3.26.94 \(\int (5-x) (3+2 x)^{5/2} (2+5 x+3 x^2)^{5/2} \, dx\) [2594]

Optimal. Leaf size=288 \[ \frac {25 \sqrt {3+2 x} (749099+216603 x) \sqrt {2+5 x+3 x^2}}{942809868}-\frac {125 \sqrt {3+2 x} (64006+79583 x) \left (2+5 x+3 x^2\right )^{3/2}}{52378326}+\frac {25 \sqrt {3+2 x} (72737+86493 x) \left (2+5 x+3 x^2\right )^{5/2}}{1247103}+\frac {2350 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{7/2}}{2907}+\frac {430}{969} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac {2}{57} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac {16503475 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{269374248 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {142149125 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{1885619736 \sqrt {3} \sqrt {2+5 x+3 x^2}} \]

[Out]

430/969*(3+2*x)^(3/2)*(3*x^2+5*x+2)^(7/2)-2/57*(3+2*x)^(5/2)*(3*x^2+5*x+2)^(7/2)-125/52378326*(64006+79583*x)*
(3*x^2+5*x+2)^(3/2)*(3+2*x)^(1/2)+25/1247103*(72737+86493*x)*(3*x^2+5*x+2)^(5/2)*(3+2*x)^(1/2)+2350/2907*(3*x^
2+5*x+2)^(7/2)*(3+2*x)^(1/2)-16503475/808122744*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)[phone]/5656859208*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)+25/942809868*(749099+216603*x)*(3+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)

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Rubi [A]
time = 0.15, antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {846, 828, 857, 732, 435, 430} \begin {gather*} \frac {142149125 \sqrt {-3 x^2-5 x-2} F\left (\text {ArcSin}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{1885619736 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {16503475 \sqrt {-3 x^2-5 x-2} E\left (\text {ArcSin}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{269374248 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {2}{57} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{7/2}+\frac {430}{969} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}+\frac {2350 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{7/2}}{2907}+\frac {25 \sqrt {2 x+3} (86493 x+72737) \left (3 x^2+5 x+2\right )^{5/2}}{1247103}-\frac {125 \sqrt {2 x+3} (79583 x+64006) \left (3 x^2+5 x+2\right )^{3/2}}{52378326}+\frac {25 \sqrt {2 x+3} (216603 x+749099) \sqrt {3 x^2+5 x+2}}{942809868} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(25*Sqrt[3 + 2*x]*(749099 + 216603*x)*Sqrt[2 + 5*x + 3*x^2])/942809868 - (125*Sqrt[3 + 2*x]*(64006 + 79583*x)*
(2 + 5*x + 3*x^2)^(3/2))/52378326 + (25*Sqrt[3 + 2*x]*(72737 + 86493*x)*(2 + 5*x + 3*x^2)^(5/2))/1247103 + (23
50*Sqrt[3 + 2*x]*(2 + 5*x + 3*x^2)^(7/2))/2907 + (430*(3 + 2*x)^(3/2)*(2 + 5*x + 3*x^2)^(7/2))/969 - (2*(3 + 2
*x)^(5/2)*(2 + 5*x + 3*x^2)^(7/2))/57 - (16503475*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]]
, -2/3])/(269374248*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (142149125*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3
]*Sqrt[1 + x]], -2/3])/(1885619736*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 828

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 846

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m
- 1)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m*(c*e*f + c*d*g - b*e*g) + e*(p
+ 1)*(2*c*f - b*g))*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] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 0])

