3.2269 \(\int \sqrt{1-2 x} (2+3 x)^4 (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=201 \[ -\frac{3}{80} (1-2 x)^{3/2} (3 x+2)^3 (5 x+3)^{7/2}-\frac{1419 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{7/2}}{11200}-\frac{3 (1-2 x)^{3/2} (522420 x+899099) (5 x+3)^{7/2}}{1280000}-\frac{135817609 (1-2 x)^{3/2} (5 x+3)^{5/2}}{20480000}-\frac{1493993699 (1-2 x)^{3/2} (5 x+3)^{3/2}}{49152000}-\frac{16433930689 (1-2 x)^{3/2} \sqrt{5 x+3}}{131072000}+\frac{180773237579 \sqrt{1-2 x} \sqrt{5 x+3}}{1310720000}+\frac{1988505613369 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1310720000 \sqrt{10}} \]

[Out]

(180773237579*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1310720000 - (16433930689*(1 - 2*x)^(
3/2)*Sqrt[3 + 5*x])/131072000 - (1493993699*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/491
52000 - (135817609*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/20480000 - (1419*(1 - 2*x)^(
3/2)*(2 + 3*x)^2*(3 + 5*x)^(7/2))/11200 - (3*(1 - 2*x)^(3/2)*(2 + 3*x)^3*(3 + 5*
x)^(7/2))/80 - (3*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2)*(899099 + 522420*x))/1280000 +
(1988505613369*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1310720000*Sqrt[10])

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Rubi [A]  time = 0.276063, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{3}{80} (1-2 x)^{3/2} (3 x+2)^3 (5 x+3)^{7/2}-\frac{1419 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{7/2}}{11200}-\frac{3 (1-2 x)^{3/2} (522420 x+899099) (5 x+3)^{7/2}}{1280000}-\frac{135817609 (1-2 x)^{3/2} (5 x+3)^{5/2}}{20480000}-\frac{1493993699 (1-2 x)^{3/2} (5 x+3)^{3/2}}{49152000}-\frac{16433930689 (1-2 x)^{3/2} \sqrt{5 x+3}}{131072000}+\frac{180773237579 \sqrt{1-2 x} \sqrt{5 x+3}}{1310720000}+\frac{1988505613369 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1310720000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - 2*x]*(2 + 3*x)^4*(3 + 5*x)^(5/2),x]

[Out]

(180773237579*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1310720000 - (16433930689*(1 - 2*x)^(
3/2)*Sqrt[3 + 5*x])/131072000 - (1493993699*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/491
52000 - (135817609*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/20480000 - (1419*(1 - 2*x)^(
3/2)*(2 + 3*x)^2*(3 + 5*x)^(7/2))/11200 - (3*(1 - 2*x)^(3/2)*(2 + 3*x)^3*(3 + 5*
x)^(7/2))/80 - (3*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2)*(899099 + 522420*x))/1280000 +
(1988505613369*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1310720000*Sqrt[10])

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Rubi in Sympy [A]  time = 26.9205, size = 187, normalized size = 0.93 \[ - \frac{3 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac{7}{2}}}{80} - \frac{1419 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac{7}{2}}}{11200} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{7}{2}} \left (\frac{41140575 x}{2} + \frac{283216185}{8}\right )}{16800000} + \frac{135817609 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{7}{2}}}{51200000} - \frac{1493993699 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{614400000} - \frac{16433930689 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{983040000} - \frac{180773237579 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1310720000} + \frac{1988505613369 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{13107200000} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**4*(3+5*x)**(5/2)*(1-2*x)**(1/2),x)

[Out]

-3*(-2*x + 1)**(3/2)*(3*x + 2)**3*(5*x + 3)**(7/2)/80 - 1419*(-2*x + 1)**(3/2)*(
3*x + 2)**2*(5*x + 3)**(7/2)/11200 - (-2*x + 1)**(3/2)*(5*x + 3)**(7/2)*(4114057
5*x/2 + 283216185/8)/16800000 + 135817609*sqrt(-2*x + 1)*(5*x + 3)**(7/2)/512000
00 - 1493993699*sqrt(-2*x + 1)*(5*x + 3)**(5/2)/614400000 - 16433930689*sqrt(-2*
x + 1)*(5*x + 3)**(3/2)/983040000 - 180773237579*sqrt(-2*x + 1)*sqrt(5*x + 3)/13
10720000 + 1988505613369*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/13107200000

