3.23.14 \(\int \frac {(5-x) (2+5 x+3 x^2)^{7/2}}{(3+2 x)^4} \, dx\)

Optimal. Leaf size=190 \[ -\frac {(3 x+37) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^3}+\frac {7 (414 x+1171) \left (3 x^2+5 x+2\right )^{5/2}}{960 (2 x+3)^2}-\frac {7 (1652 x+5713) \left (3 x^2+5 x+2\right )^{3/2}}{768 (2 x+3)}-\frac {7 (37375-78054 x) \sqrt {3 x^2+5 x+2}}{6144}+\frac {2776697 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{12288 \sqrt {3}}-\frac {59745 \sqrt {5} \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{1024} \]

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Rubi [A]  time = 0.13, antiderivative size = 190, 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 {(3 x+37) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^3}+\frac {7 (414 x+1171) \left (3 x^2+5 x+2\right )^{5/2}}{960 (2 x+3)^2}-\frac {7 (1652 x+5713) \left (3 x^2+5 x+2\right )^{3/2}}{768 (2 x+3)}-\frac {7 (37375-78054 x) \sqrt {3 x^2+5 x+2}}{6144}+\frac {2776697 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{12288 \sqrt {3}}-\frac {59745 \sqrt {5} \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{1024} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^4,x]

[Out]

(-7*(37375 - 78054*x)*Sqrt[2 + 5*x + 3*x^2])/6144 - (7*(5713 + 1652*x)*(2 + 5*x + 3*x^2)^(3/2))/(768*(3 + 2*x)
) + (7*(1171 + 414*x)*(2 + 5*x + 3*x^2)^(5/2))/(960*(3 + 2*x)^2) - ((37 + 3*x)*(2 + 5*x + 3*x^2)^(7/2))/(30*(3
+ 2*x)^3) + (2776697*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(12288*Sqrt[3]) - (59745*Sqrt[5]*A
rcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/1024

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)^4} \, dx &=-\frac {(37+3 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^3}-\frac {7}{120} \int \frac {(-346-414 x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^3} \, dx\\ &=\frac {7 (1171+414 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^2}-\frac {(37+3 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^3}+\frac {7 \int \frac {(-16796-19824 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^2} \, dx}{1536}\\ &=-\frac {7 (5713+1652 x) \left (2+5 x+3 x^2\right )^{3/2}}{768 (3+2 x)}+\frac {7 (1171+414 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^2}-\frac {(37+3 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^3}-\frac {7 \int \frac {(-526968-624432 x) \sqrt {2+5 x+3 x^2}}{3+2 x} \, dx}{12288}\\ &=-\frac {7 (37375-78054 x) \sqrt {2+5 x+3 x^2}}{6144}-\frac {7 (5713+1652 x) \left (2+5 x+3 x^2\right )^{3/2}}{768 (3+2 x)}+\frac {7 (1171+414 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^2}-\frac {(37+3 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^3}+\frac {7 \int \frac {32539824+38080416 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{589824}\\ &=-\frac {7 (37375-78054 x) \sqrt {2+5 x+3 x^2}}{6144}-\frac {7 (5713+1652 x) \left (2+5 x+3 x^2\right )^{3/2}}{768 (3+2 x)}+\frac {7 (1171+414 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^2}-\frac {(37+3 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^3}+\frac {2776697 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{12288}-\frac {298725 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{1024}\\ &=-\frac {7 (37375-78054 x) \sqrt {2+5 x+3 x^2}}{6144}-\frac {7 (5713+1652 x) \left (2+5 x+3 x^2\right )^{3/2}}{768 (3+2 x)}+\frac {7 (1171+414 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^2}-\frac {(37+3 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^3}+\frac {2776697 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{6144}+\frac {298725}{512} \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=-\frac {7 (37375-78054 x) \sqrt {2+5 x+3 x^2}}{6144}-\frac {7 (5713+1652 x) \left (2+5 x+3 x^2\right )^{3/2}}{768 (3+2 x)}+\frac {7 (1171+414 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^2}-\frac {(37+3 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^3}+\frac {2776697 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{12288 \sqrt {3}}-\frac {59745 \sqrt {5} \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{1024}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 130, normalized size = 0.68 \begin {gather*} \frac {10754100 \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 (82944 x^7-231552 x^6-1266816 x^5-3277520 x^4+746240 x^3+44770416 x^2+98927312 x+61268351\right )}{(2 x+3)^3}}{184320} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^4,x]

[Out]

((-6*Sqrt[2 + 5*x + 3*x^2]*(61268351 + 98927312*x + 44770416*x^2 + 746240*x^3 - 3277520*x^4 - 1266816*x^5 - 23
1552*x^6 + 82944*x^7))/(3 + 2*x)^3 + 10754100*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])] +
13883485*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/184320

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IntegrateAlgebraic [A]  time = 0.95, size = 131, normalized size = 0.69 \begin {gather*} \frac {2776697 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{6144 \sqrt {3}}-\frac {59745}{512} \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 (-82944 x^7+231552 x^6+1266816 x^5+3277520 x^4-746240 x^3-44770416 x^2-98927312 x-61268351\right )}{30720 (2 x+3)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^4,x]

[Out]

(Sqrt[2 + 5*x + 3*x^2]*(-61268351 - 98927312*x - 44770416*x^2 - 746240*x^3 + 3277520*x^4 + 1266816*x^5 + 23155
2*x^6 - 82944*x^7))/(30720*(3 + 2*x)^3) + (2776697*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[3]*(1 + x))])/(6144*Sqr
t[3]) - (59745*Sqrt[5]*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[5]*(1 + x))])/512

