3.2839 \(\int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x} (2+3 x)^{11/2}} \, dx\)

Optimal. Leaf size=222 \[ -\frac{2036756 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{47647845}+\frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{189 (3 x+2)^{9/2}}+\frac{32098184 \sqrt{1-2 x} \sqrt{5 x+3}}{47647845 \sqrt{3 x+2}}-\frac{43094 \sqrt{1-2 x} \sqrt{5 x+3}}{6806835 (3 x+2)^{3/2}}-\frac{168034 \sqrt{1-2 x} \sqrt{5 x+3}}{972405 (3 x+2)^{5/2}}+\frac{808 \sqrt{1-2 x} \sqrt{5 x+3}}{27783 (3 x+2)^{7/2}}-\frac{32098184 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47647845} \]

[Out]

(808*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(27783*(2 + 3*x)^(7/2)) - (168034*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(972405*(2 +
3*x)^(5/2)) - (43094*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(6806835*(2 + 3*x)^(3/2)) + (32098184*Sqrt[1 - 2*x]*Sqrt[3 +
5*x])/(47647845*Sqrt[2 + 3*x]) + (2*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(189*(2 + 3*x)^(9/2)) - (32098184*Sqrt[11/
3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/47647845 - (2036756*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7
]*Sqrt[1 - 2*x]], 35/33])/47647845

________________________________________________________________________________________

Rubi [A]  time = 0.0795626, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{189 (3 x+2)^{9/2}}+\frac{32098184 \sqrt{1-2 x} \sqrt{5 x+3}}{47647845 \sqrt{3 x+2}}-\frac{43094 \sqrt{1-2 x} \sqrt{5 x+3}}{6806835 (3 x+2)^{3/2}}-\frac{168034 \sqrt{1-2 x} \sqrt{5 x+3}}{972405 (3 x+2)^{5/2}}+\frac{808 \sqrt{1-2 x} \sqrt{5 x+3}}{27783 (3 x+2)^{7/2}}-\frac{2036756 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47647845}-\frac{32098184 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47647845} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^(5/2)/(Sqrt[1 - 2*x]*(2 + 3*x)^(11/2)),x]

[Out]

(808*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(27783*(2 + 3*x)^(7/2)) - (168034*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(972405*(2 +
3*x)^(5/2)) - (43094*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(6806835*(2 + 3*x)^(3/2)) + (32098184*Sqrt[1 - 2*x]*Sqrt[3 +
5*x])/(47647845*Sqrt[2 + 3*x]) + (2*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(189*(2 + 3*x)^(9/2)) - (32098184*Sqrt[11/
3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/47647845 - (2036756*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7
]*Sqrt[1 - 2*x]], 35/33])/47647845

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{(3+5 x)^{5/2}}{\sqrt{1-2 x} (2+3 x)^{11/2}} \, dx &=\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{189 (2+3 x)^{9/2}}-\frac{2}{189} \int \frac{\left (-441-\frac{1525 x}{2}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^{9/2}} \, dx\\ &=\frac{808 \sqrt{1-2 x} \sqrt{3+5 x}}{27783 (2+3 x)^{7/2}}+\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{189 (2+3 x)^{9/2}}-\frac{4 \int \frac{-\frac{207611}{4}-\frac{353425 x}{4}}{\sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx}{27783}\\ &=\frac{808 \sqrt{1-2 x} \sqrt{3+5 x}}{27783 (2+3 x)^{7/2}}-\frac{168034 \sqrt{1-2 x} \sqrt{3+5 x}}{972405 (2+3 x)^{5/2}}+\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{189 (2+3 x)^{9/2}}-\frac{8 \int \frac{-\frac{1658793}{8}-\frac{1260255 x}{4}}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx}{972405}\\ &=\frac{808 \sqrt{1-2 x} \sqrt{3+5 x}}{27783 (2+3 x)^{7/2}}-\frac{168034 \sqrt{1-2 x} \sqrt{3+5 x}}{972405 (2+3 x)^{5/2}}-\frac{43094 \sqrt{1-2 x} \sqrt{3+5 x}}{6806835 (2+3 x)^{3/2}}+\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{189 (2+3 x)^{9/2}}-\frac{16 \int \frac{-1030002-\frac{323205 x}{8}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{20420505}\\ &=\frac{808 \sqrt{1-2 x} \sqrt{3+5 x}}{27783 (2+3 x)^{7/2}}-\frac{168034 \sqrt{1-2 x} \sqrt{3+5 x}}{972405 (2+3 x)^{5/2}}-\frac{43094 \sqrt{1-2 x} \sqrt{3+5 x}}{6806835 (2+3 x)^{3/2}}+\frac{32098184 \sqrt{1-2 x} \sqrt{3+5 x}}{47647845 \sqrt{2+3 x}}+\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{189 (2+3 x)^{9/2}}-\frac{32 \int \frac{-\frac{161245065}{16}-\frac{60184095 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{142943535}\\ &=\frac{808 \sqrt{1-2 x} \sqrt{3+5 x}}{27783 (2+3 x)^{7/2}}-\frac{168034 \sqrt{1-2 x} \sqrt{3+5 x}}{972405 (2+3 x)^{5/2}}-\frac{43094 \sqrt{1-2 x} \sqrt{3+5 x}}{6806835 (2+3 x)^{3/2}}+\frac{32098184 \sqrt{1-2 x} \sqrt{3+5 x}}{47647845 \sqrt{2+3 x}}+\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{189 (2+3 x)^{9/2}}+\frac{11202158 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{47647845}+\frac{32098184 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{47647845}\\ &=\frac{808 \sqrt{1-2 x} \sqrt{3+5 x}}{27783 (2+3 x)^{7/2}}-\frac{168034 \sqrt{1-2 x} \sqrt{3+5 x}}{972405 (2+3 x)^{5/2}}-\frac{43094 \sqrt{1-2 x} \sqrt{3+5 x}}{6806835 (2+3 x)^{3/2}}+\frac{32098184 \sqrt{1-2 x} \sqrt{3+5 x}}{47647845 \sqrt{2+3 x}}+\frac{2 \sqrt{1-2 x} (3+5 x)^{3/2}}{189 (2+3 x)^{9/2}}-\frac{32098184 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47647845}-\frac{2036756 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47647845}\\ \end{align*}