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Mathematica [A]  time = 0.160535, size = 85, normalized size = 0.42 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (6967296000000 x^7+30838579200000 x^6+57746856960000 x^5+58346097408000 x^4+32457421737600 x^3+6882844528480 x^2-3991703112140 x-5973304472091\right )-41758617880749 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{275251200000} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 - 2*x]*(2 + 3*x)^4*(3 + 5*x)^(5/2),x]

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-5973304472091 - 3991703112140*x + 688284452848
0*x^2 + 32457421737600*x^3 + 58346097408000*x^4 + 57746856960000*x^5 + 308385792
00000*x^6 + 6967296000000*x^7) - 41758617880749*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[
1 - 2*x]])/275251200000

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Maple [A]  time = 0.017, size = 172, normalized size = 0.9 \[{\frac{1}{550502400000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 139345920000000\,{x}^{7}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{6}\sqrt{-10\,{x}^{2}-x+3}+1154937139200000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}+1166921948160000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -79834062242800\,x\sqrt{-10\,{x}^{2}-x+3}-119466089441820\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^4*(3+5*x)^(5/2)*(1-2*x)^(1/2),x)

[Out]

1/550502400000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(139345920000000*x^7*(-10*x^2-x+3)^(1
/2)[phone]*x^6*(-10*x^2-x+3)^(1/2)+1154937139200000*x^5*(-10*x^2-x+3)^(
1/2)+1166921948160000*x^4*(-10*x^2-x+3)^(1/2)[phone]*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)-79834062242800*x*(-10*x^2-x+3)^(1/2)-119466089441820*(-10*x^2-x+3)^(1
/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.48985, size = 186, normalized size = 0.93 \[ -\frac{405}{16} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{5} - \frac{49059}{448} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{4} - \frac{739881}{3584} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} - \frac{80346831}{358400} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} - \frac{4513921183}{28672000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{26326737569}{344064000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{16433930689}{65536000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{1988505613369}{26214400000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{16433930689}{1310720000} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(3*x + 2)^4*sqrt(-2*x + 1),x, algorithm="maxima")

[Out]

-405/16*(-10*x^2 - x + 3)^(3/2)*x^5 - 49059/448*(-10*x^2 - x + 3)^(3/2)*x^4 - 73
9881/3584*(-10*x^2 - x + 3)^(3/2)*x^3 - 80346831/358400*(-10*x^2 - x + 3)^(3/2)*
x^2 - 4513921183/28672000*(-10*x^2 - x + 3)^(3/2)*x - 26326737569/344064000*(-10
*x^2 - x + 3)^(3/2) + 16433930689/65536000*sqrt(-10*x^2 - x + 3)*x - 19885056133
69/26214400000*sqrt(10)*arcsin(-20/11*x - 1/11) + 16433930689/1310720000*sqrt(-1
0*x^2 - x + 3)

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Fricas [A]  time = 0.219694, size = 117, normalized size = 0.58 \[ \frac{1}{550502400000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (6967296000000 \, x^{7} + 30838579200000 \, x^{6} + 57746856960000 \, x^{5} + 58346097408000 \, x^{4} + 32457421737600 \, x^{3} + 6882844528480 \, x^{2} - 3991703112140 \, x - 5973304472091\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 41758617880749 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(3*x + 2)^4*sqrt(-2*x + 1),x, algorithm="fricas")

[Out]

1/550502400000*sqrt(10)*(2*sqrt(10)*(6967296000000*x^7 + 30838579200000*x^6 + 57
746856960000*x^5 + 58346097408000*x^4 + 32457421737600*x^3 + 6882844528480*x^2 -
3991703112140*x - 5973304472091)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 41758617880749*
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((2+3*x)**4*(3+5*x)**(5/2)*(1-2*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.276704, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(3*x + 2)^4*sqrt(-2*x + 1),x, algorithm="giac")

[Out]

Done