3.2587 \(\int \frac {(5-x) (2+5 x+3 x^2)^{3/2}}{\sqrt {3+2 x}} \, dx\)

Optimal. Leaf size=175 \[ \frac {1}{63} (52-7 x) \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}-\frac {\sqrt {2 x+3} (12429 x+107) \sqrt {3 x^2+5 x+2}}{5670}+\frac {20501 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{2268 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {11123 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{1620 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]

[Out]

1/63*(52-7*x)*(3*x^2+5*x+2)^(3/2)*(3+2*x)^(1/2)-11123/4860*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)+20501/6804*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)-1/5670*(107+12429*x)*(3+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)

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Rubi [A]  time = 0.11, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {814, 843, 718, 424, 419} \[ \frac {1}{63} (52-7 x) \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}-\frac {\sqrt {2 x+3} (12429 x+107) \sqrt {3 x^2+5 x+2}}{5670}+\frac {20501 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{2268 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {11123 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{1620 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[3 + 2*x]*(107 + 12429*x)*Sqrt[2 + 5*x + 3*x^2])/5670 + ((52 - 7*x)*Sqrt[3 + 2*x]*(2 + 5*x + 3*x^2)^(3/2
))/63 - (11123*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(1620*Sqrt[3]*Sqrt[2 + 5*x
+ 3*x^2]) + (20501*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(2268*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 814

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 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 )^{3/2}}{\sqrt {3+2 x}} \, dx &=\frac {1}{63} (52-7 x) \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}-\frac {1}{126} \int \frac {(1204+1381 x) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx\\ &=-\frac {\sqrt {3+2 x} (107+12429 x) \sqrt {2+5 x+3 x^2}}{5670}+\frac {1}{63} (52-7 x) \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}+\frac {\int \frac {-65539-77861 x}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{11340}\\ &=-\frac {\sqrt {3+2 x} (107+12429 x) \sqrt {2+5 x+3 x^2}}{5670}+\frac {1}{63} (52-7 x) \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}-\frac {11123 \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx}{3240}+\frac {20501 \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{4536}\\ &=-\frac {\sqrt {3+2 x} (107+12429 x) \sqrt {2+5 x+3 x^2}}{5670}+\frac {1}{63} (52-7 x) \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}-\frac {\left (11123 \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 )}{1620 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {\left (20501 \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 )}{2268 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ &=-\frac {\sqrt {3+2 x} (107+12429 x) \sqrt {2+5 x+3 x^2}}{5670}+\frac {1}{63} (52-7 x) \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}-\frac {11123 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{1620 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {20501 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{2268 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.32, size = 203, normalized size = 1.16 \[ -\frac {2 \left (34020 x^6-88290 x^5-687798 x^4-1306791 x^3-1043385 x^2-312914 x-10832\right ) \sqrt {2 x+3}-16358 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^2 F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )+77861 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )}{34020 (2 x+3) \sqrt {3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/34020*(2*Sqrt[3 + 2*x]*(-10832 - 312914*x - 1043385*x^2 - 1306791*x^3 - 687798*x^4 - 88290*x^5 + 34020*x^6)
+ 77861*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] - 16358*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^2*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSi
n[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])/((3 + 2*x)*Sqrt[2 + 5*x + 3*x^2])

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fricas [F]  time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (3 \, x^{3} - 10 \, x^{2} - 23 \, x - 10\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{\sqrt {2 \, x + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(3*x^3 - 10*x^2 - 23*x - 10)*sqrt(3*x^2 + 5*x + 2)/sqrt(2*x + 3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 151, normalized size = 0.86 \[ \frac {\sqrt {3 x^{2}+5 x +2}\, \sqrt {2 x +3}\, \left (-680400 x^{6}+1765800 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x +77861 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+24644 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+3331080\right )}{2041200 x^{3}+6463800 x^{2}+6463800 x +2041200} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/340200*(3*x^2+5*x+2)^(1/2)*(2*x+3)^(1/2)*(-680400*x^6+1765800*x^5+24644*(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))+77861*(2*x+3)^(1/2)*15^(1/2)*(-2*x-2)^(1/2)*(-3
0*x-20)^(1/2)*EllipticE(1/5*(30*x+45)^(1/2),1/3*15^(1/2))[phone]*x^4+26135820*x^3+25539360*x^2+14044380*x+33
31080)/(6*x^3+19*x^2+19*x+6)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{\sqrt {2\,x+3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- \frac {10 \sqrt {3 x^{2} + 5 x + 2}}{\sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {23 x \sqrt {3 x^{2} + 5 x + 2}}{\sqrt {2 x + 3}}\right )\, dx - \int \left (- \frac {10 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{\sqrt {2 x + 3}}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{\sqrt {2 x + 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-10*sqrt(3*x**2 + 5*x + 2)/sqrt(2*x + 3), x) - Integral(-23*x*sqrt(3*x**2 + 5*x + 2)/sqrt(2*x + 3),
x) - Integral(-10*x**2*sqrt(3*x**2 + 5*x + 2)/sqrt(2*x + 3), x) - Integral(3*x**3*sqrt(3*x**2 + 5*x + 2)/sqrt(
2*x + 3), x)

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