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fricas [A]  time = 0.42, size = 189, normalized size = 0.99 \begin {gather*} \frac {13883485 \, \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 ) + 10754100 \, \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 ) - 12 \, {\left (82944 \, x^{7} - 231552 \, x^{6} - 1266816 \, x^{5} - 3277520 \, x^{4} + 746240 \, x^{3} + 44770416 \, x^{2} + 98927312 \, x + 61268351\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{368640 \, {\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)^(7/2)/(3+2*x)^4,x, algorithm="fricas")

[Out]

1/368640*(13883485*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
+ 120*x + 49) + 10754100*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)) - 12*(82944*x^7 - 231552*x^6 - 1266816*x^5 - 3277520*x^4 + 746240
*x^3 + 44770416*x^2 + 98927312*x + 61268351)*sqrt(3*x^2 + 5*x + 2))/(8*x^3 + 36*x^2 + 54*x + 27)

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giac [B]  time = 0.48, size = 325, normalized size = 1.71 \begin {gather*} -\frac {1}{30720} \, {\left (2 \, {\left (12 \, {\left (18 \, {\left (24 \, x - 175\right )} x + 4661\right )} x - 218885\right )} x + 1563313\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {59745}{1024} \, \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 {2776697}{36864} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac {5 \, {\left (424596 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 2828550 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 21565510 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 26086815 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 45375675 \, \sqrt {3} x + 10164786 \, \sqrt {3} - 45375675 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}}{1536 \, {\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)^(7/2)/(3+2*x)^4,x, algorithm="giac")

[Out]

-1/30720*(2*(12*(18*(24*x - 175)*x + 4661)*x - 218885)*x + 1563313)*sqrt(3*x^2 + 5*x + 2) - 59745/1024*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))) - 2776697/36864*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))
- 5)) - 5/1536*(424596*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 2828550*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x
+ 2))^4 + 21565510*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 26086815*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)
)^2 + 45375675*sqrt(3)*x + 10164786*sqrt(3) - 45375675*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 = 274, normalized size = 1.44 \begin {gather*} \frac {59745 \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 )}{1024}+\frac {2776697 \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 )}{36864}+\frac {57 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{200 \left (x +\frac {3}{2}\right )^{2}}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{120 \left (x +\frac {3}{2}\right )^{3}}+\frac {48 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{25}+\frac {1253 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{400}-\frac {96 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{25 \left (x +\frac {3}{2}\right )}+\frac {4529 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{768}+\frac {91063 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{6144}-\frac {59745 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{1024}-\frac {3983 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{128}-\frac {11949 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{800}-\frac {1707 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{200} \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)^4,x)

[Out]

57/200/(x+3/2)^2*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-13/120/(x+3/2)^3*(-4*x+3*(x+3/2)^2-19/4)^(9/2)+48/25*(6*x+5)*(-
4*x+3*(x+3/2)^2-19/4)^(7/2)+1253/400*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(5/2)-96/25/(x+3/2)*(-4*x+3*(x+3/2)^2-19/
4)^(9/2)+4529/768*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(3/2)+91063/6144*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(1/2)+59745
/1024*5^(1/2)*arctanh(2/5*(-4*x-7/2)*5^(1/2)/(-16*x+12*(x+3/2)^2-19)^(1/2))+2776697/36864*3^(1/2)*ln(1/3*(3*x+
5/2)*3^(1/2)+(-4*x+3*(x+3/2)^2-19/4)^(1/2))-59745/1024*(-16*x+12*(x+3/2)^2-19)^(1/2)-3983/128*(-4*x+3*(x+3/2)^
2-19/4)^(3/2)-11949/800*(-4*x+3*(x+3/2)^2-19/4)^(5/2)-1707/200*(-4*x+3*(x+3/2)^2-19/4)^(7/2)

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maxima [A]  time = 1.25, size = 249, normalized size = 1.31 \begin {gather*} -\frac {171}{200} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{15 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} + \frac {57 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{50 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {3759}{200} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {581}{800} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {48 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{5 \, {\left (2 \, x + 3\right )}} + \frac {4529}{128} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {1253}{768} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {91063}{1024} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {2776697}{36864} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) + \frac {59745}{1024} \, \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 {261625}{6144} \, \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)^4,x, algorithm="maxima")

[Out]

-171/200*(3*x^2 + 5*x + 2)^(7/2) - 13/15*(3*x^2 + 5*x + 2)^(9/2)/(8*x^3 + 36*x^2 + 54*x + 27) + 57/50*(3*x^2 +
5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) + 3759/200*(3*x^2 + 5*x + 2)^(5/2)*x + 581/800*(3*x^2 + 5*x + 2)^(5/2) - 48
/5*(3*x^2 + 5*x + 2)^(7/2)/(2*x + 3) + 4529/128*(3*x^2 + 5*x + 2)^(3/2)*x - 1253/768*(3*x^2 + 5*x + 2)^(3/2) +
91063/1024*sqrt(3*x^2 + 5*x + 2)*x + 2776697/36864*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) + 5
9745/1024*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 261625/6144*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 )}^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)^(7/2))/(2*x + 3)^4,x)

[Out]

-int(((x - 5)*(5*x + 3*x^2 + 2)^(7/2))/(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 {40 \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac {292 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 {870 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 {1339 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 {1090 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac {396 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \frac {27 x^{7} \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)**(7/2)/(3+2*x)**4,x)

[Out]

-Integral(-40*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x) - Integral(-292*x*sqrt(3*
x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x) - Integral(-870*x**2*sqrt(3*x**2 + 5*x + 2)/(1
6*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x) - Integral(-1339*x**3*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3
+ 216*x**2 + 216*x + 81), x) - Integral(-1090*x**4*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*
x + 81), x) - Integral(-396*x**5*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x) - Inte
gral(27*x**7*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x)

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