3.27.60 \(\int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^{9/2}} \, dx\) [2660]

Optimal. Leaf size=191 \[ -\frac {214 \sqrt {1-2 x} \sqrt {3+5 x}}{2205 (2+3 x)^{5/2}}+\frac {8578 \sqrt {1-2 x} \sqrt {3+5 x}}{46305 (2+3 x)^{3/2}}+\frac {475592 \sqrt {1-2 x} \sqrt {3+5 x}}{324135 \sqrt {2+3 x}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}-\frac {475592 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{324135}-\frac {10628 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{324135} \]

[Out]

-475592/972405*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-10628/972405*EllipticF(1/7*21^(1
/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-2/21*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^(7/2)-214/2205*(1-2*x)^(1
/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)+8578/46305*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+475592/324135*(1-2*x)^(1/
2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.04, antiderivative size = 191, normalized size of antiderivative = 1.00, 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 = {99, 155, 157, 164, 114, 120} \begin {gather*} -\frac {10628 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{324135}-\frac {475592 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{324135}-\frac {2 \sqrt {1-2 x} (5 x+3)^{3/2}}{21 (3 x+2)^{7/2}}+\frac {475592 \sqrt {1-2 x} \sqrt {5 x+3}}{324135 \sqrt {3 x+2}}+\frac {8578 \sqrt {1-2 x} \sqrt {5 x+3}}{46305 (3 x+2)^{3/2}}-\frac {214 \sqrt {1-2 x} \sqrt {5 x+3}}{2205 (3 x+2)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-214*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2205*(2 + 3*x)^(5/2)) + (8578*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(46305*(2 + 3*x
)^(3/2)) + (475592*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(324135*Sqrt[2 + 3*x]) - (2*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(21
*(2 + 3*x)^(7/2)) - (475592*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/324135 - (10628*Sqrt
[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/324135

Rule 99

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

Rule 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2/b)*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)))], 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 120

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[2*(Rt[-b/d,
2]/(b*Sqrt[(b*e - a*f)/b]))*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)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ[(b*e - a*f)/b, 0] && Po
sQ[-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]))

Rule 155

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 157

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 164

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]

Rubi steps

\begin {align*} \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^{9/2}} \, dx &=-\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac {2}{21} \int \frac {\left (\frac {9}{2}-20 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx\\ &=-\frac {214 \sqrt {1-2 x} \sqrt {3+5 x}}{2205 (2+3 x)^{5/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac {4 \int \frac {-\frac {1097}{4}-\frac {1895 x}{2}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{2205}\\ &=-\frac {214 \sqrt {1-2 x} \sqrt {3+5 x}}{2205 (2+3 x)^{5/2}}+\frac {8578 \sqrt {1-2 x} \sqrt {3+5 x}}{46305 (2+3 x)^{3/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac {8 \int \frac {6334-\frac {21445 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{46305}\\ &=-\frac {214 \sqrt {1-2 x} \sqrt {3+5 x}}{2205 (2+3 x)^{5/2}}+\frac {8578 \sqrt {1-2 x} \sqrt {3+5 x}}{46305 (2+3 x)^{3/2}}+\frac {475592 \sqrt {1-2 x} \sqrt {3+5 x}}{324135 \sqrt {2+3 x}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac {16 \int \frac {\frac {742615}{8}+\frac {297245 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{324135}\\ &=-\frac {214 \sqrt {1-2 x} \sqrt {3+5 x}}{2205 (2+3 x)^{5/2}}+\frac {8578 \sqrt {1-2 x} \sqrt {3+5 x}}{46305 (2+3 x)^{3/2}}+\frac {475592 \sqrt {1-2 x} \sqrt {3+5 x}}{324135 \sqrt {2+3 x}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}+\frac {58454 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{324135}+\frac {475592 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{324135}\\ &=-\frac {214 \sqrt {1-2 x} \sqrt {3+5 x}}{2205 (2+3 x)^{5/2}}+\frac {8578 \sqrt {1-2 x} \sqrt {3+5 x}}{46305 (2+3 x)^{3/2}}+\frac {475592 \sqrt {1-2 x} \sqrt {3+5 x}}{324135 \sqrt {2+3 x}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{21 (2+3 x)^{7/2}}-\frac {475592 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{324135}-\frac {10628 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{324135}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 4.83, size = 104, normalized size = 0.54 \begin {gather*} \frac {2 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (1944697+8796570 x+13111191 x^2+6420492 x^3\right )}{(2+3 x)^{7/2}}+\sqrt {2} \left (237796 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-150115 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{972405} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(1944697 + 8796570*x + 13111191*x^2 + 6420492*x^3))/(2 + 3*x)^(7/2) + Sqrt[
2]*(237796*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 150115*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*
x]], -33/2])))/972405

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(400\) vs. \(2(139)=278\).
time = 0.10, size = 401, normalized size = 2.10

method	result	size
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {188 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{19845 \left (\frac {2}{3}+x \right )^{3}}+\frac {8578 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{416745 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {951184}{64827} x^{2}-\frac {475592}{324135} x +\frac {475592}{108045}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {297046 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{1361367 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {475592 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{1361367 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{5103 \left (\frac {2}{3}+x \right )^{4}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(273\)
default	\(-\frac {2 \left (2367387 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-6420492 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+4734774 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-12840984 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+3156516 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-8560656 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+701448 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-1902368 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-192614760 x^{5}-412597206 x^{4}-245446245 x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{972405 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {7}{2}}}\)	\(401\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/972405*(2367387*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^3*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x
)^(1/2)-6420492*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^3*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(
1/2)+4734774*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^2*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2
)-12840984*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^2*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)+
3156516*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)-85606
56*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)+701448*2^(
1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-1902368*2^(1/2)*(2
+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-192614760*x^5-412597206*x
^4-245446245*x^3+33270099*x^2+73335039*x+17502273)*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(7/2)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.13, size = 60, normalized size = 0.31 \begin {gather*} \frac {2 \, {\left (6420492 \, x^{3} + 13111191 \, x^{2} + 8796570 \, x + 1944697\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{324135 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/324135*(6420492*x^3 + 13111191*x^2 + 8796570*x + 1944697)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(81*x^4
+ 216*x^3 + 216*x^2 + 96*x + 16)

________________________________________________________________________________________

Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3877 deep

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^{9/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________