3.2830 \(\int \frac{(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=249 \[ -\frac{2295970088 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{4606875 \sqrt{33}}-\frac{1}{13} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^{7/2}-\frac{41}{143} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^{5/2}-\frac{14303 \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^{3/2}}{12870}-\frac{221673 \sqrt{1-2 x} (5 x+3)^{5/2} \sqrt{3 x+2}}{50050}-\frac{138809831 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{4504500}-\frac{2295970088 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{10135125}-\frac{610627101631 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{36855000 \sqrt{33}} \]

[Out]

(-2295970088*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/10135125 - (138809831*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 +
 5*x)^(3/2))/4504500 - (221673*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/50050 - (14303*Sqrt[1 - 2*x]*(2 +
3*x)^(3/2)*(3 + 5*x)^(5/2))/12870 - (41*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/143 - (Sqrt[1 - 2*x]*(2
 + 3*x)^(7/2)*(3 + 5*x)^(5/2))/13 - (610627101631*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(36855000
*Sqrt[33]) - (2295970088*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(4606875*Sqrt[33])

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Rubi [A]  time = 0.100438, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {101, 154, 158, 113, 119} \[ -\frac{1}{13} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^{7/2}-\frac{41}{143} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^{5/2}-\frac{14303 \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^{3/2}}{12870}-\frac{221673 \sqrt{1-2 x} (5 x+3)^{5/2} \sqrt{3 x+2}}{50050}-\frac{138809831 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{4504500}-\frac{2295970088 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{10135125}-\frac{2295970088 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{4606875 \sqrt{33}}-\frac{610627101631 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{36855000 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2295970088*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/10135125 - (138809831*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 +
 5*x)^(3/2))/4504500 - (221673*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/50050 - (14303*Sqrt[1 - 2*x]*(2 +
3*x)^(3/2)*(3 + 5*x)^(5/2))/12870 - (41*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/143 - (Sqrt[1 - 2*x]*(2
 + 3*x)^(7/2)*(3 + 5*x)^(5/2))/13 - (610627101631*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(36855000
*Sqrt[33]) - (2295970088*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(4606875*Sqrt[33])

Rule 101

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a +
b*x)^m*(c + d*x)^n*(e + f*x)^(p + 1))/(f*(m + n + p + 1)), x] - Dist[1/(f*(m + n + p + 1)), Int[(a + b*x)^(m -
 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a*f) + b*n*(d*e - c*f))
*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (Integ
ersQ[2*m, 2*n, 2*p] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))

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)^{7/2} (3+5 x)^{5/2}}{\sqrt{1-2 x}} \, dx &=-\frac{1}{13} \sqrt{1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}+\frac{1}{13} \int \frac{(2+3 x)^{5/2} (3+5 x)^{3/2} \left (\frac{257}{2}+205 x\right )}{\sqrt{1-2 x}} \, dx\\ &=-\frac{41}{143} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{1}{13} \sqrt{1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}-\frac{1}{715} \int \frac{\left (-\frac{45285}{2}-\frac{71515 x}{2}\right ) (2+3 x)^{3/2} (3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{14303 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{12870}-\frac{41}{143} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{1}{13} \sqrt{1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}+\frac{\int \frac{\sqrt{2+3 x} (3+5 x)^{3/2} \left (\frac{12799775}{4}+\frac{9975285 x}{2}\right )}{\sqrt{1-2 x}} \, dx}{32175}\\ &=-\frac{221673 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{50050}-\frac{14303 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{12870}-\frac{41}{143} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{1}{13} \sqrt{1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}-\frac{\int \frac{\left (-\frac{1364822645}{4}-\frac{2082147465 x}{4}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{1126125}\\ &=-\frac{138809831 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{4504500}-\frac{221673 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{50050}-\frac{14303 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{12870}-\frac{41}{143} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{1}{13} \sqrt{1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}+\frac{\int \frac{\sqrt{3+5 x} \left (\frac{179052019605}{8}+34439551320 x\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{16891875}\\ &=-\frac{2295970088 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{10135125}-\frac{138809831 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{4504500}-\frac{221673 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{50050}-\frac{14303 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{12870}-\frac{41}{143} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{1}{13} \sqrt{1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}-\frac{\int \frac{-\frac{5798711966295}{8}-\frac{9159406524465 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{152026875}\\ &=-\frac{2295970088 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{10135125}-\frac{138809831 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{4504500}-\frac{221673 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{50050}-\frac{14303 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{12870}-\frac{41}{143} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{1}{13} \sqrt{1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}+\frac{1147985044 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{4606875}+\frac{610627101631 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{405405000}\\ &=-\frac{2295970088 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{10135125}-\frac{138809831 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{4504500}-\frac{221673 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{50050}-\frac{14303 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{12870}-\frac{41}{143} \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac{1}{13} \sqrt{1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}-\frac{610627101631 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{36855000 \sqrt{33}}-\frac{2295970088 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{4606875 \sqrt{33}}\\ \end{align*}

Mathematica [A]  time = 0.28098, size = 115, normalized size = 0.46 \[ \frac{610627101631 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (61511810003 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+3 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (2104987500 x^5+9351247500 x^4+18620894250 x^3+22592085750 x^2+19961825445 x+16001700059\right )\right )}{608107500 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(610627101631*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 5*(3*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5
*x]*(16001700059 + 19961825445*x + 22592085750*x^2 + 18620894250*x^3 + 9351247500*x^4 + 2104987500*x^5) + 6151
1810003*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/(608107500*Sqrt[2])

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Maple [C]  time = 0.013, size = 165, normalized size = 0.7 \begin{align*}{\frac{1}{36486450000\,{x}^{3}+27972945000\,{x}^{2}-8513505000\,x-7297290000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -1894488750000\,{x}^{8}-9868564125000\,{x}^{7}-22769118225000\,{x}^{6}+307559050015\,\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 ) -610627101631\,\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 ) -30838634482500\,{x}^{5}-27960569725500\,{x}^{4}-20079090637650\,{x}^{3}-2782614262260\,{x}^{2}+6953485592490\,x+2880306010620 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1216215000*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(-1894488750000*x^8-9868564125000*x^7-22769118225000*x^
6+307559050015*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
))-610627101631*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))-30838634482500*x^5-27960569725500*x^4-20079090637650*x^3-2782614262260*x^2+6953485592490*x+2880306010620)/
(30*x^3+23*x^2-7*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{7}{2}}}{\sqrt{-2 \, x + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((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{{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{2 \, x - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(
2*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)**(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{{\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((2+3*x)^(7/2)*(3+5*x)^(5/2)/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

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