3.2414 \(\int (1-2 x)^{5/2} (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=160 \[ -\frac{1}{12} (5 x+3)^{5/2} (1-2 x)^{7/2}-\frac{11}{48} (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac{121}{256} \sqrt{5 x+3} (1-2 x)^{7/2}+\frac{1331 \sqrt{5 x+3} (1-2 x)^{5/2}}{7680}+\frac{14641 \sqrt{5 x+3} (1-2 x)^{3/2}}{30720}+\frac{161051 \sqrt{5 x+3} \sqrt{1-2 x}}{102400}+\frac{1771561 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{102400 \sqrt{10}} \]

[Out]

(161051*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/102400 + (14641*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/30720 + (1331*(1 - 2*x)^(5
/2)*Sqrt[3 + 5*x])/7680 - (121*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/256 - (11*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2))/48 -
((1 - 2*x)^(7/2)*(3 + 5*x)^(5/2))/12 + (1771561*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(102400*Sqrt[10])

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Rubi [A]  time = 0.0489823, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {50, 54, 216} \[ -\frac{1}{12} (5 x+3)^{5/2} (1-2 x)^{7/2}-\frac{11}{48} (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac{121}{256} \sqrt{5 x+3} (1-2 x)^{7/2}+\frac{1331 \sqrt{5 x+3} (1-2 x)^{5/2}}{7680}+\frac{14641 \sqrt{5 x+3} (1-2 x)^{3/2}}{30720}+\frac{161051 \sqrt{5 x+3} \sqrt{1-2 x}}{102400}+\frac{1771561 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{102400 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(161051*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/102400 + (14641*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/30720 + (1331*(1 - 2*x)^(5
/2)*Sqrt[3 + 5*x])/7680 - (121*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/256 - (11*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2))/48 -
((1 - 2*x)^(7/2)*(3 + 5*x)^(5/2))/12 + (1771561*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(102400*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)^{5/2} \, dx &=-\frac{1}{12} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac{55}{24} \int (1-2 x)^{5/2} (3+5 x)^{3/2} \, dx\\ &=-\frac{11}{48} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac{1}{12} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac{121}{32} \int (1-2 x)^{5/2} \sqrt{3+5 x} \, dx\\ &=-\frac{121}{256} (1-2 x)^{7/2} \sqrt{3+5 x}-\frac{11}{48} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac{1}{12} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac{1331}{512} \int \frac{(1-2 x)^{5/2}}{\sqrt{3+5 x}} \, dx\\ &=\frac{1331 (1-2 x)^{5/2} \sqrt{3+5 x}}{7680}-\frac{121}{256} (1-2 x)^{7/2} \sqrt{3+5 x}-\frac{11}{48} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac{1}{12} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac{14641 \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx}{3072}\\ &=\frac{14641 (1-2 x)^{3/2} \sqrt{3+5 x}}{30720}+\frac{1331 (1-2 x)^{5/2} \sqrt{3+5 x}}{7680}-\frac{121}{256} (1-2 x)^{7/2} \sqrt{3+5 x}-\frac{11}{48} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac{1}{12} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac{161051 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{20480}\\ &=\frac{161051 \sqrt{1-2 x} \sqrt{3+5 x}}{102400}+\frac{14641 (1-2 x)^{3/2} \sqrt{3+5 x}}{30720}+\frac{1331 (1-2 x)^{5/2} \sqrt{3+5 x}}{7680}-\frac{121}{256} (1-2 x)^{7/2} \sqrt{3+5 x}-\frac{11}{48} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac{1}{12} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac{1771561 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{204800}\\ &=\frac{161051 \sqrt{1-2 x} \sqrt{3+5 x}}{102400}+\frac{14641 (1-2 x)^{3/2} \sqrt{3+5 x}}{30720}+\frac{1331 (1-2 x)^{5/2} \sqrt{3+5 x}}{7680}-\frac{121}{256} (1-2 x)^{7/2} \sqrt{3+5 x}-\frac{11}{48} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac{1}{12} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac{1771561 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{102400 \sqrt{5}}\\ &=\frac{161051 \sqrt{1-2 x} \sqrt{3+5 x}}{102400}+\frac{14641 (1-2 x)^{3/2} \sqrt{3+5 x}}{30720}+\frac{1331 (1-2 x)^{5/2} \sqrt{3+5 x}}{7680}-\frac{121}{256} (1-2 x)^{7/2} \sqrt{3+5 x}-\frac{11}{48} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac{1}{12} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac{1771561 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{102400 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.073279, size = 84, normalized size = 0.52 \[ -\frac{10 \sqrt{5 x+3} \left (10240000 x^6-2560000 x^5-11091200 x^4+3408320 x^3+4538680 x^2-1703014 x-96003\right )+5314683 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3072000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(10*Sqrt[3 + 5*x]*(-96003 - 1703014*x + 4538680*x^2 + 3408320*x^3 - 11091200*x^4 - 2560000*x^5 + 10240000*x^6
) + 5314683*Sqrt[10 - 20*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(3072000*Sqrt[1 - 2*x])

