3.2796 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^{7/2}}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=249 \[ -\frac{7261561 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{14765625 \sqrt{33}}-\frac{48}{275} (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^{7/2}-\frac{2972 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{7/2}}{7425}-\frac{2 (1-2 x)^{5/2} (3 x+2)^{7/2}}{5 \sqrt{5 x+3}}+\frac{346636 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}}{259875}+\frac{2020841 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{6496875}-\frac{703672 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{32484375}-\frac{264260033 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{29531250 \sqrt{33}} \]

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^(7/2))/(5*Sqrt[3 + 5*x]) - (703672*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/32
484375 + (2020841*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/6496875 + (346636*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)
*Sqrt[3 + 5*x])/259875 - (2972*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x])/7425 - (48*(1 - 2*x)^(3/2)*(2 + 3*
x)^(7/2)*Sqrt[3 + 5*x])/275 - (264260033*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(29531250*Sqrt[33]
) - (7261561*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(14765625*Sqrt[33])

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Rubi [A]  time = 0.103185, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {97, 154, 158, 113, 119} \[ -\frac{48}{275} (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^{7/2}-\frac{2972 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{7/2}}{7425}-\frac{2 (1-2 x)^{5/2} (3 x+2)^{7/2}}{5 \sqrt{5 x+3}}+\frac{346636 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}}{259875}+\frac{2020841 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{6496875}-\frac{703672 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{32484375}-\frac{7261561 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{14765625 \sqrt{33}}-\frac{264260033 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{29531250 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(5/2)*(2 + 3*x)^(7/2))/(3 + 5*x)^(3/2),x]

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^(7/2))/(5*Sqrt[3 + 5*x]) - (703672*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/32
484375 + (2020841*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/6496875 + (346636*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)
*Sqrt[3 + 5*x])/259875 - (2972*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x])/7425 - (48*(1 - 2*x)^(3/2)*(2 + 3*
x)^(7/2)*Sqrt[3 + 5*x])/275 - (264260033*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(29531250*Sqrt[33]
) - (7261561*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(14765625*Sqrt[33])

Rule 97

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

Rule 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

Rule 158

Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol]
:> Dist[h/f, Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]), x], x] + Dist[(f*g - e*h)/f, Int[1/(Sqrt[a + b*
x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && SimplerQ[a + b*x, e + f*x] &&
SimplerQ[c + d*x, e + f*x]

Rule 113

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-((b*e
- a*f)/d), 2]*EllipticE[ArcSin[Sqrt[a + b*x]/Rt[-((b*c - a*d)/d), 2]], (f*(b*c - a*d))/(d*(b*e - a*f))])/b, x
] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !LtQ[-((b*c - a*d)/d),
0] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[-(d/(b*c - a*d)), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)
/b, 0])

Rule 119

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d
), 2]*EllipticF[ArcSin[Sqrt[a + b*x]/(Rt[-(b/d), 2]*Sqrt[(b*c - a*d)/b])], (f*(b*c - a*d))/(d*(b*e - a*f))])/(
b*Sqrt[(b*e - a*f)/b]), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ[(b*e - a*f)/b, 0] &
& PosQ[-(b/d)] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[(d*e - c*f)/d, 0] && GtQ[-(d/b), 0]) &&  !(SimplerQ[c +
d*x, a + b*x] && GtQ[(-(b*e) + a*f)/f, 0] && GtQ[-(f/b), 0]) &&  !(SimplerQ[e + f*x, a + b*x] && GtQ[(-(d*e)
+ c*f)/f, 0] && GtQ[(-(b*e) + a*f)/f, 0] && (PosQ[-(f/d)] || PosQ[-(f/b)]))

