3.2490 \(\int \frac {(2+3 x)^3}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\)

Optimal. Leaf size=84 \[ -\frac {1}{10} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2-\frac {\sqrt {1-2 x} \sqrt {5 x+3} (2220 x+5363)}{1600}+\frac {44437 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1600 \sqrt {10}} \]

[Out]

44437/16000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-1/10*(2+3*x)^2*(1-2*x)^(1/2)*(3+5*x)^(1/2)-1/1600*(53
63+2220*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 = {100, 147, 54, 216} \[ -\frac {1}{10} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2-\frac {\sqrt {1-2 x} \sqrt {5 x+3} (2220 x+5363)}{1600}+\frac {44437 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1600 \sqrt {10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/10 - (Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(5363 + 2220*x))/1600 + (44437*Ar
cSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1600*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 100

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

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} \sqrt {3+5 x}} \, dx &=-\frac {1}{10} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {1}{30} \int \frac {\left (-171-\frac {555 x}{2}\right ) (2+3 x)}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {1}{10} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (5363+2220 x)}{1600}+\frac {44437 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{3200}\\ &=-\frac {1}{10} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (5363+2220 x)}{1600}+\frac {44437 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{1600 \sqrt {5}}\\ &=-\frac {1}{10} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (5363+2220 x)}{1600}+\frac {44437 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1600 \sqrt {10}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.11, size = 78, normalized size = 0.93 \[ -\frac {\sqrt {1-2 x} \left (90 \sqrt {2 x-1} \sqrt {5 x+3} \left (160 x^2+460 x+667\right )+44437 \sqrt {10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )\right )}{16000 \sqrt {2 x-1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/16000*(Sqrt[1 - 2*x]*(90*Sqrt[-1 + 2*x]*Sqrt[3 + 5*x]*(667 + 460*x + 160*x^2) + 44437*Sqrt[10]*ArcSinh[Sqrt
[5/11]*Sqrt[-1 + 2*x]]))/Sqrt[-1 + 2*x]

________________________________________________________________________________________

fricas [A]  time = 0.99, size = 67, normalized size = 0.80 \[ -\frac {9}{1600} \, {\left (160 \, x^{2} + 460 \, x + 667\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {44437}{32000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/1600*(160*x^2 + 460*x + 667)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 44437/32000*sqrt(10)*arctan(1/20*sqrt(10)*(20*x
+ 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))

________________________________________________________________________________________

giac [A]  time = 0.85, size = 54, normalized size = 0.64 \[ -\frac {1}{80000} \, \sqrt {5} {\left (18 \, {\left (4 \, {\left (40 \, x + 91\right )} {\left (5 \, x + 3\right )} + 2243\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 222185 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/80000*sqrt(5)*(18*(4*(40*x + 91)*(5*x + 3) + 2243)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 222185*sqrt(2)*arcsin(1/
11*sqrt(22)*sqrt(5*x + 3)))

________________________________________________________________________________________

maple [A]  time = 0.02, size = 87, normalized size = 1.04 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-28800 \sqrt {-10 x^{2}-x +3}\, x^{2}-82800 \sqrt {-10 x^{2}-x +3}\, x +44437 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-120060 \sqrt {-10 x^{2}-x +3}\right )}{32000 \sqrt {-10 x^{2}-x +3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32000*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(-28800*(-10*x^2-x+3)^(1/2)*x^2+44437*10^(1/2)*arcsin(20/11*x+1/11)-82800
*(-10*x^2-x+3)^(1/2)*x-120060*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

________________________________________________________________________________________

maxima [A]  time = 1.37, size = 58, normalized size = 0.69 \[ -\frac {9}{10} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} - \frac {207}{80} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {44437}{32000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {6003}{1600} \, \sqrt {-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/10*sqrt(-10*x^2 - x + 3)*x^2 - 207/80*sqrt(-10*x^2 - x + 3)*x - 44437/32000*sqrt(10)*arcsin(-20/11*x - 1/11
) - 6003/1600*sqrt(-10*x^2 - x + 3)

