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

Optimal. Leaf size=179 \[ -\frac {5}{18} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac {247}{324} \sqrt {1-2 x} (5 x+3)^{5/2}-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{3 (3 x+2)}+\frac {1453}{288} \sqrt {1-2 x} (5 x+3)^{3/2}-\frac {155777 \sqrt {1-2 x} \sqrt {5 x+3}}{31104}-\frac {660959 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{93312 \sqrt {10}}-\frac {1295}{729} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \]

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Rubi [A]  time = 0.08, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {97, 154, 157, 54, 216, 93, 204} \begin {gather*} -\frac {5}{18} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac {247}{324} \sqrt {1-2 x} (5 x+3)^{5/2}-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{3 (3 x+2)}+\frac {1453}{288} \sqrt {1-2 x} (5 x+3)^{3/2}-\frac {155777 \sqrt {1-2 x} \sqrt {5 x+3}}{31104}-\frac {660959 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{93312 \sqrt {10}}-\frac {1295}{729} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-155777*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/31104 + (1453*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/288 - (247*Sqrt[1 - 2*x]*(3
+ 5*x)^(5/2))/324 - (5*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/18 - ((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(3*(2 + 3*x))
- (660959*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(93312*Sqrt[10]) - (1295*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqr
t[3 + 5*x])])/729

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 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)^2} \, dx &=-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{3 (2+3 x)}+\frac {1}{3} \int \frac {\left (-\frac {5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{2+3 x} \, dx\\ &=-\frac {5}{18} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{3 (2+3 x)}+\frac {1}{180} \int \frac {(200-6175 x) \sqrt {1-2 x} (3+5 x)^{3/2}}{2+3 x} \, dx\\ &=-\frac {247}{324} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {5}{18} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{3 (2+3 x)}+\frac {\int \frac {\left (126325-\frac {980775 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)} \, dx}{8100}\\ &=\frac {1453}{288} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {247}{324} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {5}{18} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{3 (2+3 x)}-\frac {\int \frac {\left (-\frac {268425}{2}-\frac {11683275 x}{4}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx}{97200}\\ &=-\frac {155777 \sqrt {1-2 x} \sqrt {3+5 x}}{31104}+\frac {1453}{288} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {247}{324} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {5}{18} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{3 (2+3 x)}+\frac {\int \frac {-\frac {2019975}{4}-\frac {49571925 x}{8}}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{583200}\\ &=-\frac {155777 \sqrt {1-2 x} \sqrt {3+5 x}}{31104}+\frac {1453}{288} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {247}{324} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {5}{18} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{3 (2+3 x)}-\frac {660959 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{186624}+\frac {9065 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{1458}\\ &=-\frac {155777 \sqrt {1-2 x} \sqrt {3+5 x}}{31104}+\frac {1453}{288} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {247}{324} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {5}{18} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{3 (2+3 x)}+\frac {9065}{729} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )-\frac {660959 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{93312 \sqrt {5}}\\ &=-\frac {155777 \sqrt {1-2 x} \sqrt {3+5 x}}{31104}+\frac {1453}{288} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {247}{324} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {5}{18} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{3 (2+3 x)}-\frac {660959 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{93312 \sqrt {10}}-\frac {1295}{729} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.28, size = 140, normalized size = 0.78 \begin {gather*} \frac {30 \sqrt {-(1-2 x)^2} \sqrt {5 x+3} \left (259200 x^4-214560 x^3-60348 x^2+72849 x-45658\right )-1657600 (3 x+2) \sqrt {14 x-7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )+660959 \sqrt {10-20 x} (3 x+2) \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{933120 \sqrt {2 x-1} (3 x+2)} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(30*Sqrt[-(1 - 2*x)^2]*Sqrt[3 + 5*x]*(-45658 + 72849*x - 60348*x^2 - 214560*x^3 + 259200*x^4) + 660959*Sqrt[10
- 20*x]*(2 + 3*x)*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]] - 1657600*(2 + 3*x)*Sqrt[-7 + 14*x]*ArcTan[Sqrt[1 - 2*x]
/(Sqrt[7]*Sqrt[3 + 5*x])])/(933120*Sqrt[-1 + 2*x]*(2 + 3*x))

