1024000*5^(1/2)*a
rctanh(2/5*(-4*x-7/2)*5^(1/2)/(-16*x+12*(x+3/2)^2-19)^(1/2))-8547/1024*3^(1/2)*ln(1/3*(3*x+5/2)*3^(1/2)+(-4*x+
3*(x+3/2)^2-19/4)^(1/2))+6620481/1024000*(-16*x+12*(x+3/2)^2-19)^(1/2)+2206827/640000*(-4*x+3*(x+3/2)^2-19/4)^
(3/2)+6620481/4000000*(-4*x+3*(x+3/2)^2-19/4)^(5/2)+945783/1000000*(-4*x+3*(x+3/2)^2-19/4)^(7/2)-21/4000/(x+3/
2)^5*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-13/1920/(x+3/2)^6*(-4*x+3*(x+3/2)^2-19/4)^(9/2)

________________________________________________________________________________________

maxima [B]  time = 1.35, size = 372, normalized size = 1.89 \begin {gather*} \frac {191079}{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}}}{30 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {21 \, {\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 {1143 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{5000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {459 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{6250 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {63693 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{250000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {1048383}{500000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {368739}{4000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {47169 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{50000 \, {\left (2 \, x + 3\right )}} - \frac {313551}{80000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {116487}{640000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {630693}{64000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {8547}{1024} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) - \frac {6620481}{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 {2415861}{512000} \, \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)^(7/2)/(3+2*x)^7,x, algorithm="maxima")

[Out]

191079/1000000*(3*x^2 + 5*x + 2)^(7/2) - 13/30*(3*x^2 + 5*x + 2)^(9/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3
+ 4860*x^2 + 2916*x + 729) - 21/125*(3*x^2 + 5*x + 2)^(9/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x +
243) - 1143/5000*(3*x^2 + 5*x + 2)^(9/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 459/6250*(3*x^2 + 5*x + 2)
^(9/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 63693/250000*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) - 1048383/500000
*(3*x^2 + 5*x + 2)^(5/2)*x - 368739/4000000*(3*x^2 + 5*x + 2)^(5/2) + 47169/50000*(3*x^2 + 5*x + 2)^(7/2)/(2*x
+ 3) - 313551/80000*(3*x^2 + 5*x + 2)^(3/2)*x + 116487/640000*(3*x^2 + 5*x + 2)^(3/2) - 630693/64000*sqrt(3*x
^2 + 5*x + 2)*x - 8547/1024*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) - 6620481/1024000*sqrt(5)*l
og(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 2415861/512000*sqrt(3*x^2 + 5*x + 2)

________________________________________________________________________________________

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 )}^{7/2}}{{\left (2\,x+3\right )}^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {40 \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 {292 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 {870 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 {1339 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 {1090 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 \left (- \frac {396 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}\right )\, dx - \int \frac {27 x^{7} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-40*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(-292*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(-870*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(-1339*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) - I
ntegral(-1090*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(-396*x**5*sqrt(3*x**2 + 5*x + 2)/(128*x**7 + 1344*x**6 + 6048*x**5 + 151
20*x**4 + 22680*x**3 + 20412*x**2 + 10206*x + 2187), x) - Integral(27*x**7*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)

________________________________________________________________________________________