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

Optimal. Leaf size=195 \[ -\frac {10385 \sqrt {1-2 x} (5 x+3)^{5/2}}{648 (3 x+2)}+\frac {185 (1-2 x)^{3/2} (5 x+3)^{5/2}}{108 (3 x+2)^2}-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}+\frac {2075}{72} \sqrt {1-2 x} (5 x+3)^{3/2}-\frac {48625 \sqrt {1-2 x} \sqrt {5 x+3}}{1944}-\frac {21935 \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1458}-\frac {408665 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{5832 \sqrt {7}} \]

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Rubi [A]  time = 0.08, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {97, 149, 154, 157, 54, 216, 93, 204} \begin {gather*} -\frac {10385 \sqrt {1-2 x} (5 x+3)^{5/2}}{648 (3 x+2)}+\frac {185 (1-2 x)^{3/2} (5 x+3)^{5/2}}{108 (3 x+2)^2}-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}+\frac {2075}{72} \sqrt {1-2 x} (5 x+3)^{3/2}-\frac {48625 \sqrt {1-2 x} \sqrt {5 x+3}}{1944}-\frac {21935 \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1458}-\frac {408665 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{5832 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-48625*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1944 + (2075*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/72 - ((1 - 2*x)^(5/2)*(3 + 5*
x)^(5/2))/(9*(2 + 3*x)^3) + (185*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(108*(2 + 3*x)^2) - (10385*Sqrt[1 - 2*x]*(3
+ 5*x)^(5/2))/(648*(2 + 3*x)) - (21935*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/1458 - (408665*ArcTan[Sqrt[
1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(5832*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 154

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 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)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx &=-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {1}{9} \int \frac {\left (-\frac {5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^3} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {1}{54} \int \frac {\left (-\frac {2005}{4}-2050 x\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^2} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {10385 \sqrt {1-2 x} (3+5 x)^{5/2}}{648 (2+3 x)}+\frac {1}{162} \int \frac {\left (\frac {109865}{8}-56025 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {2075}{72} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {10385 \sqrt {1-2 x} (3+5 x)^{5/2}}{648 (2+3 x)}-\frac {\int \frac {\left (\frac {19665}{2}-291750 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx}{1944}\\ &=-\frac {48625 \sqrt {1-2 x} \sqrt {3+5 x}}{1944}+\frac {2075}{72} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {10385 \sqrt {1-2 x} (3+5 x)^{5/2}}{648 (2+3 x)}+\frac {\int \frac {-468735-1316100 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{11664}\\ &=-\frac {48625 \sqrt {1-2 x} \sqrt {3+5 x}}{1944}+\frac {2075}{72} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {10385 \sqrt {1-2 x} (3+5 x)^{5/2}}{648 (2+3 x)}+\frac {408665 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{11664}-\frac {109675 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{2916}\\ &=-\frac {48625 \sqrt {1-2 x} \sqrt {3+5 x}}{1944}+\frac {2075}{72} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {10385 \sqrt {1-2 x} (3+5 x)^{5/2}}{648 (2+3 x)}+\frac {408665 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{5832}-\frac {\left (21935 \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{1458}\\ &=-\frac {48625 \sqrt {1-2 x} \sqrt {3+5 x}}{1944}+\frac {2075}{72} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac {10385 \sqrt {1-2 x} (3+5 x)^{5/2}}{648 (2+3 x)}-\frac {21935 \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1458}-\frac {408665 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{5832 \sqrt {7}}\\ \end {align*}

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Mathematica [A]  time = 0.32, size = 144, normalized size = 0.74 \begin {gather*} \frac {21 \sqrt {-(1-2 x)^2} \sqrt {5 x+3} \left (32400 x^4-93420 x^3-420531 x^2-391014 x-107984\right )-408665 \sqrt {14 x-7} (3 x+2)^3 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )+307090 \sqrt {10-20 x} (3 x+2)^3 \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{40824 \sqrt {2 x-1} (3 x+2)^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(21*Sqrt[-(1 - 2*x)^2]*Sqrt[3 + 5*x]*(-107984 - 391014*x - 420531*x^2 - 93420*x^3 + 32400*x^4) + 307090*Sqrt[1
0 - 20*x]*(2 + 3*x)^3*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]] - 408665*(2 + 3*x)^3*Sqrt[-7 + 14*x]*ArcTan[Sqrt[1 -
2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(40824*Sqrt[-1 + 2*x]*(2 + 3*x)^3)

