3.29.6 \(\int \frac {(1-2 x)^{5/2} (2+3 x)^{7/2}}{(3+5 x)^{5/2}} \, dx\) [2806]

Optimal. Leaf size=253 \[ -\frac {2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {442 (1-2 x)^{3/2} (2+3 x)^{7/2}}{75 \sqrt {3+5 x}}+\frac {500501 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{984375}+\frac {373022 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{196875}+\frac {59662 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{7875}-\frac {524}{225} \sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}-\frac {1065118 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{4921875}-\frac {595387 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{4921875} \]

[Out]

-2/15*(1-2*x)^(5/2)*(2+3*x)^(7/2)/(3+5*x)^(3/2)-1065118/14765625*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*115
5^(1/2))*33^(1/2)-595387/14765625*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-442/75*(1-2*x
)^(3/2)*(2+3*x)^(7/2)/(3+5*x)^(1/2)+373022/196875*(2+3*x)^(3/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)+59662/7875*(2+3*x)
^(5/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)-524/225*(2+3*x)^(7/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)+500501/984375*(1-2*x)^(1/
2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 253, normalized size of antiderivative = 1.00, 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 = {99, 155, 159, 164, 114, 120} \begin {gather*} -\frac {595387 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{4921875}-\frac {1065118 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{4921875}-\frac {524}{225} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{7/2}-\frac {442 (1-2 x)^{3/2} (3 x+2)^{7/2}}{75 \sqrt {5 x+3}}-\frac {2 (1-2 x)^{5/2} (3 x+2)^{7/2}}{15 (5 x+3)^{3/2}}+\frac {59662 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}}{7875}+\frac {373022 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}}{196875}+\frac {500501 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{984375} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(5/2)*(2 + 3*x)^(7/2))/(3 + 5*x)^(5/2),x]

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^(7/2))/(15*(3 + 5*x)^(3/2)) - (442*(1 - 2*x)^(3/2)*(2 + 3*x)^(7/2))/(75*Sqrt[3 +
5*x]) + (500501*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/984375 + (373022*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqr
t[3 + 5*x])/196875 + (59662*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/7875 - (524*Sqrt[1 - 2*x]*(2 + 3*x)^(
7/2)*Sqrt[3 + 5*x])/225 - (1065118*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/4921875 - (59
5387*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/4921875

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 159

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 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)^{5/2} (2+3 x)^{7/2}}{(3+5 x)^{5/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}+\frac {2}{15} \int \frac {\left (\frac {1}{2}-36 x\right ) (1-2 x)^{3/2} (2+3 x)^{5/2}}{(3+5 x)^{3/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {442 (1-2 x)^{3/2} (2+3 x)^{7/2}}{75 \sqrt {3+5 x}}+\frac {4}{75} \int \frac {\left (\frac {627}{2}-\frac {5895 x}{2}\right ) \sqrt {1-2 x} (2+3 x)^{5/2}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {442 (1-2 x)^{3/2} (2+3 x)^{7/2}}{75 \sqrt {3+5 x}}-\frac {524}{225} \sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}+\frac {8 \int \frac {\left (111060-\frac {1342395 x}{4}\right ) (2+3 x)^{5/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{10125}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {442 (1-2 x)^{3/2} (2+3 x)^{7/2}}{75 \sqrt {3+5 x}}+\frac {59662 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{7875}-\frac {524}{225} \sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}-\frac {8 \int \frac {(2+3 x)^{3/2} \left (-\frac {4470615}{8}+\frac {8392995 x}{4}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{354375}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {442 (1-2 x)^{3/2} (2+3 x)^{7/2}}{75 \sqrt {3+5 x}}+\frac {373022 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{196875}+\frac {59662 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{7875}-\frac {524}{225} \sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}+\frac {8 \int \frac {\left (\frac {13705875}{8}-\frac {67567635 x}{8}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{8859375}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {442 (1-2 x)^{3/2} (2+3 x)^{7/2}}{75 \sqrt {3+5 x}}+\frac {500501 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{984375}+\frac {373022 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{196875}+\frac {59662 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{7875}-\frac {524}{225} \sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}-\frac {8 \int \frac {-\frac {349379055}{16}-\frac {71895465 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{132890625}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {442 (1-2 x)^{3/2} (2+3 x)^{7/2}}{75 \sqrt {3+5 x}}+\frac {500501 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{984375}+\frac {373022 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{196875}+\frac {59662 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{7875}-\frac {524}{225} \sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}+\frac {1065118 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{4921875}+\frac {6549257 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{9843750}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac {442 (1-2 x)^{3/2} (2+3 x)^{7/2}}{75 \sqrt {3+5 x}}+\frac {500501 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{984375}+\frac {373022 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{196875}+\frac {59662 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{7875}-\frac {524}{225} \sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}-\frac {1065118 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{4921875}-\frac {595387 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{4921875}\\ \end {align*}