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 (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2} \, dx &=-\frac {2}{57} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{7/2}+\frac {2}{57} \int (3+2 x)^{3/2} \left (490+\frac {645 x}{2}\right ) \left (2+5 x+3 x^2\right )^{5/2} \, dx\\ &=\frac {430}{969} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac {2}{57} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{7/2}+\frac {4 \int \sqrt {3+2 x} \left (\frac {74475}{4}+\frac {52875 x}{4}\right ) \left (2+5 x+3 x^2\right )^{5/2} \, dx}{2907}\\ &=\frac {2350 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{7/2}}{2907}+\frac {430}{969} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac {2}{57} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{7/2}+\frac {8 \int \frac {\left (\frac {2145375}{4}+\frac {2948625 x}{8}\right ) \left (2+5 x+3 x^2\right )^{5/2}}{\sqrt {3+2 x}} \, dx}{130815}\\ &=\frac {25 \sqrt {3+2 x} (72737+86493 x) \left (2+5 x+3 x^2\right )^{5/2}}{1247103}+\frac {2350 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{7/2}}{2907}+\frac {430}{969} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac {2}{57} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac {4 \int \frac {\left (\frac {53845875}{8}+\frac {38370375 x}{8}\right ) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt {3+2 x}} \, dx}{11223927}\\ &=-\frac {125 \sqrt {3+2 x} (64006+79583 x) \left (2+5 x+3 x^2\right )^{3/2}}{52378326}+\frac {25 \sqrt {3+2 x} (72737+86493 x) \left (2+5 x+3 x^2\right )^{5/2}}{1247103}+\frac {2350 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{7/2}}{2907}+\frac {430}{969} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac {2}{57} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{7/2}+\frac {2 \int \frac {\left (\frac {119471625}{4}+\frac {81226125 x}{8}\right ) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx}{707107401}\\ &=\frac {25 \sqrt {3+2 x} (749099+216603 x) \sqrt {2+5 x+3 x^2}}{942809868}-\frac {125 \sqrt {3+2 x} (64006+79583 x) \left (2+5 x+3 x^2\right )^{3/2}}{52378326}+\frac {25 \sqrt {3+2 x} (72737+86493 x) \left (2+5 x+3 x^2\right )^{5/2}}{1247103}+\frac {2350 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{7/2}}{2907}+\frac {430}{969} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac {2}{57} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac {\int \frac {\frac {13798609875}{8}+\frac {15595783875 x}{8}}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{31819833045}\\ &=\frac {25 \sqrt {3+2 x} (749099+216603 x) \sqrt {2+5 x+3 x^2}}{942809868}-\frac {125 \sqrt {3+2 x} (64006+79583 x) \left (2+5 x+3 x^2\right )^{3/2}}{52378326}+\frac {25 \sqrt {3+2 x} (72737+86493 x) \left (2+5 x+3 x^2\right )^{5/2}}{1247103}+\frac {2350 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{7/2}}{2907}+\frac {430}{969} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac {2}{57} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac {16503475 \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx}{538748496}+\frac {142149125 \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{3771239472}\\ &=\frac {25 \sqrt {3+2 x} (749099+216603 x) \sqrt {2+5 x+3 x^2}}{942809868}-\frac {125 \sqrt {3+2 x} (64006+79583 x) \left (2+5 x+3 x^2\right )^{3/2}}{52378326}+\frac {25 \sqrt {3+2 x} (72737+86493 x) \left (2+5 x+3 x^2\right )^{5/2}}{1247103}+\frac {2350 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{7/2}}{2907}+\frac {430}{969} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac {2}{57} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac {\left (16503475 \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 )}{269374248 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {\left (142149125 \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 )}{1885619736 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ &=\frac {25 \sqrt {3+2 x} (749099+216603 x) \sqrt {2+5 x+3 x^2}}{942809868}-\frac {125 \sqrt {3+2 x} (64006+79583 x) \left (2+5 x+3 x^2\right )^{3/2}}{52378326}+\frac {25 \sqrt {3+2 x} (72737+86493 x) \left (2+5 x+3 x^2\right )^{5/2}}{1247103}+\frac {2350 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{7/2}}{2907}+\frac {430}{969} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac {2}{57} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac {16503475 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{269374248 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {142149125 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{1885619736 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ \end {align*}

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Mathematica [A]
time = 30.32, size = 228, normalized size = 0.79 \begin {gather*} -\frac {2 \sqrt {3+2 x} \left (-341519551612-3595384785664 x-16735272462363 x^2-45255052994607 x^3-78460508136978 x^4-90580760151282 x^5-69684837178068 x^6-34294970344572 x^7-9445976815968 x^8-694795413312 x^9+311460012864 x^{10}[phone] x^{11}\right )[phone] \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^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 )-30234850 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^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 )}{5656859208 (3+2 x) \sqrt {2+5 x+3 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5656859208*(2*Sqrt[3 + 2*x]*(-341519551612 - 3595384785664*x - 16735272462363*x^2 - 45255052994607*x^3 - 78
460508136978*x^4 - 90580760151282*x^5 - 69684837178068*x^6 - 34294970344572*x^7 - 9445976815968*x^8 - 69479541
3312*x^9 + 311460012864*x^10 + 64309557312*x^11) + 115524325*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^2*Sqrt[
(2 + 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] - 30234850*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)
]*(3 + 2*x)^2*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])/((3 + 2*x)*Sqrt[2 + 5
*x + 3*x^2])

