\(\int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx\) [2579]

Optimal result

Integrand size = 29, antiderivative size = 143 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx=\frac {1}{45} (88-9 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}-\frac {761 \sqrt {-2-5 x-3 x^2} E\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{90 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {191 \sqrt {-2-5 x-3 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {1+x}\right ),-\frac {2}{3}\right )}{18 \sqrt {3} \sqrt {2+5 x+3 x^2}} \]

[Out]

-761/270*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)+191/54*
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)+1/45*(88-9*x)*(3
+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {828, 857, 732, 435, 430} \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx=\frac {191 \sqrt {-3 x^2-5 x-2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right ),-\frac {2}{3}\right )}{18 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {761 \sqrt {-3 x^2-5 x-2} E\left (\arcsin \left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{90 \sqrt {3} \sqrt {3 x^2+5 x+2}}+\frac {1}{45} \sqrt {2 x+3} \sqrt {3 x^2+5 x+2} (88-9 x) \]

[In]

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

[Out]

((88 - 9*x)*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2])/45 - (761*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqr
t[1 + x]], -2/3])/(90*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (191*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sq
rt[1 + x]], -2/3])/(18*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 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*} \text {integral}& = \frac {1}{45} (88-9 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}-\frac {1}{90} \int \frac {664+761 x}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx \\ & = \frac {1}{45} (88-9 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}-\frac {761}{180} \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx+\frac {191}{36} \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx \\ & = \frac {1}{45} (88-9 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}-\frac {\left (761 \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 )}{90 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {\left (191 \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 )}{18 \sqrt {3} \sqrt {2+5 x+3 x^2}} \\ & = \frac {1}{45} (88-9 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}-\frac {761 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{90 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {191 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{18 \sqrt {3} \sqrt {2+5 x+3 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 25.77 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.35 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx=-\frac {2 \sqrt {3+2 x} \left (-62-1049 x-2220 x^2-1071 x^3+162 x^4\right )+761 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )-188 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^2 \sqrt {\frac {2+3 x}{3+2 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right ),\frac {3}{5}\right )}{270 (3+2 x) \sqrt {2+5 x+3 x^2}} \]

[In]

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

[Out]

-1/270*(2*Sqrt[3 + 2*x]*(-62 - 1049*x - 2220*x^2 - 1071*x^3 + 162*x^4) + 761*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] - 188*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])

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.99

method	result	size
default	\(-\frac {\sqrt {3 x^{2}+5 x +2}\, \sqrt {3+2 x}\, \left (291 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )-761 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {15}\, \sqrt {3+2 x}\, E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )+4860 x^{4}-32130 x^{3}-135090 x^{2}-145620 x -47520\right )}{4050 \left (6 x^{3}+19 x^{2}+19 x +6\right )}\)	\(141\)
risch	\(-\frac {\left (-88+9 x \right ) \sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}{45}-\frac {\left (\frac {332 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, F\left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{675 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {761 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \left (-\frac {E\left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{2}-F\left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )\right )}{1350 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right ) \sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(198\)
elliptic	\(\frac {\sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {x \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{5}+\frac {88 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{45}+\frac {332 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{675 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {761 \sqrt {-20-30 x}\, \sqrt {3+3 x}\, \sqrt {45+30 x}\, \left (\frac {E\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )}{3}-F\left (\frac {\sqrt {-20-30 x}}{5}, \frac {\sqrt {10}}{2}\right )\right )}{1350 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(209\)

[In]

int((5-x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/4050*(3*x^2+5*x+2)^(1/2)*(3+2*x)^(1/2)*(291*(-20-30*x)^(1/2)*(3+3*x)^(1/2)*15^(1/2)*(3+2*x)^(1/2)*EllipticF
(1/5*(-20-30*x)^(1/2),1/2*10^(1/2))-761*(-20-30*x)^(1/2)*(3+3*x)^(1/2)*15^(1/2)*(3+2*x)^(1/2)*EllipticE(1/5*(-
20-30*x)^(1/2),1/2*10^(1/2))+4860*x^4-32130*x^3-135090*x^2-145620*x-47520)/(6*x^3+19*x^2+19*x+6)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.08 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.36 \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx=-\frac {1}{45} \, \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (9 \, x - 88\right )} \sqrt {2 \, x + 3} + \frac {2507}{4860} \, \sqrt {6} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + \frac {761}{270} \, \sqrt {6} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) \]

[In]

integrate((5-x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/45*sqrt(3*x^2 + 5*x + 2)*(9*x - 88)*sqrt(2*x + 3) + 2507/4860*sqrt(6)*weierstrassPInverse(19/27, -28/729, x
+ 19/18) + 761/270*sqrt(6)*weierstrassZeta(19/27, -28/729, weierstrassPInverse(19/27, -28/729, x + 19/18))

Sympy [F]

\[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx=- \int \left (- \frac {5 \sqrt {3 x^{2} + 5 x + 2}}{\sqrt {2 x + 3}}\right )\, dx - \int \frac {x \sqrt {3 x^{2} + 5 x + 2}}{\sqrt {2 x + 3}}\, dx \]

[In]

integrate((5-x)*(3*x**2+5*x+2)**(1/2)/(3+2*x)**(1/2),x)

[Out]

-Integral(-5*sqrt(3*x**2 + 5*x + 2)/sqrt(2*x + 3), x) - Integral(x*sqrt(3*x**2 + 5*x + 2)/sqrt(2*x + 3), x)

Maxima [F]

\[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx=\int { -\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (x - 5\right )}}{\sqrt {2 \, x + 3}} \,d x } \]

[In]

integrate((5-x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx=\int { -\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (x - 5\right )}}{\sqrt {2 \, x + 3}} \,d x } \]

[In]

integrate((5-x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(5-x) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx=-\int \frac {\left (x-5\right )\,\sqrt {3\,x^2+5\,x+2}}{\sqrt {2\,x+3}} \,d x \]

[In]

int(-((x - 5)*(5*x + 3*x^2 + 2)^(1/2))/(2*x + 3)^(1/2),x)

[Out]

-int(((x - 5)*(5*x + 3*x^2 + 2)^(1/2))/(2*x + 3)^(1/2), x)