3.2813 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=222 \[ -\frac{2912}{5} \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )+\frac{14 (1-2 x)^{3/2}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}+\frac{96808 \sqrt{3 x+2} \sqrt{1-2 x}}{3 \sqrt{5 x+3}}-\frac{16016 \sqrt{3 x+2} \sqrt{1-2 x}}{3 (5 x+3)^{3/2}}+\frac{35948 \sqrt{1-2 x}}{45 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{1232 \sqrt{1-2 x}}{45 (3 x+2)^{3/2} (5 x+3)^{3/2}}-\frac{96808}{5} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(14*(1 - 2*x)^(3/2))/(15*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (1232*Sqrt[1 - 2*x])/(45*(2 + 3*x)^(3/2)*(3 + 5*x)
^(3/2)) + (35948*Sqrt[1 - 2*x])/(45*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (16016*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(3*(3
+ 5*x)^(3/2)) + (96808*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(3*Sqrt[3 + 5*x]) - (96808*Sqrt[11/3]*EllipticE[ArcSin[Sq
rt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (2912*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5

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Rubi [A]  time = 0.0802233, 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 = {98, 150, 152, 158, 113, 119} \[ \frac{14 (1-2 x)^{3/2}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}+\frac{96808 \sqrt{3 x+2} \sqrt{1-2 x}}{3 \sqrt{5 x+3}}-\frac{16016 \sqrt{3 x+2} \sqrt{1-2 x}}{3 (5 x+3)^{3/2}}+\frac{35948 \sqrt{1-2 x}}{45 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{1232 \sqrt{1-2 x}}{45 (3 x+2)^{3/2} (5 x+3)^{3/2}}-\frac{2912}{5} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{96808}{5} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(14*(1 - 2*x)^(3/2))/(15*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (1232*Sqrt[1 - 2*x])/(45*(2 + 3*x)^(3/2)*(3 + 5*x)
^(3/2)) + (35948*Sqrt[1 - 2*x])/(45*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (16016*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(3*(3
+ 5*x)^(3/2)) + (96808*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(3*Sqrt[3 + 5*x]) - (96808*Sqrt[11/3]*EllipticE[ArcSin[Sq
rt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (2912*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5

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 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 152

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 + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && 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)^{5/2}} \, dx &=\frac{14 (1-2 x)^{3/2}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{2}{15} \int \frac{(198-165 x) \sqrt{1-2 x}}{(2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx\\ &=\frac{14 (1-2 x)^{3/2}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{1232 \sqrt{1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}-\frac{4}{135} \int \frac{-\frac{32769}{2}+22605 x}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=\frac{14 (1-2 x)^{3/2}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{1232 \sqrt{1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{35948 \sqrt{1-2 x}}{45 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{8}{945} \int \frac{-\frac{2463615}{2}+\frac{2830905 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac{14 (1-2 x)^{3/2}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{1232 \sqrt{1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{35948 \sqrt{1-2 x}}{45 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{16016 \sqrt{1-2 x} \sqrt{2+3 x}}{3 (3+5 x)^{3/2}}+\frac{16 \int \frac{-\frac{201818925}{4}[phone] x}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx}{31185}\\ &=\frac{14 (1-2 x)^{3/2}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{1232 \sqrt{1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{35948 \sqrt{1-2 x}}{45 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{16016 \sqrt{1-2 x} \sqrt{2+3 x}}{3 (3+5 x)^{3/2}}+\frac{96808 \sqrt{1-2 x} \sqrt{2+3 x}}{3 \sqrt{3+5 x}}-\frac{32 \int \frac{-\frac{2627991135}{4}-\frac{4151066535 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{343035}\\ &=\frac{14 (1-2 x)^{3/2}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{1232 \sqrt{1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{35948 \sqrt{1-2 x}}{45 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{16016 \sqrt{1-2 x} \sqrt{2+3 x}}{3 (3+5 x)^{3/2}}+\frac{96808 \sqrt{1-2 x} \sqrt{2+3 x}}{3 \sqrt{3+5 x}}+\frac{16016}{5} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx+\frac{96808}{5} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{14 (1-2 x)^{3/2}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{1232 \sqrt{1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{35948 \sqrt{1-2 x}}{45 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{16016 \sqrt{1-2 x} \sqrt{2+3 x}}{3 (3+5 x)^{3/2}}+\frac{96808 \sqrt{1-2 x} \sqrt{2+3 x}}{3 \sqrt{3+5 x}}-\frac{96808}{5} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{2912}{5} \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 = 0.261202, size = 109, normalized size = 0.49 \[ \frac{2}{15} \left (4 \sqrt{2} \left (12101 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-6095 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{\sqrt{1-2 x} \left (32672700 x^4+83867940 x^3+80662602 x^2+34450018 x+5512543\right )}{(3 x+2)^{5/2} (5 x+3)^{3/2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((Sqrt[1 - 2*x]*(5512543 + 34450018*x + 80662602*x^2 + 83867940*x^3 + 32672700*x^4))/((2 + 3*x)^(5/2)*(3 +
5*x)^(3/2)) + 4*Sqrt[2]*(12101*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 6095*EllipticF[ArcSin[Sqrt
[2/11]*Sqrt[3 + 5*x]], -33/2])))/15

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Maple [C]  time = 0.024, size = 406, normalized size = 1.8 \begin{align*} -{\frac{2}{30\,x-15}\sqrt{1-2\,x} \left ( 2178180\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{3}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-1097100\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{3}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+4211148\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-2121060\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+2710624\,\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}-1365280\,\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}+580848\,\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 ) -292560\,\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 ) -65345400\,{x}^{5}-135063180\,{x}^{4}-77457264\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+5512543 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \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)^(5/2),x)

[Out]

-2/15*(1-2*x)^(1/2)*(2178180*2^(1/2)*EllipticE(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x^3*(3+5*x)^(1/2)*(2+3*x)
^(1/2)*(1-2*x)^(1/2)-1097100*2^(1/2)*EllipticF(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x^3*(3+5*x)^(1/2)*(2+3*x)
^(1/2)*(1-2*x)^(1/2)+4211148*2^(1/2)*EllipticE(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x^2*(3+5*x)^(1/2)*(2+3*x)
^(1/2)*(1-2*x)^(1/2)-2121060*2^(1/2)*EllipticF(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x^2*(3+5*x)^(1/2)*(2+3*x)
^(1/2)*(1-2*x)^(1/2)+2710624*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)-1365280*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)+580848*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))-292560*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))-65345400*x^5-135063180*x^4-77457264*x^3+11762566*x^2+23424932*x+5512543)/(2+3*x)^(5/2)/(3+5*x)^(3/2)/(2
*x-1)

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{10125 \, x^{7} + 45225 \, x^{6} + 86535 \, x^{5} + 91947 \, x^{4} + 58592 \, x^{3} + 22392 \, x^{2} + 4752 \, x + 432}, 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)^(5/2),x, algorithm="fricas")

[Out]

integral((4*x^2 - 4*x + 1)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(10125*x^7 + 45225*x^6 + 86535*x^5 + 919
47*x^4 + 58592*x^3 + 22392*x^2 + 4752*x + 432), 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)**(5/2),x)

[Out]

Timed out

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

[Out]

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