3.26.99 \(\int \frac {(5-x) (2+5 x+3 x^2)^{5/2}}{(3+2 x)^{5/2}} \, dx\) [2599]

Optimal. Leaf size=205 \[ \frac {5}{756} (326-6957 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}+\frac {5 (745+121 x) \left (2+5 x+3 x^2\right )^{3/2}}{126 \sqrt {3+2 x}}-\frac {(21+x) \left (2+5 x+3 x^2\right )^{5/2}}{9 (3+2 x)^{3/2}}-\frac {33335 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{216 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {306175 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{1512 \sqrt {3} \sqrt {2+5 x+3 x^2}} \]

[Out]

-1/9*(21+x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(3/2)+5/126*(745+121*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^(1/2)-33335/648*El
lipticE(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)+306175/4536*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)+5/756*(326-6957*x)*(3+
2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)

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Rubi [A]
time = 0.08, antiderivative size = 205, 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 = {826, 828, 857, 732, 435, 430} \begin {gather*} \frac {306175 \sqrt {-3 x^2-5 x-2} F\left (\text {ArcSin}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{1512 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {33335 \sqrt {-3 x^2-5 x-2} E\left (\text {ArcSin}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{216 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {(x+21) \left (3 x^2+5 x+2\right )^{5/2}}{9 (2 x+3)^{3/2}}+\frac {5 (121 x+745) \left (3 x^2+5 x+2\right )^{3/2}}{126 \sqrt {2 x+3}}+\frac {5}{756} (326-6957 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(326 - 6957*x)*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2])/756 + (5*(745 + 121*x)*(2 + 5*x + 3*x^2)^(3/2))/(126*Sq
rt[3 + 2*x]) - ((21 + x)*(2 + 5*x + 3*x^2)^(5/2))/(9*(3 + 2*x)^(3/2)) - (33335*Sqrt[-2 - 5*x - 3*x^2]*Elliptic
E[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(216*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (306175*Sqrt[-2 - 5*x - 3*x^2]*Ell
ipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(1512*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 826

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 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 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)^{5/2}} \, dx &=-\frac {(21+x) \left (2+5 x+3 x^2\right )^{5/2}}{9 (3+2 x)^{3/2}}-\frac {5}{54} \int \frac {(-303-363 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{3/2}} \, dx\\ &=\frac {5 (745+121 x) \left (2+5 x+3 x^2\right )^{3/2}}{126 \sqrt {3+2 x}}-\frac {(21+x) \left (2+5 x+3 x^2\right )^{5/2}}{9 (3+2 x)^{3/2}}+\frac {5}{252} \int \frac {(-9723-11595 x) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx\\ &=\frac {5}{756} (326-6957 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}+\frac {5 (745+121 x) \left (2+5 x+3 x^2\right )^{3/2}}{126 \sqrt {3+2 x}}-\frac {(21+x) \left (2+5 x+3 x^2\right )^{5/2}}{9 (3+2 x)^{3/2}}-\frac {\int \frac {590790+700035 x}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{4536}\\ &=\frac {5}{756} (326-6957 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}+\frac {5 (745+121 x) \left (2+5 x+3 x^2\right )^{3/2}}{126 \sqrt {3+2 x}}-\frac {(21+x) \left (2+5 x+3 x^2\right )^{5/2}}{9 (3+2 x)^{3/2}}-\frac {33335}{432} \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx+\frac {306175 \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{3024}\\ &=\frac {5}{756} (326-6957 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}+\frac {5 (745+121 x) \left (2+5 x+3 x^2\right )^{3/2}}{126 \sqrt {3+2 x}}-\frac {(21+x) \left (2+5 x+3 x^2\right )^{5/2}}{9 (3+2 x)^{3/2}}-\frac {\left (33335 \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 )}{216 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {\left (306175 \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 )}{1512 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ &=\frac {5}{756} (326-6957 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}+\frac {5 (745+121 x) \left (2+5 x+3 x^2\right )^{3/2}}{126 \sqrt {3+2 x}}-\frac {(21+x) \left (2+5 x+3 x^2\right )^{5/2}}{9 (3+2 x)^{3/2}}-\frac {33335 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{216 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {306175 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{1512 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ \end {align*}

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Mathematica [A]
time = 30.28, size = 202, normalized size = 0.99 \begin {gather*} -\frac {1099572+3207982 x+2607724 x^2+120594 x^3-561564 x^4-234684 x^5-38232 x^6+13608 x^7+233345 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{5/2} \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 )-49640 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{5/2} \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 )}{4536 (3+2 x)^{3/2} \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)^(5/2),x]

[Out]