Rubi steps

\begin{align*} \int \frac{(1-2 x)^{5/2} (2+3 x)^{7/2}}{(3+5 x)^{3/2}} \, dx &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{5 \sqrt{3+5 x}}+\frac{2}{5} \int \frac{\left (\frac{1}{2}-36 x\right ) (1-2 x)^{3/2} (2+3 x)^{5/2}}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{5 \sqrt{3+5 x}}-\frac{48}{275} (1-2 x)^{3/2} (2+3 x)^{7/2} \sqrt{3+5 x}+\frac{4}{825} \int \frac{\left (\frac{2829}{4}-\frac{11145 x}{2}\right ) \sqrt{1-2 x} (2+3 x)^{5/2}}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{5 \sqrt{3+5 x}}-\frac{2972 \sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}}{7425}-\frac{48}{275} (1-2 x)^{3/2} (2+3 x)^{7/2} \sqrt{3+5 x}+\frac{8 \int \frac{\left (\frac{1741605}{8}-\frac{1299885 x}{2}\right ) (2+3 x)^{5/2}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{111375}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{5 \sqrt{3+5 x}}+\frac{346636 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}{259875}-\frac{2972 \sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}}{7425}-\frac{48}{275} (1-2 x)^{3/2} (2+3 x)^{7/2} \sqrt{3+5 x}-\frac{8 \int \frac{(2+3 x)^{3/2} \left (-1265280+\frac{30312615 x}{8}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{3898125}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{5 \sqrt{3+5 x}}+\frac{2020841 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{6496875}+\frac{346636 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}{259875}-\frac{2972 \sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}}{7425}-\frac{48}{275} (1-2 x)^{3/2} (2+3 x)^{7/2} \sqrt{3+5 x}+\frac{8 \int \frac{\sqrt{2+3 x} \left (\frac{254408625}{16}+3958155 x\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{97453125}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{5 \sqrt{3+5 x}}-\frac{703672 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{32484375}+\frac{2020841 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{6496875}+\frac{346636 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}{259875}-\frac{2972 \sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}}{7425}-\frac{48}{275} (1-2 x)^{3/2} (2+3 x)^{7/2} \sqrt{3+5 x}-\frac{8 \int \frac{-\frac{3926957715}{8}-\frac{11891701485 x}{16}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1461796875}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{5 \sqrt{3+5 x}}-\frac{703672 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{32484375}+\frac{2020841 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{6496875}+\frac{346636 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}{259875}-\frac{2972 \sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}}{7425}-\frac{48}{275} (1-2 x)^{3/2} (2+3 x)^{7/2} \sqrt{3+5 x}+\frac{7261561 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{29531250}+\frac{264260033 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{324843750}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{5 \sqrt{3+5 x}}-\frac{703672 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{32484375}+\frac{2020841 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{6496875}+\frac{346636 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}{259875}-\frac{2972 \sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}}{7425}-\frac{48}{275} (1-2 x)^{3/2} (2+3 x)^{7/2} \sqrt{3+5 x}-\frac{264260033 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{29531250 \sqrt{33}}-\frac{7261561 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{14765625 \sqrt{33}}\\ \end{align*}

Mathematica [A]  time = 0.302857, size = 125, normalized size = 0.5 \[ \frac{-24628520 \sqrt{10 x+6} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+30 \sqrt{1-2 x} \sqrt{3 x+2} \left (127575000 x^5+56227500 x^4-141221250 x^3-32807925 x^2+71568535 x+26378214\right )[phone] \sqrt{10 x+6} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{974531250 \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(5/2)*(2 + 3*x)^(7/2))/(3 + 5*x)^(3/2),x]

[Out]

(30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(26378214 + 71568535*x - 32807925*x^2 - 141221250*x^3 + 56227500*x^4 + 1275750
00*x^5) + 264260033*Sqrt[6 + 10*x]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 24628520*Sqrt[6 + 10*x
]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(974531250*Sqrt[3 + 5*x])

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Maple [C]  time = 0.017, size = 160, normalized size = 0.6 \begin{align*}{\frac{1}{29235937500\,{x}^{3}[phone]\,{x}^{2}-6821718750\,x-5847187500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 22963500000\,{x}^{7}[phone]\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) -264260033\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) -31387500000\,{x}^{5}-13515714000\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-3502765680\,x-1582692840 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/974531250*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(22963500000*x^7+13948200000*x^6+24628520*2^(1/2)*(3+5*x
)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))-264260033*2^(1/2)*(3+5*x)^
(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))-31387500000*x^5-13515714000*
x^4+20371373550*x^3+8863610070*x^2-3502765680*x-1582692840)/(30*x^3+23*x^2-7*x-6)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (3 \, x + 2\right )}^{\frac{7}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{25 \, x^{2} + 30 \, x + 9}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(25*x^2 +
30*x + 9), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (3 \, x + 2\right )}^{\frac{7}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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