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

Optimal. Leaf size=174 \[ -\frac {(x+47) \left (3 x^2+5 x+2\right )^{7/2}}{14 (2 x+3)}+\frac {(8310 x+283) \left (3 x^2+5 x+2\right )^{5/2}}{1440}-\frac {(6925-151098 x) \left (3 x^2+5 x+2\right )^{3/2}}{13824}-\frac {(1454315-3037062 x) \sqrt {3 x^2+5 x+2}}{110592}+\frac {15434623 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{221184 \sqrt {3}}-\frac {9225}{512} \sqrt {5} \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right ) \]

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Rubi [A]  time = 0.12, antiderivative size = 174, 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, 814, 843, 621, 206, 724} \begin {gather*} -\frac {(x+47) \left (3 x^2+5 x+2\right )^{7/2}}{14 (2 x+3)}+\frac {(8310 x+283) \left (3 x^2+5 x+2\right )^{5/2}}{1440}-\frac {(6925-151098 x) \left (3 x^2+5 x+2\right )^{3/2}}{13824}-\frac {(1454315-3037062 x) \sqrt {3 x^2+5 x+2}}{110592}+\frac {15434623 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{221184 \sqrt {3}}-\frac {9225}{512} \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)^(7/2))/(3 + 2*x)^2,x]

[Out]

-((1454315 - 3037062*x)*Sqrt[2 + 5*x + 3*x^2])/110592 - ((6925 - 151098*x)*(2 + 5*x + 3*x^2)^(3/2))/13824 + ((
283 + 8310*x)*(2 + 5*x + 3*x^2)^(5/2))/1440 - ((47 + x)*(2 + 5*x + 3*x^2)^(7/2))/(14*(3 + 2*x)) + (15434623*Ar
cTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(221184*Sqrt[3]) - (9225*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt
[5]*Sqrt[2 + 5*x + 3*x^2])])/512

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 814

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*(a + b*x + c*x^
2)^p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*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] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 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)^2} \, dx &=-\frac {(47+x) \left (2+5 x+3 x^2\right )^{7/2}}{14 (3+2 x)}-\frac {1}{8} \int \frac {(-462-554 x) \left (2+5 x+3 x^2\right )^{5/2}}{3+2 x} \, dx\\ &=\frac {(283+8310 x) \left (2+5 x+3 x^2\right )^{5/2}}{1440}-\frac {(47+x) \left (2+5 x+3 x^2\right )^{7/2}}{14 (3+2 x)}+\frac {\int \frac {(84678+100732 x) \left (2+5 x+3 x^2\right )^{3/2}}{3+2 x} \, dx}{1152}\\ &=-\frac {(6925-151098 x) \left (2+5 x+3 x^2\right )^{3/2}}{13824}+\frac {(283+8310 x) \left (2+5 x+3 x^2\right )^{5/2}}{1440}-\frac {(47+x) \left (2+5 x+3 x^2\right )^{7/2}}{14 (3+2 x)}-\frac {\int \frac {(-10251972-12148248 x) \sqrt {2+5 x+3 x^2}}{3+2 x} \, dx}{110592}\\ &=-\frac {(1454315-3037062 x) \sqrt {2+5 x+3 x^2}}{110592}-\frac {(6925-151098 x) \left (2+5 x+3 x^2\right )^{3/2}}{13824}+\frac {(283+8310 x) \left (2+5 x+3 x^2\right )^{5/2}}{1440}-\frac {(47+x) \left (2+5 x+3 x^2\right )^{7/2}}{14 (3+2 x)}+\frac {\int \frac {633068856+740861904 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{5308416}\\ &=-\frac {(1454315-3037062 x) \sqrt {2+5 x+3 x^2}}{110592}-\frac {(6925-151098 x) \left (2+5 x+3 x^2\right )^{3/2}}{13824}+\frac {(283+8310 x) \left (2+5 x+3 x^2\right )^{5/2}}{1440}-\frac {(47+x) \left (2+5 x+3 x^2\right )^{7/2}}{14 (3+2 x)}+\frac {15434623 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{221184}-\frac {46125}{512} \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {(1454315-3037062 x) \sqrt {2+5 x+3 x^2}}{110592}-\frac {(6925-151098 x) \left (2+5 x+3 x^2\right )^{3/2}}{13824}+\frac {(283+8310 x) \left (2+5 x+3 x^2\right )^{5/2}}{1440}-\frac {(47+x) \left (2+5 x+3 x^2\right )^{7/2}}{14 (3+2 x)}+\frac {15434623 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{110592}+\frac {46125}{256} \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=-\frac {(1454315-3037062 x) \sqrt {2+5 x+3 x^2}}{110592}-\frac {(6925-151098 x) \left (2+5 x+3 x^2\right )^{3/2}}{13824}+\frac {(283+8310 x) \left (2+5 x+3 x^2\right )^{5/2}}{1440}-\frac {(47+x) \left (2+5 x+3 x^2\right )^{7/2}}{14 (3+2 x)}+\frac {15434623 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{221184 \sqrt {3}}-\frac {9225}{512} \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 = 0.11, size = 130, normalized size = 0.75 \begin {gather*} \frac {418446000 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )[phone] \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )-\frac {6 \sqrt {3 x^2+5 x+2} \left (7464960 x^7-13893120 x^6-125632512 x^5-273531168 x^4-275126016 x^3-179819084 x^2+28017108 x+259165107\right )}{2 x+3}}{23224320} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-6*Sqrt[2 + 5*x + 3*x^2]*(259165107 + 28017108*x - 179819084*x^2 - 275126016*x^3 - 273531168*x^4 - 125632512
*x^5 - 13893120*x^6 + 7464960*x^7))/(3 + 2*x) + 418446000*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x +
3*x^2])] + 540211805*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/23224320

