3.2883 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=218 \[ -\frac{10614544 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{18865 \sqrt{33}}+\frac{352875016 \sqrt{1-2 x} \sqrt{3 x+2}}{124509 \sqrt{5 x+3}}-\frac{5307272 \sqrt{1-2 x} \sqrt{3 x+2}}{11319 (5 x+3)^{3/2}}+\frac{120324 \sqrt{1-2 x}}{1715 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{576 \sqrt{1-2 x}}{245 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac{6 \sqrt{1-2 x}}{35 (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac{352875016 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{18865 \sqrt{33}} \]

[Out]

(6*Sqrt[1 - 2*x])/(35*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (576*Sqrt[1 - 2*x])/(245*(2 + 3*x)^(3/2)*(3 + 5*x)^(3
/2)) + (120324*Sqrt[1 - 2*x])/(1715*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (5307272*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(11
319*(3 + 5*x)^(3/2)) + (352875016*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(124509*Sqrt[3 + 5*x]) - (352875016*EllipticE[A
rcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(18865*Sqrt[33]) - (10614544*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]]
, 35/33])/(18865*Sqrt[33])

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Rubi [A]  time = 0.0839212, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {104, 152, 158, 113, 119} \[ \frac{352875016 \sqrt{1-2 x} \sqrt{3 x+2}}{124509 \sqrt{5 x+3}}-\frac{5307272 \sqrt{1-2 x} \sqrt{3 x+2}}{11319 (5 x+3)^{3/2}}+\frac{120324 \sqrt{1-2 x}}{1715 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{576 \sqrt{1-2 x}}{245 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac{6 \sqrt{1-2 x}}{35 (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac{10614544 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{18865 \sqrt{33}}-\frac{352875016 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{18865 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*Sqrt[1 - 2*x])/(35*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (576*Sqrt[1 - 2*x])/(245*(2 + 3*x)^(3/2)*(3 + 5*x)^(3
/2)) + (120324*Sqrt[1 - 2*x])/(1715*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (5307272*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(11
319*(3 + 5*x)^(3/2)) + (352875016*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(124509*Sqrt[3 + 5*x]) - (352875016*EllipticE[A
rcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(18865*Sqrt[33]) - (10614544*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]]
, 35/33])/(18865*Sqrt[33])

Rule 104

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(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*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
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}{\sqrt{1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx &=\frac{6 \sqrt{1-2 x}}{35 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{2}{35} \int \frac{74-105 x}{\sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx\\ &=\frac{6 \sqrt{1-2 x}}{35 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{576 \sqrt{1-2 x}}{245 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{4}{735} \int \frac{\frac{15681}{2}-10800 x}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=\frac{6 \sqrt{1-2 x}}{35 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{576 \sqrt{1-2 x}}{245 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{120324 \sqrt{1-2 x}}{1715 \sqrt{2+3 x} (3+5 x)^{3/2}}+\frac{8 \int \frac{589020-\frac{1353645 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}} \, dx}{5145}\\ &=\frac{6 \sqrt{1-2 x}}{35 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{576 \sqrt{1-2 x}}{245 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{120324 \sqrt{1-2 x}}{1715 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{5307272 \sqrt{1-2 x} \sqrt{2+3 x}}{11319 (3+5 x)^{3/2}}-\frac{16 \int \frac{\frac{96504045}{4}-\frac{29853405 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx}{169785}\\ &=\frac{6 \sqrt{1-2 x}}{35 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{576 \sqrt{1-2 x}}{245 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{120324 \sqrt{1-2 x}}{1715 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{5307272 \sqrt{1-2 x} \sqrt{2+3 x}}{11319 (3+5 x)^{3/2}}+\frac{352875016 \sqrt{1-2 x} \sqrt{2+3 x}}{124509 \sqrt{3+5 x}}+\frac{32 \int \frac{\frac{628315335}{2}+\frac{1984921965 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1867635}\\ &=\frac{6 \sqrt{1-2 x}}{35 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{576 \sqrt{1-2 x}}{245 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{120324 \sqrt{1-2 x}}{1715 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{5307272 \sqrt{1-2 x} \sqrt{2+3 x}}{11319 (3+5 x)^{3/2}}+\frac{352875016 \sqrt{1-2 x} \sqrt{2+3 x}}{124509 \sqrt{3+5 x}}+\frac{5307272 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{18865}+\frac{352875016 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{207515}\\ &=\frac{6 \sqrt{1-2 x}}{35 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac{576 \sqrt{1-2 x}}{245 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac{120324 \sqrt{1-2 x}}{1715 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{5307272 \sqrt{1-2 x} \sqrt{2+3 x}}{11319 (3+5 x)^{3/2}}+\frac{352875016 \sqrt{1-2 x} \sqrt{2+3 x}}{124509 \sqrt{3+5 x}}-\frac{352875016 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{18865 \sqrt{33}}-\frac{10614544 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{18865 \sqrt{33}}\\ \end{align*}

Mathematica [A]  time = 0.242963, size = 109, normalized size = 0.5 \[ \frac{2 \left (4 \sqrt{2} \left (44109377 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-22216880 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{\sqrt{1-2 x} \left (119095317900 x^4+305707177080 x^3+294023389014 x^2+125573817736 x+20093773321\right )}{(3 x+2)^{5/2} (5 x+3)^{3/2}}\right )}{622545} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((Sqrt[1 - 2*x]*(20093773321 + 125573817736*x + 294023389014*x^2 + 305707177080*x^3 + 119095317900*x^4))/((
2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + 4*Sqrt[2]*(44109377*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 222
16880*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/622545

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Maple [C]  time = 0.027, size = 406, normalized size = 1.9 \begin{align*} -{\frac{2}{1245090\,x-622545}\sqrt{1-2\,x} \left ( 7939687860\,\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}-3999038400\,\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}[phone]\,\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}-7731474240\,\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}[phone]\,\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}-4976581120\,\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}[phone]\,\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 ) -1066410240\,\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 ) -238190635800\,{x}^{5}-492319036260\,{x}^{4}-282339600948\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+20093773321 \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+3*x)^(7/2)/(3+5*x)^(5/2)/(1-2*x)^(1/2),x)

[Out]

-2/622545*(1-2*x)^(1/2)*(7939687860*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)-3999038400*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)[phone]*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)-7731474240*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)[phone]*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)-4976581120*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)[phone]*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))-1066410240*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))-238190635800*x^5-492319036260*x^4-282339600948*x^3+42875753542*x^2+85386
271094*x+20093773321)/(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{1}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{2}} \sqrt{-2 \, x + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*x)^(7/2)/(3+5*x)^(5/2)/(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{20250 \, x^{8} + 80325 \, x^{7} + 127845 \, x^{6} + 97359 \, x^{5} + 25237 \, x^{4} - 13808 \, x^{3} - 12888 \, x^{2} - 3888 \, x - 432}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*x)^(7/2)/(3+5*x)^(5/2)/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(20250*x^8 + 80325*x^7 + 127845*x^6 + 97359*x^5 + 25237*x
^4 - 13808*x^3 - 12888*x^2 - 3888*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+3*x)**(7/2)/(3+5*x)**(5/2)/(1-2*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*x)^(7/2)/(3+5*x)^(5/2)/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

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