3.2604 \(\int \frac {(5-x) (2+5 x+3 x^2)^{5/2}}{(3+2 x)^{15/2}} \, dx\)

Optimal. Leaf size=234 \[ \frac {(9 x+8) \left (3 x^2+5 x+2\right )^{5/2}}{11 (2 x+3)^{13/2}}+\frac {(73-33 x) \left (3 x^2+5 x+2\right )^{3/2}}{6930 (2 x+3)^{9/2}}+\frac {(17833 x+21492) \sqrt {3 x^2+5 x+2}}{346500 (2 x+3)^{5/2}}-\frac {5083 \sqrt {3 x^2+5 x+2}}{247500 \sqrt {2 x+3}}-\frac {9421 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{231000 \sqrt {3} \sqrt {3 x^2+5 x+2}}+\frac {5083 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{165000 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]

[Out]

1/6930*(73-33*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(9/2)+1/11*(8+9*x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(13/2)+5083/495000
*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)-9421/693000*Ell
ipticF(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)+1/346500*(21492+178
33*x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(5/2)-5083/247500*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(1/2)

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Rubi [A]  time = 0.16, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {810, 834, 843, 718, 424, 419} \[ \frac {(9 x+8) \left (3 x^2+5 x+2\right )^{5/2}}{11 (2 x+3)^{13/2}}+\frac {(73-33 x) \left (3 x^2+5 x+2\right )^{3/2}}{6930 (2 x+3)^{9/2}}+\frac {(17833 x+21492) \sqrt {3 x^2+5 x+2}}{346500 (2 x+3)^{5/2}}-\frac {5083 \sqrt {3 x^2+5 x+2}}{247500 \sqrt {2 x+3}}-\frac {9421 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{231000 \sqrt {3} \sqrt {3 x^2+5 x+2}}+\frac {5083 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{165000 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^(15/2),x]

[Out]

(-5083*Sqrt[2 + 5*x + 3*x^2])/(247500*Sqrt[3 + 2*x]) + ((21492 + 17833*x)*Sqrt[2 + 5*x + 3*x^2])/(346500*(3 +
2*x)^(5/2)) + ((73 - 33*x)*(2 + 5*x + 3*x^2)^(3/2))/(6930*(3 + 2*x)^(9/2)) + ((8 + 9*x)*(2 + 5*x + 3*x^2)^(5/2
))/(11*(3 + 2*x)^(13/2)) + (5083*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(165000*
Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) - (9421*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(2
31000*Sqrt[3]*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 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (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)^{15/2}} \, dx &=\frac {(8+9 x) \left (2+5 x+3 x^2\right )^{5/2}}{11 (3+2 x)^{13/2}}-\frac {1}{286} \int \frac {(104+39 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{11/2}} \, dx\\ &=\frac {(73-33 x) \left (2+5 x+3 x^2\right )^{3/2}}{6930 (3+2 x)^{9/2}}+\frac {(8+9 x) \left (2+5 x+3 x^2\right )^{5/2}}{11 (3+2 x)^{13/2}}+\frac {\int \frac {(-4979-6357 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^{7/2}} \, dx}{60060}\\ &=\frac {(21492+17833 x) \sqrt {2+5 x+3 x^2}}{346500 (3+2 x)^{5/2}}+\frac {(73-33 x) \left (2+5 x+3 x^2\right )^{3/2}}{6930 (3+2 x)^{9/2}}+\frac {(8+9 x) \left (2+5 x+3 x^2\right )^{5/2}}{11 (3+2 x)^{13/2}}-\frac {\int \frac {319852+367419 x}{(3+2 x)^{3/2} \sqrt {2+5 x+3 x^2}} \, dx}{9009000}\\ &=-\frac {5083 \sqrt {2+5 x+3 x^2}}{247500 \sqrt {3+2 x}}+\frac {(21492+17833 x) \sqrt {2+5 x+3 x^2}}{346500 (3+2 x)^{5/2}}+\frac {(73-33 x) \left (2+5 x+3 x^2\right )^{3/2}}{6930 (3+2 x)^{9/2}}+\frac {(8+9 x) \left (2+5 x+3 x^2\right )^{5/2}}{11 (3+2 x)^{13/2}}+\frac {\int \frac {\frac {1162941}{2}+\frac {1387659 x}{2}}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{22522500}\\ &=-\frac {5083 \sqrt {2+5 x+3 x^2}}{247500 \sqrt {3+2 x}}+\frac {(21492+17833 x) \sqrt {2+5 x+3 x^2}}{346500 (3+2 x)^{5/2}}+\frac {(73-33 x) \left (2+5 x+3 x^2\right )^{3/2}}{6930 (3+2 x)^{9/2}}+\frac {(8+9 x) \left (2+5 x+3 x^2\right )^{5/2}}{11 (3+2 x)^{13/2}}+\frac {5083 \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx}{330000}-\frac {9421 \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{462000}\\ &=-\frac {5083 \sqrt {2+5 x+3 x^2}}{247500 \sqrt {3+2 x}}+\frac {(21492+17833 x) \sqrt {2+5 x+3 x^2}}{346500 (3+2 x)^{5/2}}+\frac {(73-33 x) \left (2+5 x+3 x^2\right )^{3/2}}{6930 (3+2 x)^{9/2}}+\frac {(8+9 x) \left (2+5 x+3 x^2\right )^{5/2}}{11 (3+2 x)^{13/2}}+\frac {\left (5083 \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 )}{165000 \sqrt {3} \sqrt {2+5 x+3 x^2}}-\frac {\left (9421 \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 )}{231000 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ &=-\frac {5083 \sqrt {2+5 x+3 x^2}}{247500 \sqrt {3+2 x}}+\frac {(21492+17833 x) \sqrt {2+5 x+3 x^2}}{346500 (3+2 x)^{5/2}}+\frac {(73-33 x) \left (2+5 x+3 x^2\right )^{3/2}}{6930 (3+2 x)^{9/2}}+\frac {(8+9 x) \left (2+5 x+3 x^2\right )^{5/2}}{11 (3+2 x)^{13/2}}+\frac {5083 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{165000 \sqrt {3} \sqrt {2+5 x+3 x^2}}-\frac {9421 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{231000 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.51, size = 232, normalized size = 0.99 \[ -\frac {8 \left (3 x^2+5 x+2\right ) \left (2277184 x^6+6409516 x^5+12953760 x^4+33648370 x^3+54318160 x^2+41339721 x+11865789\right )-4 (2 x+3)^6 \left (71162 \left (3 x^2+5 x+2\right )-7318 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^{3/2} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )+35581 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^{3/2} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )\right )}{13860000 (2 x+3)^{13/2} \sqrt {3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^(15/2),x]