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IntegrateAlgebraic [A]  time = 1.04, size = 131, normalized size = 0.75 \begin {gather*} \frac {15434623 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{110592 \sqrt {3}}-\frac {9225}{256} \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 (-7464960 x^7+13893120 x^6+125632512 x^5+273531168 x^4+275126016 x^3+179819084 x^2-28017108 x-259165107\right )}{3870720 (2 x+3)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + 5*x + 3*x^2]*(-259165107 - 28017108*x + 179819084*x^2 + 275126016*x^3 + 273531168*x^4 + 125632512*x^
5 + 13893120*x^6 - 7464960*x^7))/(3870720*(3 + 2*x)) + (15434623*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[3]*(1 + x
))])/(110592*Sqrt[3]) - (9225*Sqrt[5]*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[5]*(1 + x))])/256

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fricas [A]  time = 0.43, size = 159, normalized size = 0.91 \begin {gather*} \frac {540211805 \, \sqrt {3} {\left (2 \, x + 3\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 ) + 418446000 \, \sqrt {5} {\left (2 \, x + 3\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 ) - 12 \, {\left (7464960 \, x^{7} - 13893120 \, x^{6} - 125632512 \, x^{5} - 273531168 \, x^{4} - 275126016 \, x^{3} - 179819084 \, x^{2} + 28017108 \, x + 259165107\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{46448640 \, {\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)^(7/2)/(3+2*x)^2,x, algorithm="fricas")

[Out]

1/46448640*(540211805*sqrt(3)*(2*x + 3)*log(4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) +
418446000*sqrt(5)*(2*x + 3)*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)) - 12*(7464960*x^7 - 13893120*x^6 - 125632512*x^5 - 273531168*x^4 - 275126016*x^3 - 179819084*x^2 +
28017108*x + 259165107)*sqrt(3*x^2 + 5*x + 2))/(2*x + 3)

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giac [B]  time = 1.45, size = 861, normalized size = 4.95

result too large to display

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)^2,x, algorithm="giac")

[Out]