________________________________________________________________________________________

mupad [B]  time = 7.75, size = 534, normalized size = 6.36 \[ \frac {44437\,\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,\left (\sqrt {1-2\,x}-1\right )}{2\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}\right )}{8000}-\frac {\frac {18837\,\left (\sqrt {1-2\,x}-1\right )}{390625\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {154377\,{\left (\sqrt {1-2\,x}-1\right )}^3}{156250\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {226251\,{\left (\sqrt {1-2\,x}-1\right )}^5}{156250\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}+\frac {226251\,{\left (\sqrt {1-2\,x}-1\right )}^7}{62500\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}+\frac {154377\,{\left (\sqrt {1-2\,x}-1\right )}^9}{10000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^9}-\frac {18837\,{\left (\sqrt {1-2\,x}-1\right )}^{11}}{4000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{11}}+\frac {4608\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {59904\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^4}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {107136\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^6}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {14976\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^8}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {288\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}}{\frac {192\,{\left (\sqrt {1-2\,x}-1\right )}^2}{3125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {48\,{\left (\sqrt {1-2\,x}-1\right )}^4}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {32\,{\left (\sqrt {1-2\,x}-1\right )}^6}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {12\,{\left (\sqrt {1-2\,x}-1\right )}^8}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {12\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^{12}}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{12}}+\frac {64}{15625}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(44437*10^(1/2)*atan((10^(1/2)*((1 - 2*x)^(1/2) - 1))/(2*(3^(1/2) - (5*x + 3)^(1/2)))))/8000 - ((18837*((1 - 2
*x)^(1/2) - 1))/(390625*(3^(1/2) - (5*x + 3)^(1/2))) - (154377*((1 - 2*x)^(1/2) - 1)^3)/(156250*(3^(1/2) - (5*
x + 3)^(1/2))^3) - (226251*((1 - 2*x)^(1/2) - 1)^5)/(156250*(3^(1/2) - (5*x + 3)^(1/2))^5) + (226251*((1 - 2*x
)^(1/2) - 1)^7)/(62500*(3^(1/2) - (5*x + 3)^(1/2))^7) + (154377*((1 - 2*x)^(1/2) - 1)^9)/(10000*(3^(1/2) - (5*
x + 3)^(1/2))^9) - (18837*((1 - 2*x)^(1/2) - 1)^11)/(4000*(3^(1/2) - (5*x + 3)^(1/2))^11) + (4608*3^(1/2)*((1
- 2*x)^(1/2) - 1)^2)/(15625*(3^(1/2) - (5*x + 3)^(1/2))^2) + (59904*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(15625*(3
^(1/2) - (5*x + 3)^(1/2))^4) + (107136*3^(1/2)*((1 - 2*x)^(1/2) - 1)^6)/(15625*(3^(1/2) - (5*x + 3)^(1/2))^6)
+ (14976*3^(1/2)*((1 - 2*x)^(1/2) - 1)^8)/(625*(3^(1/2) - (5*x + 3)^(1/2))^8) + (288*3^(1/2)*((1 - 2*x)^(1/2)
- 1)^10)/(25*(3^(1/2) - (5*x + 3)^(1/2))^10))/((192*((1 - 2*x)^(1/2) - 1)^2)/(3125*(3^(1/2) - (5*x + 3)^(1/2))
^2) + (48*((1 - 2*x)^(1/2) - 1)^4)/(125*(3^(1/2) - (5*x + 3)^(1/2))^4) + (32*((1 - 2*x)^(1/2) - 1)^6)/(25*(3^(
1/2) - (5*x + 3)^(1/2))^6) + (12*((1 - 2*x)^(1/2) - 1)^8)/(5*(3^(1/2) - (5*x + 3)^(1/2))^8) + (12*((1 - 2*x)^(
1/2) - 1)^10)/(5*(3^(1/2) - (5*x + 3)^(1/2))^10) + ((1 - 2*x)^(1/2) - 1)^12/(3^(1/2) - (5*x + 3)^(1/2))^12 + 6
4/15625)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________