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

Optimal. Leaf size=94 \[ -\frac {9}{100} \sqrt {5 x+3} (1-2 x)^{3/2}-\frac {2 (1-2 x)^{3/2}}{275 \sqrt {5 x+3}}+\frac {317 \sqrt {5 x+3} \sqrt {1-2 x}}{2200}+\frac {317 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{200 \sqrt {10}} \]

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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, 80, 50, 54, 216} \begin {gather*} -\frac {9}{100} \sqrt {5 x+3} (1-2 x)^{3/2}-\frac {2 (1-2 x)^{3/2}}{275 \sqrt {5 x+3}}+\frac {317 \sqrt {5 x+3} \sqrt {1-2 x}}{2200}+\frac {317 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{200 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(3/2))/(275*Sqrt[3 + 5*x]) + (317*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/2200 - (9*(1 - 2*x)^(3/2)*Sqrt[3
+ 5*x])/100 + (317*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(200*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 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 {\sqrt {1-2 x} (2+3 x)^2}{(3+5 x)^{3/2}} \, dx &=-\frac {2 (1-2 x)^{3/2}}{275 \sqrt {3+5 x}}+\frac {2}{275} \int \frac {\sqrt {1-2 x} \left (\frac {359}{2}+\frac {495 x}{2}\right )}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2}}{275 \sqrt {3+5 x}}-\frac {9}{100} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {317}{440} \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2}}{275 \sqrt {3+5 x}}+\frac {317 \sqrt {1-2 x} \sqrt {3+5 x}}{2200}-\frac {9}{100} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {317}{400} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2}}{275 \sqrt {3+5 x}}+\frac {317 \sqrt {1-2 x} \sqrt {3+5 x}}{2200}-\frac {9}{100} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {317 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{200 \sqrt {5}}\\ &=-\frac {2 (1-2 x)^{3/2}}{275 \sqrt {3+5 x}}+\frac {317 \sqrt {1-2 x} \sqrt {3+5 x}}{2200}-\frac {9}{100} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {317 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{200 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 78, normalized size = 0.83 \begin {gather*} \frac {10 \left (-360 x^3-150 x^2+103 x+31\right )+317 \sqrt {5 x+3} \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{2000 \sqrt {1-2 x} \sqrt {5 x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10*(31 + 103*x - 150*x^2 - 360*x^3) + 317*Sqrt[3 + 5*x]*Sqrt[-10 + 20*x]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/
(2000*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])

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IntegrateAlgebraic [A]  time = 0.15, size = 109, normalized size = 1.16 \begin {gather*} -\frac {\sqrt {1-2 x} \left (\frac {80 (1-2 x)^2}{(5 x+3)^2}+\frac {625 (1-2 x)}{5 x+3}-634\right )}{200 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^2}-\frac {317 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{200 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/200*(Sqrt[1 - 2*x]*(-634 + (80*(1 - 2*x)^2)/(3 + 5*x)^2 + (625*(1 - 2*x))/(3 + 5*x)))/(Sqrt[3 + 5*x]*(2 + (
5*(1 - 2*x))/(3 + 5*x))^2) - (317*ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])/Sqrt[3 + 5*x]])/(200*Sqrt[10])

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fricas [A]  time = 1.30, size = 81, normalized size = 0.86 \begin {gather*} -\frac {317 \, \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 (180 \, x^{2} + 165 \, x + 31\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{4000 \, {\left (5 \, x + 3\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)^(3/2),x, algorithm="fricas")

[Out]

-1/4000*(317*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*(180*x^2 + 165*x + 31)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(5*x + 3)

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giac [A]  time = 1.36, size = 111, normalized size = 1.18 \begin {gather*} \frac {3}{5000} \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} - 17 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {317}{2000} \, \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 )}}{1250 \, \sqrt {5 \, x + 3}} + \frac {2 \, \sqrt {10} \sqrt {5 \, x + 3}}{625 \, {\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*(1-2*x)^(1/2)/(3+5*x)^(3/2),x, algorithm="giac")

[Out]

3/5000*(12*sqrt(5)*(5*x + 3) - 17*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) + 317/2000*sqrt(10)*arcsin(1/11*sqrt(
22)*sqrt(5*x + 3)) - 1/1250*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 2/625*sqrt(10)*sqrt(
5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))

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maple [A]  time = 0.01, size = 99, normalized size = 1.05 \begin {gather*} \frac {\left (3600 \sqrt {-10 x^{2}-x +3}\, x^{2}+1585 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+3300 \sqrt {-10 x^{2}-x +3}\, x +951 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+620 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{4000 \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*(-2*x+1)^(1/2)/(5*x+3)^(3/2),x)

[Out]

1/4000*(1585*10^(1/2)*x*arcsin(20/11*x+1/11)+3600*(-10*x^2-x+3)^(1/2)*x^2+951*10^(1/2)*arcsin(20/11*x+1/11)+33
00*(-10*x^2-x+3)^(1/2)*x+620*(-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 = 1.29, size = 65, normalized size = 0.69 \begin {gather*} \frac {317}{4000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {9}{50} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {57}{1000} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {2 \, \sqrt {-10 \, x^{2} - x + 3}}{125 \, {\left (5 \, x + 3\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)^(3/2),x, algorithm="maxima")

[Out]

317/4000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 9/50*sqrt(-10*x^2 - x + 3)*x + 57/1000*sqrt(-10*x^2 - x + 3)
- 2/125*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 {\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^2}{{\left (5\,x+3\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(((1 - 2*x)^(1/2)*(3*x + 2)^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 {\sqrt {1 - 2 x} \left (3 x + 2\right )^{2}}{\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*(1-2*x)**(1/2)/(3+5*x)**(3/2),x)

[Out]

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

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