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

Optimal. Leaf size=72 \[ -\frac {9}{50} \sqrt {1-2 x} \sqrt {5 x+3}-\frac {2 \sqrt {1-2 x}}{275 \sqrt {5 x+3}}+\frac {123 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{50 \sqrt {10}} \]

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Rubi [A]  time = 0.02, antiderivative size = 72, 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 = {89, 80, 54, 216} \begin {gather*} -\frac {9}{50} \sqrt {1-2 x} \sqrt {5 x+3}-\frac {2 \sqrt {1-2 x}}{275 \sqrt {5 x+3}}+\frac {123 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{50 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x])/(275*Sqrt[3 + 5*x]) - (9*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/50 + (123*ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]])/(50*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 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 {(2+3 x)^2}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx &=-\frac {2 \sqrt {1-2 x}}{275 \sqrt {3+5 x}}+\frac {2}{275} \int \frac {\frac {363}{2}+\frac {495 x}{2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x}}{275 \sqrt {3+5 x}}-\frac {9}{50} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {123}{100} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x}}{275 \sqrt {3+5 x}}-\frac {9}{50} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {123 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{50 \sqrt {5}}\\ &=-\frac {2 \sqrt {1-2 x}}{275 \sqrt {3+5 x}}-\frac {9}{50} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {123 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{50 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 77, normalized size = 1.07 \begin {gather*} -\frac {\sqrt {1-2 x} \left (10 \sqrt {2 x-1} (495 x+301)+1353 \sqrt {50 x+30} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )\right )}{5500 \sqrt {2 x-1} \sqrt {5 x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5500*(Sqrt[1 - 2*x]*(10*Sqrt[-1 + 2*x]*(301 + 495*x) + 1353*Sqrt[30 + 50*x]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*
x]]))/(Sqrt[-1 + 2*x]*Sqrt[3 + 5*x])

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IntegrateAlgebraic [A]  time = 0.11, size = 93, normalized size = 1.29 \begin {gather*} -\frac {\sqrt {1-2 x} \left (\frac {20 (1-2 x)}{5 x+3}+1097\right )}{550 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )}-\frac {123 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{50 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/550*(Sqrt[1 - 2*x]*(1097 + (20*(1 - 2*x))/(3 + 5*x)))/(Sqrt[3 + 5*x]*(2 + (5*(1 - 2*x))/(3 + 5*x))) - (123*
ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])/Sqrt[3 + 5*x]])/(50*Sqrt[10])

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fricas [A]  time = 0.79, size = 76, normalized size = 1.06 \begin {gather*} -\frac {1353 \, \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 (495 \, x + 301\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{11000 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/11000*(1353*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*(495*x + 301)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(5*x + 3)

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giac [A]  time = 1.23, size = 98, normalized size = 1.36 \begin {gather*} -\frac {9}{250} \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {123}{500} \, \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 )}}{2750 \, \sqrt {5 \, x + 3}} + \frac {2 \, \sqrt {10} \sqrt {5 \, x + 3}}{1375 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/250*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 123/500*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/2750*s
qrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 2/1375*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x
+ 5) - sqrt(22))

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maple [A]  time = 0.01, size = 82, normalized size = 1.14 \begin {gather*} \frac {\left (6765 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-9900 \sqrt {-10 x^{2}-x +3}\, x +4059 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-6020 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{11000 \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11000*(6765*10^(1/2)*x*arcsin(20/11*x+1/11)+4059*10^(1/2)*arcsin(20/11*x+1/11)-9900*(-10*x^2-x+3)^(1/2)*x-60
20*(-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 [A]  time = 1.37, size = 50, normalized size = 0.69 \begin {gather*} \frac {123}{1000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {9}{50} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {2 \, \sqrt {-10 \, x^{2} - x + 3}}{275 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

123/1000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 9/50*sqrt(-10*x^2 - x + 3) - 2/275*sqrt(-10*x^2 - x + 3)/(5*
x + 3)

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

[Out]

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

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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} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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