3.2990 \(\int \frac{(2+3 x)^{13/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=249 \[ -\frac{776112041 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{41593750 \sqrt{33}}+\frac{7 (3 x+2)^{11/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{294 (3 x+2)^{9/2}}{121 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{4373 \sqrt{1-2 x} (3 x+2)^{7/2}}{19965 (5 x+3)^{3/2}}+\frac{150812 \sqrt{1-2 x} (3 x+2)^{5/2}}{1098075 \sqrt{5 x+3}}-\frac{31887029 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{18301250}-\frac{371279941 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{45753125}-\frac{51601293223 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{83187500 \sqrt{33}} \]

[Out]

(4373*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2))/(19965*(3 + 5*x)^(3/2)) - (294*(2 + 3*x)^(9/2))/(121*Sqrt[1 - 2*x]*(3 + 5
*x)^(3/2)) + (7*(2 + 3*x)^(11/2))/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + (150812*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)
)/(1098075*Sqrt[3 + 5*x]) - (371279941*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/45753125 - (31887029*Sqrt[1
- 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/18301250 - (51601293223*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33
])/(83187500*Sqrt[33]) - (776112041*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(41593750*Sqrt[33])

________________________________________________________________________________________

Rubi [A]  time = 0.0986754, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {98, 150, 154, 158, 113, 119} \[ \frac{7 (3 x+2)^{11/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{294 (3 x+2)^{9/2}}{121 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{4373 \sqrt{1-2 x} (3 x+2)^{7/2}}{19965 (5 x+3)^{3/2}}+\frac{150812 \sqrt{1-2 x} (3 x+2)^{5/2}}{1098075 \sqrt{5 x+3}}-\frac{31887029 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{18301250}-\frac{371279941 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{45753125}-\frac{776112041 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{41593750 \sqrt{33}}-\frac{51601293223 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{83187500 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^(13/2)/((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)),x]

[Out]

(4373*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2))/(19965*(3 + 5*x)^(3/2)) - (294*(2 + 3*x)^(9/2))/(121*Sqrt[1 - 2*x]*(3 + 5
*x)^(3/2)) + (7*(2 + 3*x)^(11/2))/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + (150812*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)
)/(1098075*Sqrt[3 + 5*x]) - (371279941*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/45753125 - (31887029*Sqrt[1
- 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/18301250 - (51601293223*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33
])/(83187500*Sqrt[33]) - (776112041*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(41593750*Sqrt[33])

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 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)^{13/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx &=\frac{7 (2+3 x)^{11/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{1}{33} \int \frac{(2+3 x)^{9/2} \left (\frac{471}{2}+411 x\right )}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=-\frac{294 (2+3 x)^{9/2}}{121 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{11/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{1}{363} \int \frac{\left (-22602-\frac{79713 x}{2}\right ) (2+3 x)^{7/2}}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx\\ &=\frac{4373 \sqrt{1-2 x} (2+3 x)^{7/2}}{19965 (3+5 x)^{3/2}}-\frac{294 (2+3 x)^{9/2}}{121 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{11/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{2 \int \frac{\left (-\frac{4880607}{4}-\frac{8285157 x}{4}\right ) (2+3 x)^{5/2}}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx}{59895}\\ &=\frac{4373 \sqrt{1-2 x} (2+3 x)^{7/2}}{19965 (3+5 x)^{3/2}}-\frac{294 (2+3 x)^{9/2}}{121 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{11/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{150812 \sqrt{1-2 x} (2+3 x)^{5/2}}{1098075 \sqrt{3+5 x}}-\frac{4 \int \frac{\left (-\frac{87743457}{4}-\frac{286983261 x}{8}\right ) (2+3 x)^{3/2}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{3294225}\\ &=\frac{4373 \sqrt{1-2 x} (2+3 x)^{7/2}}{19965 (3+5 x)^{3/2}}-\frac{294 (2+3 x)^{9/2}}{121 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{11/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{150812 \sqrt{1-2 x} (2+3 x)^{5/2}}{1098075 \sqrt{3+5 x}}-\frac{31887029 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{18301250}+\frac{4 \int \frac{\sqrt{2+3 x} \left (\frac{24723272925}{16}+\frac{10024558407 x}{4}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{82355625}\\ &=\frac{4373 \sqrt{1-2 x} (2+3 x)^{7/2}}{19965 (3+5 x)^{3/2}}-\frac{294 (2+3 x)^{9/2}}{121 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{11/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{150812 \sqrt{1-2 x} (2+3 x)^{5/2}}{1098075 \sqrt{3+5 x}}-\frac{371279941 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{45753125}-\frac{31887029 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{18301250}-\frac{4 \int \frac{-\frac{110255250681}{2}-\frac{1393234917021 x}{16}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1235334375}\\ &=\frac{4373 \sqrt{1-2 x} (2+3 x)^{7/2}}{19965 (3+5 x)^{3/2}}-\frac{294 (2+3 x)^{9/2}}{121 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{11/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{150812 \sqrt{1-2 x} (2+3 x)^{5/2}}{1098075 \sqrt{3+5 x}}-\frac{371279941 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{45753125}-\frac{31887029 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{18301250}+\frac{776112041 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{83187500}+\frac{51601293223 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{915062500}\\ &=\frac{4373 \sqrt{1-2 x} (2+3 x)^{7/2}}{19965 (3+5 x)^{3/2}}-\frac{294 (2+3 x)^{9/2}}{121 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{11/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{150812 \sqrt{1-2 x} (2+3 x)^{5/2}}{1098075 \sqrt{3+5 x}}-\frac{371279941 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{45753125}-\frac{31887029 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{18301250}-\frac{51601293223 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{83187500 \sqrt{33}}-\frac{776112041 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{41593750 \sqrt{33}}\\ \end{align*}

