3.2352 \(\int (1-2 x)^{3/2} (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=138 \[ -\frac{1}{10} (5 x+3)^{5/2} (1-2 x)^{5/2}-\frac{11}{32} (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac{121}{128} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{1331 \sqrt{5 x+3} (1-2 x)^{3/2}}{2560}+\frac{43923 \sqrt{5 x+3} \sqrt{1-2 x}}{25600}+\frac{483153 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{25600 \sqrt{10}} \]

[Out]

(43923*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/25600 + (1331*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/2560 - (121*(1 - 2*x)^(5/2)*S
qrt[3 + 5*x])/128 - (11*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/32 - ((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/10 + (483153*A
rcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(25600*Sqrt[10])

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Rubi [A]  time = 0.038584, 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} \[ -\frac{1}{10} (5 x+3)^{5/2} (1-2 x)^{5/2}-\frac{11}{32} (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac{121}{128} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{1331 \sqrt{5 x+3} (1-2 x)^{3/2}}{2560}+\frac{43923 \sqrt{5 x+3} \sqrt{1-2 x}}{25600}+\frac{483153 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{25600 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(43923*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/25600 + (1331*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/2560 - (121*(1 - 2*x)^(5/2)*S
qrt[3 + 5*x])/128 - (11*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/32 - ((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/10 + (483153*A
rcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(25600*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)^{5/2} \, dx &=-\frac{1}{10} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac{11}{4} \int (1-2 x)^{3/2} (3+5 x)^{3/2} \, dx\\ &=-\frac{11}{32} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac{1}{10} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac{363}{64} \int (1-2 x)^{3/2} \sqrt{3+5 x} \, dx\\ &=-\frac{121}{128} (1-2 x)^{5/2} \sqrt{3+5 x}-\frac{11}{32} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac{1}{10} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac{1331}{256} \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx\\ &=\frac{1331 (1-2 x)^{3/2} \sqrt{3+5 x}}{2560}-\frac{121}{128} (1-2 x)^{5/2} \sqrt{3+5 x}-\frac{11}{32} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac{1}{10} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac{43923 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{5120}\\ &=\frac{43923 \sqrt{1-2 x} \sqrt{3+5 x}}{25600}+\frac{1331 (1-2 x)^{3/2} \sqrt{3+5 x}}{2560}-\frac{121}{128} (1-2 x)^{5/2} \sqrt{3+5 x}-\frac{11}{32} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac{1}{10} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac{483153 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{51200}\\ &=\frac{43923 \sqrt{1-2 x} \sqrt{3+5 x}}{25600}+\frac{1331 (1-2 x)^{3/2} \sqrt{3+5 x}}{2560}-\frac{121}{128} (1-2 x)^{5/2} \sqrt{3+5 x}-\frac{11}{32} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac{1}{10} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac{483153 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{25600 \sqrt{5}}\\ &=\frac{43923 \sqrt{1-2 x} \sqrt{3+5 x}}{25600}+\frac{1331 (1-2 x)^{3/2} \sqrt{3+5 x}}{2560}-\frac{121}{128} (1-2 x)^{5/2} \sqrt{3+5 x}-\frac{11}{32} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac{1}{10} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac{483153 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{25600 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0720846, size = 79, normalized size = 0.57 \[ \frac{10 \sqrt{5 x+3} \left (512000 x^5+198400 x^4-476480 x^3-169640 x^2+179954 x-16407\right )-483153 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{256000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[3 + 5*x]*(-16407 + 179954*x - 169640*x^2 - 476480*x^3 + 198400*x^4 + 512000*x^5) - 483153*Sqrt[10 - 2
0*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(256000*Sqrt[1 - 2*x])

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Maple [A]  time = 0.004, size = 120, normalized size = 0.9 \begin{align*}{\frac{1}{25} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}} \left ( 3+5\,x \right ) ^{{\frac{7}{2}}}}+{\frac{33}{1000} \left ( 3+5\,x \right ) ^{{\frac{7}{2}}}\sqrt{1-2\,x}}-{\frac{121}{4000} \left ( 3+5\,x \right ) ^{{\frac{5}{2}}}\sqrt{1-2\,x}}-{\frac{1331}{6400} \left ( 3+5\,x \right ) ^{{\frac{3}{2}}}\sqrt{1-2\,x}}-{\frac{43923}{25600}\sqrt{1-2\,x}\sqrt{3+5\,x}}+{\frac{483153\,\sqrt{10}}{512000}\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)^(5/2),x)

[Out]

