3.2449 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=138 \[ -\frac{76 (1-2 x)^{7/2}}{1815 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{7/2}}{825 (5 x+3)^{3/2}}+\frac{329 \sqrt{5 x+3} (1-2 x)^{5/2}}{45375}+\frac{329 \sqrt{5 x+3} (1-2 x)^{3/2}}{16500}+\frac{329 \sqrt{5 x+3} \sqrt{1-2 x}}{5000}+\frac{3619 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{5000 \sqrt{10}} \]

[Out]

(-2*(1 - 2*x)^(7/2))/(825*(3 + 5*x)^(3/2)) - (76*(1 - 2*x)^(7/2))/(1815*Sqrt[3 + 5*x]) + (329*Sqrt[1 - 2*x]*Sq
rt[3 + 5*x])/5000 + (329*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/16500 + (329*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/45375 + (3
619*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5000*Sqrt[10])

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Rubi [A]  time = 0.0386716, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {89, 78, 50, 54, 216} \[ -\frac{76 (1-2 x)^{7/2}}{1815 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{7/2}}{825 (5 x+3)^{3/2}}+\frac{329 \sqrt{5 x+3} (1-2 x)^{5/2}}{45375}+\frac{329 \sqrt{5 x+3} (1-2 x)^{3/2}}{16500}+\frac{329 \sqrt{5 x+3} \sqrt{1-2 x}}{5000}+\frac{3619 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{5000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(7/2))/(825*(3 + 5*x)^(3/2)) - (76*(1 - 2*x)^(7/2))/(1815*Sqrt[3 + 5*x]) + (329*Sqrt[1 - 2*x]*Sq
rt[3 + 5*x])/5000 + (329*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/16500 + (329*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/45375 + (3
619*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5000*Sqrt[10])

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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 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} (2+3 x)^2}{(3+5 x)^{5/2}} \, dx &=-\frac{2 (1-2 x)^{7/2}}{825 (3+5 x)^{3/2}}+\frac{2}{825} \int \frac{(1-2 x)^{5/2} \left (\frac{1081}{2}+\frac{1485 x}{2}\right )}{(3+5 x)^{3/2}} \, dx\\ &=-\frac{2 (1-2 x)^{7/2}}{825 (3+5 x)^{3/2}}-\frac{76 (1-2 x)^{7/2}}{1815 \sqrt{3+5 x}}+\frac{329 \int \frac{(1-2 x)^{5/2}}{\sqrt{3+5 x}} \, dx}{3025}\\ &=-\frac{2 (1-2 x)^{7/2}}{825 (3+5 x)^{3/2}}-\frac{76 (1-2 x)^{7/2}}{1815 \sqrt{3+5 x}}+\frac{329 (1-2 x)^{5/2} \sqrt{3+5 x}}{45375}+\frac{329 \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx}{1650}\\ &=-\frac{2 (1-2 x)^{7/2}}{825 (3+5 x)^{3/2}}-\frac{76 (1-2 x)^{7/2}}{1815 \sqrt{3+5 x}}+\frac{329 (1-2 x)^{3/2} \sqrt{3+5 x}}{16500}+\frac{329 (1-2 x)^{5/2} \sqrt{3+5 x}}{45375}+\frac{329 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{1000}\\ &=-\frac{2 (1-2 x)^{7/2}}{825 (3+5 x)^{3/2}}-\frac{76 (1-2 x)^{7/2}}{1815 \sqrt{3+5 x}}+\frac{329 \sqrt{1-2 x} \sqrt{3+5 x}}{5000}+\frac{329 (1-2 x)^{3/2} \sqrt{3+5 x}}{16500}+\frac{329 (1-2 x)^{5/2} \sqrt{3+5 x}}{45375}+\frac{3619 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{10000}\\ &=-\frac{2 (1-2 x)^{7/2}}{825 (3+5 x)^{3/2}}-\frac{76 (1-2 x)^{7/2}}{1815 \sqrt{3+5 x}}+\frac{329 \sqrt{1-2 x} \sqrt{3+5 x}}{5000}+\frac{329 (1-2 x)^{3/2} \sqrt{3+5 x}}{16500}+\frac{329 (1-2 x)^{5/2} \sqrt{3+5 x}}{45375}+\frac{3619 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{5000 \sqrt{5}}\\ &=-\frac{2 (1-2 x)^{7/2}}{825 (3+5 x)^{3/2}}-\frac{76 (1-2 x)^{7/2}}{1815 \sqrt{3+5 x}}+\frac{329 \sqrt{1-2 x} \sqrt{3+5 x}}{5000}+\frac{329 (1-2 x)^{3/2} \sqrt{3+5 x}}{16500}+\frac{329 (1-2 x)^{5/2} \sqrt{3+5 x}}{45375}+\frac{3619 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{5000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0547056, size = 88, normalized size = 0.64 \[ \frac{-10 \left (72000 x^5-106200 x^4+42270 x^3+78275 x^2-19664 x-10633\right )-10857 \sqrt{10-20 x} (5 x+3)^{3/2} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{150000 \sqrt{1-2 x} (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*(-10633 - 19664*x + 78275*x^2 + 42270*x^3 - 106200*x^4 + 72000*x^5) - 10857*Sqrt[10 - 20*x]*(3 + 5*x)^(3/
2)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(150000*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))

