3.2402 \(\int (1-2 x)^{5/2} (3+5 x)^{3/2} \, dx\)

Optimal. Leaf size=138 \[ -\frac{1}{10} (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac{33}{160} \sqrt{5 x+3} (1-2 x)^{7/2}+\frac{121 \sqrt{5 x+3} (1-2 x)^{5/2}}{1600}+\frac{1331 \sqrt{5 x+3} (1-2 x)^{3/2}}{6400}+\frac{43923 \sqrt{5 x+3} \sqrt{1-2 x}}{64000}+\frac{483153 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{64000 \sqrt{10}} \]

[Out]

(43923*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/64000 + (1331*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/6400 + (121*(1 - 2*x)^(5/2)*S
qrt[3 + 5*x])/1600 - (33*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/160 - ((1 - 2*x)^(7/2)*(3 + 5*x)^(3/2))/10 + (483153*A
rcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(64000*Sqrt[10])

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Rubi [A]  time = 0.0389041, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {50, 54, 216} \[ -\frac{1}{10} (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac{33}{160} \sqrt{5 x+3} (1-2 x)^{7/2}+\frac{121 \sqrt{5 x+3} (1-2 x)^{5/2}}{1600}+\frac{1331 \sqrt{5 x+3} (1-2 x)^{3/2}}{6400}+\frac{43923 \sqrt{5 x+3} \sqrt{1-2 x}}{64000}+\frac{483153 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{64000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(43923*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/64000 + (1331*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/6400 + (121*(1 - 2*x)^(5/2)*S
qrt[3 + 5*x])/1600 - (33*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/160 - ((1 - 2*x)^(7/2)*(3 + 5*x)^(3/2))/10 + (483153*A
rcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(64000*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)^{5/2} (3+5 x)^{3/2} \, dx &=-\frac{1}{10} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac{33}{20} \int (1-2 x)^{5/2} \sqrt{3+5 x} \, dx\\ &=-\frac{33}{160} (1-2 x)^{7/2} \sqrt{3+5 x}-\frac{1}{10} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac{363}{320} \int \frac{(1-2 x)^{5/2}}{\sqrt{3+5 x}} \, dx\\ &=\frac{121 (1-2 x)^{5/2} \sqrt{3+5 x}}{1600}-\frac{33}{160} (1-2 x)^{7/2} \sqrt{3+5 x}-\frac{1}{10} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac{1331}{640} \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx\\ &=\frac{1331 (1-2 x)^{3/2} \sqrt{3+5 x}}{6400}+\frac{121 (1-2 x)^{5/2} \sqrt{3+5 x}}{1600}-\frac{33}{160} (1-2 x)^{7/2} \sqrt{3+5 x}-\frac{1}{10} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac{43923 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{12800}\\ &=\frac{43923 \sqrt{1-2 x} \sqrt{3+5 x}}{64000}+\frac{1331 (1-2 x)^{3/2} \sqrt{3+5 x}}{6400}+\frac{121 (1-2 x)^{5/2} \sqrt{3+5 x}}{1600}-\frac{33}{160} (1-2 x)^{7/2} \sqrt{3+5 x}-\frac{1}{10} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac{483153 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{128000}\\ &=\frac{43923 \sqrt{1-2 x} \sqrt{3+5 x}}{64000}+\frac{1331 (1-2 x)^{3/2} \sqrt{3+5 x}}{6400}+\frac{121 (1-2 x)^{5/2} \sqrt{3+5 x}}{1600}-\frac{33}{160} (1-2 x)^{7/2} \sqrt{3+5 x}-\frac{1}{10} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac{483153 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{64000 \sqrt{5}}\\ &=\frac{43923 \sqrt{1-2 x} \sqrt{3+5 x}}{64000}+\frac{1331 (1-2 x)^{3/2} \sqrt{3+5 x}}{6400}+\frac{121 (1-2 x)^{5/2} \sqrt{3+5 x}}{1600}-\frac{33}{160} (1-2 x)^{7/2} \sqrt{3+5 x}-\frac{1}{10} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac{483153 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{64000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0698316, size = 79, normalized size = 0.57 \[ \frac{10 \sqrt{5 x+3} \left (-512000 x^5+505600 x^4+230080 x^3-410280 x^2+57074 x+29673\right )-483153 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{640000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[3 + 5*x]*(29673 + 57074*x - 410280*x^2 + 230080*x^3 + 505600*x^4 - 512000*x^5) - 483153*Sqrt[10 - 20*
x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(640000*Sqrt[1 - 2*x])

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Maple [A]  time = 0.005, size = 120, normalized size = 0.9 \begin{align*}{\frac{1}{25} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}} \left ( 3+5\,x \right ) ^{{\frac{5}{2}}}}+{\frac{11}{200} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}} \left ( 3+5\,x \right ) ^{{\frac{5}{2}}}}+{\frac{121}{2000} \left ( 3+5\,x \right ) ^{{\frac{5}{2}}}\sqrt{1-2\,x}}-{\frac{1331}{16000} \left ( 3+5\,x \right ) ^{{\frac{3}{2}}}\sqrt{1-2\,x}}-{\frac{43923}{64000}\sqrt{1-2\,x}\sqrt{3+5\,x}}+{\frac{483153\,\sqrt{10}}{1280000}\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)^(5/2)*(3+5*x)^(3/2),x)

