\(\int (1-2 x)^{3/2} (3+5 x)^{5/2} \, dx\) [2352]

Optimal result

Integrand size = 19, antiderivative size = 138 \[ \int (1-2 x)^{3/2} (3+5 x)^{5/2} \, dx=\frac {43923 \sqrt {1-2 x} \sqrt {3+5 x}}{25600}+\frac {1331 (1-2 x)^{3/2} \sqrt {3+5 x}}{2560}-\frac {121}{128} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {11}{32} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac {1}{10} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac {483153 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{25600 \sqrt {10}} \]

[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)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {52, 56, 222} \[ \int (1-2 x)^{3/2} (3+5 x)^{5/2} \, dx=\frac {483153 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{25600 \sqrt {10}}-\frac {1}{10} (5 x+3)^{5/2} (1-2 x)^{5/2}-\frac {11}{32} (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac {121}{128} \sqrt {5 x+3} (1-2 x)^{5/2}+\frac {1331 \sqrt {5 x+3} (1-2 x)^{3/2}}{2560}+\frac {43923 \sqrt {5 x+3} \sqrt {1-2 x}}{25600} \]

[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 52

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 56

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 222

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*} \text {integral}& = -\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 \text {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] (verified)

Time = 0.18 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.54 \[ \int (1-2 x)^{3/2} (3+5 x)^{5/2} \, dx=\frac {-10 \sqrt {1-2 x} \sqrt {3+5 x} \left (16407-147140 x-124640 x^2+227200 x^3+256000 x^4\right )+483153 \sqrt {10} \arctan \left (\frac {\sqrt {\frac {6}{5}+2 x}}{\sqrt {1-2 x}}\right )}{256000} \]

[In]

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

[Out]

(-10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(16407 - 147140*x - 124640*x^2 + 227200*x^3 + 256000*x^4) + 483153*Sqrt[10]*A
rcTan[Sqrt[6/5 + 2*x]/Sqrt[1 - 2*x]])/256000

Maple [A] (verified)

Time = 1.12 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.78

method	result	size
risch	\(\frac {\left (256000 x^{4}+227200 x^{3}-124640 x^{2}-147140 x +16407\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{25600 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {483153 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{512000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(108\)
default	\(\frac {\left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {7}{2}}}{25}+\frac {33 \left (3+5 x \right )^{\frac {7}{2}} \sqrt {1-2 x}}{1000}-\frac {121 \left (3+5 x \right )^{\frac {5}{2}} \sqrt {1-2 x}}{4000}-\frac {1331 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}{6400}-\frac {43923 \sqrt {1-2 x}\, \sqrt {3+5 x}}{25600}+\frac {483153 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{512000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(120\)

[In]

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

[Out]

1/25600*(256000*x^4+227200*x^3-124640*x^2-147140*x+16407)*(-1+2*x)*(3+5*x)^(1/2)/(-(-1+2*x)*(3+5*x))^(1/2)*((1
-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)+483153/512000*10^(1/2)*arcsin(20/11*x+1/11)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)
^(1/2)/(3+5*x)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.56 \[ \int (1-2 x)^{3/2} (3+5 x)^{5/2} \, dx=-\frac {1}{25600} \, {\left (256000 \, x^{4} + 227200 \, x^{3} - 124640 \, x^{2} - 147140 \, x + 16407\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {483153}{512000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]

[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))

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 56.49 (sec) , antiderivative size = 309, normalized size of antiderivative = 2.24 \[ \int (1-2 x)^{3/2} (3+5 x)^{5/2} \, dx=\begin {cases} - \frac {100 i \left (x + \frac {3}{5}\right )^{\frac {11}{2}}}{\sqrt {10 x - 5}} + \frac {1045 i \left (x + \frac {3}{5}\right )^{\frac {9}{2}}}{4 \sqrt {10 x - 5}} - \frac {2783 i \left (x + \frac {3}{5}\right )^{\frac {7}{2}}}{16 \sqrt {10 x - 5}} - \frac {1331 i \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{640 \sqrt {10 x - 5}} - \frac {14641 i \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{2560 \sqrt {10 x - 5}} + \frac {483153 i \sqrt {x + \frac {3}{5}}}{25600 \sqrt {10 x - 5}} - \frac {483153 \sqrt {10} i \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{256000} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\\frac {483153 \sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{256000} + \frac {100 \left (x + \frac {3}{5}\right )^{\frac {11}{2}}}{\sqrt {5 - 10 x}} - \frac {1045 \left (x + \frac {3}{5}\right )^{\frac {9}{2}}}{4 \sqrt {5 - 10 x}} + \frac {2783 \left (x + \frac {3}{5}\right )^{\frac {7}{2}}}{16 \sqrt {5 - 10 x}} + \frac {1331 \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{640 \sqrt {5 - 10 x}} + \frac {14641 \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{2560 \sqrt {5 - 10 x}} - \frac {483153 \sqrt {x + \frac {3}{5}}}{25600 \sqrt {5 - 10 x}} & \text {otherwise} \end {cases} \]

[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, Abs(x + 3/5) > 11/10), (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 - 10*
x)) + 1331*(x + 3/5)**(5/2)/(640*sqrt(5 - 10*x)) + 14641*(x + 3/5)**(3/2)/(2560*sqrt(5 - 10*x)) - 483153*sqrt(
x + 3/5)/(25600*sqrt(5 - 10*x)), True))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.61 \[ \int (1-2 x)^{3/2} (3+5 x)^{5/2} \, dx=-\frac {1}{10} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {11}{16} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {11}{320} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {3993}{1280} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {483153}{512000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {3993}{25600} \, \sqrt {-10 \, x^{2} - x + 3} \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (99) = 198\).

Time = 0.33 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.99 \[ \int (1-2 x)^{3/2} (3+5 x)^{5/2} \, dx=-\frac {1}{3840000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {13}{384000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {3}{8000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {81}{2000} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {27}{50} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]

[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))

Mupad [F(-1)]

Timed out. \[ \int (1-2 x)^{3/2} (3+5 x)^{5/2} \, dx=\int {\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2} \,d x \]

[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)