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

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

[Out]

-264260033/974531250*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-7261561/487265625*Elliptic
F(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-2/5*(1-2*x)^(5/2)*(2+3*x)^(7/2)/(3+5*x)^(1/2)-48/275*(1
-2*x)^(3/2)*(2+3*x)^(7/2)*(3+5*x)^(1/2)+2020841/6496875*(2+3*x)^(3/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)+346636/25987
5*(2+3*x)^(5/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)-2972/7425*(2+3*x)^(7/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)-703672/3248437
5*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]  time = 0.10, antiderivative size = 249, normalized size of antiderivative = 1.00, 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 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)]))

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]

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*}

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Mathematica [A]  time = 0.41, size = 125, normalized size = 0.50 \[ \frac {-24628520 \sqrt {10 x+6} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+30 \sqrt {1-2 x} \sqrt {3 x+2} \left (127575000 x^5+56227500 x^4-141221250 x^3-32807925 x^2+71568535 x+26378214\right )[phone] \sqrt {10 x+6} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{974531250 \sqrt {5 x+3}} \]

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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fricas [F]  time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{25 \, x^{2} + 30 \, x + 9}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}}\,{d x} \]

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)

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maple [C]  time = 0.02, size = 160, normalized size = 0.64 \[ \frac {\sqrt {-2 x +1}\, \sqrt {3 x +2}\, \sqrt {5 x +3}\, \left (22963500000 x^{7}[phone] x^{6}-31387500000 x^{5}-13515714000 x^{4}[phone] x^{3}[phone] x^{2}-3502765680 x -264260033 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-1582692840\right )}{29235937500 x^{3}[phone] x^{2}-6821718750 x -5847187500} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/974531250*(-2*x+1)^(1/2)*(3*x+2)^(1/2)*(5*x+3)^(1/2)*(22963500000*x^7+13948200000*x^6+24628520*2^(1/2)*(5*x+
3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)*EllipticF(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))-264260033*2^(1/2)*(5*x+3
)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)*EllipticE(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))-31387500000*x^5-135157140
00*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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}}\,{d x} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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