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

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

[Out]

4501/1875*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)-391/37
5*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)-26/25*(3*x^2+5
*x+2)^(1/2)/(3+2*x)^(5/2)-782/375*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(3/2)-9002/1875*(3*x^2+5*x+2)^(1/2)/(3+2*x)^(1/2
)

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Rubi [A]  time = 0.13, antiderivative size = 192, 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 = {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 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 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]

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*}

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Mathematica [A]  time = 0.29, size = 182, normalized size = 0.95 \[ -\frac {23460 x^3+80140 x^2+84040 x+3328 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^{7/2} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )-4501 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^{7/2} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )+27360}{1875 (2 x+3)^{5/2} \sqrt {3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/1875*(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])/((3 + 2*x)^(5/2)*Sqrt[
2 + 5*x + 3*x^2])

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fricas [F]  time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3} {\left (x - 5\right )}}{48 \, x^{6} + 368 \, x^{5} + 1160 \, x^{4} + 1920 \, x^{3} + 1755 \, x^{2} + 837 \, x + 162}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {x - 5}{\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (2 \, x + 3\right )}^{\frac {7}{2}}}\,{d x} \]

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)

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maple [A]  time = 0.02, size = 296, normalized size = 1.54 \[ \frac {-1080240 x^{4}-5275720 x^{3}-18004 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{2} \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+10184 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x^{2} \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )-9353300 x^{2}-54012 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+30552 \sqrt {15}\, \sqrt {2 x +3}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, x \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )-7051780 x -40509 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+22914 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )-1893960}{18750 \sqrt {3 x^{2}+5 x +2}\, \left (2 x +3\right )^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {x - 5}{\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (2 \, x + 3\right )}^{\frac {7}{2}}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {x-5}{{\left (2\,x+3\right )}^{7/2}\,\sqrt {3\,x^2+5\,x+2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {x}{8 x^{3} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 36 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 54 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 27 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{8 x^{3} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 36 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 54 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 27 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \]

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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