3.27.5 \(\int \frac {(5-x) (2+5 x+3 x^2)^{5/2}}{(3+2 x)^{17/2}} \, dx\) [2605]

Optimal. Leaf size=261 \[ \frac {594851 \sqrt {2+5 x+3 x^2}}{112612500 (3+2 x)^{3/2}}+\frac {335723 \sqrt {2+5 x+3 x^2}}{80437500 \sqrt {3+2 x}}-\frac {(386846+328339 x) \sqrt {2+5 x+3 x^2}}{7507500 (3+2 x)^{7/2}}-\frac {(8901+8399 x) \left (2+5 x+3 x^2\right )^{3/2}}{64350 (3+2 x)^{11/2}}+\frac {(94+119 x) \left (2+5 x+3 x^2\right )^{5/2}}{195 (3+2 x)^{15/2}}-\frac {335723 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{53625000 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {594851 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{75075000 \sqrt {3} \sqrt {2+5 x+3 x^2}} \]

[Out]

-1/64350*(8901+8399*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(11/2)+1/195*(94+119*x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(15/2)-
335723/160875000*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)
+594851/225225000*EllipticF(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
)+594851/112612500*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(3/2)-1/7507500*(386846+328339*x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(
7/2)+335723/80437500*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(1/2)

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Rubi [A]
time = 0.12, antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {824, 848, 857, 732, 435, 430} \begin {gather*} \frac {594851 \sqrt {-3 x^2-5 x-2} F\left (\text {ArcSin}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{75075000 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {335723 \sqrt {-3 x^2-5 x-2} E\left (\text {ArcSin}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{53625000 \sqrt {3} \sqrt {3 x^2+5 x+2}}+\frac {(119 x+94) \left (3 x^2+5 x+2\right )^{5/2}}{195 (2 x+3)^{15/2}}-\frac {(8399 x+8901) \left (3 x^2+5 x+2\right )^{3/2}}{64350 (2 x+3)^{11/2}}-\frac {(328339 x+386846) \sqrt {3 x^2+5 x+2}}{7507500 (2 x+3)^{7/2}}+\frac {335723 \sqrt {3 x^2+5 x+2}}{80437500 \sqrt {2 x+3}}+\frac {594851 \sqrt {3 x^2+5 x+2}}{112612500 (2 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^(17/2),x]

[Out]

(594851*Sqrt[2 + 5*x + 3*x^2])/(112612500*(3 + 2*x)^(3/2)) + (335723*Sqrt[2 + 5*x + 3*x^2])/(80437500*Sqrt[3 +
2*x]) - ((386846 + 328339*x)*Sqrt[2 + 5*x + 3*x^2])/(7507500*(3 + 2*x)^(7/2)) - ((8901 + 8399*x)*(2 + 5*x + 3
*x^2)^(3/2))/(64350*(3 + 2*x)^(11/2)) + ((94 + 119*x)*(2 + 5*x + 3*x^2)^(5/2))/(195*(3 + 2*x)^(15/2)) - (33572
3*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(53625000*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]
) + (594851*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(75075000*Sqrt[3]*Sqrt[2 + 5*
x + 3*x^2])

Rule 430

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

Rule 435

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

Rule 732

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 824

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((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), 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 848

