3.2963 \(\int \frac{(2+3 x)^{5/2} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=220 \[ -\frac{12601}{140} \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )+\frac{(3 x+2)^{5/2} (5 x+3)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{170 (3 x+2)^{3/2} (5 x+3)^{5/2}}{33 \sqrt{1-2 x}}-\frac{1355}{154} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{28283}{462} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{12601}{28} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{69819}{70} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(-12601*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/28 - (28283*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/46
2 - (1355*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/154 - (170*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/(33*Sqrt[1
- 2*x]) + ((2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/(3*(1 - 2*x)^(3/2)) - (69819*Sqrt[33]*EllipticE[ArcSin[Sqrt[3/7]*S
qrt[1 - 2*x]], 35/33])/70 - (12601*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/140

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Rubi [A]  time = 0.0812684, antiderivative size = 220, 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} \[ \frac{(3 x+2)^{5/2} (5 x+3)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{170 (3 x+2)^{3/2} (5 x+3)^{5/2}}{33 \sqrt{1-2 x}}-\frac{1355}{154} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{28283}{462} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{12601}{28} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{12601}{140} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{69819}{70} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-12601*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/28 - (28283*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/46
2 - (1355*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/154 - (170*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/(33*Sqrt[1
- 2*x]) + ((2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/(3*(1 - 2*x)^(3/2)) - (69819*Sqrt[33]*EllipticE[ArcSin[Sqrt[3/7]*S
qrt[1 - 2*x]], 35/33])/70 - (12601*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/140

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{(2+3 x)^{5/2} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx &=\frac{(2+3 x)^{5/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{1}{3} \int \frac{(2+3 x)^{3/2} (3+5 x)^{3/2} \left (\frac{95}{2}+75 x\right )}{(1-2 x)^{3/2}} \, dx\\ &=-\frac{170 (2+3 x)^{3/2} (3+5 x)^{5/2}}{33 \sqrt{1-2 x}}+\frac{(2+3 x)^{5/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{1}{33} \int \frac{\left (-6520-\frac{20325 x}{2}\right ) \sqrt{2+3 x} (3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{1355}{154} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{170 (2+3 x)^{3/2} (3+5 x)^{5/2}}{33 \sqrt{1-2 x}}+\frac{(2+3 x)^{5/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}+\frac{\int \frac{(3+5 x)^{3/2} \left (\frac{2780875}{4}+\frac{2121225 x}{2}\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{1155}\\ &=-\frac{28283}{462} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{1355}{154} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{170 (2+3 x)^{3/2} (3+5 x)^{5/2}}{33 \sqrt{1-2 x}}+\frac{(2+3 x)^{5/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{\int \frac{\left (-\frac{91206225}{2}-\frac{280687275 x}{4}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{17325}\\ &=-\frac{12601}{28} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{28283}{462} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{1355}{154} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{170 (2+3 x)^{3/2} (3+5 x)^{5/2}}{33 \sqrt{1-2 x}}+\frac{(2+3 x)^{5/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}+\frac{\int \frac{\frac{11815083225}{8}+\frac{4665654675 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{155925}\\ &=-\frac{12601}{28} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{28283}{462} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{1355}{154} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{170 (2+3 x)^{3/2} (3+5 x)^{5/2}}{33 \sqrt{1-2 x}}+\frac{(2+3 x)^{5/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}+\frac{138611}{280} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx+\frac{209457}{70} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=-\frac{12601}{28} \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{28283}{462} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}-\frac{1355}{154} \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}-\frac{170 (2+3 x)^{3/2} (3+5 x)^{5/2}}{33 \sqrt{1-2 x}}+\frac{(2+3 x)^{5/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{69819}{70} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{12601}{140} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )\\ \end{align*}

Mathematica [A]  time = 0.233754, size = 130, normalized size = 0.59 \[ -\frac{-421995 \sqrt{2-4 x} (2 x-1) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+10 \sqrt{3 x+2} \sqrt{5 x+3} \left (2700 x^4+12960 x^3+36606 x^2-175958 x+66663\right )+837828 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{840 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(10*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(66663 - 175958*x + 36606*x^2 + 12960*x^3 + 2700*x^4) + 837828*Sqrt[2 - 4*x]*
(-1 + 2*x)*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 421995*Sqrt[2 - 4*x]*(-1 + 2*x)*EllipticF[ArcS
in[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(840*(1 - 2*x)^(3/2))

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Maple [C]  time = 0.021, size = 243, normalized size = 1.1 \begin{align*}{\frac{1}{840\, \left ( 2\,x-1 \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( 843990\,\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}-1675656\,\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}-405000\,{x}^{6}-421995\,\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 ) +837828\,\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 ) -2457000\,{x}^{5}-8115300\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-2108490\,x-3999780 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/840*(843990*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)-1675656*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
)-405000*x^6-421995*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))+837828*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
))-2457000*x^5-8115300*x^4+18660960*x^3+21236210*x^2-2108490*x-3999780)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1
/2)/(2*x-1)^2/(15*x^2+19*x+6)

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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 (3 \, x + 2\right )}^{\frac{5}{2}}}{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(8*x^3 - 12*x^
2 + 6*x - 1), 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((2+3*x)**(5/2)*(3+5*x)**(5/2)/(1-2*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 (3 \, x + 2\right )}^{\frac{5}{2}}}{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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