3.2471 \(\int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=72 \[ -\frac{1}{4} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{33}{16} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{363 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{16 \sqrt{10}} \]

[Out]

(-33*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/16 - (Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/4 + (363*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x
]])/(16*Sqrt[10])

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Rubi [A]  time = 0.0165265, antiderivative size = 72, normalized size of antiderivative = 1., 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} \[ -\frac{1}{4} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{33}{16} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{363 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{16 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^(3/2)/Sqrt[1 - 2*x],x]

[Out]

(-33*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/16 - (Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/4 + (363*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x
]])/(16*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{(3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx &=-\frac{1}{4} \sqrt{1-2 x} (3+5 x)^{3/2}+\frac{33}{8} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{33}{16} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{1}{4} \sqrt{1-2 x} (3+5 x)^{3/2}+\frac{363}{32} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{33}{16} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{1}{4} \sqrt{1-2 x} (3+5 x)^{3/2}+\frac{363 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{16 \sqrt{5}}\\ &=-\frac{33}{16} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{1}{4} \sqrt{1-2 x} (3+5 x)^{3/2}+\frac{363 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{16 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0208689, size = 55, normalized size = 0.76 \[ \frac{1}{160} \left (-50 \sqrt{1-2 x} \sqrt{5 x+3} (4 x+9)-363 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-50*Sqrt[1 - 2*x]*(9 + 4*x)*Sqrt[3 + 5*x] - 363*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/160

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Maple [A]  time = 0.003, size = 72, normalized size = 1. \begin{align*} -{\frac{1}{4} \left ( 3+5\,x \right ) ^{{\frac{3}{2}}}\sqrt{1-2\,x}}-{\frac{33}{16}\sqrt{1-2\,x}\sqrt{3+5\,x}}+{\frac{363\,\sqrt{10}}{320}\sqrt{ \left ( 1-2\,x \right ) \left ( 3+5\,x \right ) }\arcsin \left ({\frac{20\,x}{11}}+{\frac{1}{11}} \right ){\frac{1}{\sqrt{1-2\,x}}}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(3+5*x)^(3/2)*(1-2*x)^(1/2)-33/16*(1-2*x)^(1/2)*(3+5*x)^(1/2)+363/320*((1-2*x)*(3+5*x))^(1/2)/(3+5*x)^(1/
2)/(1-2*x)^(1/2)*10^(1/2)*arcsin(20/11*x+1/11)

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Maxima [A]  time = 3.25601, size = 55, normalized size = 0.76 \begin{align*} -\frac{5}{4} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{363}{320} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) - \frac{45}{16} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/4*sqrt(-10*x^2 - x + 3)*x - 363/320*sqrt(10)*arcsin(-20/11*x - 1/11) - 45/16*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.76167, size = 194, normalized size = 2.69 \begin{align*} -\frac{5}{16} \, \sqrt{5 \, x + 3}{\left (4 \, x + 9\right )} \sqrt{-2 \, x + 1} - \frac{363}{320} \, \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/16*sqrt(5*x + 3)*(4*x + 9)*sqrt(-2*x + 1) - 363/320*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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Sympy [A]  time = 3.60871, size = 187, normalized size = 2.6 \begin{align*} \begin{cases} - \frac{25 i \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{2 \sqrt{10 x - 5}} - \frac{55 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{8 \sqrt{10 x - 5}} + \frac{363 i \sqrt{x + \frac{3}{5}}}{16 \sqrt{10 x - 5}} - \frac{363 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{160} & \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 )}}{160} + \frac{25 \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{2 \sqrt{5 - 10 x}} + \frac{55 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{8 \sqrt{5 - 10 x}} - \frac{363 \sqrt{x + \frac{3}{5}}}{16 \sqrt{5 - 10 x}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-25*I*(x + 3/5)**(5/2)/(2*sqrt(10*x - 5)) - 55*I*(x + 3/5)**(3/2)/(8*sqrt(10*x - 5)) + 363*I*sqrt(x
+ 3/5)/(16*sqrt(10*x - 5)) - 363*sqrt(10)*I*acosh(sqrt(110)*sqrt(x + 3/5)/11)/160, 10*Abs(x + 3/5)/11 > 1), (
363*sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/160 + 25*(x + 3/5)**(5/2)/(2*sqrt(5 - 10*x)) + 55*(x + 3/5)**(3/
2)/(8*sqrt(5 - 10*x)) - 363*sqrt(x + 3/5)/(16*sqrt(5 - 10*x)), True))

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Giac [A]  time = 2.52228, size = 61, normalized size = 0.85 \begin{align*} -\frac{1}{160} \, \sqrt{5}{\left (10 \, \sqrt{5 \, x + 3}{\left (4 \, x + 9\right )} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/160*sqrt(5)*(10*sqrt(5*x + 3)*(4*x + 9)*sqrt(-10*x + 5) - 363*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))