3.2612 \(\int \frac{5-x}{(3+2 x)^{7/2} \sqrt{2+5 x+3 x^2}} \, dx\)

Optimal. Leaf size=192 \[ -\frac{391 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{125 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{9002 \sqrt{3 x^2+5 x+2}}{1875 \sqrt{2 x+3}}-\frac{782 \sqrt{3 x^2+5 x+2}}{375 (2 x+3)^{3/2}}-\frac{26 \sqrt{3 x^2+5 x+2}}{25 (2 x+3)^{5/2}}+\frac{4501 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{625 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

(-26*Sqrt[2 + 5*x + 3*x^2])/(25*(3 + 2*x)^(5/2)) - (782*Sqrt[2 + 5*x + 3*x^2])/(375*(3 + 2*x)^(3/2)) - (9002*S
qrt[2 + 5*x + 3*x^2])/(1875*Sqrt[3 + 2*x]) + (4501*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]
], -2/3])/(625*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) - (391*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 +
x]], -2/3])/(125*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.132433, antiderivative size = 192, normalized size of antiderivative = 1., 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 = {834, 843, 718, 424, 419} \[ -\frac{9002 \sqrt{3 x^2+5 x+2}}{1875 \sqrt{2 x+3}}-\frac{782 \sqrt{3 x^2+5 x+2}}{375 (2 x+3)^{3/2}}-\frac{26 \sqrt{3 x^2+5 x+2}}{25 (2 x+3)^{5/2}}-\frac{391 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{125 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{4501 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{625 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

Int[(5 - x)/((3 + 2*x)^(7/2)*Sqrt[2 + 5*x + 3*x^2]),x]

[Out]

(-26*Sqrt[2 + 5*x + 3*x^2])/(25*(3 + 2*x)^(5/2)) - (782*Sqrt[2 + 5*x + 3*x^2])/(375*(3 + 2*x)^(3/2)) - (9002*S
qrt[2 + 5*x + 3*x^2])/(1875*Sqrt[3 + 2*x]) + (4501*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]
], -2/3])/(625*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) - (391*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 +
x]], -2/3])/(125*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

Rule 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*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] && LtQ[m, -1] && (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 \frac{5-x}{(3+2 x)^{7/2} \sqrt{2+5 x+3 x^2}} \, dx &=-\frac{26 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)^{5/2}}-\frac{2}{25} \int \frac{-10+\frac{117 x}{2}}{(3+2 x)^{5/2} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{26 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)^{5/2}}-\frac{782 \sqrt{2+5 x+3 x^2}}{375 (3+2 x)^{3/2}}+\frac{4}{375} \int \frac{\frac{491}{4}-\frac{1173 x}{4}}{(3+2 x)^{3/2} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{26 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)^{5/2}}-\frac{782 \sqrt{2+5 x+3 x^2}}{375 (3+2 x)^{3/2}}-\frac{9002 \sqrt{2+5 x+3 x^2}}{1875 \sqrt{3+2 x}}-\frac{8 \int \frac{-\frac{8661}{4}-\frac{13503 x}{8}}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{1875}\\ &=-\frac{26 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)^{5/2}}-\frac{782 \sqrt{2+5 x+3 x^2}}{375 (3+2 x)^{3/2}}-\frac{9002 \sqrt{2+5 x+3 x^2}}{1875 \sqrt{3+2 x}}-\frac{391}{250} \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx+\frac{4501 \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx}{1250}\\ &=-\frac{26 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)^{5/2}}-\frac{782 \sqrt{2+5 x+3 x^2}}{375 (3+2 x)^{3/2}}-\frac{9002 \sqrt{2+5 x+3 x^2}}{1875 \sqrt{3+2 x}}-\frac{\left (391 \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 )}{125 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{\left (4501 \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 )}{625 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=-\frac{26 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)^{5/2}}-\frac{782 \sqrt{2+5 x+3 x^2}}{375 (3+2 x)^{3/2}}-\frac{9002 \sqrt{2+5 x+3 x^2}}{1875 \sqrt{3+2 x}}+\frac{4501 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{625 \sqrt{3} \sqrt{2+5 x+3 x^2}}-\frac{391 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{125 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.302428, size = 182, normalized size = 0.95 \[ -\frac{3328 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{7/2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+23460 x^3+80140 x^2+84040 x-4501 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{7/2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+27360}{1875 (2 x+3)^{5/2} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(5 - x)/((3 + 2*x)^(7/2)*Sqrt[2 + 5*x + 3*x^2]),x]

