3.2372 \(\int \frac {(1-2 x)^{3/2} (2+3 x)^2}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=116 \[ -\frac {3}{50} \sqrt {5 x+3} (1-2 x)^{5/2}-\frac {2 (1-2 x)^{5/2}}{275 \sqrt {5 x+3}}+\frac {119 \sqrt {5 x+3} (1-2 x)^{3/2}}{2200}+\frac {357 \sqrt {5 x+3} \sqrt {1-2 x}}{2000}+\frac {3927 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{2000 \sqrt {10}} \]

[Out]

3927/20000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-2/275*(1-2*x)^(5/2)/(3+5*x)^(1/2)+119/2200*(1-2*x)^(3/
2)*(3+5*x)^(1/2)-3/50*(1-2*x)^(5/2)*(3+5*x)^(1/2)+357/2000*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 116, normalized size of antiderivative = 1.00, 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, 80, 50, 54, 216} \[ -\frac {3}{50} \sqrt {5 x+3} (1-2 x)^{5/2}-\frac {2 (1-2 x)^{5/2}}{275 \sqrt {5 x+3}}+\frac {119 \sqrt {5 x+3} (1-2 x)^{3/2}}{2200}+\frac {357 \sqrt {5 x+3} \sqrt {1-2 x}}{2000}+\frac {3927 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{2000 \sqrt {10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(5/2))/(275*Sqrt[3 + 5*x]) + (357*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/2000 + (119*(1 - 2*x)^(3/2)*Sqrt[
3 + 5*x])/2200 - (3*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/50 + (3927*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(2000*Sqrt[10]
)

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 80

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

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 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)^{3/2}} \, dx &=-\frac {2 (1-2 x)^{5/2}}{275 \sqrt {3+5 x}}+\frac {2}{275} \int \frac {(1-2 x)^{3/2} \left (\frac {355}{2}+\frac {495 x}{2}\right )}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2}}{275 \sqrt {3+5 x}}-\frac {3}{50} (1-2 x)^{5/2} \sqrt {3+5 x}+\frac {119}{220} \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2}}{275 \sqrt {3+5 x}}+\frac {119 (1-2 x)^{3/2} \sqrt {3+5 x}}{2200}-\frac {3}{50} (1-2 x)^{5/2} \sqrt {3+5 x}+\frac {357}{400} \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2}}{275 \sqrt {3+5 x}}+\frac {357 \sqrt {1-2 x} \sqrt {3+5 x}}{2000}+\frac {119 (1-2 x)^{3/2} \sqrt {3+5 x}}{2200}-\frac {3}{50} (1-2 x)^{5/2} \sqrt {3+5 x}+\frac {3927 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{4000}\\ &=-\frac {2 (1-2 x)^{5/2}}{275 \sqrt {3+5 x}}+\frac {357 \sqrt {1-2 x} \sqrt {3+5 x}}{2000}+\frac {119 (1-2 x)^{3/2} \sqrt {3+5 x}}{2200}-\frac {3}{50} (1-2 x)^{5/2} \sqrt {3+5 x}+\frac {3927 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{2000 \sqrt {5}}\\ &=-\frac {2 (1-2 x)^{5/2}}{275 \sqrt {3+5 x}}+\frac {357 \sqrt {1-2 x} \sqrt {3+5 x}}{2000}+\frac {119 (1-2 x)^{3/2} \sqrt {3+5 x}}{2200}-\frac {3}{50} (1-2 x)^{5/2} \sqrt {3+5 x}+\frac {3927 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{2000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 83, normalized size = 0.72 \[ \frac {10 \left (4800 x^4-2040 x^3-5330 x^2+533 x+1021\right )+3927 \sqrt {5 x+3} \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{20000 \sqrt {1-2 x} \sqrt {5 x+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*(1021 + 533*x - 5330*x^2 - 2040*x^3 + 4800*x^4) + 3927*Sqrt[3 + 5*x]*Sqrt[-10 + 20*x]*ArcSinh[Sqrt[5/11]*S
qrt[-1 + 2*x]])/(20000*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])

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fricas [A]  time = 1.07, size = 86, normalized size = 0.74 \[ -\frac {3927 \, \sqrt {10} {\left (5 \, x + 3\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (2400 \, x^{3} + 180 \, x^{2} - 2575 \, x - 1021\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{40000 \, {\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/40000*(3927*sqrt(10)*(5*x + 3)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3
)) + 20*(2400*x^3 + 180*x^2 - 2575*x - 1021)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(5*x + 3)

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giac [A]  time = 1.35, size = 124, normalized size = 1.07 \[ -\frac {1}{50000} \, {\left (12 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} - 69 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 199 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {3927}{20000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {11 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{6250 \, \sqrt {5 \, x + 3}} + \frac {22 \, \sqrt {10} \sqrt {5 \, x + 3}}{3125 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/50000*(12*(8*sqrt(5)*(5*x + 3) - 69*sqrt(5))*(5*x + 3) - 199*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) + 3927/
20000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 11/6250*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqr
t(5*x + 3) + 22/3125*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))

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maple [A]  time = 0.01, size = 116, normalized size = 1.00 \[ \frac {\left (-48000 \sqrt {-10 x^{2}-x +3}\, x^{3}-3600 \sqrt {-10 x^{2}-x +3}\, x^{2}+19635 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+51500 \sqrt {-10 x^{2}-x +3}\, x +11781 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+20420 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{40000 \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/40000*(-48000*(-10*x^2-x+3)^(1/2)*x^3+19635*10^(1/2)*x*arcsin(20/11*x+1/11)-3600*(-10*x^2-x+3)^(1/2)*x^2+117
81*10^(1/2)*arcsin(20/11*x+1/11)+51500*(-10*x^2-x+3)^(1/2)*x+20420*(-10*x^2-x+3)^(1/2))*(-2*x+1)^(1/2)/(-10*x^
2-x+3)^(1/2)/(5*x+3)^(1/2)

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maxima [C]  time = 1.32, size = 154, normalized size = 1.33 \[ -\frac {11979}{200000} i \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {23}{11}\right ) + \frac {957}{25000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {3}{125} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {99}{500} \, \sqrt {10 \, x^{2} + 23 \, x + \frac {51}{5}} x + \frac {2277}{10000} \, \sqrt {10 \, x^{2} + 23 \, x + \frac {51}{5}} + \frac {99}{1250} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{125 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{125 \, {\left (5 \, x + 3\right )}} - \frac {33 \, \sqrt {-10 \, x^{2} - x + 3}}{625 \, {\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-11979/200000*I*sqrt(5)*sqrt(2)*arcsin(20/11*x + 23/11) + 957/25000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 3
/125*(-10*x^2 - x + 3)^(3/2) + 99/500*sqrt(10*x^2 + 23*x + 51/5)*x + 2277/10000*sqrt(10*x^2 + 23*x + 51/5) + 9
9/1250*sqrt(-10*x^2 - x + 3) + 1/125*(-10*x^2 - x + 3)^(3/2)/(25*x^2 + 30*x + 9) + 3/125*(-10*x^2 - x + 3)^(3/
2)/(5*x + 3) - 33/625*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{2}}{\left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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