3.2320 \(\int \frac {\sqrt {1-2 x} (2+3 x)^2}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=94 \[ -\frac {12 (1-2 x)^{3/2}}{275 \sqrt {5 x+3}}-\frac {2 (1-2 x)^{3/2}}{825 (5 x+3)^{3/2}}+\frac {3}{55} \sqrt {5 x+3} \sqrt {1-2 x}+\frac {3 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}} \]

[Out]

-2/825*(1-2*x)^(3/2)/(3+5*x)^(3/2)+3/50*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-12/275*(1-2*x)^(3/2)/(3+5
*x)^(1/2)+3/55*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 5, 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} \[ -\frac {12 (1-2 x)^{3/2}}{275 \sqrt {5 x+3}}-\frac {2 (1-2 x)^{3/2}}{825 (5 x+3)^{3/2}}+\frac {3}{55} \sqrt {5 x+3} \sqrt {1-2 x}+\frac {3 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(3/2))/(825*(3 + 5*x)^(3/2)) - (12*(1 - 2*x)^(3/2))/(275*Sqrt[3 + 5*x]) + (3*Sqrt[1 - 2*x]*Sqrt[
3 + 5*x])/55 + (3*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5*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 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 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 {\sqrt {1-2 x} (2+3 x)^2}{(3+5 x)^{5/2}} \, dx &=-\frac {2 (1-2 x)^{3/2}}{825 (3+5 x)^{3/2}}+\frac {2}{825} \int \frac {\sqrt {1-2 x} \left (\frac {1089}{2}+\frac {1485 x}{2}\right )}{(3+5 x)^{3/2}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2}}{825 (3+5 x)^{3/2}}-\frac {12 (1-2 x)^{3/2}}{275 \sqrt {3+5 x}}+\frac {3}{11} \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2}}{825 (3+5 x)^{3/2}}-\frac {12 (1-2 x)^{3/2}}{275 \sqrt {3+5 x}}+\frac {3}{55} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {3}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2}}{825 (3+5 x)^{3/2}}-\frac {12 (1-2 x)^{3/2}}{275 \sqrt {3+5 x}}+\frac {3}{55} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{5 \sqrt {5}}\\ &=-\frac {2 (1-2 x)^{3/2}}{825 (3+5 x)^{3/2}}-\frac {12 (1-2 x)^{3/2}}{275 \sqrt {3+5 x}}+\frac {3}{55} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {3 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{5 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 78, normalized size = 0.83 \[ \frac {10 \left (-594 x^3-259 x^2+160 x+59\right )+99 \sqrt {20 x-10} (5 x+3)^{3/2} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{1650 \sqrt {1-2 x} (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*(59 + 160*x - 259*x^2 - 594*x^3) + 99*(3 + 5*x)^(3/2)*Sqrt[-10 + 20*x]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])
/(1650*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))

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fricas [A]  time = 1.13, size = 91, normalized size = 0.97 \[ -\frac {99 \, \sqrt {10} {\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 20 \, {\left (297 \, x^{2} + 278 \, x + 59\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{3300 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3300*(99*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*(297*x^2 + 278*x + 59)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)

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giac [B]  time = 1.70, size = 158, normalized size = 1.68 \[ -\frac {1}{330000} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {1572 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} + \frac {9}{625} \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {3}{50} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {393 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{20625 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/330000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 1572*(sqrt(2)*sqrt(-10*x + 5) - s
qrt(22))/sqrt(5*x + 3)) + 9/625*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 3/50*sqrt(10)*arcsin(1/11*sqrt(22)*sqr
t(5*x + 3)) + 1/20625*sqrt(10)*(5*x + 3)^(3/2)*(393*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4)/(sqr
t(2)*sqrt(-10*x + 5) - sqrt(22))^3

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maple [A]  time = 0.01, size = 113, normalized size = 1.20 \[ \frac {\left (2475 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+5940 \sqrt {-10 x^{2}-x +3}\, x^{2}+2970 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+5560 \sqrt {-10 x^{2}-x +3}\, x +891 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1180 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{3300 \sqrt {-10 x^{2}-x +3}\, \left (5 x +3\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3300*(2475*10^(1/2)*x^2*arcsin(20/11*x+1/11)+2970*10^(1/2)*x*arcsin(20/11*x+1/11)+5940*(-10*x^2-x+3)^(1/2)*x
^2+891*10^(1/2)*arcsin(20/11*x+1/11)+5560*(-10*x^2-x+3)^(1/2)*x+1180*(-10*x^2-x+3)^(1/2))*(-2*x+1)^(1/2)/(-10*
x^2-x+3)^(1/2)/(5*x+3)^(3/2)

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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