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

Optimal. Leaf size=77 \[ -\frac {3}{20} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)-\frac {333}{400} \sqrt {1-2 x} \sqrt {5 x+3}+\frac {3827 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{400 \sqrt {10}} \]

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Rubi [A]  time = 0.02, antiderivative size = 77, 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 = {90, 80, 54, 216} \begin {gather*} -\frac {3}{20} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)-\frac {333}{400} \sqrt {1-2 x} \sqrt {5 x+3}+\frac {3827 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{400 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-333*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/400 - (3*Sqrt[1 - 2*x]*(2 + 3*x)*Sqrt[3 + 5*x])/20 + (3827*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 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 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 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)^2}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx &=-\frac {3}{20} \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}-\frac {1}{20} \int \frac {-104-\frac {333 x}{2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {333}{400} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {3}{20} \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}+\frac {3827}{800} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {333}{400} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {3}{20} \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}+\frac {3827 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{400 \sqrt {5}}\\ &=-\frac {333}{400} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {3}{20} \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}+\frac {3827 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{400 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 73, normalized size = 0.95 \begin {gather*} -\frac {\sqrt {1-2 x} \left (30 \sqrt {2 x-1} \sqrt {5 x+3} (60 x+151)+3827 \sqrt {10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )\right )}{4000 \sqrt {2 x-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4000*(Sqrt[1 - 2*x]*(30*Sqrt[-1 + 2*x]*Sqrt[3 + 5*x]*(151 + 60*x) + 3827*Sqrt[10]*ArcSinh[Sqrt[5/11]*Sqrt[-
1 + 2*x]]))/Sqrt[-1 + 2*x]

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IntegrateAlgebraic [A]  time = 0.13, size = 93, normalized size = 1.21 \begin {gather*} -\frac {33 \sqrt {1-2 x} \left (\frac {575 (1-2 x)}{5 x+3}+362\right )}{400 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^2}-\frac {3827 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{400 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-33*Sqrt[1 - 2*x]*(362 + (575*(1 - 2*x))/(3 + 5*x)))/(400*Sqrt[3 + 5*x]*(2 + (5*(1 - 2*x))/(3 + 5*x))^2) - (3
827*ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])/Sqrt[3 + 5*x]])/(400*Sqrt[10])

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fricas [A]  time = 1.21, size = 62, normalized size = 0.81 \begin {gather*} -\frac {3}{400} \, {\left (60 \, x + 151\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {3827}{8000} \, \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/400*(60*x + 151)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 3827/8000*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x
+ 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))

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giac [A]  time = 1.01, size = 45, normalized size = 0.58 \begin {gather*} -\frac {1}{4000} \, \sqrt {5} {\left (6 \, {\left (60 \, x + 151\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 3827 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4000*sqrt(5)*(6*(60*x + 151)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 3827*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3
)))

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maple [A]  time = 0.01, size = 70, normalized size = 0.91 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-3600 \sqrt {-10 x^{2}-x +3}\, x +3827 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-9060 \sqrt {-10 x^{2}-x +3}\right )}{8000 \sqrt {-10 x^{2}-x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8000*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(3827*10^(1/2)*arcsin(20/11*x+1/11)-3600*(-10*x^2-x+3)^(1/2)*x-9060*(-10*x
^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.35, size = 41, normalized size = 0.53 \begin {gather*} -\frac {9}{20} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {3827}{8000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {453}{400} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/20*sqrt(-10*x^2 - x + 3)*x - 3827/8000*sqrt(10)*arcsin(-20/11*x - 1/11) - 453/400*sqrt(-10*x^2 - x + 3)

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mupad [B]  time = 10.63, size = 360, normalized size = 4.68 \begin {gather*} \frac {3827\,\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 )}{2000}-\frac {\frac {627\,\left (\sqrt {1-2\,x}-1\right )}{15625\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {7941\,{\left (\sqrt {1-2\,x}-1\right )}^3}{6250\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}+\frac {7941\,{\left (\sqrt {1-2\,x}-1\right )}^5}{2500\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}-\frac {627\,{\left (\sqrt {1-2\,x}-1\right )}^7}{1000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}+\frac {384\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {1632\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^4}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {96\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^6}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}}{\frac {32\,{\left (\sqrt {1-2\,x}-1\right )}^2}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {24\,{\left (\sqrt {1-2\,x}-1\right )}^4}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {8\,{\left (\sqrt {1-2\,x}-1\right )}^6}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^8}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {16}{625}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3827*10^(1/2)*atan((10^(1/2)*((1 - 2*x)^(1/2) - 1))/(2*(3^(1/2) - (5*x + 3)^(1/2)))))/2000 - ((627*((1 - 2*x)
^(1/2) - 1))/(15625*(3^(1/2) - (5*x + 3)^(1/2))) - (7941*((1 - 2*x)^(1/2) - 1)^3)/(6250*(3^(1/2) - (5*x + 3)^(
1/2))^3) + (7941*((1 - 2*x)^(1/2) - 1)^5)/(2500*(3^(1/2) - (5*x + 3)^(1/2))^5) - (627*((1 - 2*x)^(1/2) - 1)^7)
/(1000*(3^(1/2) - (5*x + 3)^(1/2))^7) + (384*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(625*(3^(1/2) - (5*x + 3)^(1/2))
^2) + (1632*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(625*(3^(1/2) - (5*x + 3)^(1/2))^4) + (96*3^(1/2)*((1 - 2*x)^(1/2
) - 1)^6)/(25*(3^(1/2) - (5*x + 3)^(1/2))^6))/((32*((1 - 2*x)^(1/2) - 1)^2)/(125*(3^(1/2) - (5*x + 3)^(1/2))^2
) + (24*((1 - 2*x)^(1/2) - 1)^4)/(25*(3^(1/2) - (5*x + 3)^(1/2))^4) + (8*((1 - 2*x)^(1/2) - 1)^6)/(5*(3^(1/2)
- (5*x + 3)^(1/2))^6) + ((1 - 2*x)^(1/2) - 1)^8/(3^(1/2) - (5*x + 3)^(1/2))^8 + 16/625)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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