3.2991 \(\int \frac {(2+3 x)^{11/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=218 \[ -\frac {5442127 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1663750 \sqrt {33}}+\frac {7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {217 (3 x+2)^{7/2}}{121 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {3218 \sqrt {1-2 x} (3 x+2)^{5/2}}{19965 (5 x+3)^{3/2}}+\frac {110519 \sqrt {1-2 x} (3 x+2)^{3/2}}{1098075 \sqrt {5 x+3}}-\frac {5199979 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{3660250}-\frac {90397364 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{831875 \sqrt {33}} \]

[Out]

7/33*(2+3*x)^(9/2)/(1-2*x)^(3/2)/(3+5*x)^(3/2)-90397364/27451875*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*115
5^(1/2))*33^(1/2)-5442127/54903750*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-217/121*(2+3
*x)^(7/2)/(3+5*x)^(3/2)/(1-2*x)^(1/2)+3218/19965*(2+3*x)^(5/2)*(1-2*x)^(1/2)/(3+5*x)^(3/2)+110519/1098075*(2+3
*x)^(3/2)*(1-2*x)^(1/2)/(3+5*x)^(1/2)-5199979/3660250*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]  time = 0.08, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {98, 150, 154, 158, 113, 119} \[ \frac {7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {217 (3 x+2)^{7/2}}{121 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {3218 \sqrt {1-2 x} (3 x+2)^{5/2}}{19965 (5 x+3)^{3/2}}+\frac {110519 \sqrt {1-2 x} (3 x+2)^{3/2}}{1098075 \sqrt {5 x+3}}-\frac {5199979 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{3660250}-\frac {5442127 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1663750 \sqrt {33}}-\frac {90397364 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{831875 \sqrt {33}} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^(11/2)/((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)),x]

[Out]

(3218*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2))/(19965*(3 + 5*x)^(3/2)) - (217*(2 + 3*x)^(7/2))/(121*Sqrt[1 - 2*x]*(3 + 5
*x)^(3/2)) + (7*(2 + 3*x)^(9/2))/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + (110519*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2))
/(1098075*Sqrt[3 + 5*x]) - (5199979*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/3660250 - (90397364*EllipticE[A
rcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(831875*Sqrt[33]) - (5442127*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]]
, 35/33])/(1663750*Sqrt[33])

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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 150

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

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 {(2+3 x)^{11/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx &=\frac {7 (2+3 x)^{9/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1}{33} \int \frac {(2+3 x)^{7/2} \left (\frac {345}{2}+306 x\right )}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=-\frac {217 (2+3 x)^{7/2}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{9/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1}{363} \int \frac {\left (-\frac {21705}{2}-\frac {39393 x}{2}\right ) (2+3 x)^{5/2}}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {3218 \sqrt {1-2 x} (2+3 x)^{5/2}}{19965 (3+5 x)^{3/2}}-\frac {217 (2+3 x)^{7/2}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{9/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {2 \int \frac {\left (-594474-\frac {4073679 x}{4}\right ) (2+3 x)^{3/2}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx}{59895}\\ &=\frac {3218 \sqrt {1-2 x} (2+3 x)^{5/2}}{19965 (3+5 x)^{3/2}}-\frac {217 (2+3 x)^{7/2}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{9/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {110519 \sqrt {1-2 x} (2+3 x)^{3/2}}{1098075 \sqrt {3+5 x}}-\frac {4 \int \frac {\left (-\frac {86636925}{8}-\frac {140399433 x}{8}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{3294225}\\ &=\frac {3218 \sqrt {1-2 x} (2+3 x)^{5/2}}{19965 (3+5 x)^{3/2}}-\frac {217 (2+3 x)^{7/2}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{9/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {110519 \sqrt {1-2 x} (2+3 x)^{3/2}}{1098075 \sqrt {3+5 x}}-\frac {5199979 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3660250}+\frac {4 \int \frac {\frac {6181011531}{16}[phone] x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{49413375}\\ &=\frac {3218 \sqrt {1-2 x} (2+3 x)^{5/2}}{19965 (3+5 x)^{3/2}}-\frac {217 (2+3 x)^{7/2}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{9/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {110519 \sqrt {1-2 x} (2+3 x)^{3/2}}{1098075 \sqrt {3+5 x}}-\frac {5199979 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3660250}+\frac {5442127 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{3327500}+\frac {90397364 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{9150625}\\ &=\frac {3218 \sqrt {1-2 x} (2+3 x)^{5/2}}{19965 (3+5 x)^{3/2}}-\frac {217 (2+3 x)^{7/2}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{9/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {110519 \sqrt {1-2 x} (2+3 x)^{3/2}}{1098075 \sqrt {3+5 x}}-\frac {5199979 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3660250}-\frac {90397364 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{831875 \sqrt {33}}-\frac {5442127 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1663750 \sqrt {33}}\\ \end {align*}