Mathematica [A]  time = 0.186393, size = 107, normalized size = 0.48 \[ \frac{12066320 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{24 \sqrt{2-4 x} \sqrt{5 x+3} \left (1299976452 x^4+3462531489 x^3+3421407609 x^2+1489220097 x+241253543\right )}{(3 x+2)^{9/2}}[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{571774140 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)^(5/2)/(Sqrt[1 - 2*x]*(2 + 3*x)^(11/2)),x]

[Out]

((24*Sqrt[2 - 4*x]*Sqrt[3 + 5*x]*(241253543 + 1489220097*x + 3421407609*x^2 + 3462531489*x^3 + 1299976452*x^4)
)/(2 + 3*x)^(9/2) + 256785472*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 12066320*EllipticF[ArcSin[S
qrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(571774140*Sqrt[2])

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Maple [C]  time = 0.023, size = 504, normalized size = 2.3 \begin{align*} -{\frac{2}{1429435350\,{x}^{2}[phone]\,x-428830605} \left ( 61085745\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{4}\sqrt{2+3\,x}\sqrt{1-2\,x}\sqrt{3+5\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{4}\sqrt{2+3\,x}\sqrt{1-2\,x}\sqrt{3+5\,x}[phone]\,\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}^{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}[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}[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}[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}-38999293560\,{x}^{6}[phone]\,\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 ) -107775874026\,{x}^{5}-101330034669\,{x}^{4}-23778042336\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+2171281887 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2+3\,x \right ) ^{-{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+5*x)^(5/2)/(2+3*x)^(11/2)/(1-2*x)^(1/2),x)

[Out]

-2/142943535*(61085745*2^(1/2)*EllipticF(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x^4*(2+3*x)^(1/2)*(1-2*x)^(1/2)
*(3+5*x)^(1/2)[phone]*2^(1/2)*EllipticE(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x^4*(2+3*x)^(1/2)*(1-2*x)^(1
/2)*(3+5*x)^(1/2)[phone]*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^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)[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)[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)[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)-38999293560*x^6+12066320*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Ellipti
cF(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))-107775874026*x^5-101330034669*x^4-23778042336*x^3+19087401900*x^2+12679
220244*x+2171281887)*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(9/2)

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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{11}{2}} \sqrt{-2 \, x + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(5/2)/(2+3*x)^(11/2)/(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)/((3*x + 2)^(11/2)*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 (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{1458 \, x^{7} + 5103 \, x^{6} + 6804 \, x^{5} + 3780 \, x^{4} - 1008 \, x^{2} - 448 \, x - 64}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(5/2)/(2+3*x)^(11/2)/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

integral(-(25*x^2 + 30*x + 9)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(1458*x^7 + 5103*x^6 + 6804*x^5 + 378
0*x^4 - 1008*x^2 - 448*x - 64), 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((3+5*x)**(5/2)/(2+3*x)**(11/2)/(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{11}{2}} \sqrt{-2 \, x + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(5/2)/(2+3*x)^(11/2)/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(5/2)/((3*x + 2)^(11/2)*sqrt(-2*x + 1)), x)