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Maple [A]  time = 0.003, size = 136, normalized size = 0.9 \begin{align*}{\frac{1}{30} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}} \left ( 3+5\,x \right ) ^{{\frac{7}{2}}}}+{\frac{11}{300} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}} \left ( 3+5\,x \right ) ^{{\frac{7}{2}}}}+{\frac{121}{4000} \left ( 3+5\,x \right ) ^{{\frac{7}{2}}}\sqrt{1-2\,x}}-{\frac{1331}{48000} \left ( 3+5\,x \right ) ^{{\frac{5}{2}}}\sqrt{1-2\,x}}-{\frac{14641}{76800} \left ( 3+5\,x \right ) ^{{\frac{3}{2}}}\sqrt{1-2\,x}}-{\frac{161051}{102400}\sqrt{1-2\,x}\sqrt{3+5\,x}}+{\frac{1771561\,\sqrt{10}}{2048000}\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)^(5/2),x)

[Out]

1/30*(1-2*x)^(5/2)*(3+5*x)^(7/2)+11/300*(1-2*x)^(3/2)*(3+5*x)^(7/2)+121/4000*(3+5*x)^(7/2)*(1-2*x)^(1/2)-1331/
48000*(3+5*x)^(5/2)*(1-2*x)^(1/2)-14641/76800*(3+5*x)^(3/2)*(1-2*x)^(1/2)-161051/102400*(1-2*x)^(1/2)*(3+5*x)^
(1/2)+1771561/2048000*((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 = 3.63594, size = 134, normalized size = 0.84 \begin{align*} \frac{1}{6} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x + \frac{1}{120} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{121}{192} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{121}{3840} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{14641}{5120} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{1771561}{2048000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{14641}{102400} \, \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)^(5/2),x, algorithm="maxima")

[Out]

1/6*(-10*x^2 - x + 3)^(5/2)*x + 1/120*(-10*x^2 - x + 3)^(5/2) + 121/192*(-10*x^2 - x + 3)^(3/2)*x + 121/3840*(
-10*x^2 - x + 3)^(3/2) + 14641/5120*sqrt(-10*x^2 - x + 3)*x - 1771561/2048000*sqrt(10)*arcsin(-20/11*x - 1/11)
+ 14641/102400*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.70228, size = 297, normalized size = 1.86 \begin{align*} \frac{1}{307200} \,{\left (5120000 \, x^{5} + 1280000 \, x^{4} - 4905600 \, x^{3} - 748640 \, x^{2} + 1895020 \, x + 96003\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{1771561}{2048000} \, \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)^(5/2),x, algorithm="fricas")

[Out]