[Out]

-(27360 + 84040*x + 80140*x^2 + 23460*x^3 - 4501*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(7/2)*Sqrt[(2 + 3*x
)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] + 3328*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^
(7/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])/(1875*(3 + 2*x)^(5/2)*Sqrt[2
+ 5*x + 3*x^2])

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Maple [A]  time = 0.02, size = 296, normalized size = 1.5 \begin{align*}{\frac{1}{18750} \left ( 10184\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}-18004\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}+30552\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) x\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}-54012\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) x\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}+22914\,\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 ) -40509\,\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 ) -1080240\,{x}^{4}-5275720\,{x}^{3}-9353300\,{x}^{2}-7051780\,x-1893960 \right ) \left ( 3+2\,x \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)/(3+2*x)^(7/2)/(3*x^2+5*x+2)^(1/2),x)

[Out]

1/18750*(10184*15^(1/2)*EllipticF(1/5*(30*x+45)^(1/2),1/3*15^(1/2))*x^2*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-20-30*x
)^(1/2)-18004*15^(1/2)*EllipticE(1/5*(30*x+45)^(1/2),1/3*15^(1/2))*x^2*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-20-30*x)
^(1/2)+30552*15^(1/2)*EllipticF(1/5*(30*x+45)^(1/2),1/3*15^(1/2))*x*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-20-30*x)^(1
/2)-54012*15^(1/2)*EllipticE(1/5*(30*x+45)^(1/2),1/3*15^(1/2))*x*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-20-30*x)^(1/2)
+22914*(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))-4050
9*(3+2*x)^(1/2)*15^(1/2)*(-2-2*x)^(1/2)*(-20-30*x)^(1/2)*EllipticE(1/5*(30*x+45)^(1/2),1/3*15^(1/2))-1080240*x
^4-5275720*x^3-9353300*x^2-7051780*x-1893960)/(3*x^2+5*x+2)^(1/2)/(3+2*x)^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{x - 5}{\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (2 \, x + 3\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^(7/2)/(3*x^2+5*x+2)^(1/2),x, algorithm="maxima")

[Out]

-integrate((x - 5)/(sqrt(3*x^2 + 5*x + 2)*(2*x + 3)^(7/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{2 \, x + 3}{\left (x - 5\right )}}{48 \, x^{6} + 368 \, x^{5} + 1160 \, x^{4} + 1920 \, x^{3} + 1755 \, x^{2} + 837 \, x + 162}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^(7/2)/(3*x^2+5*x+2)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3)*(x - 5)/(48*x^6 + 368*x^5 + 1160*x^4 + 1920*x^3 + 1755*x^2 + 837
*x + 162), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x}{8 x^{3} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 36 x^{2} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 54 x \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 27 \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int - \frac{5}{8 x^{3} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 36 x^{2} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 54 x \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 27 \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)**(7/2)/(3*x**2+5*x+2)**(1/2),x)

[Out]

-Integral(x/(8*x**3*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 36*x**2*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 54*x
*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 27*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-5/(8*x**3*sqr
t(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 36*x**2*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 54*x*sqrt(2*x + 3)*sqrt(3*x
**2 + 5*x + 2) + 27*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 -\frac{x - 5}{\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (2 \, x + 3\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)/(3+2*x)^(7/2)/(3*x^2+5*x+2)^(1/2),x, algorithm="giac")

[Out]

integrate(-(x - 5)/(sqrt(3*x^2 + 5*x + 2)*(2*x + 3)^(7/2)), x)