-1/4536*(1099572 + 3207982*x + 2607724*x^2 + 120594*x^3 - 561564*x^4 - 234684*x^5 - 38232*x^6 + 13608*x^7 + 23
3345*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(5/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt
[3 + 2*x]], 3/5] - 49640*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(5/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[A
rcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])/((3 + 2*x)^(3/2)*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]
time = 0.05, size = 223, normalized size = 1.09

method	result	size
default	\(\frac {-27216 x^{7}+93338 \sqrt {15}\, \EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x \sqrt {-20-30 x}\, \sqrt {3+2 x}\, \sqrt {-2-2 x}+29132 \sqrt {15}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x \sqrt {-20-30 x}\, \sqrt {3+2 x}\, \sqrt {-2-2 x}+76464 x^{6}+140007 \sqrt {15}\, \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )+43698 \sqrt {15}\, \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )+469368 x^{5}+1123128 x^{4}+5359092 x^{3}[phone] x^{2}[phone] x +3401136}{9072 \sqrt {3 x^{2}+5 x +2}\, \left (3+2 x \right )^{\frac {3}{2}}}\)	\(223\)
elliptic	\(\frac {\sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {325 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{384 \left (x +\frac {3}{2}\right )^{2}}+\frac {\frac {3715}{32} x^{2}+\frac {18575}{96} x +\frac {3715}{48}}{\sqrt {\left (x +\frac {3}{2}\right ) \left (6 x^{2}+10 x +4\right )}}-\frac {x^{3} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{4}+\frac {157 x^{2} \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{84}-\frac {817 x \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{336}+\frac {12557 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{1512}-\frac {19693 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{2268 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}-\frac {6667 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{2}-\EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )\right )}{648 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(305\)

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)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/9072*(-27216*x^7+93338*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)+29132*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)+76464*x^6+140007*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))+43698*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))+469368*x^5+1123128*x^4+5359092*x^3+12518772*x^2+11318256*x+3401136)/(3*x^2+5*x+2)^(1/2)/(3+2*x)^
(3/2)

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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)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.19, size = 106, normalized size = 0.52 \begin {gather*} \frac {888815 \, \sqrt {6} {\left (4 \, x^{2} + 12 \, x + 9\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 4200210 \, \sqrt {6} {\left (4 \, x^{2} + 12 \, x + 9\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 ) - 108 \, {\left (756 \, x^{5} - 3384 \, x^{4} - 7902 \, x^{3} - 15772 \, x^{2} - 117309 \, x - 141714\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{81648 \, {\left (4 \, x^{2} + 12 \, x + 9\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)^(5/2),x, algorithm="fricas")

[Out]

1/81648*(888815*sqrt(6)*(4*x^2 + 12*x + 9)*weierstrassPInverse(19/27, -28/729, x + 19/18) + 4200210*sqrt(6)*(4
*x^2 + 12*x + 9)*weierstrassZeta(19/27, -28/729, weierstrassPInverse(19/27, -28/729, x + 19/18)) - 108*(756*x^
5 - 3384*x^4 - 7902*x^3 - 15772*x^2 - 117309*x - 141714)*sqrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3))/(4*x^2 + 12*x +
9)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {20 \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} \sqrt {2 x + 3} + 12 x \sqrt {2 x + 3} + 9 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {96 x \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} \sqrt {2 x + 3} + 12 x \sqrt {2 x + 3} + 9 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {165 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} \sqrt {2 x + 3} + 12 x \sqrt {2 x + 3} + 9 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {113 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} \sqrt {2 x + 3} + 12 x \sqrt {2 x + 3} + 9 \sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {15 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} \sqrt {2 x + 3} + 12 x \sqrt {2 x + 3} + 9 \sqrt {2 x + 3}}\right )\, dx - \int \frac {9 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} \sqrt {2 x + 3} + 12 x \sqrt {2 x + 3} + 9 \sqrt {2 x + 3}}\, dx \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)**(5/2),x)

[Out]

-Integral(-20*sqrt(3*x**2 + 5*x + 2)/(4*x**2*sqrt(2*x + 3) + 12*x*sqrt(2*x + 3) + 9*sqrt(2*x + 3)), x) - Integ
ral(-96*x*sqrt(3*x**2 + 5*x + 2)/(4*x**2*sqrt(2*x + 3) + 12*x*sqrt(2*x + 3) + 9*sqrt(2*x + 3)), x) - Integral(
-165*x**2*sqrt(3*x**2 + 5*x + 2)/(4*x**2*sqrt(2*x + 3) + 12*x*sqrt(2*x + 3) + 9*sqrt(2*x + 3)), x) - Integral(
-113*x**3*sqrt(3*x**2 + 5*x + 2)/(4*x**2*sqrt(2*x + 3) + 12*x*sqrt(2*x + 3) + 9*sqrt(2*x + 3)), x) - Integral(
-15*x**4*sqrt(3*x**2 + 5*x + 2)/(4*x**2*sqrt(2*x + 3) + 12*x*sqrt(2*x + 3) + 9*sqrt(2*x + 3)), x) - Integral(9
*x**5*sqrt(3*x**2 + 5*x + 2)/(4*x**2*sqrt(2*x + 3) + 12*x*sqrt(2*x + 3) + 9*sqrt(2*x + 3)), 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*x^2+5*x+2)^(5/2)/(3+2*x)^(5/2),x, algorithm="giac")

[Out]

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

[Out]

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

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