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Maple [A]
time = 0.05, size = 176, normalized size = 0.61

method	result	size
default	\(\frac {\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}\, \left (-257238229248 x^{11}-1245840051456 x^{10}[phone] x^{9}[phone] x^{8}[phone] x^{7}[phone] x^{6}[phone] x^{5}[phone] \sqrt {15}\, \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )+5324960 \sqrt {15}\, \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right )}{67882310496 x^{3}[phone] x^{2}[phone] x [phone]}\)	\(176\)
risch	\(-\frac {\left (3572753184 x^{8}[phone] x^{7}-68880579768 x^{6}-329194523196 x^{5}-650694586500 x^{4}-699517082754 x^{3}-427399643682 x^{2}-139652898507 x -18986144459\right ) \sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}{942809868}-\frac {\left (\frac {1075915 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{297729432 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {3300695 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{2}-\EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )\right )}{808122744 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right ) \sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(233\)
elliptic	\(\frac {\sqrt {3 x^{2}+5 x +2}\, \sqrt {3+2 x}\, \sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {72 x^{8} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{19}-\frac {108 \sqrt {6 x^{3}+19 x^{2}+19 x +6}\, x^{7}}{17}+\frac {1242 x^{6} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{17}+\frac {4398409 \sqrt {6 x^{3}+19 x^{2}+19 x +6}\, x^{5}}{12597}+\frac {5033375 x^{4} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{7293}+\frac {97398647 x^{3} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{131274}+\frac {2158584059 x^{2} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{4761666}+\frac {816683617 x \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{5513508}+\frac {58780633 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{2918916}-\frac {1075915 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{297729432 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}-\frac {3300695 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{2}-\EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )\right )}{808122744 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{6 x^{3}+19 x^{2}+19 x +6}\)	\(380\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11313718416*(3+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)*(-257238229248*x^11-1245840051456*x^10+2779181653248*x^9+37783
907263872*x^8+137179881378288*x^7+278739348712272*x^6+362323040605128*x^5+23104865*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))+5324960*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]*x^4+181020211978428*x^3+6
6942476141352*x^2+14383849629156*x+1367002401048)/(6*x^3+19*x^2+19*x+6)

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.90, size = 87, normalized size = 0.30 \begin {gather*} -\frac {1}{942809868} \, {\left (3572753184 \, x^{8} + 5989615632 \, x^{7} - 68880579768 \, x^{6} - 329194523196 \, x^{5} - 650694586500 \, x^{4} - 699517082754 \, x^{3} - 427399643682 \, x^{2} - 139652898507 \, x - 18986144459\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} + \frac {18691975}{5359129776} \, \sqrt {6} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + \frac {16503475}{808122744} \, \sqrt {6} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/942809868*(3572753184*x^8 + 5989615632*x^7 - 68880579768*x^6 - 329194523196*x^5 - 650694586500*x^4 - 699517
082754*x^3 - 427399643682*x^2 - 139652898507*x - 18986144459)*sqrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3) + 18691975/5
359129776*sqrt(6)*weierstrassPInverse(19/27, -28/729, x + 19/18) + 16503475/808122744*sqrt(6)*weierstrassZeta(
19/27, -28/729, weierstrassPInverse(19/27, -28/729, x + 19/18))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- 180 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 1104 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 2717 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 3381 x^{3} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 2151 x^{4} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 551 x^{5} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 48 x^{6} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int 36 x^{7} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-180*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(-1104*x*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2
), x) - Integral(-2717*x**2*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(-3381*x**3*sqrt(2*x + 3)*sqrt(
3*x**2 + 5*x + 2), x) - Integral(-2151*x**4*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(-551*x**5*sqrt
(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(48*x**6*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(36
*x**7*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x)

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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