3.2499 \(\int \frac {(2+3 x)^3}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=84 \[ -\frac {2 \sqrt {1-2 x} (3 x+2)^2}{55 \sqrt {5 x+3}}-\frac {3 \sqrt {1-2 x} \sqrt {5 x+3} (300 x+979)}{4400}+\frac {2493 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{400 \sqrt {10}} \]

[Out]

2493/4000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-2/55*(2+3*x)^2*(1-2*x)^(1/2)/(3+5*x)^(1/2)-3/4400*(979+
300*x)*(1-2*x)^(1/2)*(3+5*x)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {98, 147, 54, 216} \[ -\frac {2 \sqrt {1-2 x} (3 x+2)^2}{55 \sqrt {5 x+3}}-\frac {3 \sqrt {1-2 x} \sqrt {5 x+3} (300 x+979)}{4400}+\frac {2493 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{400 \sqrt {10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(55*Sqrt[3 + 5*x]) - (3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(979 + 300*x))/4400 + (2493
*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(400*Sqrt[10])

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 98

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

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 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 {(2+3 x)^3}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx &=-\frac {2 \sqrt {1-2 x} (2+3 x)^2}{55 \sqrt {3+5 x}}-\frac {2}{55} \int \frac {\left (-39-\frac {75 x}{2}\right ) (2+3 x)}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^2}{55 \sqrt {3+5 x}}-\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} (979+300 x)}{4400}+\frac {2493}{800} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^2}{55 \sqrt {3+5 x}}-\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} (979+300 x)}{4400}+\frac {2493 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{400 \sqrt {5}}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^2}{55 \sqrt {3+5 x}}-\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} (979+300 x)}{4400}+\frac {2493 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{400 \sqrt {10}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 82, normalized size = 0.98 \[ -\frac {\sqrt {1-2 x} \left (10 \sqrt {2 x-1} \left (5940 x^2+19305 x+9451\right )+27423 \sqrt {50 x+30} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )\right )}{44000 \sqrt {2 x-1} \sqrt {5 x+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/44000*(Sqrt[1 - 2*x]*(10*Sqrt[-1 + 2*x]*(9451 + 19305*x + 5940*x^2) + 27423*Sqrt[30 + 50*x]*ArcSinh[Sqrt[5/
11]*Sqrt[-1 + 2*x]]))/(Sqrt[-1 + 2*x]*Sqrt[3 + 5*x])

________________________________________________________________________________________

fricas [A]  time = 0.94, size = 81, normalized size = 0.96 \[ -\frac {27423 \, \sqrt {10} {\left (5 \, x + 3\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (5940 \, x^{2} + 19305 \, x + 9451\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{88000 \, {\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/88000*(27423*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*(5940*x^2 + 19305*x + 9451)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(5*x + 3)

________________________________________________________________________________________

giac [A]  time = 1.10, size = 111, normalized size = 1.32 \[ -\frac {27}{10000} \, {\left (4 \, \sqrt {5} {\left (5 \, x + 3\right )} + 41 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {2493}{4000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{13750 \, \sqrt {5 \, x + 3}} + \frac {2 \, \sqrt {10} \sqrt {5 \, x + 3}}{6875 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/10000*(4*sqrt(5)*(5*x + 3) + 41*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) + 2493/4000*sqrt(10)*arcsin(1/11*sq
rt(22)*sqrt(5*x + 3)) - 1/13750*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 2/6875*sqrt(10)*
sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))

________________________________________________________________________________________

maple [A]  time = 0.02, size = 99, normalized size = 1.18 \[ \frac {\left (-118800 \sqrt {-10 x^{2}-x +3}\, x^{2}+137115 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-386100 \sqrt {-10 x^{2}-x +3}\, x +82269 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-189020 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{88000 \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/88000*(137115*10^(1/2)*x*arcsin(20/11*x+1/11)-118800*(-10*x^2-x+3)^(1/2)*x^2+82269*10^(1/2)*arcsin(20/11*x+1
/11)-386100*(-10*x^2-x+3)^(1/2)*x-189020*(-10*x^2-x+3)^(1/2))*(-2*x+1)^(1/2)/(-10*x^2-x+3)^(1/2)/(5*x+3)^(1/2)

________________________________________________________________________________________

maxima [A]  time = 1.28, size = 65, normalized size = 0.77 \[ \frac {2493}{8000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {27}{100} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {1431}{2000} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {2 \, \sqrt {-10 \, x^{2} - x + 3}}{1375 \, {\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2493/8000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 27/100*sqrt(-10*x^2 - x + 3)*x - 1431/2000*sqrt(-10*x^2 - x
 + 3) - 2/1375*sqrt(-10*x^2 - x + 3)/(5*x + 3)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________