3.27.12 \(\int \frac {5-x}{(3+2 x)^{7/2} \sqrt {2+5 x+3 x^2}} \, dx\) [2612]

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

[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 = 0.08, 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 = {848, 857, 732, 435, 430} \begin {gather*} -\frac {391 \sqrt {-3 x^2-5 x-2} F\left (\text {ArcSin}\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 (\text {ArcSin}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{625 \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}} \end {gather*}

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 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]
))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 732

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 848

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 857

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 ) \text {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 ) \text {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 = 30.22, size = 182, normalized size = 0.95 \begin {gather*} -\frac {27360+84040 x+80140 x^2+23460 x^3-4501 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{7/2} \sqrt {\frac {2+3 x}{3+2 x}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )+3328 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} (3+2 x)^{7/2} \sqrt {\frac {2+3 x}{3+2 x}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )}{1875 (3+2 x)^{5/2} \sqrt {2+5 x+3 x^2}} \end {gather*}

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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Maple [A]
time = 0.08, size = 296, normalized size = 1.54

method	result	size
elliptic	\(\frac {\sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {13 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{100 \left (x +\frac {3}{2}\right )^{3}}-\frac {391 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{750 \left (x +\frac {3}{2}\right )^{2}}-\frac {4501 \left (6 x^{2}+10 x +4\right )}{1875 \sqrt {\left (x +\frac {3}{2}\right ) \left (6 x^{2}+10 x +4\right )}}+\frac {5774 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{9375 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}+\frac {4501 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{2}-\EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )\right )}{9375 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(246\)
default	\(\frac {10184 \sqrt {15}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x^{2} \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}-18004 \sqrt {15}\, \EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x^{2} \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}+30552 \sqrt {15}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x \sqrt {-20-30 x}\, \sqrt {3+2 x}\, \sqrt {-2-2 x}-54012 \sqrt {15}\, \EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x \sqrt {-20-30 x}\, \sqrt {3+2 x}\, \sqrt {-2-2 x}+22914 \sqrt {15}\, \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )-40509 \sqrt {15}\, \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )-1080240 x^{4}-5275720 x^{3}-9353300 x^{2}-7051780 x -1893960}{18750 \sqrt {3 x^{2}+5 x +2}\, \left (3+2 x \right )^{\frac {5}{2}}}\)	\(296\)

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,method=_RETURNVERBOSE)

[Out]

1/18750*(10184*15^(1/2)*EllipticF(1/5*(45+30*x)^(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*(45+30*x)^(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*(45+30*x)^(1/2),1/3*15^(1/2))*x*(-20-30*x)^(1/2)*(3+2*x)^(1/2)*(-2-2*x)^(1
/2)-54012*15^(1/2)*EllipticE(1/5*(45+30*x)^(1/2),1/3*15^(1/2))*x*(-20-30*x)^(1/2)*(3+2*x)^(1/2)*(-2-2*x)^(1/2)
+22914*15^(1/2)*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-20-30*x)^(1/2)*EllipticF(1/5*(45+30*x)^(1/2),1/3*15^(1/2))-4050
9*15^(1/2)*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-20-30*x)^(1/2)*EllipticE(1/5*(45+30*x)^(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.90, size = 106, normalized size = 0.55 \begin {gather*} \frac {18413 \, \sqrt {6} {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) - 81018 \, \sqrt {6} {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) - 36 \, {\left (18004 \, x^{2} + 57922 \, x + 47349\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}}{33750 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \end {gather*}

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]

1/33750*(18413*sqrt(6)*(8*x^3 + 36*x^2 + 54*x + 27)*weierstrassPInverse(19/27, -28/729, x + 19/18) - 81018*sqr
t(6)*(8*x^3 + 36*x^2 + 54*x + 27)*weierstrassZeta(19/27, -28/729, weierstrassPInverse(19/27, -28/729, x + 19/1
8)) - 36*(18004*x^2 + 57922*x + 47349)*sqrt(3*x^2 + 5*x + 2)*sqrt(2*x + 3))/(8*x^3 + 36*x^2 + 54*x + 27)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \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 \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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