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

Optimal. Leaf size=195 \[ -\frac {(x+8) \left (3 x^2+5 x+2\right )^{7/2}}{8 (2 x+3)^4}+\frac {7 (43 x+93) \left (3 x^2+5 x+2\right )^{5/2}}{96 (2 x+3)^3}-\frac {35 (343 x+736) \left (3 x^2+5 x+2\right )^{3/2}}{768 (2 x+3)^2}+\frac {35 (2701 x+5795) \sqrt {3 x^2+5 x+2}}{1024 (2 x+3)}-\frac {744275 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{4096 \sqrt {3}}+\frac {192171 \sqrt {5} \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{4096} \]

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Rubi [A]  time = 0.13, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {812, 843, 621, 206, 724} \begin {gather*} -\frac {(x+8) \left (3 x^2+5 x+2\right )^{7/2}}{8 (2 x+3)^4}+\frac {7 (43 x+93) \left (3 x^2+5 x+2\right )^{5/2}}{96 (2 x+3)^3}-\frac {35 (343 x+736) \left (3 x^2+5 x+2\right )^{3/2}}{768 (2 x+3)^2}+\frac {35 (2701 x+5795) \sqrt {3 x^2+5 x+2}}{1024 (2 x+3)}-\frac {744275 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{4096 \sqrt {3}}+\frac {192171 \sqrt {5} \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{4096} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(35*(5795 + 2701*x)*Sqrt[2 + 5*x + 3*x^2])/(1024*(3 + 2*x)) - (35*(736 + 343*x)*(2 + 5*x + 3*x^2)^(3/2))/(768*
(3 + 2*x)^2) + (7*(93 + 43*x)*(2 + 5*x + 3*x^2)^(5/2))/(96*(3 + 2*x)^3) - ((8 + x)*(2 + 5*x + 3*x^2)^(7/2))/(8
*(3 + 2*x)^4) - (744275*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(4096*Sqrt[3]) + (192171*Sqrt[5]
*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/4096

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 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)^5} \, dx &=-\frac {(8+x) \left (2+5 x+3 x^2\right )^{7/2}}{8 (3+2 x)^4}-\frac {7}{128} \int \frac {(-288-344 x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^4} \, dx\\ &=\frac {7 (93+43 x) \left (2+5 x+3 x^2\right )^{5/2}}{96 (3+2 x)^3}-\frac {(8+x) \left (2+5 x+3 x^2\right )^{7/2}}{8 (3+2 x)^4}+\frac {35 \int \frac {(-14064-16464 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^3} \, dx}{9216}\\ &=-\frac {35 (736+343 x) \left (2+5 x+3 x^2\right )^{3/2}}{768 (3+2 x)^2}+\frac {7 (93+43 x) \left (2+5 x+3 x^2\right )^{5/2}}{96 (3+2 x)^3}-\frac {(8+x) \left (2+5 x+3 x^2\right )^{7/2}}{8 (3+2 x)^4}-\frac {35 \int \frac {(-443136-518592 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^2} \, dx}{98304}\\ &=\frac {35 (5795+2701 x) \sqrt {2+5 x+3 x^2}}{1024 (3+2 x)}-\frac {35 (736+343 x) \left (2+5 x+3 x^2\right )^{3/2}}{768 (3+2 x)^2}+\frac {7 (93+43 x) \left (2+5 x+3 x^2\right )^{5/2}}{96 (3+2 x)^3}-\frac {(8+x) \left (2+5 x+3 x^2\right )^{7/2}}{8 (3+2 x)^4}+\frac {35 \int \frac {-6977664-8165760 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{786432}\\ &=\frac {35 (5795+2701 x) \sqrt {2+5 x+3 x^2}}{1024 (3+2 x)}-\frac {35 (736+343 x) \left (2+5 x+3 x^2\right )^{3/2}}{768 (3+2 x)^2}+\frac {7 (93+43 x) \left (2+5 x+3 x^2\right )^{5/2}}{96 (3+2 x)^3}-\frac {(8+x) \left (2+5 x+3 x^2\right )^{7/2}}{8 (3+2 x)^4}-\frac {744275 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{4096}+\frac {960855 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{4096}\\ &=\frac {35 (5795+2701 x) \sqrt {2+5 x+3 x^2}}{1024 (3+2 x)}-\frac {35 (736+343 x) \left (2+5 x+3 x^2\right )^{3/2}}{768 (3+2 x)^2}+\frac {7 (93+43 x) \left (2+5 x+3 x^2\right )^{5/2}}{96 (3+2 x)^3}-\frac {(8+x) \left (2+5 x+3 x^2\right )^{7/2}}{8 (3+2 x)^4}-\frac {744275 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{2048}-\frac {960855 \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{2048}\\ &=\frac {35 (5795+2701 x) \sqrt {2+5 x+3 x^2}}{1024 (3+2 x)}-\frac {35 (736+343 x) \left (2+5 x+3 x^2\right )^{3/2}}{768 (3+2 x)^2}+\frac {7 (93+43 x) \left (2+5 x+3 x^2\right )^{5/2}}{96 (3+2 x)^3}-\frac {(8+x) \left (2+5 x+3 x^2\right )^{7/2}}{8 (3+2 x)^4}-\frac {744275 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{4096 \sqrt {3}}+\frac {192171 \sqrt {5} \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{4096}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 130, normalized size = 0.67 \begin {gather*} \frac {-576513 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )-744275 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )-\frac {12 \sqrt {3 x^2+5 x+2} \left (3456 x^7-12864 x^6-38288 x^5-253688 x^4-2869312 x^3-9107922 x^2-11295211 x-4933171\right )}{(2 x+3)^4}}{12288} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-12*Sqrt[2 + 5*x + 3*x^2]*(-4933171 - 11295211*x - 9107922*x^2 - 2869312*x^3 - 253688*x^4 - 38288*x^5 - 1286
4*x^6 + 3456*x^7))/(3 + 2*x)^4 - 576513*Sqrt[5]*ArcTanh[(-7 - 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])] - 744275
*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])])/12288

