3.2340 \(\int (1-2 x)^{3/2} (3+5 x)^{3/2} \, dx\)

Optimal. Leaf size=116 \[ -\frac{1}{8} (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac{11}{32} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{121}{640} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{3993 \sqrt{5 x+3} \sqrt{1-2 x}}{6400}+\frac{43923 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{6400 \sqrt{10}} \]

[Out]

(3993*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/6400 + (121*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/640 - (11*(1 - 2*x)^(5/2)*Sqrt[3
+ 5*x])/32 - ((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/8 + (43923*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(6400*Sqrt[10])

________________________________________________________________________________________

Rubi [A]  time = 0.0294395, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {50, 54, 216} \[ -\frac{1}{8} (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac{11}{32} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{121}{640} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{3993 \sqrt{5 x+3} \sqrt{1-2 x}}{6400}+\frac{43923 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{6400 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3993*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/6400 + (121*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/640 - (11*(1 - 2*x)^(5/2)*Sqrt[3
+ 5*x])/32 - ((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/8 + (43923*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(6400*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} (3+5 x)^{3/2} \, dx &=-\frac{1}{8} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac{33}{16} \int (1-2 x)^{3/2} \sqrt{3+5 x} \, dx\\ &=-\frac{11}{32} (1-2 x)^{5/2} \sqrt{3+5 x}-\frac{1}{8} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac{121}{64} \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx\\ &=\frac{121}{640} (1-2 x)^{3/2} \sqrt{3+5 x}-\frac{11}{32} (1-2 x)^{5/2} \sqrt{3+5 x}-\frac{1}{8} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac{3993 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{1280}\\ &=\frac{3993 \sqrt{1-2 x} \sqrt{3+5 x}}{6400}+\frac{121}{640} (1-2 x)^{3/2} \sqrt{3+5 x}-\frac{11}{32} (1-2 x)^{5/2} \sqrt{3+5 x}-\frac{1}{8} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac{43923 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{12800}\\ &=\frac{3993 \sqrt{1-2 x} \sqrt{3+5 x}}{6400}+\frac{121}{640} (1-2 x)^{3/2} \sqrt{3+5 x}-\frac{11}{32} (1-2 x)^{5/2} \sqrt{3+5 x}-\frac{1}{8} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac{43923 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{6400 \sqrt{5}}\\ &=\frac{3993 \sqrt{1-2 x} \sqrt{3+5 x}}{6400}+\frac{121}{640} (1-2 x)^{3/2} \sqrt{3+5 x}-\frac{11}{32} (1-2 x)^{5/2} \sqrt{3+5 x}-\frac{1}{8} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac{43923 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{6400 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0621182, size = 74, normalized size = 0.64 \[ \frac{10 \sqrt{5 x+3} \left (32000 x^4-11200 x^3-26360 x^2+10774 x+603\right )-43923 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{64000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[3 + 5*x]*(603 + 10774*x - 26360*x^2 - 11200*x^3 + 32000*x^4) - 43923*Sqrt[10 - 20*x]*ArcSin[Sqrt[5/11
]*Sqrt[1 - 2*x]])/(64000*Sqrt[1 - 2*x])

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 104, normalized size = 0.9 \begin{align*}{\frac{1}{20} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}} \left ( 3+5\,x \right ) ^{{\frac{5}{2}}}}+{\frac{11}{200} \left ( 3+5\,x \right ) ^{{\frac{5}{2}}}\sqrt{1-2\,x}}-{\frac{121}{1600} \left ( 3+5\,x \right ) ^{{\frac{3}{2}}}\sqrt{1-2\,x}}-{\frac{3993}{6400}\sqrt{1-2\,x}\sqrt{3+5\,x}}+{\frac{43923\,\sqrt{10}}{128000}\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((1-2*x)^(3/2)*(3+5*x)^(3/2),x)

[Out]

1/20*(1-2*x)^(3/2)*(3+5*x)^(5/2)+11/200*(3+5*x)^(5/2)*(1-2*x)^(1/2)-121/1600*(3+5*x)^(3/2)*(1-2*x)^(1/2)-3993/
6400*(1-2*x)^(1/2)*(3+5*x)^(1/2)+43923/128000*((1-2*x)*(3+5*x))^(1/2)/(3+5*x)^(1/2)/(1-2*x)^(1/2)*10^(1/2)*arc
sin(20/11*x+1/11)