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 857

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)^{17/2}} \, dx &=\frac {(94+119 x) \left (2+5 x+3 x^2\right )^{5/2}}{195 (3+2 x)^{15/2}}-\frac {1}{390} \int \frac {(-118-243 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{13/2}} \, dx\\ &=-\frac {(8901+8399 x) \left (2+5 x+3 x^2\right )^{3/2}}{64350 (3+2 x)^{11/2}}+\frac {(94+119 x) \left (2+5 x+3 x^2\right )^{5/2}}{195 (3+2 x)^{15/2}}+\frac {\int \frac {(20841+22347 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^{9/2}} \, dx}{128700}\\ &=-\frac {(386846+328339 x) \sqrt {2+5 x+3 x^2}}{7507500 (3+2 x)^{7/2}}-\frac {(8901+8399 x) \left (2+5 x+3 x^2\right )^{3/2}}{64350 (3+2 x)^{11/2}}+\frac {(94+119 x) \left (2+5 x+3 x^2\right )^{5/2}}{195 (3+2 x)^{15/2}}-\frac {\int \frac {-1257990-1433511 x}{(3+2 x)^{5/2} \sqrt {2+5 x+3 x^2}} \, dx}{45045000}\\ &=\frac {594851 \sqrt {2+5 x+3 x^2}}{112612500 (3+2 x)^{3/2}}-\frac {(386846+328339 x) \sqrt {2+5 x+3 x^2}}{7507500 (3+2 x)^{7/2}}-\frac {(8901+8399 x) \left (2+5 x+3 x^2\right )^{3/2}}{64350 (3+2 x)^{11/2}}+\frac {(94+119 x) \left (2+5 x+3 x^2\right )^{5/2}}{195 (3+2 x)^{15/2}}+\frac {\int \frac {\frac {4505397}{2}+\frac {5353659 x}{2}}{(3+2 x)^{3/2} \sqrt {2+5 x+3 x^2}} \, dx}{337837500}\\ &=\frac {594851 \sqrt {2+5 x+3 x^2}}{112612500 (3+2 x)^{3/2}}+\frac {335723 \sqrt {2+5 x+3 x^2}}{80437500 \sqrt {3+2 x}}-\frac {(386846+328339 x) \sqrt {2+5 x+3 x^2}}{7507500 (3+2 x)^{7/2}}-\frac {(8901+8399 x) \left (2+5 x+3 x^2\right )^{3/2}}{64350 (3+2 x)^{11/2}}+\frac {(94+119 x) \left (2+5 x+3 x^2\right )^{5/2}}{195 (3+2 x)^{15/2}}-\frac {\int \frac {4585419+\frac {21150549 x}{4}}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{844593750}\\ &=\frac {594851 \sqrt {2+5 x+3 x^2}}{112612500 (3+2 x)^{3/2}}+\frac {335723 \sqrt {2+5 x+3 x^2}}{80437500 \sqrt {3+2 x}}-\frac {(386846+328339 x) \sqrt {2+5 x+3 x^2}}{7507500 (3+2 x)^{7/2}}-\frac {(8901+8399 x) \left (2+5 x+3 x^2\right )^{3/2}}{64350 (3+2 x)^{11/2}}+\frac {(94+119 x) \left (2+5 x+3 x^2\right )^{5/2}}{195 (3+2 x)^{15/2}}-\frac {335723 \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx}{107250000}+\frac {594851 \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{150150000}\\ &=\frac {594851 \sqrt {2+5 x+3 x^2}}{112612500 (3+2 x)^{3/2}}+\frac {335723 \sqrt {2+5 x+3 x^2}}{80437500 \sqrt {3+2 x}}-\frac {(386846+328339 x) \sqrt {2+5 x+3 x^2}}{7507500 (3+2 x)^{7/2}}-\frac {(8901+8399 x) \left (2+5 x+3 x^2\right )^{3/2}}{64350 (3+2 x)^{11/2}}+\frac {(94+119 x) \left (2+5 x+3 x^2\right )^{5/2}}{195 (3+2 x)^{15/2}}-\frac {\left (335723 \sqrt {-2-5 x-3 x^2}\right ) \text {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 )}{53625000 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {\left (594851 \sqrt {-2-5 x-3 x^2}\right ) \text {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 )}{75075000 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ &=\frac {594851 \sqrt {2+5 x+3 x^2}}{112612500 (3+2 x)^{3/2}}+\frac {335723 \sqrt {2+5 x+3 x^2}}{80437500 \sqrt {3+2 x}}-\frac {(386846+328339 x) \sqrt {2+5 x+3 x^2}}{7507500 (3+2 x)^{7/2}}-\frac {(8901+8399 x) \left (2+5 x+3 x^2\right )^{3/2}}{64350 (3+2 x)^{11/2}}+\frac {(94+119 x) \left (2+5 x+3 x^2\right )^{5/2}}{195 (3+2 x)^{15/2}}-\frac {335723 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{53625000 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {594851 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{75075000 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ \end {align*}

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Mathematica [A]
time = 20.51, size = 237, normalized size = 0.91 \begin {gather*} -\frac {-8 \left (2+5 x+3 x^2\right ) \left (4641518352+24502214271 x+55283449932 x^2+67557035830 x^3+46830142120 x^4+17742950508 x^5+3348834304 x^6+300807808 x^7\right )+2 (3+2 x)^7 \left (9400244 \left (2+5 x+3 x^2\right )+4700122 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{3/2} \sqrt {\frac {2+3 x}{3+2 x}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )-1131016 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{3/2} \sqrt {\frac {2+3 x}{3+2 x}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )\right )}{4504500000 (3+2 x)^{15/2} \sqrt {2+5 x+3 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^(17/2),x]

