3.2736 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^{9/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=191 \[ -\frac{37768 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{5145}+\frac{1255552 \sqrt{1-2 x} \sqrt{5 x+3}}{5145 \sqrt{3 x+2}}+\frac{18068 \sqrt{1-2 x} \sqrt{5 x+3}}{735 (3 x+2)^{3/2}}+\frac{388 \sqrt{1-2 x} \sqrt{5 x+3}}{105 (3 x+2)^{5/2}}+\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{3 (3 x+2)^{7/2}}-\frac{1255552 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5145} \]

[Out]

(2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)^(7/2)) + (388*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(105*(2 + 3*x)^(5/2))
+ (18068*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(735*(2 + 3*x)^(3/2)) + (1255552*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(5145*Sqrt
[2 + 3*x]) - (1255552*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5145 - (37768*Sqrt[11/3]*E
llipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5145

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Rubi [A]  time = 0.065468, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {98, 152, 158, 113, 119} \[ \frac{1255552 \sqrt{1-2 x} \sqrt{5 x+3}}{5145 \sqrt{3 x+2}}+\frac{18068 \sqrt{1-2 x} \sqrt{5 x+3}}{735 (3 x+2)^{3/2}}+\frac{388 \sqrt{1-2 x} \sqrt{5 x+3}}{105 (3 x+2)^{5/2}}+\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{3 (3 x+2)^{7/2}}-\frac{37768 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5145}-\frac{1255552 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5145} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(3/2)/((2 + 3*x)^(9/2)*Sqrt[3 + 5*x]),x]

[Out]

(2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)^(7/2)) + (388*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(105*(2 + 3*x)^(5/2))
+ (18068*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(735*(2 + 3*x)^(3/2)) + (1255552*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(5145*Sqrt
[2 + 3*x]) - (1255552*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5145 - (37768*Sqrt[11/3]*E
llipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5145

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 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)^{3/2}}{(2+3 x)^{9/2} \sqrt{3+5 x}} \, dx &=\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{3 (2+3 x)^{7/2}}+\frac{2}{21} \int \frac{119-161 x}{\sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{3 (2+3 x)^{7/2}}+\frac{388 \sqrt{1-2 x} \sqrt{3+5 x}}{105 (2+3 x)^{5/2}}+\frac{4}{735} \int \frac{\frac{18039}{2}-10185 x}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{3 (2+3 x)^{7/2}}+\frac{388 \sqrt{1-2 x} \sqrt{3+5 x}}{105 (2+3 x)^{5/2}}+\frac{18068 \sqrt{1-2 x} \sqrt{3+5 x}}{735 (2+3 x)^{3/2}}+\frac{8 \int \frac{391209-\frac{474285 x}{2}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{15435}\\ &=\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{3 (2+3 x)^{7/2}}+\frac{388 \sqrt{1-2 x} \sqrt{3+5 x}}{105 (2+3 x)^{5/2}}+\frac{18068 \sqrt{1-2 x} \sqrt{3+5 x}}{735 (2+3 x)^{3/2}}+\frac{1255552 \sqrt{1-2 x} \sqrt{3+5 x}}{5145 \sqrt{2+3 x}}+\frac{16 \int \frac{\frac{20865495}{4}+8239560 x}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{108045}\\ &=\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{3 (2+3 x)^{7/2}}+\frac{388 \sqrt{1-2 x} \sqrt{3+5 x}}{105 (2+3 x)^{5/2}}+\frac{18068 \sqrt{1-2 x} \sqrt{3+5 x}}{735 (2+3 x)^{3/2}}+\frac{1255552 \sqrt{1-2 x} \sqrt{3+5 x}}{5145 \sqrt{2+3 x}}+\frac{207724 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{5145}+\frac{1255552 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{5145}\\ &=\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{3 (2+3 x)^{7/2}}+\frac{388 \sqrt{1-2 x} \sqrt{3+5 x}}{105 (2+3 x)^{5/2}}+\frac{18068 \sqrt{1-2 x} \sqrt{3+5 x}}{735 (2+3 x)^{3/2}}+\frac{1255552 \sqrt{1-2 x} \sqrt{3+5 x}}{5145 \sqrt{2+3 x}}-\frac{1255552 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5145}-\frac{37768 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5145}\\ \end{align*}

Mathematica [A]  time = 0.244477, size = 106, normalized size = 0.55 \[ \frac{4 \left (\sqrt{2} \left (313888 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-158095 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (16949952 x^3+34469046 x^2+23387310 x+5295887\right )}{2 (3 x+2)^{7/2}}\right )}{15435} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^(3/2)/((2 + 3*x)^(9/2)*Sqrt[3 + 5*x]),x]

[Out]

(4*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(5295887 + 23387310*x + 34469046*x^2 + 16949952*x^3))/(2*(2 + 3*x)^(7/2)) +
Sqrt[2]*(313888*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 158095*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[
3 + 5*x]], -33/2])))/15435

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Maple [C]  time = 0.022, size = 409, normalized size = 2.1 \begin{align*}{\frac{2}{154350\,{x}^{2}+15435\,x-46305} \left ( 8537130\,\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}-16949952\,\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}[phone]\,\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}-33899904\,\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}[phone]\,\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}-22599936\,\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}+2529520\,\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 ) -5022208\,\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 ) [phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}-81182874\,{x}^{2}-194598129\,x-47662983 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(3/2)/(2+3*x)^(9/2)/(3+5*x)^(1/2),x)

[Out]

2/15435*(8537130*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)-16949952*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)[phone]*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)-33899904*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)[phone]*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)-22599936*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)+2529520*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))-5022208*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)
)[phone]*x^5+1084921236*x^4+652476870*x^3-81182874*x^2-194598129*x-47662983)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(1
0*x^2+x-3)/(2+3*x)^(7/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{9}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)/(2+3*x)^(9/2)/(3+5*x)^(1/2),x, algorithm="maxima")

[Out]

integrate((-2*x + 1)^(3/2)/(sqrt(5*x + 3)*(3*x + 2)^(9/2)), 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}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)/(2+3*x)^(9/2)/(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(5*x + 3)*sqrt(3*x + 2)*(-2*x + 1)^(3/2)/(1215*x^6 + 4779*x^5 + 7830*x^4 + 6840*x^3 + 3360*x^2 +
880*x + 96), 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)**(3/2)/(2+3*x)**(9/2)/(3+5*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 (-2 \, x + 1\right )}^{\frac{3}{2}}}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{9}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)/(2+3*x)^(9/2)/(3+5*x)^(1/2),x, algorithm="giac")

[Out]

integrate((-2*x + 1)^(3/2)/(sqrt(5*x + 3)*(3*x + 2)^(9/2)), x)