3.2577 \(\int (5-x) (3+2 x)^{3/2} \sqrt{2+5 x+3 x^2} \, dx\)

Optimal. Leaf size=197 \[ \frac{5773 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{3402 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{2}{27} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{3/2}+\frac{202}{189} \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{3/2}+\frac{\sqrt{2 x+3} (30033 x+27914) \sqrt{3 x^2+5 x+2}}{8505}-\frac{4729 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{2430 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

(Sqrt[3 + 2*x]*(27914 + 30033*x)*Sqrt[2 + 5*x + 3*x^2])/8505 + (202*Sqrt[3 + 2*x]*(2 + 5*x + 3*x^2)^(3/2))/189
- (2*(3 + 2*x)^(3/2)*(2 + 5*x + 3*x^2)^(3/2))/27 - (4729*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt
[1 + x]], -2/3])/(2430*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (5773*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*
Sqrt[1 + x]], -2/3])/(3402*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.133377, antiderivative size = 197, normalized size of antiderivative = 1., 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 = {832, 814, 843, 718, 424, 419} \[ -\frac{2}{27} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{3/2}+\frac{202}{189} \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{3/2}+\frac{\sqrt{2 x+3} (30033 x+27914) \sqrt{3 x^2+5 x+2}}{8505}+\frac{5773 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{3402 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{4729 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{2430 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[3 + 2*x]*(27914 + 30033*x)*Sqrt[2 + 5*x + 3*x^2])/8505 + (202*Sqrt[3 + 2*x]*(2 + 5*x + 3*x^2)^(3/2))/189
- (2*(3 + 2*x)^(3/2)*(2 + 5*x + 3*x^2)^(3/2))/27 - (4729*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt
[1 + x]], -2/3])/(2430*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (5773*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*
Sqrt[1 + x]], -2/3])/(3402*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

Rule 832

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 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]

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 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 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)])

Rubi steps

\begin{align*} \int (5-x) (3+2 x)^{3/2} \sqrt{2+5 x+3 x^2} \, dx &=-\frac{2}{27} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac{2}{27} \int \sqrt{3+2 x} \left (231+\frac{303 x}{2}\right ) \sqrt{2+5 x+3 x^2} \, dx\\ &=\frac{202}{189} \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{3/2}-\frac{2}{27} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac{4}{567} \int \frac{\left (\frac{14259}{4}+\frac{10011 x}{4}\right ) \sqrt{2+5 x+3 x^2}}{\sqrt{3+2 x}} \, dx\\ &=\frac{\sqrt{3+2 x} (27914+30033 x) \sqrt{2+5 x+3 x^2}}{8505}+\frac{202}{189} \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{3/2}-\frac{2}{27} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac{2 \int \frac{\frac{52833}{2}+\frac{99309 x}{4}}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{25515}\\ &=\frac{\sqrt{3+2 x} (27914+30033 x) \sqrt{2+5 x+3 x^2}}{8505}+\frac{202}{189} \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{3/2}-\frac{2}{27} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac{5773 \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{6804}-\frac{4729 \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx}{4860}\\ &=\frac{\sqrt{3+2 x} (27914+30033 x) \sqrt{2+5 x+3 x^2}}{8505}+\frac{202}{189} \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{3/2}-\frac{2}{27} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac{\left (5773 \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 )}{3402 \sqrt{3} \sqrt{2+5 x+3 x^2}}-\frac{\left (4729 \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 )}{2430 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=\frac{\sqrt{3+2 x} (27914+30033 x) \sqrt{2+5 x+3 x^2}}{8505}+\frac{202}{189} \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{3/2}-\frac{2}{27} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac{4729 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{2430 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{5773 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{3402 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.343406, size = 203, normalized size = 1.03 \[ -\frac{-15784 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+2 \left (68040 x^6-59940 x^5-1799874 x^4-5185953 x^3-6208230 x^2-3389617 x-695446\right ) \sqrt{2 x+3}+33103 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{51030 (2 x+3) \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(2*Sqrt[3 + 2*x]*(-695446 - 3389617*x - 6208230*x^2 - 5185953*x^3 - 1799874*x^4 - 59940*x^5 + 68040*x^6) + 33
103*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] - 15784*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^2*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSin[Sqr
t[5/3]/Sqrt[3 + 2*x]], 3/5])/(51030*(3 + 2*x)*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.012, size = 151, normalized size = 0.8 \begin{align*} -{\frac{1}{3061800\,{x}^{3}+9695700\,{x}^{2}+9695700\,x+3061800}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( 1360800\,{x}^{6}-1198800\,{x}^{5}+4238\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-20-30\,x}{\it EllipticF} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) -33103\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-20-30\,x}{\it EllipticE} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) -35997480\,{x}^{4}-103719060\,{x}^{3}-126150780\,{x}^{2}-71102640\,x-15233040 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/510300*(3+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)*(1360800*x^6-1198800*x^5+4238*(3+2*x)^(1/2)*15^(1/2)*(-2-2*x)^(1/2
)*(-20-30*x)^(1/2)*EllipticF(1/5*(30*x+45)^(1/2),1/3*15^(1/2))-33103*(3+2*x)^(1/2)*15^(1/2)*(-2-2*x)^(1/2)*(-2
0-30*x)^(1/2)*EllipticE(1/5*(30*x+45)^(1/2),1/3*15^(1/2))-35997480*x^4-103719060*x^3-126150780*x^2-71102640*x-
15233040)/(6*x^3+19*x^2+19*x+6)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (2 \, x + 3\right )}^{\frac{3}{2}}{\left (x - 5\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (2 \, x^{2} - 7 \, x - 15\right )} \sqrt{2 \, x + 3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(3*x^2 + 5*x + 2)*(2*x^2 - 7*x - 15)*sqrt(2*x + 3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - 15 \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 7 x \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 2 x^{2} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-15*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x) - Integral(-7*x*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x
) - Integral(2*x**2*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (2 \, x + 3\right )}^{\frac{3}{2}}{\left (x - 5\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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