[Out]

-1/4504500000*(-8*(2 + 5*x + 3*x^2)*(4641518352 + 24502214271*x + 55283449932*x^2 + 67557035830*x^3 + 46830142
120*x^4 + 17742950508*x^5 + 3348834304*x^6 + 300807808*x^7) + 2*(3 + 2*x)^7*(9400244*(2 + 5*x + 3*x^2) + 47001
22*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] - 1131016*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(3/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[A
rcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5]))/((3 + 2*x)^(15/2)*Sqrt[2 + 5*x + 3*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(760\) vs. \(2(211)=422\).
time = 0.10, size = 761, normalized size = 2.92

method	result	size
elliptic	\(\frac {\sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {65 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{24576 \left (x +\frac {3}{2}\right )^{8}}+\frac {3299 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{159744 \left (x +\frac {3}{2}\right )^{7}}-\frac {11439 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{183040 \left (x +\frac {3}{2}\right )^{6}}+\frac {149093 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{1647360 \left (x +\frac {3}{2}\right )^{5}}-\frac {636491 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{10483200 \left (x +\frac {3}{2}\right )^{4}}+\frac {6055369 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{480480000 \left (x +\frac {3}{2}\right )^{3}}+\frac {594851 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{450450000 \left (x +\frac {3}{2}\right )^{2}}+\frac {\frac {335723}{26812500} x^{2}+\frac {335723}{16087500} x +\frac {335723}{40218750}}{\sqrt {\left (x +\frac {3}{2}\right ) \left (6 x^{2}+10 x +4\right )}}-\frac {509491 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{1407656250 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}-\frac {335723 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{2}-\EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )\right )}{804375000 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(366\)
default	\(\frac {185660734080+1444240406040 x [phone] \sqrt {15}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x^{4} \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}[phone] \sqrt {15}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x^{7} \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}[phone] \sqrt {15}\, \EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x^{7} \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}[phone] \sqrt {15}\, \EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x^{6} \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}[phone] \sqrt {15}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x^{6} \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}[phone] \sqrt {15}\, \EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x^{5} \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}[phone] \sqrt {15}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x^{5} \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}[phone] \sqrt {15}\, \EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x^{4} \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}[phone] \sqrt {15}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x^{3} \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \sqrt {3+2 x}[phone] \sqrt {15}\, \EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x^{3} \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \sqrt {3+2 x}[phone] \sqrt {15}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x^{2} \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}[phone] \sqrt {15}\, \EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x^{2} \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}[phone] \sqrt {15}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x \sqrt {-20-30 x}\, \sqrt {3+2 x}\, \sqrt {-2-2 x}[phone] \sqrt {15}\, \EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x \sqrt {-20-30 x}\, \sqrt {3+2 x}\, \sqrt {-2-2 x}[phone] x^{2}[phone] x^{5}[phone] x^{6}[phone] x^{7}[phone] x^{4}[phone] x^{3}[phone] x^{8}[phone] x^{9}[phone] \sqrt {15}\, \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )[phone] \sqrt {15}\, \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{11261250000 \sqrt {3 x^{2}+5 x +2}\, \left (3+2 x \right )^{\frac {15}{2}}}\)	\(761\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(17/2),x,method=_RETURNVERBOSE)

[Out]