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Maple [A]  time = 0.011, size = 147, normalized size = 1.1 \begin{align*}{\frac{1}{300000} \left ( 720000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+271425\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-702000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+325710\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+71700\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+97713\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +818600\,x\sqrt{-10\,{x}^{2}-x+3}+212660\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/300000*(720000*x^4*(-10*x^2-x+3)^(1/2)+271425*10^(1/2)*arcsin(20/11*x+1/11)*x^2-702000*x^3*(-10*x^2-x+3)^(1/
2)+325710*10^(1/2)*arcsin(20/11*x+1/11)*x+71700*x^2*(-10*x^2-x+3)^(1/2)+97713*10^(1/2)*arcsin(20/11*x+1/11)+81
8600*x*(-10*x^2-x+3)^(1/2)+212660*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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Maxima [B]  time = 2.45459, size = 333, normalized size = 2.41 \begin{align*} \frac{3619}{100000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{125 \,{\left (625 \, x^{4} + 1500 \, x^{3} + 1350 \, x^{2} + 540 \, x + 81\right )}} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{125 \,{\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{125 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{1089}{5000} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{11 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{750 \,{\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac{33 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{33 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{500 \,{\left (5 \, x + 3\right )}} - \frac{121 \, \sqrt{-10 \, x^{2} - x + 3}}{3750 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac{3113 \, \sqrt{-10 \, x^{2} - x + 3}}{3750 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3619/100000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 1/125*(-10*x^2 - x + 3)^(5/2)/(625*x^4 + 1500*x^3 + 1350*
x^2 + 540*x + 81) + 3/125*(-10*x^2 - x + 3)^(5/2)/(125*x^3 + 225*x^2 + 135*x + 27) + 3/125*(-10*x^2 - x + 3)^(
5/2)/(25*x^2 + 30*x + 9) + 1089/5000*sqrt(-10*x^2 - x + 3) - 11/750*(-10*x^2 - x + 3)^(3/2)/(125*x^3 + 225*x^2
+ 135*x + 27) + 33/250*(-10*x^2 - x + 3)^(3/2)/(25*x^2 + 30*x + 9) + 33/500*(-10*x^2 - x + 3)^(3/2)/(5*x + 3)
- 121/3750*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x + 9) - 3113/3750*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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Fricas [A]  time = 1.85035, size = 316, normalized size = 2.29 \begin{align*} -\frac{10857 \, \sqrt{10}{\left (25 \, x^{2} + 30 \, x + 9\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 ) - 20 \,{\left (36000 \, x^{4} - 35100 \, x^{3} + 3585 \, x^{2} + 40930 \, x + 10633\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{300000 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/300000*(10857*sqrt(10)*(25*x^2 + 30*x + 9)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10
*x^2 + x - 3)) - 20*(36000*x^4 - 35100*x^3 + 3585*x^2 + 40930*x + 10633)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(25*x^2
+ 30*x + 9)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 2.54897, size = 255, normalized size = 1.85 \begin{align*} \frac{1}{125000} \,{\left (12 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} - 135 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 9635 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{11 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{750000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} + \frac{3619}{50000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{1353 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{62500 \, \sqrt{5 \, x + 3}} + \frac{11 \,{\left (\frac{369 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{46875 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/125000*(12*(8*sqrt(5)*(5*x + 3) - 135*sqrt(5))*(5*x + 3) + 9635*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) - 11/
750000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 3619/50000*sqrt(10)*arcsin(1/11*sqrt(
22)*sqrt(5*x + 3)) - 1353/62500*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 11/46875*(369*sq
rt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4*sqrt(10))*(5*x + 3)^(3/2)/(sqrt(2)*sqrt(-10*x + 5)
- sqrt(22))^3