3.2330 \(\int (1-2 x)^{3/2} \sqrt {3+5 x} \, dx\)

Optimal. Leaf size=94 \[ -\frac {1}{6} \sqrt {5 x+3} (1-2 x)^{5/2}+\frac {11}{120} \sqrt {5 x+3} (1-2 x)^{3/2}+\frac {121}{400} \sqrt {5 x+3} \sqrt {1-2 x}+\frac {1331 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{400 \sqrt {10}} \]

[Out]

1331/4000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+11/120*(1-2*x)^(3/2)*(3+5*x)^(1/2)-1/6*(1-2*x)^(5/2)*(3
+5*x)^(1/2)+121/400*(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 = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {50, 54, 216} \[ -\frac {1}{6} \sqrt {5 x+3} (1-2 x)^{5/2}+\frac {11}{120} \sqrt {5 x+3} (1-2 x)^{3/2}+\frac {121}{400} \sqrt {5 x+3} \sqrt {1-2 x}+\frac {1331 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{400 \sqrt {10}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(3/2)*Sqrt[3 + 5*x],x]

[Out]

(121*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/400 + (11*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/120 - ((1 - 2*x)^(5/2)*Sqrt[3 + 5*x
])/6 + (1331*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(400*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 (1-2 x)^{3/2} \sqrt {3+5 x} \, dx &=-\frac {1}{6} (1-2 x)^{5/2} \sqrt {3+5 x}+\frac {11}{12} \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx\\ &=\frac {11}{120} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1}{6} (1-2 x)^{5/2} \sqrt {3+5 x}+\frac {121}{80} \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx\\ &=\frac {121}{400} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {11}{120} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1}{6} (1-2 x)^{5/2} \sqrt {3+5 x}+\frac {1331}{800} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {121}{400} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {11}{120} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1}{6} (1-2 x)^{5/2} \sqrt {3+5 x}+\frac {1331 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{400 \sqrt {5}}\\ &=\frac {121}{400} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {11}{120} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1}{6} (1-2 x)^{5/2} \sqrt {3+5 x}+\frac {1331 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{400 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 69, normalized size = 0.73 \[ \frac {10 \sqrt {5 x+3} \left (1600 x^3-1960 x^2+34 x+273\right )+3993 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{12000 \sqrt {1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[3 + 5*x]*(273 + 34*x - 1960*x^2 + 1600*x^3) + 3993*Sqrt[-10 + 20*x]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]
])/(12000*Sqrt[1 - 2*x])

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fricas [A]  time = 0.97, size = 67, normalized size = 0.71 \[ -\frac {1}{1200} \, {\left (800 \, x^{2} - 580 \, x - 273\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {1331}{8000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]

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]

-1/1200*(800*x^2 - 580*x - 273)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 1331/8000*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 [B]  time = 1.10, size = 140, normalized size = 1.49 \[ -\frac {1}{12000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {1}{2000} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {3}{50} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]

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]

-1/12000*sqrt(5)*(2*(4*(40*x - 59)*(5*x + 3) + 1293)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 4785*sqrt(2)*arcsin(1/11*
sqrt(22)*sqrt(5*x + 3))) - 1/2000*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))) + 3/50*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 2*sqrt(5*x + 3)*s
qrt(-10*x + 5))

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maple [A]  time = 0.00, size = 88, normalized size = 0.94 \[ \frac {1331 \sqrt {\left (-2 x +1\right ) \left (5 x +3\right )}\, \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{8000 \sqrt {5 x +3}\, \sqrt {-2 x +1}}+\frac {\left (-2 x +1\right )^{\frac {3}{2}} \left (5 x +3\right )^{\frac {3}{2}}}{15}+\frac {11 \left (5 x +3\right )^{\frac {3}{2}} \sqrt {-2 x +1}}{100}-\frac {121 \sqrt {-2 x +1}\, \sqrt {5 x +3}}{400} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(-2*x+1)^(3/2)*(5*x+3)^(3/2)+11/100*(5*x+3)^(3/2)*(-2*x+1)^(1/2)-121/400*(-2*x+1)^(1/2)*(5*x+3)^(1/2)+133
1/8000*((-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 = 1.31, size = 55, normalized size = 0.59 \[ \frac {1}{15} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {11}{20} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {1331}{8000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {11}{400} \, \sqrt {-10 \, x^{2} - x + 3} \]

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]

1/15*(-10*x^2 - x + 3)^(3/2) + 11/20*sqrt(-10*x^2 - x + 3)*x - 1331/8000*sqrt(10)*arcsin(-20/11*x - 1/11) + 11
/400*sqrt(-10*x^2 - x + 3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (1-2\,x\right )}^{3/2}\,\sqrt {5\,x+3} \,d x \]

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 = 4.37, size = 230, normalized size = 2.45 \[ \begin {cases} - \frac {20 i \left (x + \frac {3}{5}\right )^{\frac {7}{2}}}{3 \sqrt {10 x - 5}} + \frac {121 i \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{6 \sqrt {10 x - 5}} - \frac {2057 i \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{120 \sqrt {10 x - 5}} + \frac {1331 i \sqrt {x + \frac {3}{5}}}{400 \sqrt {10 x - 5}} - \frac {1331 \sqrt {10} i \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{4000} & \text {for}\: \frac {10 \left |{x + \frac {3}{5}}\right |}{11} > 1 \\\frac {1331 \sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{4000} + \frac {20 \left (x + \frac {3}{5}\right )^{\frac {7}{2}}}{3 \sqrt {5 - 10 x}} - \frac {121 \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{6 \sqrt {5 - 10 x}} + \frac {2057 \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{120 \sqrt {5 - 10 x}} - \frac {1331 \sqrt {x + \frac {3}{5}}}{400 \sqrt {5 - 10 x}} & \text {otherwise} \end {cases} \]

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((-20*I*(x + 3/5)**(7/2)/(3*sqrt(10*x - 5)) + 121*I*(x + 3/5)**(5/2)/(6*sqrt(10*x - 5)) - 2057*I*(x +
3/5)**(3/2)/(120*sqrt(10*x - 5)) + 1331*I*sqrt(x + 3/5)/(400*sqrt(10*x - 5)) - 1331*sqrt(10)*I*acosh(sqrt(110
)*sqrt(x + 3/5)/11)/4000, 10*Abs(x + 3/5)/11 > 1), (1331*sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/4000 + 20*(
x + 3/5)**(7/2)/(3*sqrt(5 - 10*x)) - 121*(x + 3/5)**(5/2)/(6*sqrt(5 - 10*x)) + 2057*(x + 3/5)**(3/2)/(120*sqrt
(5 - 10*x)) - 1331*sqrt(x + 3/5)/(400*sqrt(5 - 10*x)), True))

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