[Out]

1/25*(1-2*x)^(5/2)*(3+5*x)^(5/2)+11/200*(1-2*x)^(3/2)*(3+5*x)^(5/2)+121/2000*(3+5*x)^(5/2)*(1-2*x)^(1/2)-1331/
16000*(3+5*x)^(3/2)*(1-2*x)^(1/2)-43923/64000*(1-2*x)^(1/2)*(3+5*x)^(1/2)+483153/1280000*((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 = 3.62138, size = 113, normalized size = 0.82 \begin{align*} \frac{1}{25} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{11}{40} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{11}{800} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{3993}{3200} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{483153}{1280000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{3993}{64000} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25*(-10*x^2 - x + 3)^(5/2) + 11/40*(-10*x^2 - x + 3)^(3/2)*x + 11/800*(-10*x^2 - x + 3)^(3/2) + 3993/3200*sq
rt(-10*x^2 - x + 3)*x - 483153/1280000*sqrt(10)*arcsin(-20/11*x - 1/11) + 3993/64000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.5398, size = 271, normalized size = 1.96 \begin{align*} \frac{1}{64000} \,{\left (256000 \, x^{4} - 124800 \, x^{3} - 177440 \, x^{2} + 116420 \, x + 29673\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{483153}{1280000} \, \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)^(5/2)*(3+5*x)^(3/2),x, algorithm="fricas")

[Out]

1/64000*(256000*x^4 - 124800*x^3 - 177440*x^2 + 116420*x + 29673)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 483153/128000
0*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 = 54.7691, size = 311, normalized size = 2.25 \begin{align*} \begin{cases} \frac{40 i \left (x + \frac{3}{5}\right )^{\frac{11}{2}}}{\sqrt{10 x - 5}} - \frac{319 i \left (x + \frac{3}{5}\right )^{\frac{9}{2}}}{2 \sqrt{10 x - 5}} + \frac{8833 i \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{40 \sqrt{10 x - 5}} - \frac{171699 i \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{1600 \sqrt{10 x - 5}} - \frac{14641 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{6400 \sqrt{10 x - 5}} + \frac{483153 i \sqrt{x + \frac{3}{5}}}{64000 \sqrt{10 x - 5}} - \frac{483153 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{640000} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{483153 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{640000} - \frac{40 \left (x + \frac{3}{5}\right )^{\frac{11}{2}}}{\sqrt{5 - 10 x}} + \frac{319 \left (x + \frac{3}{5}\right )^{\frac{9}{2}}}{2 \sqrt{5 - 10 x}} - \frac{8833 \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{40 \sqrt{5 - 10 x}} + \frac{171699 \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{1600 \sqrt{5 - 10 x}} + \frac{14641 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{6400 \sqrt{5 - 10 x}} - \frac{483153 \sqrt{x + \frac{3}{5}}}{64000 \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)**(5/2)*(3+5*x)**(3/2),x)

[Out]

Piecewise((40*I*(x + 3/5)**(11/2)/sqrt(10*x - 5) - 319*I*(x + 3/5)**(9/2)/(2*sqrt(10*x - 5)) + 8833*I*(x + 3/5
)**(7/2)/(40*sqrt(10*x - 5)) - 171699*I*(x + 3/5)**(5/2)/(1600*sqrt(10*x - 5)) - 14641*I*(x + 3/5)**(3/2)/(640
0*sqrt(10*x - 5)) + 483153*I*sqrt(x + 3/5)/(64000*sqrt(10*x - 5)) - 483153*sqrt(10)*I*acosh(sqrt(110)*sqrt(x +
3/5)/11)/640000, 10*Abs(x + 3/5)/11 > 1), (483153*sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/640000 - 40*(x +
3/5)**(11/2)/sqrt(5 - 10*x) + 319*(x + 3/5)**(9/2)/(2*sqrt(5 - 10*x)) - 8833*(x + 3/5)**(7/2)/(40*sqrt(5 - 10*
x)) + 171699*(x + 3/5)**(5/2)/(1600*sqrt(5 - 10*x)) + 14641*(x + 3/5)**(3/2)/(6400*sqrt(5 - 10*x)) - 483153*sq
rt(x + 3/5)/(64000*sqrt(5 - 10*x)), True))

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Giac [B]  time = 2.14258, size = 317, normalized size = 2.3 \begin{align*} \frac{1}{9600000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (12 \,{\left (80 \, x - 143\right )}{\left (5 \, x + 3\right )} + 9773\right )}{\left (5 \, x + 3\right )} - 136405\right )}{\left (5 \, x + 3\right )} + 60555\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 666105 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{1}{240000} \, \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{7}{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)^(5/2)*(3+5*x)^(3/2),x, algorithm="giac")

[Out]

1/9600000*sqrt(5)*(2*(4*(8*(12*(80*x - 143)*(5*x + 3) + 9773)*(5*x + 3) - 136405)*(5*x + 3) + 60555)*sqrt(5*x
+ 3)*sqrt(-10*x + 5) - 666105*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 1/240000*sqrt(5)*(2*(4*(8*(60*x -
71)*(5*x + 3) + 2179)*(5*x + 3) - 4125)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 45375*sqrt(2)*arcsin(1/11*sqrt(22)*sq
rt(5*x + 3))) - 7/24000*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)))