[Out]

-1/13860000*(8*(2 + 5*x + 3*x^2)*(11865789 + 41339721*x + 54318160*x^2 + 33648370*x^3 + 12953760*x^4 + 6409516
*x^5 + 2277184*x^6) - 4*(3 + 2*x)^6*(71162*(2 + 5*x + 3*x^2) + 35581*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)
^(3/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] - 7318*Sqrt[5]*Sqrt[(1 + x)/(
3 + 2*x)]*(3 + 2*x)^(3/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5]))/((3 + 2*
x)^(13/2)*Sqrt[2 + 5*x + 3*x^2])

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fricas [F]  time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (9 \, x^{5} - 15 \, x^{4} - 113 \, x^{3} - 165 \, x^{2} - 96 \, x - 20\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561}, x\right ) \]

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)^(15/2),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)/(256*x^8 + 3072
*x^7 + 16128*x^6 + 48384*x^5 + 90720*x^4 + 108864*x^3 + 81648*x^2 + 34992*x + 6561), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {15}{2}}}\,{d x} \]

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)^(15/2),x, algorithm="giac")

[Out]

integrate(-(3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(15/2), x)

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maple [B]  time = 0.04, size = 668, normalized size = 2.85 \[ -\frac {136631040 x^{8}[phone] x^{7}+2277184 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{6} \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+737536 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{6} \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )[phone] x^{6}[phone] \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{5} \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+6637824 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{5} \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )[phone] x^{5}[phone] \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \sqrt {2 x +3}\, x^{4} \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )[phone] \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \sqrt {2 x +3}\, x^{4} \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )[phone] x^{4}[phone] \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 )[phone] \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}[phone] \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 )[phone] \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}[phone] \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 )[phone] \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 [phone] \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 )+8400996 \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]}{34650000 \sqrt {3 x^{2}+5 x +2}\, \left (2 x +3\right )^{\frac {13}{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)^(15/2),x)

[Out]

-1/34650000*(737536*15^(1/2)*EllipticF(1/5*(30*x+45)^(1/2),1/3*15^(1/2))*x^6*(2*x+3)^(1/2)*(-2*x-2)^(1/2)*(-30
*x-20)^(1/2)+2277184*15^(1/2)*EllipticE(1/5*(30*x+45)^(1/2),1/3*15^(1/2))*x^6*(2*x+3)^(1/2)*(-2*x-2)^(1/2)*(-3
0*x-20)^(1/2)+6637824*15^(1/2)*EllipticF(1/5*(30*x+45)^(1/2),1/3*15^(1/2))*x^5*(2*x+3)^(1/2)*(-2*x-2)^(1/2)*(-
30*x-20)^(1/2)[phone]*15^(1/2)*EllipticE(1/5*(30*x+45)^(1/2),1/3*15^(1/2))*x^5*(2*x+3)^(1/2)*(-2*x-2)^(1/2)*
(-30*x-20)^(1/2)[phone]*15^(1/2)*EllipticF(1/5*(30*x+45)^(1/2),1/3*15^(1/2))*x^4*(-2*x-2)^(1/2)*(-30*x-20)^(
1/2)*(2*x+3)^(1/2)[phone]*15^(1/2)*EllipticE(1/5*(30*x+45)^(1/2),1/3*15^(1/2))*x^4*(-2*x-2)^(1/2)*(-30*x-20)
^(1/2)*(2*x+3)^(1/2)[phone]*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)[phone]*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)[phone]*x^8+56006640*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)[phone]*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)[phone]*x^7+33603984*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)[phone]*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)[phone]*x^6+8400996*(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))[phone]*(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+7142077000*x^4+9258134060
*x^3+7018645840*x^2+2840167740*x+474631560)/(3*x^2+5*x+2)^(1/2)/(2*x+3)^(13/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac {15}{2}}}\,{d x} \]

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)^(15/2),x, algorithm="maxima")

[Out]

-integrate((3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(15/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ -\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2}}{{\left (2\,x+3\right )}^{15/2}} \,d x \]

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)^(15/2),x)

[Out]

-int(((x - 5)*(5*x + 3*x^2 + 2)^(5/2))/(2*x + 3)^(15/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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)**(15/2),x)

[Out]

Timed out

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