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IntegrateAlgebraic [A]  time = 0.93, size = 131, normalized size = 0.67 \begin {gather*} -\frac {744275 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{2048 \sqrt {3}}+\frac {192171 \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{2048}+\frac {\sqrt {3 x^2+5 x+2} \left (-3456 x^7+12864 x^6+38288 x^5+253688 x^4+2869312 x^3+9107922 x^2+11295211 x+4933171\right )}{1024 (2 x+3)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + 5*x + 3*x^2]*(4933171 + 11295211*x + 9107922*x^2 + 2869312*x^3 + 253688*x^4 + 38288*x^5 + 12864*x^6
- 3456*x^7))/(1024*(3 + 2*x)^4) - (744275*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[3]*(1 + x))])/(2048*Sqrt[3]) + (
192171*Sqrt[5]*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[5]*(1 + x))])/2048

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fricas [A]  time = 0.44, size = 203, normalized size = 1.04 \begin {gather*} \frac {744275 \, \sqrt {3} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\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 ) + 576513 \, \sqrt {5} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\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 ) - 24 \, {\left (3456 \, x^{7} - 12864 \, x^{6} - 38288 \, x^{5} - 253688 \, x^{4} - 2869312 \, x^{3} - 9107922 \, x^{2} - 11295211 \, x - 4933171\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{24576 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\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)^5,x, algorithm="fricas")

[Out]

1/24576*(744275*sqrt(3)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(-4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5
) + 72*x^2 + 120*x + 49) + 576513*sqrt(5)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*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)) - 24*(3456*x^7 - 12864*x^6 - 38288*x^5 - 25368
8*x^4 - 2869312*x^3 - 9107922*x^2 - 11295211*x - 4933171)*sqrt(3*x^2 + 5*x + 2))/(16*x^4 + 96*x^3 + 216*x^2 +
216*x + 81)

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giac [B]  time = 1.31, size = 636, normalized size = 3.26 \begin {gather*} \frac {744275}{12288} \, \sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 2 \, \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {2 \, \sqrt {5}}{2 \, x + 3} \right |}}{{\left | 2 \, \sqrt {3} + 2 \, \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {2 \, \sqrt {5}}{2 \, x + 3} \right |}}\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {192171}{4096} \, \sqrt {5} \log \left ({\left | \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )} - 4 \right |}\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {1}{4096} \, {\left (\frac {5 \, {\left (\frac {50 \, {\left (\frac {13 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}{2 \, x + 3} - 88 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} + 14343 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} - 181996 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )} \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} - \frac {479709 \, {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{7} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 499296 \, \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{6} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 3133183 \, {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{5} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 3365712 \, \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{4} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 7550211 \, {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{3} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 8139744 \, \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{2} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 6574257 \, {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 6966000 \, \sqrt {5} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}{2048 \, {\left ({\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{2} - 3\right )}^{4}} \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)^5,x, algorithm="giac")

[Out]