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Mathematica [A]
time = 9.02, size = 117, normalized size = 0.46 \begin {gather*} \frac {\frac {30 \sqrt {1-2 x} \sqrt {2+3 x} \left (1215489+4026600 x+470675 x^2-5654250 x^3+1327500 x^4+4725000 x^5\right )}{(3+5 x)^{3/2}}+2130236 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )[phone] \sqrt {2} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{29531250} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(5/2)*(2 + 3*x)^(7/2))/(3 + 5*x)^(5/2),x]

[Out]

((30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(1215489 + 4026600*x + 470675*x^2 - 5654250*x^3 + 1327500*x^4 + 4725000*x^5))
/(3 + 5*x)^(3/2) + 2130236*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 17517535*Sqrt[2]*Ellip
ticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/29531250

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Maple [A]
time = 0.10, size = 235, normalized size = 0.93

method	result	size
default	\(-\frac {\left (-850500000 x^{7}[phone] \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-10651180 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-380700000 x^{6}[phone] \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-6390708 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )[phone] x^{5}[phone] x^{4}-1078163250 x^{3}-311345520 x^{2}[phone] x [phone]\right ) \sqrt {2+3 x}\, \sqrt {1-2 x}}{29531250 \left (6 x^{2}+x -2\right ) \left (3+5 x \right )^{\frac {3}{2}}}\)	\(235\)
elliptic	\(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {242 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1171875 \left (x +\frac {3}{5}\right )^{2}}-\frac {6952 \left (-30 x^{2}-5 x +10\right )}{234375 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}+\frac {24 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{125}-\frac {772 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{4375}-\frac {1906 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{21875}+\frac {184283 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{984375}+\frac {2587993 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{41343750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1065118 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{20671875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\left (6 x^{2}+x -2\right ) \sqrt {3+5 x}}\)	\(319\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(5/2)*(2+3*x)^(7/2)/(3+5*x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/29531250*(-850500000*x^7+98238855*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)-10651180*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)-380700000*x^6+58943313*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/7*(2
8+42*x)^(1/2),1/2*70^(1/2))-6390708*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))[phone]*x^5+164556000*x^4-1078163250*x^3-311345520*x^2+205131330*x+72929340)*(2+3*x)^(
1/2)*(1-2*x)^(1/2)/(6*x^2+x-2)/(3+5*x)^(3/2)

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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)^(5/2)*(2+3*x)^(7/2)/(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

integrate((3*x + 2)^(7/2)*(-2*x + 1)^(5/2)/(5*x + 3)^(5/2), x)

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Fricas [A]
time = 0.22, size = 60, normalized size = 0.24 \begin {gather*} \frac {{\left (4725000 \, x^{5} + 1327500 \, x^{4} - 5654250 \, x^{3} + 470675 \, x^{2} + 4026600 \, x + 1215489\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{984375 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(2+3*x)^(7/2)/(3+5*x)^(5/2),x, algorithm="fricas")

[Out]

1/984375*(4725000*x^5 + 1327500*x^4 - 5654250*x^3 + 470675*x^2 + 4026600*x + 1215489)*sqrt(5*x + 3)*sqrt(3*x +
2)*sqrt(-2*x + 1)/(25*x^2 + 30*x + 9)

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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)**(5/2)*(2+3*x)**(7/2)/(3+5*x)**(5/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 8856 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)^(5/2)*(2+3*x)^(7/2)/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

integrate((3*x + 2)^(7/2)*(-2*x + 1)^(5/2)/(5*x + 3)^(5/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{7/2}}{{\left (5\,x+3\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1 - 2*x)^(5/2)*(3*x + 2)^(7/2))/(5*x + 3)^(5/2),x)

[Out]

int(((1 - 2*x)^(5/2)*(3*x + 2)^(7/2))/(5*x + 3)^(5/2), x)

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