3.2597 \(\int \frac {(5-x) (2+5 x+3 x^2)^{5/2}}{\sqrt {3+2 x}} \, dx\)

Optimal. Leaf size=207 \[ \frac {1}{429} (224-33 x) \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2}-\frac {5 \sqrt {2 x+3} (4669 x+563) \left (3 x^2+5 x+2\right )^{3/2}}{18018}+\frac {(34372-676791 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}{324324}+\frac {5983645 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{648648 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {651617 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{92664 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]

[Out]

-5/18018*(563+4669*x)*(3*x^2+5*x+2)^(3/2)*(3+2*x)^(1/2)+1/429*(224-33*x)*(3*x^2+5*x+2)^(5/2)*(3+2*x)^(1/2)-651
617/277992*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)+59836
45/1945944*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)+1/324
324*(34372-676791*x)*(3+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)

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Rubi [A]  time = 0.13, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ \frac {1}{429} (224-33 x) \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2}-\frac {5 \sqrt {2 x+3} (4669 x+563) \left (3 x^2+5 x+2\right )^{3/2}}{18018}+\frac {(34372-676791 x) \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}}{324324}+\frac {5983645 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{648648 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {651617 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{92664 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((34372 - 676791*x)*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2])/324324 - (5*Sqrt[3 + 2*x]*(563 + 4669*x)*(2 + 5*x + 3
*x^2)^(3/2))/18018 + ((224 - 33*x)*Sqrt[3 + 2*x]*(2 + 5*x + 3*x^2)^(5/2))/429 - (651617*Sqrt[-2 - 5*x - 3*x^2]
*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(92664*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (5983645*Sqrt[-2 - 5*x
- 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(648648*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 )^{5/2}}{\sqrt {3+2 x}} \, dx &=\frac {1}{429} (224-33 x) \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}-\frac {5}{858} \int \frac {(1744+2001 x) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt {3+2 x}} \, dx\\ &=-\frac {5 \sqrt {3+2 x} (563+4669 x) \left (2+5 x+3 x^2\right )^{3/2}}{18018}+\frac {1}{429} (224-33 x) \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}+\frac {5 \int \frac {(-188643-225597 x) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx}{108108}\\ &=\frac {(34372-676791 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}{324324}-\frac {5 \sqrt {3+2 x} (563+4669 x) \left (2+5 x+3 x^2\right )^{3/2}}{18018}+\frac {1}{429} (224-33 x) \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}-\frac {\int \frac {11550468+13683957 x}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{1945944}\\ &=\frac {(34372-676791 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}{324324}-\frac {5 \sqrt {3+2 x} (563+4669 x) \left (2+5 x+3 x^2\right )^{3/2}}{18018}+\frac {1}{429} (224-33 x) \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}-\frac {651617 \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx}{185328}+\frac {5983645 \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{1297296}\\ &=\frac {(34372-676791 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}{324324}-\frac {5 \sqrt {3+2 x} (563+4669 x) \left (2+5 x+3 x^2\right )^{3/2}}{18018}+\frac {1}{429} (224-33 x) \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}-\frac {\left (651617 \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 )}{92664 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {\left (5983645 \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 )}{648648 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ &=\frac {(34372-676791 x) \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}{324324}-\frac {5 \sqrt {3+2 x} (563+4669 x) \left (2+5 x+3 x^2\right )^{3/2}}{18018}+\frac {1}{429} (224-33 x) \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}-\frac {651617 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{92664 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {5983645 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{648648 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.33, size = 213, normalized size = 1.03 \[ -\frac {2 \left (4041576 x^8-1163484 x^7-83553120 x^6-268524558 x^5-406647648 x^4-349849791 x^3-170798082 x^2-39284147 x-1864706\right ) \sqrt {2 x+3}-971132 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^2 F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )+4561319 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )}{1945944 (2 x+3) \sqrt {3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/1945944*(2*Sqrt[3 + 2*x]*(-1864706 - 39284147*x - 170798082*x^2 - 349849791*x^3 - 406647648*x^4 - 268524558
*x^5 - 83553120*x^6 - 1163484*x^7 + 4041576*x^8) + 4561319*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] - 971132*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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fricas [F]  time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (9 \, x^{5} - 15 \, x^{4} - 113 \, x^{3} - 165 \, x^{2} - 96 \, x - 20\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{\sqrt {2 \, x + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(9*x^5 - 15*x^4 - 113*x^3 - 165*x^2 - 96*x - 20)*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 \[ \int -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} {\left (x - 5\right )}}{\sqrt {2 \, x + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 161, normalized size = 0.78 \[ \frac {\sqrt {3 x^{2}+5 x +2}\, \sqrt {2 x +3}\, \left (-80831520 x^{8}[phone] x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x +4561319 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+1422326 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )[phone]\right )}{116756640 x^{3}[phone] x^{2}[phone] x [phone]} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/19459440*(3*x^2+5*x+2)^(1/2)*(2*x+3)^(1/2)*(-80831520*x^8+23269680*x^7+1671062400*x^6+5370491160*x^5+1422326
*(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))+4561319*(2
*x+3)^(1/2)*15^(1/2)*(-2*x-2)^(1/2)*(-30*x-20)^(1/2)*EllipticE(1/5*(30*x+45)^(1/2),1/3*15^(1/2))[phone]*x^
4+6996995820*x^3+3689640780*x^2+1241814840*x+219746880)/(6*x^3+19*x^2+19*x+6)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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)**(1/2),x)

[Out]

Timed out

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