3.2601 \(\int \frac {(5-x) (2+5 x+3 x^2)^{5/2}}{(3+2 x)^{9/2}} \, dx\)

Optimal. Leaf size=207 \[ -\frac {(7 x+43) \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^{7/2}}-\frac {(2531 x+3354) \left (3 x^2+5 x+2\right )^{3/2}}{210 (2 x+3)^{5/2}}+\frac {(1823 x+6292) \sqrt {3 x^2+5 x+2}}{140 \sqrt {2 x+3}}+\frac {2505 \sqrt {3} \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{56 \sqrt {3 x^2+5 x+2}}-\frac {4091 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{40 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]

[Out]

-1/210*(3354+2531*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(5/2)-1/35*(43+7*x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(7/2)-4091/12
0*EllipticE(3^(1/2)*(1+x)^(1/2),1/3*I*6^(1/2))*(-3*x^2-5*x-2)^(1/2)*3^(1/2)/(3*x^2+5*x+2)^(1/2)+2505/56*Ellipt
icF(3^(1/2)*(1+x)^(1/2),1/3*I*6^(1/2))*(-3*x^2-5*x-2)^(1/2)*3^(1/2)/(3*x^2+5*x+2)^(1/2)+1/140*(6292+1823*x)*(3
*x^2+5*x+2)^(1/2)/(3+2*x)^(1/2)

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Rubi [A]  time = 0.13, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {812, 810, 843, 718, 424, 419} \[ -\frac {(7 x+43) \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^{7/2}}-\frac {(2531 x+3354) \left (3 x^2+5 x+2\right )^{3/2}}{210 (2 x+3)^{5/2}}+\frac {(1823 x+6292) \sqrt {3 x^2+5 x+2}}{140 \sqrt {2 x+3}}+\frac {2505 \sqrt {3} \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{56 \sqrt {3 x^2+5 x+2}}-\frac {4091 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{40 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^(9/2),x]

[Out]

((6292 + 1823*x)*Sqrt[2 + 5*x + 3*x^2])/(140*Sqrt[3 + 2*x]) - ((3354 + 2531*x)*(2 + 5*x + 3*x^2)^(3/2))/(210*(
3 + 2*x)^(5/2)) - ((43 + 7*x)*(2 + 5*x + 3*x^2)^(5/2))/(35*(3 + 2*x)^(7/2)) - (4091*Sqrt[-2 - 5*x - 3*x^2]*Ell
ipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(40*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (2505*Sqrt[3]*Sqrt[-2 - 5*x -
3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(56*Sqrt[2 + 5*x + 3*x^2])

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 718

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(2*Rt[b^2 - 4*a*c, 2]
*(d + e*x)^m*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))])/(c*Sqrt[a + b*x + c*x^2]*((2*c*(d + e*x))/(2*c*d -
b*e - e*Rt[b^2 - 4*a*c, 2]))^m), Subst[Int[(1 + (2*e*Rt[b^2 - 4*a*c, 2]*x^2)/(2*c*d - b*e - e*Rt[b^2 - 4*a*c,
2]))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b
, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4]

