3.2332 \(\int (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=194 \[ -\frac{3}{80} (1-2 x)^{5/2} (3 x+2)^2 (5 x+3)^{7/2}-\frac{9 (1-2 x)^{5/2} (16120 x+25043) (5 x+3)^{7/2}}{448000}-\frac{306029 (1-2 x)^{5/2} (5 x+3)^{5/2}}{256000}-\frac{3366319 (1-2 x)^{5/2} (5 x+3)^{3/2}}{819200}-\frac{37029509 (1-2 x)^{5/2} \sqrt{5 x+3}}{3276800}+\frac{407324599 (1-2 x)^{3/2} \sqrt{5 x+3}}{65536000}+\frac{13441711767 \sqrt{1-2 x} \sqrt{5 x+3}}{655360000}+\frac{147858829437 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{655360000 \sqrt{10}} \]

[Out]

(13441711767*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/655360000 + (407324599*(1 - 2*x)^(3/2)
*Sqrt[3 + 5*x])/65536000 - (37029509*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/3276800 - (3
366319*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/819200 - (306029*(1 - 2*x)^(5/2)*(3 + 5*
x)^(5/2))/256000 - (3*(1 - 2*x)^(5/2)*(2 + 3*x)^2*(3 + 5*x)^(7/2))/80 - (9*(1 -
2*x)^(5/2)*(3 + 5*x)^(7/2)*(25043 + 16120*x))/448000 + (147858829437*ArcSin[Sqrt
[2/11]*Sqrt[3 + 5*x]])/(655360000*Sqrt[10])

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Rubi [A]  time = 0.244404, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{3}{80} (1-2 x)^{5/2} (3 x+2)^2 (5 x+3)^{7/2}-\frac{9 (1-2 x)^{5/2} (16120 x+25043) (5 x+3)^{7/2}}{448000}-\frac{306029 (1-2 x)^{5/2} (5 x+3)^{5/2}}{256000}-\frac{3366319 (1-2 x)^{5/2} (5 x+3)^{3/2}}{819200}-\frac{37029509 (1-2 x)^{5/2} \sqrt{5 x+3}}{3276800}+\frac{407324599 (1-2 x)^{3/2} \sqrt{5 x+3}}{65536000}+\frac{13441711767 \sqrt{1-2 x} \sqrt{5 x+3}}{655360000}+\frac{147858829437 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{655360000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(13441711767*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/655360000 + (407324599*(1 - 2*x)^(3/2)
*Sqrt[3 + 5*x])/65536000 - (37029509*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/3276800 - (3
366319*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/819200 - (306029*(1 - 2*x)^(5/2)*(3 + 5*
x)^(5/2))/256000 - (3*(1 - 2*x)^(5/2)*(2 + 3*x)^2*(3 + 5*x)^(7/2))/80 - (9*(1 -
2*x)^(5/2)*(3 + 5*x)^(7/2)*(25043 + 16120*x))/448000 + (147858829437*ArcSin[Sqrt
[2/11]*Sqrt[3 + 5*x]])/(655360000*Sqrt[10])

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Rubi in Sympy [A]  time = 21.1402, size = 178, normalized size = 0.92 \[ - \frac{3 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac{7}{2}}}{80} - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{7}{2}} \left (108810 x + \frac{676161}{4}\right )}{336000} + \frac{306029 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{7}{2}}}{640000} + \frac{10098957 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{7}{2}}}{25600000} - \frac{37029509 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{102400000} - \frac{407324599 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{163840000} - \frac{13441711767 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{655360000} + \frac{147858829437 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{6553600000} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(3/2)*(2+3*x)**3*(3+5*x)**(5/2),x)

[Out]

