3.24.73 \(\int \frac {(1-2 x)^{3/2} (2+3 x)}{(3+5 x)^{3/2}} \, dx\) [2373]

Optimal. Leaf size=94 \[ -\frac {2 (1-2 x)^{5/2}}{55 \sqrt {3+5 x}}+\frac {3}{20} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {1}{22} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {33 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{20 \sqrt {10}} \]

[Out]

33/200*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-2/55*(1-2*x)^(5/2)/(3+5*x)^(1/2)+1/22*(1-2*x)^(3/2)*(3+5*x
)^(1/2)+3/20*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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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 = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {79, 52, 56, 222} \begin {gather*} \frac {33 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{20 \sqrt {10}}-\frac {2 (1-2 x)^{5/2}}{55 \sqrt {5 x+3}}+\frac {1}{22} \sqrt {5 x+3} (1-2 x)^{3/2}+\frac {3}{20} \sqrt {5 x+3} \sqrt {1-2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(5/2))/(55*Sqrt[3 + 5*x]) + (3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/20 + ((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])
/22 + (33*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(20*Sqrt[10])

Rule 52

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 56

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 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 222

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 {(1-2 x)^{3/2} (2+3 x)}{(3+5 x)^{3/2}} \, dx &=-\frac {2 (1-2 x)^{5/2}}{55 \sqrt {3+5 x}}+\frac {5}{11} \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2}}{55 \sqrt {3+5 x}}+\frac {1}{22} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {3}{4} \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2}}{55 \sqrt {3+5 x}}+\frac {3}{20} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {1}{22} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {33}{40} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2}}{55 \sqrt {3+5 x}}+\frac {3}{20} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {1}{22} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {33 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{20 \sqrt {5}}\\ &=-\frac {2 (1-2 x)^{5/2}}{55 \sqrt {3+5 x}}+\frac {3}{20} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {1}{22} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {33 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{20 \sqrt {10}}\\ \end {align*}

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Mathematica [A]
time = 0.27, size = 76, normalized size = 0.81 \begin {gather*} \frac {5 \sqrt {1-2 x} \left (11+17 x-12 x^2\right )-33 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {6+10 x}}{\sqrt {11}-\sqrt {5-10 x}}\right )}{100 \sqrt {3+5 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*Sqrt[1 - 2*x]*(11 + 17*x - 12*x^2) - 33*Sqrt[30 + 50*x]*ArcTan[Sqrt[6 + 10*x]/(Sqrt[11] - Sqrt[5 - 10*x])])
/(100*Sqrt[3 + 5*x])

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Maple [A]
time = 0.13, size = 99, normalized size = 1.05

method	result	size
default	\(\frac {\left (165 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -240 x^{2} \sqrt {-10 x^{2}-x +3}+99 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+340 x \sqrt {-10 x^{2}-x +3}+220 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{400 \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\)	\(99\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/400*(165*10^(1/2)*arcsin(20/11*x+1/11)*x-240*x^2*(-10*x^2-x+3)^(1/2)+99*10^(1/2)*arcsin(20/11*x+1/11)+340*x*
(-10*x^2-x+3)^(1/2)+220*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]
time = 0.52, size = 97, normalized size = 1.03 \begin {gather*} \frac {33}{400} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {99}{500} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{25 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{50 \, {\left (5 \, x + 3\right )}} - \frac {33 \, \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((1-2*x)^(3/2)*(2+3*x)/(3+5*x)^(3/2),x, algorithm="maxima")

[Out]

33/400*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 99/500*sqrt(-10*x^2 - x + 3) + 1/25*(-10*x^2 - x + 3)^(3/2)/(2
5*x^2 + 30*x + 9) + 3/50*(-10*x^2 - x + 3)^(3/2)/(5*x + 3) - 33/125*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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Fricas [A]
time = 1.03, size = 81, normalized size = 0.86 \begin {gather*} -\frac {33 \, \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 (12 \, x^{2} - 17 \, x - 11\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{400 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/400*(33*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*(12*x^2 - 17*x - 11)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(5*x + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.61, size = 111, normalized size = 1.18 \begin {gather*} -\frac {1}{2500} \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} - 157 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {33}{200} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {11 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{1250 \, \sqrt {5 \, x + 3}} + \frac {22 \, \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((1-2*x)^(3/2)*(2+3*x)/(3+5*x)^(3/2),x, algorithm="giac")

[Out]

-1/2500*(12*sqrt(5)*(5*x + 3) - 157*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) + 33/200*sqrt(10)*arcsin(1/11*sqrt(
22)*sqrt(5*x + 3)) - 11/1250*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 22/625*sqrt(10)*sqr
t(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))

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

[Out]

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

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