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]

-11/32*(1-2*x)^(5/2)*(3+5*x)^(3/2)-1/10*(1-2*x)^(5/2)*(3+5*x)^(5/2)+483153/256000*arcsin(1/11*22^(1/2)*(3+5*x)
^(1/2))*10^(1/2)+1331/2560*(1-2*x)^(3/2)*(3+5*x)^(1/2)-121/128*(1-2*x)^(5/2)*(3+5*x)^(1/2)+43923/25600*(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} \[ -\frac {1}{10} (5 x+3)^{5/2} (1-2 x)^{5/2}-\frac {11}{32} (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac {121}{128} \sqrt {5 x+3} (1-2 x)^{5/2}+\frac {1331 \sqrt {5 x+3} (1-2 x)^{3/2}}{2560}+\frac {43923 \sqrt {5 x+3} \sqrt {1-2 x}}{25600}+\frac {483153 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{25600 \sqrt {10}} \]

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*}

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Mathematica [A]  time = 0.06, size = 79, normalized size = 0.57 \[ \frac {10 \sqrt {5 x+3} \left (512000 x^5+198400 x^4-476480 x^3-169640 x^2+179954 x-16407\right )+483153 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{256000 \sqrt {1-2 x}} \]

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 +
20*x]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(256000*Sqrt[1 - 2*x])

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fricas [A]  time = 1.12, size = 77, normalized size = 0.56 \[ -\frac {1}{25600} \, {\left (256000 \, x^{4} + 227200 \, x^{3} - 124640 \, x^{2} - 147140 \, x + 16407\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {483153}{512000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]

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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giac [B]  time = 1.19, size = 275, normalized size = 1.99 \[ -\frac {1}{3840000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {13}{384000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {3}{8000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {81}{2000} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {27}{50} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3840000*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))) - 13/384000*sqrt(5)*(2*(4*(8*
(60*x - 119)*(5*x + 3) + 6163)*(5*x + 3) - 66189)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 184305*sqrt(2)*arcsin(1/11*s
qrt(22)*sqrt(5*x + 3))) - 3/8000*sqrt(5)*(2*(4*(40*x - 59)*(5*x + 3) + 1293)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 4
785*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 81/2000*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))) + 27/50*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 \[ \frac {483153 \sqrt {\left (-2 x +1\right ) \left (5 x +3\right )}\, \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{512000 \sqrt {5 x +3}\, \sqrt {-2 x +1}}+\frac {\left (-2 x +1\right )^{\frac {3}{2}} \left (5 x +3\right )^{\frac {7}{2}}}{25}+\frac {33 \left (5 x +3\right )^{\frac {7}{2}} \sqrt {-2 x +1}}{1000}-\frac {121 \left (5 x +3\right )^{\frac {5}{2}} \sqrt {-2 x +1}}{4000}-\frac {1331 \left (5 x +3\right )^{\frac {3}{2}} \sqrt {-2 x +1}}{6400}-\frac {43923 \sqrt {-2 x +1}\, \sqrt {5 x +3}}{25600} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25*(-2*x+1)^(3/2)*(5*x+3)^(7/2)+33/1000*(5*x+3)^(7/2)*(-2*x+1)^(1/2)-121/4000*(5*x+3)^(5/2)*(-2*x+1)^(1/2)-1
331/6400*(5*x+3)^(3/2)*(-2*x+1)^(1/2)-43923/25600*(-2*x+1)^(1/2)*(5*x+3)^(1/2)+483153/512000*((-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 = 1.24, size = 84, normalized size = 0.61 \[ -\frac {1}{10} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {11}{16} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {11}{320} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {3993}{1280} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {483153}{512000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {3993}{25600} \, \sqrt {-10 \, x^{2} - x + 3} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 16.30, size = 311, normalized size = 2.25 \[ \begin {cases} - \frac {100 i \left (x + \frac {3}{5}\right )^{\frac {11}{2}}}{\sqrt {10 x - 5}} + \frac {1045 i \left (x + \frac {3}{5}\right )^{\frac {9}{2}}}{4 \sqrt {10 x - 5}} - \frac {2783 i \left (x + \frac {3}{5}\right )^{\frac {7}{2}}}{16 \sqrt {10 x - 5}} - \frac {1331 i \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{640 \sqrt {10 x - 5}} - \frac {14641 i \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{2560 \sqrt {10 x - 5}} + \frac {483153 i \sqrt {x + \frac {3}{5}}}{25600 \sqrt {10 x - 5}} - \frac {483153 \sqrt {10} i \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{256000} & \text {for}\: \frac {10 \left |{x + \frac {3}{5}}\right |}{11} > 1 \\\frac {483153 \sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{256000} + \frac {100 \left (x + \frac {3}{5}\right )^{\frac {11}{2}}}{\sqrt {5 - 10 x}} - \frac {1045 \left (x + \frac {3}{5}\right )^{\frac {9}{2}}}{4 \sqrt {5 - 10 x}} + \frac {2783 \left (x + \frac {3}{5}\right )^{\frac {7}{2}}}{16 \sqrt {5 - 10 x}} + \frac {1331 \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{640 \sqrt {5 - 10 x}} + \frac {14641 \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{2560 \sqrt {5 - 10 x}} - \frac {483153 \sqrt {x + \frac {3}{5}}}{25600 \sqrt {5 - 10 x}} & \text {otherwise} \end {cases} \]

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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