3.2381 \(\int \frac{(1-2 x)^{3/2} (2+3 x)^2}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=116 \[ -\frac{388 (1-2 x)^{5/2}}{9075 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{5/2}}{825 (5 x+3)^{3/2}}+\frac{343 \sqrt{5 x+3} (1-2 x)^{3/2}}{18150}+\frac{343 \sqrt{5 x+3} \sqrt{1-2 x}}{5500}+\frac{343 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{500 \sqrt{10}} \]

[Out]

(-2*(1 - 2*x)^(5/2))/(825*(3 + 5*x)^(3/2)) - (388*(1 - 2*x)^(5/2))/(9075*Sqrt[3 + 5*x]) + (343*Sqrt[1 - 2*x]*S
qrt[3 + 5*x])/5500 + (343*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/18150 + (343*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(500*S
qrt[10])

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Rubi [A]  time = 0.0288644, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{388 (1-2 x)^{5/2}}{9075 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{5/2}}{825 (5 x+3)^{3/2}}+\frac{343 \sqrt{5 x+3} (1-2 x)^{3/2}}{18150}+\frac{343 \sqrt{5 x+3} \sqrt{1-2 x}}{5500}+\frac{343 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{500 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(5/2))/(825*(3 + 5*x)^(3/2)) - (388*(1 - 2*x)^(5/2))/(9075*Sqrt[3 + 5*x]) + (343*Sqrt[1 - 2*x]*S
qrt[3 + 5*x])/5500 + (343*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/18150 + (343*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(500*S
qrt[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)^{3/2} (2+3 x)^2}{(3+5 x)^{5/2}} \, dx &=-\frac{2 (1-2 x)^{5/2}}{825 (3+5 x)^{3/2}}+\frac{2}{825} \int \frac{(1-2 x)^{3/2} \left (\frac{1085}{2}+\frac{1485 x}{2}\right )}{(3+5 x)^{3/2}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2}}{825 (3+5 x)^{3/2}}-\frac{388 (1-2 x)^{5/2}}{9075 \sqrt{3+5 x}}+\frac{343 \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx}{1815}\\ &=-\frac{2 (1-2 x)^{5/2}}{825 (3+5 x)^{3/2}}-\frac{388 (1-2 x)^{5/2}}{9075 \sqrt{3+5 x}}+\frac{343 (1-2 x)^{3/2} \sqrt{3+5 x}}{18150}+\frac{343 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{1100}\\ &=-\frac{2 (1-2 x)^{5/2}}{825 (3+5 x)^{3/2}}-\frac{388 (1-2 x)^{5/2}}{9075 \sqrt{3+5 x}}+\frac{343 \sqrt{1-2 x} \sqrt{3+5 x}}{5500}+\frac{343 (1-2 x)^{3/2} \sqrt{3+5 x}}{18150}+\frac{343 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{1000}\\ &=-\frac{2 (1-2 x)^{5/2}}{825 (3+5 x)^{3/2}}-\frac{388 (1-2 x)^{5/2}}{9075 \sqrt{3+5 x}}+\frac{343 \sqrt{1-2 x} \sqrt{3+5 x}}{5500}+\frac{343 (1-2 x)^{3/2} \sqrt{3+5 x}}{18150}+\frac{343 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{500 \sqrt{5}}\\ &=-\frac{2 (1-2 x)^{5/2}}{825 (3+5 x)^{3/2}}-\frac{388 (1-2 x)^{5/2}}{9075 \sqrt{3+5 x}}+\frac{343 \sqrt{1-2 x} \sqrt{3+5 x}}{5500}+\frac{343 (1-2 x)^{3/2} \sqrt{3+5 x}}{18150}+\frac{343 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{500 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0482475, size = 83, normalized size = 0.72 \[ \frac{10 \left (5400 x^4-6390 x^3-5375 x^2+1808 x+901\right )-1029 \sqrt{10-20 x} (5 x+3)^{3/2} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15000 \sqrt{1-2 x} (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*(901 + 1808*x - 5375*x^2 - 6390*x^3 + 5400*x^4) - 1029*Sqrt[10 - 20*x]*(3 + 5*x)^(3/2)*ArcSin[Sqrt[5/11]*S
qrt[1 - 2*x]])/(15000*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))

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Maple [A]  time = 0.012, size = 130, normalized size = 1.1 \begin{align*}{\frac{1}{30000} \left ( 25725\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-54000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+30870\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+36900\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+9261\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +72200\,x\sqrt{-10\,{x}^{2}-x+3}+18020\,\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)^(3/2)*(2+3*x)^2/(3+5*x)^(5/2),x)

[Out]

1/30000*(25725*10^(1/2)*arcsin(20/11*x+1/11)*x^2-54000*x^3*(-10*x^2-x+3)^(1/2)+30870*10^(1/2)*arcsin(20/11*x+1
/11)*x+36900*x^2*(-10*x^2-x+3)^(1/2)+9261*10^(1/2)*arcsin(20/11*x+1/11)+72200*x*(-10*x^2-x+3)^(1/2)+18020*(-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 [A]  time = 2.59529, size = 208, normalized size = 1.79 \begin{align*} \frac{343}{10000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{297}{2500} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{375 \,{\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac{6 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{125 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{250 \,{\left (5 \, x + 3\right )}} - \frac{11 \, \sqrt{-10 \, x^{2} - x + 3}}{1875 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac{116 \, \sqrt{-10 \, x^{2} - x + 3}}{375 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

343/10000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 297/2500*sqrt(-10*x^2 - x + 3) - 1/375*(-10*x^2 - x + 3)^(3
/2)/(125*x^3 + 225*x^2 + 135*x + 27) + 6/125*(-10*x^2 - x + 3)^(3/2)/(25*x^2 + 30*x + 9) + 9/250*(-10*x^2 - x
+ 3)^(3/2)/(5*x + 3) - 11/1875*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x + 9) - 116/375*sqrt(-10*x^2 - x + 3)/(5*x
+ 3)

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Fricas [A]  time = 1.54951, size = 292, normalized size = 2.52 \begin{align*} -\frac{1029 \, \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 (2700 \, x^{3} - 1845 \, x^{2} - 3610 \, x - 901\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{30000 \,{\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)^(3/2)*(2+3*x)^2/(3+5*x)^(5/2),x, algorithm="fricas")

[Out]

-1/30000*(1029*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*(2700*x^3 - 1845*x^2 - 3610*x - 901)*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)**(3/2)*(2+3*x)**2/(3+5*x)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 2.30384, size = 238, normalized size = 2.05 \begin{align*} -\frac{3}{12500} \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} - 149 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{150000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} + \frac{343}{5000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{127 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{12500 \, \sqrt{5 \, x + 3}} + \frac{{\left (\frac{381 \, \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}}}{9375 \,{\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)^(3/2)*(2+3*x)^2/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

-3/12500*(12*sqrt(5)*(5*x + 3) - 149*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) - 1/150000*sqrt(10)*(sqrt(2)*sqrt(
-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 343/5000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 127/12500*s
qrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 1/9375*(381*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - s
qrt(22))^2/(5*x + 3) + 4*sqrt(10))*(5*x + 3)^(3/2)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3