3.2779 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{5/2}} \, dx\)

Optimal. Leaf size=222 \[ \frac{135334 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{25515}+\frac{5260}{567} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{370 (1-2 x)^{3/2} (5 x+3)^{5/2}}{27 \sqrt{3 x+2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}-\frac{31298}{567} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}+\frac{135334 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{5103}-\frac{452399 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{25515} \]

[Out]

(135334*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/5103 - (31298*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/
567 - (2*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(9*(2 + 3*x)^(3/2)) + (370*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(27*Sqrt
[2 + 3*x]) + (5260*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/567 - (452399*Sqrt[11/3]*EllipticE[ArcSin[Sqrt
[3/7]*Sqrt[1 - 2*x]], 35/33])/25515 + (135334*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/25
515

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Rubi [A]  time = 0.0794737, antiderivative size = 222, normalized size of antiderivative = 1., 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 = {97, 150, 154, 158, 113, 119} \[ \frac{5260}{567} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{370 (1-2 x)^{3/2} (5 x+3)^{5/2}}{27 \sqrt{3 x+2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}-\frac{31298}{567} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}+\frac{135334 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{5103}+\frac{135334 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{25515}-\frac{452399 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{25515} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(135334*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/5103 - (31298*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/
567 - (2*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(9*(2 + 3*x)^(3/2)) + (370*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(27*Sqrt
[2 + 3*x]) + (5260*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/567 - (452399*Sqrt[11/3]*EllipticE[ArcSin[Sqrt
[3/7]*Sqrt[1 - 2*x]], 35/33])/25515 + (135334*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/25
515

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 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]

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} (3+5 x)^{5/2}}{(2+3 x)^{5/2}} \, dx &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}+\frac{2}{9} \int \frac{\left (-\frac{5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{3/2}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}+\frac{370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 \sqrt{2+3 x}}-\frac{4}{27} \int \frac{\left (\frac{235}{2}-\frac{6575 x}{2}\right ) \sqrt{1-2 x} (3+5 x)^{3/2}}{\sqrt{2+3 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}+\frac{370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 \sqrt{2+3 x}}+\frac{5260}{567} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{8 \int \frac{\left (83425-\frac{1173675 x}{4}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{2835}\\ &=-\frac{31298}{567} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}+\frac{370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 \sqrt{2+3 x}}+\frac{5260}{567} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{8 \int \frac{\left (\frac{1656225}{8}-\frac{5075025 x}{4}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{42525}\\ &=\frac{135334 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{5103}-\frac{31298}{567} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}+\frac{370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 \sqrt{2+3 x}}+\frac{5260}{567} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{8 \int \frac{-\frac{2298225}{2}-\frac{33929925 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{382725}\\ &=\frac{135334 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{5103}-\frac{31298}{567} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}+\frac{370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 \sqrt{2+3 x}}+\frac{5260}{567} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}+\frac{452399 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{25515}-\frac{744337 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{25515}\\ &=\frac{135334 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{5103}-\frac{31298}{567} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}+\frac{370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 \sqrt{2+3 x}}+\frac{5260}{567} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{452399 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{25515}+\frac{135334 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{25515}\\ \end{align*}

Mathematica [A]  time = 0.283928, size = 112, normalized size = 0.5 \[ \frac{-2685410 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{30 \sqrt{1-2 x} \sqrt{5 x+3} \left (24300 x^4-25110 x^3+5949 x^2+108285 x+56963\right )}{(3 x+2)^{3/2}}+452399 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{76545} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((30*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(56963 + 108285*x + 5949*x^2 - 25110*x^3 + 24300*x^4))/(2 + 3*x)^(3/2) + 4523
99*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 2685410*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sq
rt[3 + 5*x]], -33/2])/76545

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Maple [C]  time = 0.02, size = 234, normalized size = 1.1 \begin{align*}{\frac{1}{765450\,{x}^{2}+76545\,x-229635} \left ( 8056230\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-1357197\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+7290000\,{x}^{6}+5370820\,\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 ) -904798\,\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 ) -6804000\,{x}^{5}-1155600\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-8036760\,x-5126670 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/76545*(8056230*2^(1/2)*EllipticF(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)
^(1/2)-1357197*2^(1/2)*EllipticE(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(
1/2)+7290000*x^6+5370820*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))-904798*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))-6804000*x^5-1155600*x^4+34923870*x^3+19802040*x^2-8036760*x-5126670)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x
^2+x-3)/(2+3*x)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(27*x^3 + 54*x^2 + 3
6*x + 8), 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)*(3+5*x)**(5/2)/(2+3*x)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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