Mathematica [A]  time = 0.290895, size = 117, normalized size = 0.47 \[ \frac{-25989595870 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-\frac{10 \sqrt{3 x+2} \left (8004966750 x^5+53010668700 x^4-222254370925 x^3-215557803774 x^2+21979664649 x+36533948644\right )}{(1-2 x)^{3/2} (5 x+3)^{3/2}}[phone] \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{2745187500} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^(13/2)/((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)),x]

[Out]

((-10*Sqrt[2 + 3*x]*(36533948644 + 21979664649*x - 215557803774*x^2 - 222254370925*x^3 + 53010668700*x^4 + 800
4966750*x^5))/((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + 51601293223*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x
]], -33/2] - 25989595870*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/2745187500

________________________________________________________________________________________

Maple [C]  time = 0.024, size = 321, normalized size = 1.3 \begin{align*}{\frac{1}{2745187500\, \left ( 2\,x-1 \right ) ^{2}}\sqrt{1-2\,x} \left ( 259895958700\,\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}-516012932230\,\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}-51601293223\,\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}-240149002500\,{x}^{6}-77968787610\,\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 ) [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 ) -1750419396000\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-1535611752300\,x-730678972880 \right ) \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{2+3\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^(13/2)/(1-2*x)^(5/2)/(3+5*x)^(5/2),x)

[Out]

1/2745187500*(1-2*x)^(1/2)*(259895958700*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)-516012932230*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)-51601293223*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)-240149002500*x^6-77968787610*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))[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))-1750419396000*x^5+5607417753750*x^4+10911821531720
*x^3+3651766136010*x^2-1535611752300*x-730678972880)/(3+5*x)^(3/2)/(2*x-1)^2/(2+3*x)^(1/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (3 \, x + 2\right )}^{\frac{13}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^(13/2)/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

integrate((3*x + 2)^(13/2)/((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{1000 \, x^{6} + 300 \, x^{5} - 870 \, x^{4} - 179 \, x^{3} + 261 \, x^{2} + 27 \, x - 27}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^(13/2)/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="fricas")

[Out]

integral(-(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(
-2*x + 1)/(1000*x^6 + 300*x^5 - 870*x^4 - 179*x^3 + 261*x^2 + 27*x - 27), x)

________________________________________________________________________________________

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)**(13/2)/(1-2*x)**(5/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (3 \, x + 2\right )}^{\frac{13}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^(13/2)/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

integrate((3*x + 2)^(13/2)/((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)), x)