3.28.6 \(\int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{9/2}} \, dx\) [2706]

Optimal. Leaf size=191 \[ -\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{21 (2+3 x)^{7/2}}+\frac {82 \sqrt {1-2 x} \sqrt {3+5 x}}{315 (2+3 x)^{5/2}}+\frac {8516 \sqrt {1-2 x} \sqrt {3+5 x}}{6615 (2+3 x)^{3/2}}+\frac {595324 \sqrt {1-2 x} \sqrt {3+5 x}}{46305 \sqrt {2+3 x}}-\frac {595324 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{46305}-\frac {18016 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{46305} \]

[Out]

-595324/138915*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-18016/138915*EllipticF(1/7*21^(1
/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-2/21*(1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^(7/2)+82/315*(1-2*x)^(1/2
)*(3+5*x)^(1/2)/(2+3*x)^(5/2)+8516/6615*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+595324/46305*(1-2*x)^(1/2)*(
3+5*x)^(1/2)/(2+3*x)^(1/2)

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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 {18016 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{46305}-\frac {595324 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{46305}-\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{21 (3 x+2)^{7/2}}+\frac {595324 \sqrt {5 x+3} \sqrt {1-2 x}}{46305 \sqrt {3 x+2}}+\frac {8516 \sqrt {5 x+3} \sqrt {1-2 x}}{6615 (3 x+2)^{3/2}}+\frac {82 \sqrt {5 x+3} \sqrt {1-2 x}}{315 (3 x+2)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(21*(2 + 3*x)^(7/2)) + (82*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(315*(2 + 3*x)^(5/2
)) + (8516*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(6615*(2 + 3*x)^(3/2)) + (595324*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(46305*S
qrt[2 + 3*x]) - (595324*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/46305 - (18016*Sqrt[11/3
]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/46305

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 {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{9/2}} \, dx &=-\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{21 (2+3 x)^{7/2}}+\frac {2}{21} \int \frac {\left (-\frac {13}{2}-20 x\right ) \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}}{21 (2+3 x)^{7/2}}+\frac {82 \sqrt {1-2 x} \sqrt {3+5 x}}{315 (2+3 x)^{5/2}}-\frac {4}{315} \int \frac {-\frac {433}{2}+\frac {415 x}{2}}{\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}}{21 (2+3 x)^{7/2}}+\frac {82 \sqrt {1-2 x} \sqrt {3+5 x}}{315 (2+3 x)^{5/2}}+\frac {8516 \sqrt {1-2 x} \sqrt {3+5 x}}{6615 (2+3 x)^{3/2}}-\frac {8 \int \frac {-\frac {35417}{4}+\frac {10645 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}}{21 (2+3 x)^{7/2}}+\frac {82 \sqrt {1-2 x} \sqrt {3+5 x}}{315 (2+3 x)^{5/2}}+\frac {8516 \sqrt {1-2 x} \sqrt {3+5 x}}{6615 (2+3 x)^{3/2}}+\frac {595324 \sqrt {1-2 x} \sqrt {3+5 x}}{46305 \sqrt {2+3 x}}-\frac {16 \int \frac {-\frac {471265}{4}-\frac {744155 x}{4}}{\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}}{21 (2+3 x)^{7/2}}+\frac {82 \sqrt {1-2 x} \sqrt {3+5 x}}{315 (2+3 x)^{5/2}}+\frac {8516 \sqrt {1-2 x} \sqrt {3+5 x}}{6615 (2+3 x)^{3/2}}+\frac {595324 \sqrt {1-2 x} \sqrt {3+5 x}}{46305 \sqrt {2+3 x}}+\frac {99088 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{46305}+\frac {595324 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{46305}\\ &=-\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{21 (2+3 x)^{7/2}}+\frac {82 \sqrt {1-2 x} \sqrt {3+5 x}}{315 (2+3 x)^{5/2}}+\frac {8516 \sqrt {1-2 x} \sqrt {3+5 x}}{6615 (2+3 x)^{3/2}}+\frac {595324 \sqrt {1-2 x} \sqrt {3+5 x}}{46305 \sqrt {2+3 x}}-\frac {595324 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{46305}-\frac {18016 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{46305}\\ \end {align*}

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Mathematica [A]
time = 6.84, size = 106, normalized size = 0.55 \begin {gather*} \frac {4 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (2510369+11095995 x+16342002 x^2+8036874 x^3\right )}{2 (2+3 x)^{7/2}}+\sqrt {2} \left (148831 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-74515 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{138915} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2510369 + 11095995*x + 16342002*x^2 + 8036874*x^3))/(2*(2 + 3*x)^(7/2)) +
Sqrt[2]*(148831*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 74515*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3
+ 5*x]], -33/2])))/138915

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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	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {34 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2835 \left (\frac {2}{3}+x \right )^{3}}+\frac {8516 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{59535 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {1190648}{9261} x^{2}-\frac {595324}{46305} x +\frac {595324}{15435}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {377012 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{194481 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {595324 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{194481 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{729 \left (\frac {2}{3}+x \right )^{4}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(273\)
default	\(-\frac {2 \left (4013064 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-8036874 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+8026128 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-16073748 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+5350752 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-10715832 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+1189056 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-2381296 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-241106220 x^{5}-514370682 x^{4}-309573990 x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{138915 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {7}{2}}}\)	\(401\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/138915*(4013064*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)-8036874*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)+8026128*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
)-16073748*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)+
5350752*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)-10715
832*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)+1189056*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))-2381296*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))-241106220*x^5-514370682
*x^4-309573990*x^3+38478963*x^2+92332848*x+22593321)*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(7/2)

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

[Out]

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

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Fricas [A]
time = 0.30, size = 60, normalized size = 0.31 \begin {gather*} \frac {2 \, {\left (8036874 \, x^{3} + 16342002 \, x^{2} + 11095995 \, x + 2510369\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{46305 \, {\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((1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^(9/2),x, algorithm="fricas")

[Out]

2/46305*(8036874*x^3 + 16342002*x^2 + 11095995*x + 2510369)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(81*x^4
+ 216*x^3 + 216*x^2 + 96*x + 16)

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

[Out]

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

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

[Out]

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

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

[Out]

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

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