1/25*(1-2*x)^(3/2)*(3+5*x)^(7/2)+33/1000*(3+5*x)^(7/2)*(1-2*x)^(1/2)-121/4000*(3+5*x)^(5/2)*(1-2*x)^(1/2)-1331
/6400*(3+5*x)^(3/2)*(1-2*x)^(1/2)-43923/25600*(1-2*x)^(1/2)*(3+5*x)^(1/2)+483153/512000*((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 = 1.59569, size = 113, normalized size = 0.82 \begin{align*} -\frac{1}{10} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{11}{16} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{11}{320} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{3993}{1280} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{483153}{512000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{3993}{25600} \, \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)^(5/2),x, algorithm="maxima")

[Out]

-1/10*(-10*x^2 - x + 3)^(5/2) + 11/16*(-10*x^2 - x + 3)^(3/2)*x + 11/320*(-10*x^2 - x + 3)^(3/2) + 3993/1280*s
qrt(-10*x^2 - x + 3)*x - 483153/512000*sqrt(10)*arcsin(-20/11*x - 1/11) + 3993/25600*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.52511, size = 271, normalized size = 1.96 \begin{align*} -\frac{1}{25600} \,{\left (256000 \, x^{4} + 227200 \, x^{3} - 124640 \, x^{2} - 147140 \, x + 16407\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{483153}{512000} \, \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)^(5/2),x, algorithm="fricas")

[Out]

-1/25600*(256000*x^4 + 227200*x^3 - 124640*x^2 - 147140*x + 16407)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 483153/51200
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 = 52.2165, size = 311, normalized size = 2.25 \begin{align*} \begin{cases} - \frac{100 i \left (x + \frac{3}{5}\right )^{\frac{11}{2}}}{\sqrt{10 x - 5}} + \frac{1045 i \left (x + \frac{3}{5}\right )^{\frac{9}{2}}}{4 \sqrt{10 x - 5}} - \frac{2783 i \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{16 \sqrt{10 x - 5}} - \frac{1331 i \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{640 \sqrt{10 x - 5}} - \frac{14641 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{2560 \sqrt{10 x - 5}} + \frac{483153 i \sqrt{x + \frac{3}{5}}}{25600 \sqrt{10 x - 5}} - \frac{483153 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{256000} & \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 )}}{256000} + \frac{100 \left (x + \frac{3}{5}\right )^{\frac{11}{2}}}{\sqrt{5 - 10 x}} - \frac{1045 \left (x + \frac{3}{5}\right )^{\frac{9}{2}}}{4 \sqrt{5 - 10 x}} + \frac{2783 \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{16 \sqrt{5 - 10 x}} + \frac{1331 \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{640 \sqrt{5 - 10 x}} + \frac{14641 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{2560 \sqrt{5 - 10 x}} - \frac{483153 \sqrt{x + \frac{3}{5}}}{25600 \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)**(5/2),x)

[Out]

Piecewise((-100*I*(x + 3/5)**(11/2)/sqrt(10*x - 5) + 1045*I*(x + 3/5)**(9/2)/(4*sqrt(10*x - 5)) - 2783*I*(x +
3/5)**(7/2)/(16*sqrt(10*x - 5)) - 1331*I*(x + 3/5)**(5/2)/(640*sqrt(10*x - 5)) - 14641*I*(x + 3/5)**(3/2)/(256
0*sqrt(10*x - 5)) + 483153*I*sqrt(x + 3/5)/(25600*sqrt(10*x - 5)) - 483153*sqrt(10)*I*acosh(sqrt(110)*sqrt(x +
3/5)/11)/256000, 10*Abs(x + 3/5)/11 > 1), (483153*sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/256000 + 100*(x +
3/5)**(11/2)/sqrt(5 - 10*x) - 1045*(x + 3/5)**(9/2)/(4*sqrt(5 - 10*x)) + 2783*(x + 3/5)**(7/2)/(16*sqrt(5 - 1
0*x)) + 1331*(x + 3/5)**(5/2)/(640*sqrt(5 - 10*x)) + 14641*(x + 3/5)**(3/2)/(2560*sqrt(5 - 10*x)) - 483153*sqr
t(x + 3/5)/(25600*sqrt(5 - 10*x)), True))

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Giac [B]  time = 1.54997, size = 317, normalized size = 2.3 \begin{align*} -\frac{1}{3840000} \, \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{7}{384000} \, \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}{2000} \, \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{9}{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)^(5/2),x, algorithm="giac")

[Out]

-1/3840000*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))) - 7/384000*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)*s
qrt(5*x + 3))) + 1/2000*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))) + 9/400*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)))