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IntegrateAlgebraic [A]  time = 0.32, size = 192, normalized size = 1.07 \begin {gather*} -\frac {11 \sqrt {1-2 x} \left (\frac {19472125 (1-2 x)^4}{(5 x+3)^4}+\frac {116517325 (1-2 x)^3}{(5 x+3)^3}+\frac {13953978 (1-2 x)^2}{(5 x+3)^2}+\frac {4762444 (1-2 x)}{5 x+3}+559048\right )}{31104 \sqrt {5 x+3} \left (\frac {1-2 x}{5 x+3}+7\right ) \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^4}+\frac {660959 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{93312 \sqrt {10}}-\frac {1295}{729} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-11*Sqrt[1 - 2*x]*(559048 + (19472125*(1 - 2*x)^4)/(3 + 5*x)^4 + (116517325*(1 - 2*x)^3)/(3 + 5*x)^3 + (13953
978*(1 - 2*x)^2)/(3 + 5*x)^2 + (4762444*(1 - 2*x))/(3 + 5*x)))/(31104*Sqrt[3 + 5*x]*(7 + (1 - 2*x)/(3 + 5*x))*
(2 + (5*(1 - 2*x))/(3 + 5*x))^4) + (660959*ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])/Sqrt[3 + 5*x]])/(93312*Sqrt[10]) -
(1295*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/729

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fricas [A]  time = 1.26, size = 136, normalized size = 0.76 \begin {gather*} -\frac {1657600 \, \sqrt {7} {\left (3 \, x + 2\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 ) - 660959 \, \sqrt {10} {\left (3 \, x + 2\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 ) - 60 \, {\left (259200 \, x^{4} - 214560 \, x^{3} - 60348 \, x^{2} + 72849 \, x - 45658\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1866240 \, {\left (3 \, x + 2\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)^2,x, algorithm="fricas")

[Out]

-1/1866240*(1657600*sqrt(7)*(3*x + 2)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x
- 3)) - 660959*sqrt(10)*(3*x + 2)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x -
3)) - 60*(259200*x^4 - 214560*x^3 - 60348*x^2 + 72849*x - 45658)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(3*x + 2)

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giac [B]  time = 3.25, size = 318, normalized size = 1.78 \begin {gather*} \frac {259}{2916} \, \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}{777600} \, {\left (12 \, {\left (8 \, {\left (36 \, \sqrt {5} {\left (5 \, x + 3\right )} - 593 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 26185 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 622085 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {660959}{1866240} \, \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 {1078 \, \sqrt {10} {\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 )}}{243 \, {\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 )}} \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)^2,x, algorithm="giac")

[Out]

259/2916*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*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)))) + 1/777600*(12*(8*(36*sqrt(5)*(5*x + 3) - 593*sqrt(5))
*(5*x + 3) + 26185*sqrt(5))*(5*x + 3) - 622085*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) - 660959/1866240*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)))) - 1078/243*sqrt(10)*((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)))/(((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)

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maple [A]  time = 0.01, size = 197, normalized size = 1.10 \begin {gather*} -\frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-15552000 \sqrt {-10 x^{2}-x +3}\, x^{4}[phone] \sqrt {-10 x^{2}-x +3}\, x^{3}+3620880 \sqrt {-10 x^{2}-x +3}\, x^{2}+1982877 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-4972800 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-4370940 \sqrt {-10 x^{2}-x +3}\, x +1321918 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-3315200 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+2739480 \sqrt {-10 x^{2}-x +3}\right )}{1866240 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )} \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)^2,x)

[Out]

-1/1866240*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(-15552000*(-10*x^2-x+3)^(1/2)*x^4+12873600*(-10*x^2-x+3)^(1/2)*x^3+19
82877*10^(1/2)*x*arcsin(20/11*x+1/11)-4972800*7^(1/2)*x*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+362
0880*(-10*x^2-x+3)^(1/2)*x^2+1321918*10^(1/2)*arcsin(20/11*x+1/11)-3315200*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/
2)/(-10*x^2-x+3)^(1/2))-4370940*(-10*x^2-x+3)^(1/2)*x+2739480*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(3*x+2)

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maxima [A]  time = 1.40, size = 119, normalized size = 0.66 \begin {gather*} -\frac {25}{18} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {695}{648} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{3 \, {\left (3 \, x + 2\right )}} + \frac {11045}{2592} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {660959}{1866240} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {1295}{1458} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {76253}{31104} \, \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)^2,x, algorithm="maxima")

[Out]

-25/18*(-10*x^2 - x + 3)^(3/2)*x + 695/648*(-10*x^2 - x + 3)^(3/2) - 1/3*(-10*x^2 - x + 3)^(5/2)/(3*x + 2) + 1
1045/2592*sqrt(-10*x^2 - x + 3)*x - 660959/1866240*sqrt(10)*arcsin(20/11*x + 1/11) + 1295/1458*sqrt(7)*arcsin(
37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 76253/31104*sqrt(-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 )}^2} \,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)^2,x)

[Out]

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

[Out]

Timed out

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