1/307200*(5120000*x^5 + 1280000*x^4 - 4905600*x^3 - 748640*x^2 + 1895020*x + 96003)*sqrt(5*x + 3)*sqrt(-2*x +
1) - 1771561/2048000*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 = 113.704, size = 357, normalized size = 2.23 \begin{align*} \begin{cases} \frac{500 i \left (x + \frac{3}{5}\right )^{\frac{13}{2}}}{3 \sqrt{10 x - 5}} - \frac{1925 i \left (x + \frac{3}{5}\right )^{\frac{11}{2}}}{3 \sqrt{10 x - 5}} + \frac{40535 i \left (x + \frac{3}{5}\right )^{\frac{9}{2}}}{48 \sqrt{10 x - 5}} - \frac{73205 i \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{192 \sqrt{10 x - 5}} - \frac{14641 i \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{7680 \sqrt{10 x - 5}} - \frac{161051 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{30720 \sqrt{10 x - 5}} + \frac{1771561 i \sqrt{x + \frac{3}{5}}}{102400 \sqrt{10 x - 5}} - \frac{1771561 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{1024000} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{1771561 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{1024000} - \frac{500 \left (x + \frac{3}{5}\right )^{\frac{13}{2}}}{3 \sqrt{5 - 10 x}} + \frac{1925 \left (x + \frac{3}{5}\right )^{\frac{11}{2}}}{3 \sqrt{5 - 10 x}} - \frac{40535 \left (x + \frac{3}{5}\right )^{\frac{9}{2}}}{48 \sqrt{5 - 10 x}} + \frac{73205 \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{192 \sqrt{5 - 10 x}} + \frac{14641 \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{7680 \sqrt{5 - 10 x}} + \frac{161051 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{30720 \sqrt{5 - 10 x}} - \frac{1771561 \sqrt{x + \frac{3}{5}}}{102400 \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)**(5/2),x)

[Out]

Piecewise((500*I*(x + 3/5)**(13/2)/(3*sqrt(10*x - 5)) - 1925*I*(x + 3/5)**(11/2)/(3*sqrt(10*x - 5)) + 40535*I*
(x + 3/5)**(9/2)/(48*sqrt(10*x - 5)) - 73205*I*(x + 3/5)**(7/2)/(192*sqrt(10*x - 5)) - 14641*I*(x + 3/5)**(5/2
)/(7680*sqrt(10*x - 5)) - 161051*I*(x + 3/5)**(3/2)/(30720*sqrt(10*x - 5)) + 1771561*I*sqrt(x + 3/5)/(102400*s
qrt(10*x - 5)) - 1771561*sqrt(10)*I*acosh(sqrt(110)*sqrt(x + 3/5)/11)/1024000, 10*Abs(x + 3/5)/11 > 1), (17715
61*sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/1024000 - 500*(x + 3/5)**(13/2)/(3*sqrt(5 - 10*x)) + 1925*(x + 3/
5)**(11/2)/(3*sqrt(5 - 10*x)) - 40535*(x + 3/5)**(9/2)/(48*sqrt(5 - 10*x)) + 73205*(x + 3/5)**(7/2)/(192*sqrt(
5 - 10*x)) + 14641*(x + 3/5)**(5/2)/(7680*sqrt(5 - 10*x)) + 161051*(x + 3/5)**(3/2)/(30720*sqrt(5 - 10*x)) - 1
771561*sqrt(x + 3/5)/(102400*sqrt(5 - 10*x)), True))

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Giac [B]  time = 2.0965, size = 427, normalized size = 2.67 \begin{align*} \frac{1}{76800000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (100 \, x - 239\right )}{\left (5 \, x + 3\right )} + 27999\right )}{\left (5 \, x + 3\right )} - 318159\right )}{\left (5 \, x + 3\right )} + 3237255\right )}{\left (5 \, x + 3\right )} - 2656665\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 29223315 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \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{59}{1920000} \, \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}{4000} \, \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)^(5/2)*(3+5*x)^(5/2),x, algorithm="giac")

[Out]

1/76800000*sqrt(5)*(2*(4*(8*(4*(16*(100*x - 239)*(5*x + 3) + 27999)*(5*x + 3) - 318159)*(5*x + 3) + 3237255)*(
5*x + 3) - 2656665)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 29223315*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 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))) - 59/1920000*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))) - 1/4000*sqrt(5)*(2*(4*(40*x - 23)*(5*x + 3) + 33)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 363*sqrt(2)*a
rcsin(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)))