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 )^{5/2}}{(3+2 x)^{9/2}} \, dx &=-\frac {(43+7 x) \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^{7/2}}-\frac {1}{14} \int \frac {(-187-223 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{7/2}} \, dx\\ &=-\frac {(3354+2531 x) \left (2+5 x+3 x^2\right )^{3/2}}{210 (3+2 x)^{5/2}}-\frac {(43+7 x) \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^{7/2}}+\frac {1}{700} \int \frac {(23230+27345 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^{3/2}} \, dx\\ &=\frac {(6292+1823 x) \sqrt {2+5 x+3 x^2}}{140 \sqrt {3+2 x}}-\frac {(3354+2531 x) \left (2+5 x+3 x^2\right )^{3/2}}{210 (3+2 x)^{5/2}}-\frac {(43+7 x) \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^{7/2}}-\frac {\int \frac {362520+429555 x}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{4200}\\ &=\frac {(6292+1823 x) \sqrt {2+5 x+3 x^2}}{140 \sqrt {3+2 x}}-\frac {(3354+2531 x) \left (2+5 x+3 x^2\right )^{3/2}}{210 (3+2 x)^{5/2}}-\frac {(43+7 x) \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^{7/2}}-\frac {4091}{80} \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx+\frac {7515}{112} \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {(6292+1823 x) \sqrt {2+5 x+3 x^2}}{140 \sqrt {3+2 x}}-\frac {(3354+2531 x) \left (2+5 x+3 x^2\right )^{3/2}}{210 (3+2 x)^{5/2}}-\frac {(43+7 x) \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^{7/2}}-\frac {\left (4091 \sqrt {-2-5 x-3 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 x^2}{3}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {6+6 x}}{\sqrt {2}}\right )}{40 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {\left (2505 \sqrt {3} \sqrt {-2-5 x-3 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 x^2}{3}}} \, dx,x,\frac {\sqrt {6+6 x}}{\sqrt {2}}\right )}{56 \sqrt {2+5 x+3 x^2}}\\ &=\frac {(6292+1823 x) \sqrt {2+5 x+3 x^2}}{140 \sqrt {3+2 x}}-\frac {(3354+2531 x) \left (2+5 x+3 x^2\right )^{3/2}}{210 (3+2 x)^{5/2}}-\frac {(43+7 x) \left (2+5 x+3 x^2\right )^{5/2}}{35 (3+2 x)^{7/2}}-\frac {4091 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{40 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {2505 \sqrt {3} \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{56 \sqrt {2+5 x+3 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.39, size = 202, normalized size = 0.98 \[ -\frac {4536 x^7-29736 x^6+158172 x^5+2140148 x^4+6437058 x^3+8516152 x^2+5250234 x-6092 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} (2 x+3)^{9/2} \sqrt {\frac {3 x+2}{2 x+3}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )+28637 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} (2 x+3)^{9/2} \sqrt {\frac {3 x+2}{2 x+3}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )+1223436}{840 (2 x+3)^{7/2} \sqrt {3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^(9/2),x]

[Out]

-1/840*(1223436 + 5250234*x + 8516152*x^2 + 6437058*x^3 + 2140148*x^4 + 158172*x^5 - 29736*x^6 + 4536*x^7 + 28
637*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(9/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[
3 + 2*x]], 3/5] - 6092*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(9/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[Arc
Sin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])/((3 + 2*x)^(7/2)*Sqrt[2 + 5*x + 3*x^2])

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fricas [F]  time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (9 \, x^{5} - 15 \, x^{4} - 113 \, x^{3} - 165 \, x^{2} - 96 \, x - 20\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(9/2),x, algorithm="fricas")

[Out]

integral(-(9*x^5 - 15*x^4 - 113*x^3 - 165*x^2 - 96*x - 20)*sqrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3)/(32*x^5 + 240*x
^4 + 720*x^3 + 1080*x^2 + 810*x + 243), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(9/2),x, algorithm="giac")

[Out]

integrate(-(3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(9/2), x)

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maple [B]  time = 0.02, size = 399, normalized size = 1.93 \[ \frac {-45360 x^{7}+297360 x^{6}[phone] x^{5}[phone] x^{4}+229096 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{3} \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+71504 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{3} \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )[phone] x^{3}+1030932 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{2} \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+321768 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{2} \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )[phone] x^{2}+1546398 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+482652 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )[phone] x +773199 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+241326 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )[phone]}{8400 \sqrt {3 x^{2}+5 x +2}\, \left (2 x +3\right )^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+5*x+2)^(5/2)/(2*x+3)^(9/2),x)

[Out]

1/8400*(71504*15^(1/2)*EllipticF(1/5*(30*x+45)^(1/2),1/3*15^(1/2))*x^3*(2*x+3)^(1/2)*(-2*x-2)^(1/2)*(-30*x-20)
^(1/2)+229096*15^(1/2)*EllipticE(1/5*(30*x+45)^(1/2),1/3*15^(1/2))*x^3*(2*x+3)^(1/2)*(-2*x-2)^(1/2)*(-30*x-20)
^(1/2)+321768*15^(1/2)*EllipticF(1/5*(30*x+45)^(1/2),1/3*15^(1/2))*x^2*(2*x+3)^(1/2)*(-2*x-2)^(1/2)*(-30*x-20)
^(1/2)+1030932*15^(1/2)*EllipticE(1/5*(30*x+45)^(1/2),1/3*15^(1/2))*x^2*(2*x+3)^(1/2)*(-2*x-2)^(1/2)*(-30*x-20
)^(1/2)-45360*x^7+482652*15^(1/2)*EllipticF(1/5*(30*x+45)^(1/2),1/3*15^(1/2))*x*(2*x+3)^(1/2)*(-2*x-2)^(1/2)*(
-30*x-20)^(1/2)+1546398*15^(1/2)*EllipticE(1/5*(30*x+45)^(1/2),1/3*15^(1/2))*x*(2*x+3)^(1/2)*(-2*x-2)^(1/2)*(-
30*x-20)^(1/2)+297360*x^6+241326*(2*x+3)^(1/2)*15^(1/2)*(-2*x-2)^(1/2)*(-30*x-20)^(1/2)*EllipticF(1/5*(30*x+45
)^(1/2),1/3*15^(1/2))+773199*(2*x+3)^(1/2)*15^(1/2)*(-2*x-2)^(1/2)*(-30*x-20)^(1/2)*EllipticE(1/5*(30*x+45)^(1
/2),1/3*15^(1/2))[phone]*x^5+63364040*x^4+140670340*x^3+157107500*x^2+86673480*x+18693600)/(3*x^2+5*x+2)^(1/
2)/(2*x+3)^(7/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(9/2),x, algorithm="maxima")