1/11261250000*(185660734080+1444240406040*x+79896832*15^(1/2)*EllipticF(1/5*(45+30*x)^(1/2),1/3*15^(1/2))*x^7*
(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-20-30*x)^(1/2)[phone]*15^(1/2)*EllipticE(1/5*(45+30*x)^(1/2),1/3*15^(1/2))*x
^7*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-20-30*x)^(1/2)[phone]*15^(1/2)*EllipticE(1/5*(45+30*x)^(1/2),1/3*15^(1/2
))*x^6*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-20-30*x)^(1/2)[phone]*15^(1/2)*EllipticF(1/5*(45+30*x)^(1/2),1/3*15^(
1/2))*x^6*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-20-30*x)^(1/2)[phone]*15^(1/2)*EllipticE(1/5*(45+30*x)^(1/2),1/3
*15^(1/2))*x^5*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-20-30*x)^(1/2)[phone]*15^(1/2)*EllipticF(1/5*(45+30*x)^(1/2)
,1/3*15^(1/2))*x^5*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-20-30*x)^(1/2)[phone]*15^(1/2)*EllipticF(1/5*(45+30*x)^(
1/2),1/3*15^(1/2))*x^4*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-20-30*x)^(1/2)[phone]*15^(1/2)*EllipticE(1/5*(45+30
*x)^(1/2),1/3*15^(1/2))*x^4*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-20-30*x)^(1/2)[phone]*15^(1/2)*EllipticF(1/5*(
45+30*x)^(1/2),1/3*15^(1/2))*x^3*(-2-2*x)^(1/2)*(-20-30*x)^(1/2)*(3+2*x)^(1/2)[phone]*15^(1/2)*EllipticE(
1/5*(45+30*x)^(1/2),1/3*15^(1/2))*x^3*(-2-2*x)^(1/2)*(-20-30*x)^(1/2)*(3+2*x)^(1/2)[phone]*15^(1/2)*Ellip
ticF(1/5*(45+30*x)^(1/2),1/3*15^(1/2))*x^2*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-20-30*x)^(1/2)[phone]*15^(1/2)*
EllipticE(1/5*(45+30*x)^(1/2),1/3*15^(1/2))*x^2*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-20-30*x)^(1/2)[phone]*15^(1
/2)*EllipticF(1/5*(45+30*x)^(1/2),1/3*15^(1/2))*x*(-20-30*x)^(1/2)*(3+2*x)^(1/2)*(-2-2*x)^(1/2)[phone]*15
^(1/2)*EllipticE(1/5*(45+30*x)^(1/2),1/3*15^(1/2))*x*(-20-30*x)^(1/2)*(3+2*x)^(1/2)*(-2-2*x)^(1/2)[phone]
00*x^2+9446154382120*x^5+4718056950160*x^6+1411492773200*x^7+11945916263720*x^4+9700759282660*x^3+231010839040
*x^8+18048468480*x^9+1365112278*15^(1/2)*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-20-30*x)^(1/2)*EllipticF(1/5*(45+30*x)
^(1/2),1/3*15^(1/2))[phone]*15^(1/2)*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-20-30*x)^(1/2)*EllipticE(1/5*(45+30*x)
^(1/2),1/3*15^(1/2)))/(3*x^2+5*x+2)^(1/2)/(3+2*x)^(15/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)^(17/2),x, algorithm="maxima")

[Out]

-integrate((3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(17/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.91, size = 206, normalized size = 0.79 \begin {gather*} \frac {7967807 \, \sqrt {6} {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 42301098 \, \sqrt {6} {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) + 36 \, {\left (300807808 \, x^{7} + 3348834304 \, x^{6} + 17742950508 \, x^{5} + 46830142120 \, x^{4} + 67557035830 \, x^{3} + 55283449932 \, x^{2} + 24502214271 \, x + 4641518352\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{20270250000 \, {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )}} \end {gather*}

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)^(17/2),x, algorithm="fricas")

[Out]

1/20270250000*(7967807*sqrt(6)*(256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^5 + 90720*x^4 + 108864*x^3 + 81648*x^
2 + 34992*x + 6561)*weierstrassPInverse(19/27, -28/729, x + 19/18) + 42301098*sqrt(6)*(256*x^8 + 3072*x^7 + 16
128*x^6 + 48384*x^5 + 90720*x^4 + 108864*x^3 + 81648*x^2 + 34992*x + 6561)*weierstrassZeta(19/27, -28/729, wei
erstrassPInverse(19/27, -28/729, x + 19/18)) + 36*(300807808*x^7 + 3348834304*x^6 + 17742950508*x^5 + 46830142
120*x^4 + 67557035830*x^3 + 55283449932*x^2 + 24502214271*x + 4641518352)*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)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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)**(17/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 5985 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)^(17/2),x, algorithm="giac")

[Out]

integrate(-(3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(17/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2}}{{\left (2\,x+3\right )}^{17/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x - 5)*(5*x + 3*x^2 + 2)^(5/2))/(2*x + 3)^(17/2),x)

[Out]

-int(((x - 5)*(5*x + 3*x^2 + 2)^(5/2))/(2*x + 3)^(17/2), x)

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