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]

-1/10*(1-2*x)^(7/2)*(3+5*x)^(3/2)+483153/640000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+1331/6400*(1-2*x)
^(3/2)*(3+5*x)^(1/2)+121/1600*(1-2*x)^(5/2)*(3+5*x)^(1/2)-33/160*(1-2*x)^(7/2)*(3+5*x)^(1/2)+43923/64000*(1-2*
x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 138, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.08, size = 79, normalized size = 0.57 \[ \frac {10 \sqrt {5 x+3} \left (-512000 x^5+505600 x^4+230080 x^3-410280 x^2+57074 x+29673\right )+483153 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{640000 \sqrt {1-2 x}} \]

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]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(640000*Sqrt[1 - 2*x])

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fricas [A]  time = 1.04, size = 77, normalized size = 0.56 \[ \frac {1}{64000} \, {\left (256000 \, x^{4} - 124800 \, x^{3} - 177440 \, x^{2} + 116420 \, x + 29673\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {483153}{1280000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]

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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giac [B]  time = 1.27, size = 275, normalized size = 1.99 \[ \frac {1}{9600000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {1}{480000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {59}{120000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {3}{1000} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {9}{50} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9600000*sqrt(5)*(2*(4*(8*(12*(80*x - 203)*(5*x + 3) + 19073)*(5*x + 3) - 506185)*(5*x + 3) + 4031895)*sqrt(5
*x + 3)*sqrt(-10*x + 5) + 10392195*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 1/480000*sqrt(5)*(2*(4*(8*(6
0*x - 119)*(5*x + 3) + 6163)*(5*x + 3) - 66189)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 184305*sqrt(2)*arcsin(1/11*sqr
t(22)*sqrt(5*x + 3))) - 59/120000*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))) - 3/1000*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))) + 9/50*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x
+ 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))

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maple [A]  time = 0.00, size = 120, normalized size = 0.87 \[ \frac {483153 \sqrt {\left (-2 x +1\right ) \left (5 x +3\right )}\, \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{1280000 \sqrt {5 x +3}\, \sqrt {-2 x +1}}+\frac {\left (-2 x +1\right )^{\frac {5}{2}} \left (5 x +3\right )^{\frac {5}{2}}}{25}+\frac {11 \left (-2 x +1\right )^{\frac {3}{2}} \left (5 x +3\right )^{\frac {5}{2}}}{200}+\frac {121 \left (5 x +3\right )^{\frac {5}{2}} \sqrt {-2 x +1}}{2000}-\frac {1331 \left (5 x +3\right )^{\frac {3}{2}} \sqrt {-2 x +1}}{16000}-\frac {43923 \sqrt {-2 x +1}\, \sqrt {5 x +3}}{64000} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25*(-2*x+1)^(5/2)*(5*x+3)^(5/2)+11/200*(-2*x+1)^(3/2)*(5*x+3)^(5/2)+121/2000*(5*x+3)^(5/2)*(-2*x+1)^(1/2)-13
31/16000*(5*x+3)^(3/2)*(-2*x+1)^(1/2)-43923/64000*(-2*x+1)^(1/2)*(5*x+3)^(1/2)+483153/1280000*((-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 = 1.18, size = 84, normalized size = 0.61 \[ \frac {1}{25} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {11}{40} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {11}{800} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {3993}{3200} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {483153}{1280000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {3993}{64000} \, \sqrt {-10 \, x^{2} - x + 3} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 14.57, size = 311, normalized size = 2.25 \[ \begin {cases} \frac {40 i \left (x + \frac {3}{5}\right )^{\frac {11}{2}}}{\sqrt {10 x - 5}} - \frac {319 i \left (x + \frac {3}{5}\right )^{\frac {9}{2}}}{2 \sqrt {10 x - 5}} + \frac {8833 i \left (x + \frac {3}{5}\right )^{\frac {7}{2}}}{40 \sqrt {10 x - 5}} - \frac {171699 i \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{1600 \sqrt {10 x - 5}} - \frac {14641 i \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{6400 \sqrt {10 x - 5}} + \frac {483153 i \sqrt {x + \frac {3}{5}}}{64000 \sqrt {10 x - 5}} - \frac {483153 \sqrt {10} i \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{640000} & \text {for}\: \frac {10 \left |{x + \frac {3}{5}}\right |}{11} > 1 \\\frac {483153 \sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{640000} - \frac {40 \left (x + \frac {3}{5}\right )^{\frac {11}{2}}}{\sqrt {5 - 10 x}} + \frac {319 \left (x + \frac {3}{5}\right )^{\frac {9}{2}}}{2 \sqrt {5 - 10 x}} - \frac {8833 \left (x + \frac {3}{5}\right )^{\frac {7}{2}}}{40 \sqrt {5 - 10 x}} + \frac {171699 \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{1600 \sqrt {5 - 10 x}} + \frac {14641 \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{6400 \sqrt {5 - 10 x}} - \frac {483153 \sqrt {x + \frac {3}{5}}}{64000 \sqrt {5 - 10 x}} & \text {otherwise} \end {cases} \]

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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