-3*(-2*x + 1)**(5/2)*(3*x + 2)**2*(5*x + 3)**(7/2)/80 - (-2*x + 1)**(5/2)*(5*x +
3)**(7/2)*(108810*x + 676161/4)/336000 + 306029*(-2*x + 1)**(3/2)*(5*x + 3)**(7
/2)/640000 + 10098957*sqrt(-2*x + 1)*(5*x + 3)**(7/2)/25600000 - 37029509*sqrt(-
2*x + 1)*(5*x + 3)**(5/2)/102400000 - 407324599*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/
163840000 - 13441711767*sqrt(-2*x + 1)*sqrt(5*x + 3)/655360000 + 147858829437*sq
rt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/6553600000

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Mathematica [A]  time = 0.153072, size = 85, normalized size = 0.44 \[ \frac{-10 \sqrt{1-2 x} \sqrt{5 x+3} \left (774144000000 x^7+2394316800000 x^6+2554199040000 x^5+592093952000 x^4-910419721600 x^3-749541131680 x^2-138459209260 x+116041578381\right )-1035011806059 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{45875200000} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(116041578381 - 138459209260*x - 749541131680*x
^2 - 910419721600*x^3 + 592093952000*x^4 + 2554199040000*x^5 + 2394316800000*x^6
+ 774144000000*x^7) - 1035011806059*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/
45875200000

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Maple [A]  time = 0.016, size = 172, normalized size = 0.9 \[{\frac{1}{91750400000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -15482880000000\,{x}^{7}\sqrt{-10\,{x}^{2}-x+3}-47886336000000\,{x}^{6}\sqrt{-10\,{x}^{2}-x+3}-51083980800000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-11841879040000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-2320831567620\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/91750400000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-15482880000000*x^7*(-10*x^2-x+3)^(1/
2)-47886336000000*x^6*(-10*x^2-x+3)^(1/2)-51083980800000*x^5*(-10*x^2-x+3)^(1/2)
-11841879040000*x^4*(-10*x^2-x+3)^(1/2)[phone]*x^3*(-10*x^2-x+3)^(1/2)+1
4990822633600*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11
)[phone]*x*(-10*x^2-x+3)^(1/2)-2320831567620*(-10*x^2-x+3)^(1/2))/(-10*x^
2-x+3)^(1/2)

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Maxima [A]  time = 1.51418, size = 180, normalized size = 0.93 \[ -\frac{27}{16} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{3} - \frac{2187}{448} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{2} - \frac{100119}{17920} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x - \frac{5653247}{1792000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{3366319}{409600} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{3366319}{8192000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{1221973797}{32768000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{147858829437}{13107200000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{1221973797}{655360000} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(3*x + 2)^3*(-2*x + 1)^(3/2),x, algorithm="maxima")

[Out]

-27/16*(-10*x^2 - x + 3)^(5/2)*x^3 - 2187/448*(-10*x^2 - x + 3)^(5/2)*x^2 - 1001
19/17920*(-10*x^2 - x + 3)^(5/2)*x - 5653247/1792000*(-10*x^2 - x + 3)^(5/2) + 3
366319/409600*(-10*x^2 - x + 3)^(3/2)*x + 3366319/8192000*(-10*x^2 - x + 3)^(3/2
) + 1221973797/32768000*sqrt(-10*x^2 - x + 3)*x - 147858829437/13107200000*sqrt(
10)*arcsin(-20/11*x - 1/11) + 1221973797/655360000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.222881, size = 117, normalized size = 0.6 \[ -\frac{1}{91750400000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (774144000000 \, x^{7} + 2394316800000 \, x^{6} + 2554199040000 \, x^{5} + 592093952000 \, x^{4} - 910419721600 \, x^{3} - 749541131680 \, x^{2} - 138459209260 \, x + 116041578381\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 1035011806059 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(3*x + 2)^3*(-2*x + 1)^(3/2),x, algorithm="fricas")

[Out]

-1/91750400000*sqrt(10)*(2*sqrt(10)*(774144000000*x^7 + 2394316800000*x^6 + 2554
199040000*x^5 + 592093952000*x^4 - 910419721600*x^3 - 749541131680*x^2 - 1384592
09260*x + 116041578381)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 1035011806059*arctan(1/20
*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-2*x)**(3/2)*(2+3*x)**3*(3+5*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.275045, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done