-15434623/663552*sqrt(3)*log(abs(-2*sqrt(3) + 2*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + 2*sqrt(5)/(2*x + 3))/
abs(2*sqrt(3) + 2*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + 2*sqrt(5)/(2*x + 3)))*sgn(1/(2*x + 3)) + 9225/512*s
qrt(5)*log(abs(sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3)) - 4))*sgn(1/(2*x + 3)) - 1
625/512*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3)*sgn(1/(2*x + 3)) + 1/3870720*(1702084195*(sqrt(-8/(2*x + 3) + 5
/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^13*sgn(1/(2*x + 3)) - 3595838400*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x +
3)^2 + 3) + sqrt(5)/(2*x + 3))^12*sgn(1/(2*x + 3)) - 462583100*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(
5)/(2*x + 3))^11*sgn(1/(2*x + 3)) + 12803555520*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x
+ 3))^10*sgn(1/(2*x + 3)) + 91554292599*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^9*sgn(1/
(2*x + 3)) - 132950643840*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^8*sgn(1/(2*x +
3)) - 221215739904*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^7*sgn(1/(2*x + 3)) + 432202780
800*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^6*sgn(1/(2*x + 3)) + 252015304401*(sq
rt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^5*sgn(1/(2*x + 3)) - 680038027200*sqrt(5)*(sqrt(-8/(
2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^4*sgn(1/(2*x + 3)) - 506502404100*(sqrt(-8/(2*x + 3) + 5/(2
*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^3*sgn(1/(2*x + 3)) + 786343723200*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)
^2 + 3) + sqrt(5)/(2*x + 3))^2*sgn(1/(2*x + 3)) + 178876045845*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(
5)/(2*x + 3))*sgn(1/(2*x + 3)) - 339366412800*sqrt(5)*sgn(1/(2*x + 3)))/((sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 +
3) + sqrt(5)/(2*x + 3))^2 - 3)^7

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maple [A]  time = 0.06, size = 232, normalized size = 1.33 \begin {gather*} \frac {9225 \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 )}{512}+\frac {15434623 \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 )}{663552}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{10 \left (x +\frac {3}{2}\right )}-\frac {369 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{140}+\frac {277 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{288}+\frac {25183 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{13824}+\frac {506177 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{110592}-\frac {369 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{80}-\frac {615 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{64}-\frac {9225 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{512}+\frac {13 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{20} \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)^2,x)

[Out]

-13/10/(x+3/2)*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-369/140*(-4*x+3*(x+3/2)^2-19/4)^(7/2)+277/288*(6*x+5)*(-4*x+3*(x+
3/2)^2-19/4)^(5/2)+25183/13824*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(3/2)+506177/110592*(6*x+5)*(-4*x+3*(x+3/2)^2-1
9/4)^(1/2)[phone]/663552*3^(1/2)*ln(1/3*(3*x+5/2)*3^(1/2)+(-4*x+3*(x+3/2)^2-19/4)^(1/2))-369/80*(-4*x+3*(x+3
/2)^2-19/4)^(5/2)-615/64*(-4*x+3*(x+3/2)^2-19/4)^(3/2)-9225/512*(-16*x+12*(x+3/2)^2-19)^(1/2)+9225/512*5^(1/2)
*arctanh(2/5*(-4*x-7/2)*5^(1/2)/(-16*x+12*(x+3/2)^2-19)^(1/2))+13/20*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(7/2)

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maxima [A]  time = 1.31, size = 192, normalized size = 1.10 \begin {gather*} -\frac {1}{28} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} + \frac {277}{48} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {283}{1440} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{4 \, {\left (2 \, x + 3\right )}} + \frac {25183}{2304} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {6925}{13824} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {506177}{18432} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {15434623}{663552} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) + \frac {9225}{512} \, \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 {1454315}{110592} \, \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)^2,x, algorithm="maxima")

[Out]

-1/28*(3*x^2 + 5*x + 2)^(7/2) + 277/48*(3*x^2 + 5*x + 2)^(5/2)*x + 283/1440*(3*x^2 + 5*x + 2)^(5/2) - 13/4*(3*
x^2 + 5*x + 2)^(7/2)/(2*x + 3) + 25183/2304*(3*x^2 + 5*x + 2)^(3/2)*x - 6925/13824*(3*x^2 + 5*x + 2)^(3/2) + 5
06177/18432*sqrt(3*x^2 + 5*x + 2)*x + 15434623/663552*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) +
9225/512*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 1454315/110592*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 )}^{7/2}}{{\left (2\,x+3\right )}^2} \,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)^2,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {40 \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \left (- \frac {292 x \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \left (- \frac {870 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \left (- \frac {1339 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \left (- \frac {1090 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \left (- \frac {396 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \frac {27 x^{7} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\, 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)**2,x)

[Out]

-Integral(-40*sqrt(3*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x) - Integral(-292*x*sqrt(3*x**2 + 5*x + 2)/(4*x**2
+ 12*x + 9), x) - Integral(-870*x**2*sqrt(3*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x) - Integral(-1339*x**3*sqrt
(3*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x) - Integral(-1090*x**4*sqrt(3*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x
) - Integral(-396*x**5*sqrt(3*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x) - Integral(27*x**7*sqrt(3*x**2 + 5*x + 2
)/(4*x**2 + 12*x + 9), x)

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