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Mathematica [A]  time = 0.35, size = 112, normalized size = 0.51 \[ \frac {-181999265 \sqrt {2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )-\frac {10 \sqrt {3 x+2} \left (177888150 x^4-1825153850 x^3-1696384053 x^2+89252928 x+246962693\right )}{(1-2 x)^{3/2} (5 x+3)^{3/2}}[phone] \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{109807500} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^(11/2)/((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)),x]

[Out]

((-10*Sqrt[2 + 3*x]*(246962693 + 89252928*x - 1696384053*x^2 - 1825153850*x^3 + 177888150*x^4))/((1 - 2*x)^(3/
2)*(3 + 5*x)^(3/2)) + 361589456*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 181999265*Sqrt[2]
*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/109807500

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fricas [F]  time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{1000 \, x^{6} + 300 \, x^{5} - 870 \, x^{4} - 179 \, x^{3} + 261 \, x^{2} + 27 \, x - 27}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^(11/2)/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="fricas")

[Out]

integral(-(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(10
00*x^6 + 300*x^5 - 870*x^4 - 179*x^3 + 261*x^2 + 27*x - 27), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {11}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^(11/2)/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

integrate((3*x + 2)^(11/2)/((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)), x)

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maple [C]  time = 0.03, size = 316, normalized size = 1.45 \[ \frac {\left (-5336644500 x^{5}[phone] x^{4}[phone] x^{3}-3615894560 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] x^{2}-361589456 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-9193939350 x [phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-545997795 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-4939253860\right ) \sqrt {-2 x +1}}{109807500 \left (5 x +3\right )^{\frac {3}{2}} \left (2 x -1\right )^{2} \sqrt {3 x +2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x+2)^(11/2)/(-2*x+1)^(5/2)/(5*x+3)^(5/2),x)

[Out]

1/109807500*(1819992650*2^(1/2)*EllipticF(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))*x^2*(5*x+3)^(1/2)*(3*x+2)^(1/2
)*(-2*x+1)^(1/2)-3615894560*2^(1/2)*EllipticE(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))*x^2*(5*x+3)^(1/2)*(3*x+2)^
(1/2)*(-2*x+1)^(1/2)[phone]*2^(1/2)*EllipticF(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))*x*(5*x+3)^(1/2)*(3*x+2)
^(1/2)*(-2*x+1)^(1/2)-361589456*2^(1/2)*EllipticE(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))*x*(5*x+3)^(1/2)*(3*x+2
)^(1/2)*(-2*x+1)^(1/2)-545997795*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))[phone]*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))-5336644500*x^5+51196852500*x^4+87394598590*x^3+31250093220*x^2-9193939350*x-4939253860)*
(-2*x+1)^(1/2)/(5*x+3)^(3/2)/(2*x-1)^2/(3*x+2)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {11}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^(11/2)/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

integrate((3*x + 2)^(11/2)/((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (3\,x+2\right )}^{11/2}}{{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x + 2)^(11/2)/((1 - 2*x)^(5/2)*(5*x + 3)^(5/2)),x)

[Out]

int((3*x + 2)^(11/2)/((1 - 2*x)^(5/2)*(5*x + 3)^(5/2)), 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((2+3*x)**(11/2)/(1-2*x)**(5/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

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