3.2793 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^{9/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=191 \[ -\frac{21160 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{1323}+\frac{2 \sqrt{5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{7/2}}+\frac{703480 \sqrt{5 x+3} \sqrt{1-2 x}}{1323 \sqrt{3 x+2}}+\frac{10124 \sqrt{5 x+3} \sqrt{1-2 x}}{189 (3 x+2)^{3/2}}+\frac{76 \sqrt{5 x+3} \sqrt{1-2 x}}{9 (3 x+2)^{5/2}}-\frac{703480 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1323} \]

[Out]

(2*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(3*(2 + 3*x)^(7/2)) + (76*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(9*(2 + 3*x)^(5/2)) +
(10124*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(189*(2 + 3*x)^(3/2)) + (703480*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1323*Sqrt[2
+ 3*x]) - (703480*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1323 - (21160*Sqrt[11/3]*Elli
pticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1323

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Rubi [A]  time = 0.0665178, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{2 \sqrt{5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{7/2}}+\frac{703480 \sqrt{5 x+3} \sqrt{1-2 x}}{1323 \sqrt{3 x+2}}+\frac{10124 \sqrt{5 x+3} \sqrt{1-2 x}}{189 (3 x+2)^{3/2}}+\frac{76 \sqrt{5 x+3} \sqrt{1-2 x}}{9 (3 x+2)^{5/2}}-\frac{21160 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1323}-\frac{703480 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1323} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(3*(2 + 3*x)^(7/2)) + (76*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(9*(2 + 3*x)^(5/2)) +
(10124*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(189*(2 + 3*x)^(3/2)) + (703480*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1323*Sqrt[2
+ 3*x]) - (703480*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1323 - (21160*Sqrt[11/3]*Elli
pticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1323

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)^{9/2} \sqrt{3+5 x}} \, dx &=\frac{2 (1-2 x)^{3/2} \sqrt{3+5 x}}{3 (2+3 x)^{7/2}}+\frac{2}{21} \int \frac{(161-91 x) \sqrt{1-2 x}}{(2+3 x)^{7/2} \sqrt{3+5 x}} \, dx\\ &=\frac{2 (1-2 x)^{3/2} \sqrt{3+5 x}}{3 (2+3 x)^{7/2}}+\frac{76 \sqrt{1-2 x} \sqrt{3+5 x}}{9 (2+3 x)^{5/2}}-\frac{4}{315} \int \frac{-\frac{16835}{2}+9520 x}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx\\ &=\frac{2 (1-2 x)^{3/2} \sqrt{3+5 x}}{3 (2+3 x)^{7/2}}+\frac{76 \sqrt{1-2 x} \sqrt{3+5 x}}{9 (2+3 x)^{5/2}}+\frac{10124 \sqrt{1-2 x} \sqrt{3+5 x}}{189 (2+3 x)^{3/2}}-\frac{8 \int \frac{-\frac{730625}{2}+\frac{442925 x}{2}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{6615}\\ &=\frac{2 (1-2 x)^{3/2} \sqrt{3+5 x}}{3 (2+3 x)^{7/2}}+\frac{76 \sqrt{1-2 x} \sqrt{3+5 x}}{9 (2+3 x)^{5/2}}+\frac{10124 \sqrt{1-2 x} \sqrt{3+5 x}}{189 (2+3 x)^{3/2}}+\frac{703480 \sqrt{1-2 x} \sqrt{3+5 x}}{1323 \sqrt{2+3 x}}-\frac{16 \int \frac{-\frac{19484675}{4}-\frac{15388625 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{46305}\\ &=\frac{2 (1-2 x)^{3/2} \sqrt{3+5 x}}{3 (2+3 x)^{7/2}}+\frac{76 \sqrt{1-2 x} \sqrt{3+5 x}}{9 (2+3 x)^{5/2}}+\frac{10124 \sqrt{1-2 x} \sqrt{3+5 x}}{189 (2+3 x)^{3/2}}+\frac{703480 \sqrt{1-2 x} \sqrt{3+5 x}}{1323 \sqrt{2+3 x}}+\frac{116380 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1323}+\frac{703480 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{1323}\\ &=\frac{2 (1-2 x)^{3/2} \sqrt{3+5 x}}{3 (2+3 x)^{7/2}}+\frac{76 \sqrt{1-2 x} \sqrt{3+5 x}}{9 (2+3 x)^{5/2}}+\frac{10124 \sqrt{1-2 x} \sqrt{3+5 x}}{189 (2+3 x)^{3/2}}+\frac{703480 \sqrt{1-2 x} \sqrt{3+5 x}}{1323 \sqrt{2+3 x}}-\frac{703480 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1323}-\frac{21160 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1323}\\ \end{align*}

Mathematica [A]  time = 0.232558, size = 107, normalized size = 0.56 \[ \frac{4 \left (5 \sqrt{2} \left (35174 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-17717 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (9496980 x^3+19312866 x^2+13103724 x+2967269\right )}{2 (3 x+2)^{7/2}}\right )}{3969} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2967269 + 13103724*x + 19312866*x^2 + 9496980*x^3))/(2*(2 + 3*x)^(7/2)) +
5*Sqrt[2]*(35174*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 17717*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3
+ 5*x]], -33/2])))/3969

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Maple [C]  time = 0.02, size = 409, normalized size = 2.1 \begin{align*}{\frac{2}{39690\,{x}^{2}+3969\,x-11907} \left ( 4783590\,\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}-9496980\,\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}+9567180\,\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}-18993960\,\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}+6378120\,\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}-12662640\,\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}+1417360\,\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 ) -2813920\,\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}-45486552\,{x}^{2}-109031709\,x-26705421 \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)^(5/2)/(2+3*x)^(9/2)/(3+5*x)^(1/2),x)

[Out]

2/3969*(4783590*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)-9496980*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)+9567180*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)-18993960*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)+6378120*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)-12662640*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)+1417360*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)
)-2813920*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))+28
4909400*x^5+607876920*x^4+365577498*x^3-45486552*x^2-109031709*x-26705421)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*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{5}{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)^(5/2)/(2+3*x)^(9/2)/(3+5*x)^(1/2),x, algorithm="maxima")

[Out]

integrate((-2*x + 1)^(5/2)/(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{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{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)^(5/2)/(2+3*x)^(9/2)/(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

integral((4*x^2 - 4*x + 1)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(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)**(5/2)/(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{5}{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)^(5/2)/(2+3*x)^(9/2)/(3+5*x)^(1/2),x, algorithm="giac")

[Out]

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