3.2344 \(\int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=149 \[ \frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{36 (3 x+2)^2}-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^3}-\frac {661 \sqrt {1-2 x} \sqrt {5 x+3}}{1512 (3 x+2)}+\frac {20}{81} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {19573 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{4536 \sqrt {7}} \]

[Out]

-1/9*(1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^3-19573/31752*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)
+20/81*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+37/36*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^2-661/1512*(1-2*
x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)

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Rubi [A]  time = 0.05, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {97, 149, 157, 54, 216, 93, 204} \[ \frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{36 (3 x+2)^2}-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^3}-\frac {661 \sqrt {1-2 x} \sqrt {5 x+3}}{1512 (3 x+2)}+\frac {20}{81} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {19573 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{4536 \sqrt {7}} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^4,x]

[Out]

(-661*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1512*(2 + 3*x)) - ((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(9*(2 + 3*x)^3) + (37*
Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(36*(2 + 3*x)^2) + (20*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/81 - (19573*A
rcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(4536*Sqrt[7])

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 97

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 149

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] && IntegerQ[m]

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin {align*} \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx &=-\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac {1}{9} \int \frac {\left (-\frac {3}{2}-30 x\right ) \sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^3} \, dx\\ &=-\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{36 (2+3 x)^2}-\frac {1}{54} \int \frac {\left (-\frac {981}{4}-120 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {661 \sqrt {1-2 x} \sqrt {3+5 x}}{1512 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{36 (2+3 x)^2}-\frac {\int \frac {-\frac {41973}{8}-4200 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{1134}\\ &=-\frac {661 \sqrt {1-2 x} \sqrt {3+5 x}}{1512 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{36 (2+3 x)^2}+\frac {100}{81} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx+\frac {19573 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{9072}\\ &=-\frac {661 \sqrt {1-2 x} \sqrt {3+5 x}}{1512 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{36 (2+3 x)^2}+\frac {19573 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{4536}+\frac {1}{81} \left (40 \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=-\frac {661 \sqrt {1-2 x} \sqrt {3+5 x}}{1512 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{36 (2+3 x)^2}+\frac {20}{81} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {19573 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{4536 \sqrt {7}}\\ \end {align*}

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Mathematica [A]  time = 0.24, size = 134, normalized size = 0.90 \[ \frac {21 \sqrt {-(1-2 x)^2} \sqrt {5 x+3} \left (19041 x^2+21762 x+6176\right )-19573 \sqrt {14 x-7} (3 x+2)^3 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-7840 \sqrt {10-20 x} (3 x+2)^3 \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{31752 \sqrt {2 x-1} (3 x+2)^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^4,x]

[Out]

(21*Sqrt[-(1 - 2*x)^2]*Sqrt[3 + 5*x]*(6176 + 21762*x + 19041*x^2) - 7840*Sqrt[10 - 20*x]*(2 + 3*x)^3*ArcSinh[S
qrt[5/11]*Sqrt[-1 + 2*x]] - 19573*(2 + 3*x)^3*Sqrt[-7 + 14*x]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(
31752*Sqrt[-1 + 2*x]*(2 + 3*x)^3)

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fricas [A]  time = 1.20, size = 156, normalized size = 1.05 \[ -\frac {19573 \, \sqrt {7} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 7840 \, \sqrt {10} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 42 \, {\left (19041 \, x^{2} + 21762 \, x + 6176\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{63504 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^4,x, algorithm="fricas")

[Out]

-1/63504*(19573*sqrt(7)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x +
1)/(10*x^2 + x - 3)) + 7840*sqrt(10)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x +
3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 42*(19041*x^2 + 21762*x + 6176)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(27*x^3 +
54*x^2 + 36*x + 8)

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giac [B]  time = 2.45, size = 377, normalized size = 2.53 \[ \frac {19573}{635040} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {10}{81} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {11 \, \sqrt {10} {\left (661 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 499520 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {139630400 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {558521600 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{756 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^4,x, algorithm="giac")

[Out]

19573/635040*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(2
2))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 10/81*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)
*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 11/756*sqrt(1
0)*(661*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(
22)))^5 + 499520*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5
) - sqrt(22)))^3 - 139630400*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 558521600*sqrt(5*x + 3)/(sqr
t(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt
(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^3

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maple [B]  time = 0.01, size = 253, normalized size = 1.70 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (211680 \sqrt {10}\, x^{3} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+528471 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+423360 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1056942 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+799722 \sqrt {-10 x^{2}-x +3}\, x^{2}+282240 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+704628 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+914004 \sqrt {-10 x^{2}-x +3}\, x +62720 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+156584 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+259392 \sqrt {-10 x^{2}-x +3}\right )}{63504 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x+1)^(3/2)*(5*x+3)^(3/2)/(3*x+2)^4,x)

[Out]

1/63504*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(528471*7^(1/2)*x^3*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+21
1680*10^(1/2)*x^3*arcsin(20/11*x+1/11)+1056942*7^(1/2)*x^2*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+
423360*10^(1/2)*x^2*arcsin(20/11*x+1/11)+704628*7^(1/2)*x*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+2
82240*10^(1/2)*x*arcsin(20/11*x+1/11)+799722*(-10*x^2-x+3)^(1/2)*x^2+156584*7^(1/2)*arctan(1/14*(37*x+20)*7^(1
/2)/(-10*x^2-x+3)^(1/2))+62720*10^(1/2)*arcsin(20/11*x+1/11)+914004*(-10*x^2-x+3)^(1/2)*x+259392*(-10*x^2-x+3)
^(1/2))/(-10*x^2-x+3)^(1/2)/(3*x+2)^3

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maxima [A]  time = 1.30, size = 161, normalized size = 1.08 \[ \frac {185}{882} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{7 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {37 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{196 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {4045}{1764} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {10}{81} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {19573}{63504} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {8573}{10584} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {83 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{1176 \, {\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^4,x, algorithm="maxima")

[Out]

185/882*(-10*x^2 - x + 3)^(3/2) + 1/7*(-10*x^2 - x + 3)^(5/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 37/196*(-10*x^2 -
x + 3)^(5/2)/(9*x^2 + 12*x + 4) + 4045/1764*sqrt(-10*x^2 - x + 3)*x + 10/81*sqrt(10)*arcsin(20/11*x + 1/11) +
19573/63504*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 8573/10584*sqrt(-10*x^2 - x + 3) + 83
/1176*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1 - 2*x)^(3/2)*(5*x + 3)^(3/2))/(3*x + 2)^4,x)

[Out]

int(((1 - 2*x)^(3/2)*(5*x + 3)^(3/2))/(3*x + 2)^4, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**(3/2)*(3+5*x)**(3/2)/(2+3*x)**4,x)

[Out]

Timed out

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