3.2396 \(\int (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=182 \[ -\frac{3}{70} (5 x+3)^{7/2} (1-2 x)^{7/2}-\frac{37}{240} (5 x+3)^{5/2} (1-2 x)^{7/2}-\frac{407}{960} (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac{4477 \sqrt{5 x+3} (1-2 x)^{7/2}}{5120}+\frac{49247 \sqrt{5 x+3} (1-2 x)^{5/2}}{153600}+\frac{541717 \sqrt{5 x+3} (1-2 x)^{3/2}}{614400}+\frac{5958887 \sqrt{5 x+3} \sqrt{1-2 x}}{2048000}+\frac{65547757 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2048000 \sqrt{10}} \]

[Out]

(5958887*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/2048000 + (541717*(1 - 2*x)^(3/2)*Sqrt[3 +
5*x])/614400 + (49247*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/153600 - (4477*(1 - 2*x)^(
7/2)*Sqrt[3 + 5*x])/5120 - (407*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2))/960 - (37*(1 -
2*x)^(7/2)*(3 + 5*x)^(5/2))/240 - (3*(1 - 2*x)^(7/2)*(3 + 5*x)^(7/2))/70 + (6554
7757*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(2048000*Sqrt[10])

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Rubi [A]  time = 0.201269, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{3}{70} (5 x+3)^{7/2} (1-2 x)^{7/2}-\frac{37}{240} (5 x+3)^{5/2} (1-2 x)^{7/2}-\frac{407}{960} (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac{4477 \sqrt{5 x+3} (1-2 x)^{7/2}}{5120}+\frac{49247 \sqrt{5 x+3} (1-2 x)^{5/2}}{153600}+\frac{541717 \sqrt{5 x+3} (1-2 x)^{3/2}}{614400}+\frac{5958887 \sqrt{5 x+3} \sqrt{1-2 x}}{2048000}+\frac{65547757 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2048000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(5958887*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/2048000 + (541717*(1 - 2*x)^(3/2)*Sqrt[3 +
5*x])/614400 + (49247*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/153600 - (4477*(1 - 2*x)^(
7/2)*Sqrt[3 + 5*x])/5120 - (407*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2))/960 - (37*(1 -
2*x)^(7/2)*(3 + 5*x)^(5/2))/240 - (3*(1 - 2*x)^(7/2)*(3 + 5*x)^(7/2))/70 + (6554
7757*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(2048000*Sqrt[10])

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Rubi in Sympy [A]  time = 16.5472, size = 167, normalized size = 0.92 \[ - \frac{3 \left (- 2 x + 1\right )^{\frac{7}{2}} \left (5 x + 3\right )^{\frac{7}{2}}}{70} + \frac{37 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{7}{2}}}{600} + \frac{407 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{7}{2}}}{6000} - \frac{4477 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{32000} - \frac{49247 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{76800} - \frac{541717 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{204800} + \frac{5958887 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2048000} + \frac{65547757 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{20480000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(-2*x + 1)**(7/2)*(5*x + 3)**(7/2)/70 + 37*(-2*x + 1)**(5/2)*(5*x + 3)**(7/2)
/600 + 407*(-2*x + 1)**(3/2)*(5*x + 3)**(7/2)/6000 - 4477*(-2*x + 1)**(3/2)*(5*x
+ 3)**(5/2)/32000 - 49247*(-2*x + 1)**(3/2)*(5*x + 3)**(3/2)/76800 - 541717*(-2
*x + 1)**(3/2)*sqrt(5*x + 3)/204800 + 5958887*sqrt(-2*x + 1)*sqrt(5*x + 3)/20480
00 + 65547757*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/20480000

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Mathematica [A]  time = 0.104475, size = 80, normalized size = 0.44 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (1843200000 x^6+1879040000 x^5-1272064000 x^4-1600483200 x^3+287177440 x^2+540576580 x-24901623\right )-1376502897 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{430080000} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-24901623 + 540576580*x + 287177440*x^2 - 16004
83200*x^3 - 1272064000*x^4 + 1879040000*x^5 + 1843200000*x^6) - 1376502897*Sqrt[
10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/430080000

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Maple [A]  time = 0.013, size = 155, normalized size = 0.9 \[{\frac{1}{860160000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 36864000000\,{x}^{6}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-25441280000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-32009664000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-498032460\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/860160000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(36864000000*x^6*(-10*x^2-x+3)^(1/2)+375
80800000*x^5*(-10*x^2-x+3)^(1/2)-25441280000*x^4*(-10*x^2-x+3)^(1/2)-32009664000
*x^3*(-10*x^2-x+3)^(1/2)[phone]*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)-498032460*(-10*x^2-x+3)^(
1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.50295, size = 153, normalized size = 0.84 \[ -\frac{3}{70} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}} + \frac{37}{120} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x + \frac{37}{2400} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{4477}{3840} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{4477}{76800} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{541717}{102400} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{65547757}{40960000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{541717}{2048000} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/70*(-10*x^2 - x + 3)^(7/2) + 37/120*(-10*x^2 - x + 3)^(5/2)*x + 37/2400*(-10*
x^2 - x + 3)^(5/2) + 4477/3840*(-10*x^2 - x + 3)^(3/2)*x + 4477/76800*(-10*x^2 -
x + 3)^(3/2) + 541717/102400*sqrt(-10*x^2 - x + 3)*x - 65547757/40960000*sqrt(1
0)*arcsin(-20/11*x - 1/11) + 541717/2048000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.220898, size = 111, normalized size = 0.61 \[ \frac{1}{860160000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (1843200000 \, x^{6} + 1879040000 \, x^{5} - 1272064000 \, x^{4} - 1600483200 \, x^{3} + 287177440 \, x^{2} + 540576580 \, x - 24901623\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 1376502897 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/860160000*sqrt(10)*(2*sqrt(10)*(1843200000*x^6 + 1879040000*x^5 - 1272064000*x
^4 - 1600483200*x^3 + 287177440*x^2 + 540576580*x - 24901623)*sqrt(5*x + 3)*sqrt
(-2*x + 1) + 1376502897*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)**(5/2)*(2+3*x)*(3+5*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.273803, size = 548, normalized size = 3.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3584000000*sqrt(5)*(2*(4*(8*(4*(16*(20*(120*x - 359)*(5*x + 3) + 63769)*(5*x +
3) - 3968469)*(5*x + 3) + 33617829)*(5*x + 3) - 276044685)*(5*x + 3) + 87356115
)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 960917265*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*
x + 3))) + 13/384000000*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)*sqr
t(-10*x + 5) + 29223315*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 137/19200
0000*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))) - 17/240000*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)*sqrt(5*x + 3))) + 1/1600*sqrt(5)*(2*(4*(40*x - 23)*(5*x + 3) + 33)*sqrt(5*x
+ 3)*sqrt(-10*x + 5) - 363*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 9/200*
sqrt(5)*(2*(20*x + 1)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 121*sqrt(2)*arcsin(1/11*sq
rt(22)*sqrt(5*x + 3)))