[Out]

-integrate((3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(9/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ -\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2}}{{\left (2\,x+3\right )}^{9/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x - 5)*(5*x + 3*x^2 + 2)^(5/2))/(2*x + 3)^(9/2),x)

[Out]

-int(((x - 5)*(5*x + 3*x^2 + 2)^(5/2))/(2*x + 3)^(9/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- \frac {20 \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} \sqrt {2 x + 3} + 96 x^{3} \sqrt {2 x + 3} + 216 x^{2} \sqrt {2 x + 3} + 216 x \sqrt {2 x + 3} + 81 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {96 x \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} \sqrt {2 x + 3} + 96 x^{3} \sqrt {2 x + 3} + 216 x^{2} \sqrt {2 x + 3} + 216 x \sqrt {2 x + 3} + 81 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {165 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} \sqrt {2 x + 3} + 96 x^{3} \sqrt {2 x + 3} + 216 x^{2} \sqrt {2 x + 3} + 216 x \sqrt {2 x + 3} + 81 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {113 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} \sqrt {2 x + 3} + 96 x^{3} \sqrt {2 x + 3} + 216 x^{2} \sqrt {2 x + 3} + 216 x \sqrt {2 x + 3} + 81 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {15 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} \sqrt {2 x + 3} + 96 x^{3} \sqrt {2 x + 3} + 216 x^{2} \sqrt {2 x + 3} + 216 x \sqrt {2 x + 3} + 81 \sqrt {2 x + 3}}\right )\, dx - \int \frac {9 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} \sqrt {2 x + 3} + 96 x^{3} \sqrt {2 x + 3} + 216 x^{2} \sqrt {2 x + 3} + 216 x \sqrt {2 x + 3} + 81 \sqrt {2 x + 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x**2+5*x+2)**(5/2)/(3+2*x)**(9/2),x)

[Out]

-Integral(-20*sqrt(3*x**2 + 5*x + 2)/(16*x**4*sqrt(2*x + 3) + 96*x**3*sqrt(2*x + 3) + 216*x**2*sqrt(2*x + 3) +
216*x*sqrt(2*x + 3) + 81*sqrt(2*x + 3)), x) - Integral(-96*x*sqrt(3*x**2 + 5*x + 2)/(16*x**4*sqrt(2*x + 3) +
96*x**3*sqrt(2*x + 3) + 216*x**2*sqrt(2*x + 3) + 216*x*sqrt(2*x + 3) + 81*sqrt(2*x + 3)), x) - Integral(-165*x
**2*sqrt(3*x**2 + 5*x + 2)/(16*x**4*sqrt(2*x + 3) + 96*x**3*sqrt(2*x + 3) + 216*x**2*sqrt(2*x + 3) + 216*x*sqr
t(2*x + 3) + 81*sqrt(2*x + 3)), x) - Integral(-113*x**3*sqrt(3*x**2 + 5*x + 2)/(16*x**4*sqrt(2*x + 3) + 96*x**
3*sqrt(2*x + 3) + 216*x**2*sqrt(2*x + 3) + 216*x*sqrt(2*x + 3) + 81*sqrt(2*x + 3)), x) - Integral(-15*x**4*sqr
t(3*x**2 + 5*x + 2)/(16*x**4*sqrt(2*x + 3) + 96*x**3*sqrt(2*x + 3) + 216*x**2*sqrt(2*x + 3) + 216*x*sqrt(2*x +
3) + 81*sqrt(2*x + 3)), x) - Integral(9*x**5*sqrt(3*x**2 + 5*x + 2)/(16*x**4*sqrt(2*x + 3) + 96*x**3*sqrt(2*x
+ 3) + 216*x**2*sqrt(2*x + 3) + 216*x*sqrt(2*x + 3) + 81*sqrt(2*x + 3)), x)

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