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IntegrateAlgebraic [A]  time = 0.36, size = 194, normalized size = 0.99 \begin {gather*} -\frac {11 \sqrt {1-2 x} \left (\frac {243125 (1-2 x)^4}{(5 x+3)^4}+\frac {4586600 (1-2 x)^3}{(5 x+3)^3}+\frac {27339831 (1-2 x)^2}{(5 x+3)^2}+\frac {28846160 (1-2 x)}{5 x+3}+6692420\right )}{1944 \sqrt {5 x+3} \left (\frac {1-2 x}{5 x+3}+7\right )^3 \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^2}+\frac {21935 \sqrt {\frac {5}{2}} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{1458}-\frac {408665 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{5832 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-11*Sqrt[1 - 2*x]*(6692420 + (243125*(1 - 2*x)^4)/(3 + 5*x)^4 + (4586600*(1 - 2*x)^3)/(3 + 5*x)^3 + (27339831
*(1 - 2*x)^2)/(3 + 5*x)^2 + (28846160*(1 - 2*x))/(3 + 5*x)))/(1944*Sqrt[3 + 5*x]*(7 + (1 - 2*x)/(3 + 5*x))^3*(
2 + (5*(1 - 2*x))/(3 + 5*x))^2) + (21935*Sqrt[5/2]*ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])/Sqrt[3 + 5*x]])/1458 - (40
8665*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(5832*Sqrt[7])

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fricas [A]  time = 1.56, size = 172, normalized size = 0.88 \begin {gather*} \frac {307090 \, \sqrt {5} \sqrt {2} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 408665 \, \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 ) + 42 \, {\left (32400 \, x^{4} - 93420 \, x^{3} - 420531 \, x^{2} - 391014 \, x - 107984\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{81648 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/81648*(307090*sqrt(5)*sqrt(2)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x +
3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 408665*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)) + 42*(32400*x^4 - 93420*x^3 - 420531*x^2 - 391014*x - 1079
84)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)

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giac [B]  time = 5.14, size = 409, normalized size = 2.10 \begin {gather*} \frac {81733}{163296} \, \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 {1}{486} \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} - 329 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {21935}{5832} \, \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 (2803 \, {\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} + 1982400 \, {\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 {411208000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {1644832000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{324 \, {\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

81733/163296*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)))) + 1/486*(12*sqrt(5)*(5*x + 3) - 329*sqrt(5))*sqrt(
5*x + 3)*sqrt(-10*x + 5) - 21935/5832*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - s
qrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 11/324*sqrt(10)*(2803*((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 + 1982400*((sqrt(2)*s
qrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 411208000
*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 1644832000*sqrt(5*x + 3)/(sqrt(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 [A]  time = 0.01, size = 287, normalized size = 1.47 \begin {gather*} -\frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-1360800 \sqrt {-10 x^{2}-x +3}\, x^{4}+8291430 \sqrt {10}\, x^{3} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-11033955 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3923640 \sqrt {-10 x^{2}-x +3}\, x^{3}[phone] \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-22067910 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{2}[phone] \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-14711940 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x +2456720 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-3269320 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+4535328 \sqrt {-10 x^{2}-x +3}\right )}{81648 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/81648*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(8291430*10^(1/2)*x^3*arcsin(20/11*x+1/11)-11033955*7^(1/2)*x^3*arctan(1
/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-1360800*(-10*x^2-x+3)^(1/2)*x^4+16582860*10^(1/2)*x^2*arcsin(20/11*
x+1/11)-22067910*7^(1/2)*x^2*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+3923640*(-10*x^2-x+3)^(1/2)*x^
3+11055240*10^(1/2)*x*arcsin(20/11*x+1/11)-14711940*7^(1/2)*x*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2
))[phone]*(-10*x^2-x+3)^(1/2)*x^2+2456720*10^(1/2)*arcsin(20/11*x+1/11)-3269320*7^(1/2)*arctan(1/14*(37*x+20
)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x+4535328*(-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 = 1.44, size = 190, normalized size = 0.97 \begin {gather*} -\frac {185}{882} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{7 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} - \frac {37 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{196 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {16075}{1764} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {189865}{31752} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {6347 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{3528 \, {\left (3 \, x + 2\right )}} + \frac {41225}{2268} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {21935}{5832} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {408665}{81648} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {191965}{13608} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-185/882*(-10*x^2 - x + 3)^(5/2) + 1/7*(-10*x^2 - x + 3)^(7/2)/(27*x^3 + 54*x^2 + 36*x + 8) - 37/196*(-10*x^2
- x + 3)^(7/2)/(9*x^2 + 12*x + 4) - 16075/1764*(-10*x^2 - x + 3)^(3/2)*x + 189865/31752*(-10*x^2 - x + 3)^(3/2
) - 6347/3528*(-10*x^2 - x + 3)^(5/2)/(3*x + 2) + 41225/2268*sqrt(-10*x^2 - x + 3)*x - 21935/5832*sqrt(10)*arc
sin(20/11*x + 1/11) + 408665/81648*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 191965/13608*sq
rt(-10*x^2 - x + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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