________________________________________________________________________________________

Maxima [A]  time = 2.14259, size = 95, normalized size = 0.82 \begin{align*} \frac{1}{4} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{1}{80} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{363}{320} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{43923}{128000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{363}{6400} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(-10*x^2 - x + 3)^(3/2)*x + 1/80*(-10*x^2 - x + 3)^(3/2) + 363/320*sqrt(-10*x^2 - x + 3)*x - 43923/128000*
sqrt(10)*arcsin(-20/11*x - 1/11) + 363/6400*sqrt(-10*x^2 - x + 3)

________________________________________________________________________________________

Fricas [A]  time = 1.5143, size = 243, normalized size = 2.09 \begin{align*} -\frac{1}{6400} \,{\left (16000 \, x^{3} + 2400 \, x^{2} - 11980 \, x - 603\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{43923}{128000} \, \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((1-2*x)^(3/2)*(3+5*x)^(3/2),x, algorithm="fricas")

[Out]

-1/6400*(16000*x^3 + 2400*x^2 - 11980*x - 603)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 43923/128000*sqrt(10)*arctan(1/2
0*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))

________________________________________________________________________________________

Sympy [A]  time = 8.85933, size = 269, normalized size = 2.32 \begin{align*} \begin{cases} - \frac{25 i \left (x + \frac{3}{5}\right )^{\frac{9}{2}}}{\sqrt{10 x - 5}} + \frac{275 i \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{4 \sqrt{10 x - 5}} - \frac{1573 i \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{32 \sqrt{10 x - 5}} - \frac{1331 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{640 \sqrt{10 x - 5}} + \frac{43923 i \sqrt{x + \frac{3}{5}}}{6400 \sqrt{10 x - 5}} - \frac{43923 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{64000} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{43923 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{64000} + \frac{25 \left (x + \frac{3}{5}\right )^{\frac{9}{2}}}{\sqrt{5 - 10 x}} - \frac{275 \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{4 \sqrt{5 - 10 x}} + \frac{1573 \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{32 \sqrt{5 - 10 x}} + \frac{1331 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{640 \sqrt{5 - 10 x}} - \frac{43923 \sqrt{x + \frac{3}{5}}}{6400 \sqrt{5 - 10 x}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-25*I*(x + 3/5)**(9/2)/sqrt(10*x - 5) + 275*I*(x + 3/5)**(7/2)/(4*sqrt(10*x - 5)) - 1573*I*(x + 3/5
)**(5/2)/(32*sqrt(10*x - 5)) - 1331*I*(x + 3/5)**(3/2)/(640*sqrt(10*x - 5)) + 43923*I*sqrt(x + 3/5)/(6400*sqrt
(10*x - 5)) - 43923*sqrt(10)*I*acosh(sqrt(110)*sqrt(x + 3/5)/11)/64000, 10*Abs(x + 3/5)/11 > 1), (43923*sqrt(1
0)*asin(sqrt(110)*sqrt(x + 3/5)/11)/64000 + 25*(x + 3/5)**(9/2)/sqrt(5 - 10*x) - 275*(x + 3/5)**(7/2)/(4*sqrt(
5 - 10*x)) + 1573*(x + 3/5)**(5/2)/(32*sqrt(5 - 10*x)) + 1331*(x + 3/5)**(3/2)/(640*sqrt(5 - 10*x)) - 43923*sq
rt(x + 3/5)/(6400*sqrt(5 - 10*x)), True))

________________________________________________________________________________________

Giac [A]  time = 1.21459, size = 220, normalized size = 1.9 \begin{align*} -\frac{1}{192000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{1}{24000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{3}{400} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \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((1-2*x)^(3/2)*(3+5*x)^(3/2),x, algorithm="giac")

[Out]

-1/192000*sqrt(5)*(2*(4*(8*(60*x - 71)*(5*x + 3) + 2179)*(5*x + 3) - 4125)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 453
75*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 1/24000*sqrt(5)*(2*(4*(40*x - 23)*(5*x + 3) + 33)*sqrt(5*x +
3)*sqrt(-10*x + 5) - 363*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 3/400*sqrt(5)*(2*(20*x + 1)*sqrt(5*x
+ 3)*sqrt(-10*x + 5) + 121*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))