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

Optimal. Leaf size=72 \[ \frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}+\frac {33}{100} \sqrt {5 x+3} \sqrt {1-2 x}+\frac {363 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{100 \sqrt {10}} \]

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Rubi [A]  time = 0.01, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {50, 54, 216} \begin {gather*} \frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}+\frac {33}{100} \sqrt {5 x+3} \sqrt {1-2 x}+\frac {363 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{100 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(3/2)/Sqrt[3 + 5*x],x]

[Out]

(33*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/100 + ((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/10 + (363*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*
x]])/(100*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 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 {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx &=\frac {1}{10} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {33}{20} \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx\\ &=\frac {33}{100} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {1}{10} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {363}{200} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {33}{100} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {1}{10} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {363 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{100 \sqrt {5}}\\ &=\frac {33}{100} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {1}{10} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {363 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{100 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 64, normalized size = 0.89 \begin {gather*} \frac {10 \sqrt {5 x+3} \left (40 x^2-106 x+43\right )+363 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{1000 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[3 + 5*x]*(43 - 106*x + 40*x^2) + 363*Sqrt[-10 + 20*x]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(1000*Sqrt[
1 - 2*x])

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IntegrateAlgebraic [A]  time = 0.13, size = 93, normalized size = 1.29 \begin {gather*} \frac {121 \sqrt {1-2 x} \left (\frac {25 (1-2 x)}{5 x+3}+6\right )}{100 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^2}-\frac {363 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{100 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(121*Sqrt[1 - 2*x]*(6 + (25*(1 - 2*x))/(3 + 5*x)))/(100*Sqrt[3 + 5*x]*(2 + (5*(1 - 2*x))/(3 + 5*x))^2) - (363*
ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])/Sqrt[3 + 5*x]])/(100*Sqrt[10])

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fricas [A]  time = 0.98, size = 62, normalized size = 0.86 \begin {gather*} -\frac {1}{100} \, {\left (20 \, x - 43\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {363}{2000} \, \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((1-2*x)^(3/2)/(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

-1/100*(20*x - 43)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 363/2000*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 = 1.12, size = 86, normalized size = 1.19 \begin {gather*} -\frac {1}{1000} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {1}{50} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1000*sqrt(5)*(2*(20*x - 23)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 143*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))
) + 1/50*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))

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maple [A]  time = 0.00, size = 72, normalized size = 1.00 \begin {gather*} \frac {363 \sqrt {\left (-2 x +1\right ) \left (5 x +3\right )}\, \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{2000 \sqrt {5 x +3}\, \sqrt {-2 x +1}}+\frac {\left (-2 x +1\right )^{\frac {3}{2}} \sqrt {5 x +3}}{10}+\frac {33 \sqrt {-2 x +1}\, \sqrt {5 x +3}}{100} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(-2*x+1)^(3/2)*(5*x+3)^(1/2)+33/100*(-2*x+1)^(1/2)*(5*x+3)^(1/2)+363/2000*((-2*x+1)*(5*x+3))^(1/2)/(5*x+3
)^(1/2)/(-2*x+1)^(1/2)*10^(1/2)*arcsin(20/11*x+1/11)

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maxima [A]  time = 1.06, size = 41, normalized size = 0.57 \begin {gather*} -\frac {1}{5} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {363}{2000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {43}{100} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*sqrt(-10*x^2 - x + 3)*x - 363/2000*sqrt(10)*arcsin(-20/11*x - 1/11) + 43/100*sqrt(-10*x^2 - x + 3)

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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}}{\sqrt {5\,x+3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 2.65, size = 184, normalized size = 2.56 \begin {gather*} \begin {cases} - \frac {2 i \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{\sqrt {10 x - 5}} + \frac {77 i \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{10 \sqrt {10 x - 5}} - \frac {121 i \sqrt {x + \frac {3}{5}}}{20 \sqrt {10 x - 5}} - \frac {363 \sqrt {10} i \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{1000} & \text {for}\: \frac {10 \left |{x + \frac {3}{5}}\right |}{11} > 1 \\\frac {363 \sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{1000} + \frac {2 \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{\sqrt {5 - 10 x}} - \frac {77 \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{10 \sqrt {5 - 10 x}} + \frac {121 \sqrt {x + \frac {3}{5}}}{20 \sqrt {5 - 10 x}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*I*(x + 3/5)**(5/2)/sqrt(10*x - 5) + 77*I*(x + 3/5)**(3/2)/(10*sqrt(10*x - 5)) - 121*I*sqrt(x + 3
/5)/(20*sqrt(10*x - 5)) - 363*sqrt(10)*I*acosh(sqrt(110)*sqrt(x + 3/5)/11)/1000, 10*Abs(x + 3/5)/11 > 1), (363
*sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/1000 + 2*(x + 3/5)**(5/2)/sqrt(5 - 10*x) - 77*(x + 3/5)**(3/2)/(10*
sqrt(5 - 10*x)) + 121*sqrt(x + 3/5)/(20*sqrt(5 - 10*x)), True))

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