744275/12288*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)) - 192171/4096*sq
rt(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/
4096*(5*(50*(13*sgn(1/(2*x + 3))/(2*x + 3) - 88*sgn(1/(2*x + 3)))/(2*x + 3) + 14343*sgn(1/(2*x + 3)))/(2*x + 3
) - 181996*sgn(1/(2*x + 3)))*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) - 1/2048*(479709*(sqrt(-8/(2*x + 3) + 5/(2
*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^7*sgn(1/(2*x + 3)) - 499296*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3
) + sqrt(5)/(2*x + 3))^6*sgn(1/(2*x + 3)) - 3133183*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3
))^5*sgn(1/(2*x + 3)) + 3365712*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^4*sgn(1/(
2*x + 3)) + 7550211*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^3*sgn(1/(2*x + 3)) - 8139744*
sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^2*sgn(1/(2*x + 3)) - 6574257*(sqrt(-8/(2*
x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))*sgn(1/(2*x + 3)) + 6966000*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)^4

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maple [A]  time = 0.06, size = 295, normalized size = 1.51 \begin {gather*} -\frac {192171 \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 )}{4096}-\frac {744275 \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 )}{12288}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{320 \left (x +\frac {3}{2}\right )^{4}}-\frac {1263 \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 )^{2}}+\frac {3 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{100 \left (x +\frac {3}{2}\right )^{3}}-\frac {1479 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{1000}-\frac {10101 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{4000}+\frac {1479 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{500 \left (x +\frac {3}{2}\right )}-\frac {6069 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{1280}-\frac {24409 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{2048}+\frac {192171 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{4096}+\frac {64057 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{2560}+\frac {192171 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{16000}+\frac {27453 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{4000} \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)^5,x)

[Out]

-13/320/(x+3/2)^4*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-1263/4000/(x+3/2)^2*(-4*x+3*(x+3/2)^2-19/4)^(9/2)+3/100/(x+3/2
)^3*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-1479/1000*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(7/2)-10101/4000*(6*x+5)*(-4*x+3*(
x+3/2)^2-19/4)^(5/2)+1479/500/(x+3/2)*(-4*x+3*(x+3/2)^2-19/4)^(9/2)-6069/1280*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^
(3/2)-24409/2048*(6*x+5)*(-4*x+3*(x+3/2)^2-19/4)^(1/2)-192171/4096*5^(1/2)*arctanh(2/5*(-4*x-7/2)*5^(1/2)/(-16
*x+12*(x+3/2)^2-19)^(1/2))-744275/12288*3^(1/2)*ln(1/3*(3*x+5/2)*3^(1/2)+(-4*x+3*(x+3/2)^2-19/4)^(1/2))+192171
/4096*(-16*x+12*(x+3/2)^2-19)^(1/2)+64057/2560*(-4*x+3*(x+3/2)^2-19/4)^(3/2)+192171/16000*(-4*x+3*(x+3/2)^2-19
/4)^(5/2)+27453/4000*(-4*x+3*(x+3/2)^2-19/4)^(7/2)

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maxima [A]  time = 1.18, size = 285, normalized size = 1.46 \begin {gather*} \frac {3789}{4000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{20 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} + \frac {6 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{25 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {1263 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{1000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {30303}{2000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {9849}{16000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {1479 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{200 \, {\left (2 \, x + 3\right )}} - \frac {18207}{640} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {3367}{2560} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {73227}{1024} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {744275}{12288} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) - \frac {192171}{4096} \, \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 {35063}{1024} \, \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)^5,x, algorithm="maxima")

[Out]

3789/4000*(3*x^2 + 5*x + 2)^(7/2) - 13/20*(3*x^2 + 5*x + 2)^(9/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) + 6
/25*(3*x^2 + 5*x + 2)^(9/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 1263/1000*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9
) - 30303/2000*(3*x^2 + 5*x + 2)^(5/2)*x - 9849/16000*(3*x^2 + 5*x + 2)^(5/2) + 1479/200*(3*x^2 + 5*x + 2)^(7/
2)/(2*x + 3) - 18207/640*(3*x^2 + 5*x + 2)^(3/2)*x + 3367/2560*(3*x^2 + 5*x + 2)^(3/2) - 73227/1024*sqrt(3*x^2
+ 5*x + 2)*x - 744275/12288*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) - 192171/4096*sqrt(5)*log(
sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 35063/1024*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 )}^5} \,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)^5,x)

[Out]

-int(((x - 5)*(5*x + 3*x^2 + 2)^(7/2))/(2*x + 3)^5, 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}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac {292 x \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac {870 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac {1339 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac {1090 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac {396 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \frac {27 x^{7} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\, 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)**5,x)

[Out]

-Integral(-40*sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243), x) - Integral(
-292*x*sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243), x) - Integral(-870*x*
*2*sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243), x) - Integral(-1339*x**3*
sqrt(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243), x) - Integral(-1090*x**4*sqr
t(3*x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243), x) - Integral(-396*x**5*sqrt(3*
x**2 + 5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243), x) - Integral(27*x**7*sqrt(3*x**2 +
5*x + 2)/(32*x**5 + 240*x**4 + 720*x**3 + 1080*x**2 + 810*x + 243), x)

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