3.2481 \(\int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=96 \[ -\frac{1}{6} \sqrt{1-2 x} (5 x+3)^{5/2}-\frac{55}{48} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{605}{64} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{1331}{64} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

[Out]

(-605*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/64 - (55*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/48 - (Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)
)/6 + (1331*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/64

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Rubi [A]  time = 0.0214017, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {50, 54, 216} \[ -\frac{1}{6} \sqrt{1-2 x} (5 x+3)^{5/2}-\frac{55}{48} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{605}{64} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{1331}{64} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^(5/2)/Sqrt[1 - 2*x],x]

[Out]

(-605*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/64 - (55*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/48 - (Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)
)/6 + (1331*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/64

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)^{5/2}}{\sqrt{1-2 x}} \, dx &=-\frac{1}{6} \sqrt{1-2 x} (3+5 x)^{5/2}+\frac{55}{12} \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{55}{48} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{1}{6} \sqrt{1-2 x} (3+5 x)^{5/2}+\frac{605}{32} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{605}{64} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{55}{48} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{1}{6} \sqrt{1-2 x} (3+5 x)^{5/2}+\frac{6655}{128} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{605}{64} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{55}{48} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{1}{6} \sqrt{1-2 x} (3+5 x)^{5/2}+\frac{1}{64} \left (1331 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )\\ &=-\frac{605}{64} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{55}{48} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{1}{6} \sqrt{1-2 x} (3+5 x)^{5/2}+\frac{1331}{64} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0273195, size = 60, normalized size = 0.62 \[ \frac{1}{384} \left (-2 \sqrt{1-2 x} \sqrt{5 x+3} \left (800 x^2+2060 x+2763\right )-3993 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2763 + 2060*x + 800*x^2) - 3993*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/38
4

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Maple [A]  time = 0.004, size = 88, normalized size = 0.9 \begin{align*} -{\frac{1}{6} \left ( 3+5\,x \right ) ^{{\frac{5}{2}}}\sqrt{1-2\,x}}-{\frac{55}{48} \left ( 3+5\,x \right ) ^{{\frac{3}{2}}}\sqrt{1-2\,x}}-{\frac{605}{64}\sqrt{1-2\,x}\sqrt{3+5\,x}}+{\frac{1331\,\sqrt{10}}{256}\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)^(5/2)/(1-2*x)^(1/2),x)

[Out]

-1/6*(3+5*x)^(5/2)*(1-2*x)^(1/2)-55/48*(3+5*x)^(3/2)*(1-2*x)^(1/2)-605/64*(1-2*x)^(1/2)*(3+5*x)^(1/2)+1331/256
*((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 = 2.29382, size = 78, normalized size = 0.81 \begin{align*} -\frac{25}{6} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} - \frac{515}{48} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{1331}{256} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) - \frac{921}{64} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/6*sqrt(-10*x^2 - x + 3)*x^2 - 515/48*sqrt(-10*x^2 - x + 3)*x - 1331/256*sqrt(10)*arcsin(-20/11*x - 1/11) -
921/64*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.77028, size = 238, normalized size = 2.48 \begin{align*} -\frac{1}{192} \,{\left (800 \, x^{2} + 2060 \, x + 2763\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{1331}{256} \, \sqrt{5} \sqrt{2} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\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)^(5/2)/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/192*(800*x^2 + 2060*x + 2763)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 1331/256*sqrt(5)*sqrt(2)*arctan(1/20*sqrt(5)*s
qrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))

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Sympy [A]  time = 12.2168, size = 230, normalized size = 2.4 \begin{align*} \begin{cases} - \frac{125 i \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{3 \sqrt{10 x - 5}} - \frac{275 i \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{24 \sqrt{10 x - 5}} - \frac{3025 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{96 \sqrt{10 x - 5}} + \frac{6655 i \sqrt{x + \frac{3}{5}}}{64 \sqrt{10 x - 5}} - \frac{1331 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{128} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{1331 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{128} + \frac{125 \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{3 \sqrt{5 - 10 x}} + \frac{275 \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{24 \sqrt{5 - 10 x}} + \frac{3025 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{96 \sqrt{5 - 10 x}} - \frac{6655 \sqrt{x + \frac{3}{5}}}{64 \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)**(5/2)/(1-2*x)**(1/2),x)

[Out]

Piecewise((-125*I*(x + 3/5)**(7/2)/(3*sqrt(10*x - 5)) - 275*I*(x + 3/5)**(5/2)/(24*sqrt(10*x - 5)) - 3025*I*(x
+ 3/5)**(3/2)/(96*sqrt(10*x - 5)) + 6655*I*sqrt(x + 3/5)/(64*sqrt(10*x - 5)) - 1331*sqrt(10)*I*acosh(sqrt(110
)*sqrt(x + 3/5)/11)/128, 10*Abs(x + 3/5)/11 > 1), (1331*sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/128 + 125*(x
+ 3/5)**(7/2)/(3*sqrt(5 - 10*x)) + 275*(x + 3/5)**(5/2)/(24*sqrt(5 - 10*x)) + 3025*(x + 3/5)**(3/2)/(96*sqrt(
5 - 10*x)) - 6655*sqrt(x + 3/5)/(64*sqrt(5 - 10*x)), True))

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Giac [A]  time = 2.18854, size = 73, normalized size = 0.76 \begin{align*} -\frac{1}{1920} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x + 79\right )}{\left (5 \, x + 3\right )} + 1815\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 19965 \, \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)^(5/2)/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

-1/1920*sqrt(5)*(2*(4*(40*x + 79)*(5*x + 3) + 1815)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 19965*sqrt(2)*arcsin(1/11*
sqrt(22)*sqrt(5*x + 3)))