3.2140 \(\int \frac{(2+3 x)^5 (3+5 x)^3}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=118 \[ -\frac{30375 (1-2 x)^{13/2}}{3328}+\frac{277425 (1-2 x)^{11/2}}{1408}-\frac{246315}{128} (1-2 x)^{9/2}+\frac{10121229}{896} (1-2 x)^{7/2}-\frac{2887773}{64} (1-2 x)^{5/2}+\frac{52725715}{384} (1-2 x)^{3/2}-\frac{60160485}{128} \sqrt{1-2 x}-\frac{39220335}{128 \sqrt{1-2 x}}+\frac{22370117}{768 (1-2 x)^{3/2}} \]

[Out]

22370117/(768*(1 - 2*x)^(3/2)) - 39220335/(128*Sqrt[1 - 2*x]) - (60160485*Sqrt[1
- 2*x])/128 + (52725715*(1 - 2*x)^(3/2))/384 - (2887773*(1 - 2*x)^(5/2))/64 + (
10121229*(1 - 2*x)^(7/2))/896 - (246315*(1 - 2*x)^(9/2))/128 + (277425*(1 - 2*x)
^(11/2))/1408 - (30375*(1 - 2*x)^(13/2))/3328

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Rubi [A]  time = 0.0914841, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ -\frac{30375 (1-2 x)^{13/2}}{3328}+\frac{277425 (1-2 x)^{11/2}}{1408}-\frac{246315}{128} (1-2 x)^{9/2}+\frac{10121229}{896} (1-2 x)^{7/2}-\frac{2887773}{64} (1-2 x)^{5/2}+\frac{52725715}{384} (1-2 x)^{3/2}-\frac{60160485}{128} \sqrt{1-2 x}-\frac{39220335}{128 \sqrt{1-2 x}}+\frac{22370117}{768 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

22370117/(768*(1 - 2*x)^(3/2)) - 39220335/(128*Sqrt[1 - 2*x]) - (60160485*Sqrt[1
- 2*x])/128 + (52725715*(1 - 2*x)^(3/2))/384 - (2887773*(1 - 2*x)^(5/2))/64 + (
10121229*(1 - 2*x)^(7/2))/896 - (246315*(1 - 2*x)^(9/2))/128 + (277425*(1 - 2*x)
^(11/2))/1408 - (30375*(1 - 2*x)^(13/2))/3328

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Rubi in Sympy [A]  time = 12.398, size = 105, normalized size = 0.89 \[ - \frac{30375 \left (- 2 x + 1\right )^{\frac{13}{2}}}{3328} + \frac{277425 \left (- 2 x + 1\right )^{\frac{11}{2}}}{1408} - \frac{246315 \left (- 2 x + 1\right )^{\frac{9}{2}}}{128} + \frac{10121229 \left (- 2 x + 1\right )^{\frac{7}{2}}}{896} - \frac{2887773 \left (- 2 x + 1\right )^{\frac{5}{2}}}{64} + \frac{52725715 \left (- 2 x + 1\right )^{\frac{3}{2}}}{384} - \frac{60160485 \sqrt{- 2 x + 1}}{128} - \frac{39220335}{128 \sqrt{- 2 x + 1}} + \frac{22370117}{768 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-30375*(-2*x + 1)**(13/2)/3328 + 277425*(-2*x + 1)**(11/2)/1408 - 246315*(-2*x +
1)**(9/2)/128 + 10121229*(-2*x + 1)**(7/2)/896 - 2887773*(-2*x + 1)**(5/2)/64 +
52725715*(-2*x + 1)**(3/2)/384 - 60160485*sqrt(-2*x + 1)/128 - 39220335/(128*sq
rt(-2*x + 1)) + 22370117/(768*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.0659168, size = 53, normalized size = 0.45 \[ -\frac{7016625 x^8+47670525 x^7+153878760 x^6+324478899 x^5+540496701 x^4+905206628 x^3+2892917004 x^2-5818266408 x+1938557272}{3003 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(1938557272 - 5818266408*x + 2892917004*x^2 + 905206628*x^3 + 540496701*x^4 + 3
24478899*x^5 + 153878760*x^6 + 47670525*x^7 + 7016625*x^8)/(3003*(1 - 2*x)^(3/2)
)

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Maple [A]  time = 0.006, size = 50, normalized size = 0.4 \[ -{\frac{7016625\,{x}^{8}[phone]\,{x}^{7}[phone]\,{x}^{6}[phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-5818266408\,x+1938557272}{3003} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3003*(7016625*x^8+47670525*x^7+153878760*x^6+324478899*x^5+540496701*x^4+9052
06628*x^3+2892917004*x^2-5818266408*x+1938557272)/(1-2*x)^(3/2)

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Maxima [A]  time = 1.33201, size = 105, normalized size = 0.89 \[ -\frac{30375}{3328} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} + \frac{277425}{1408} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - \frac{246315}{128} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{10121229}{896} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{2887773}{64} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{52725715}{384} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{60160485}{128} \, \sqrt{-2 \, x + 1} + \frac{290521 \,{\left (1620 \, x - 733\right )}}{768 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-30375/3328*(-2*x + 1)^(13/2) + 277425/1408*(-2*x + 1)^(11/2) - 246315/128*(-2*x
+ 1)^(9/2) + 10121229/896*(-2*x + 1)^(7/2) - 2887773/64*(-2*x + 1)^(5/2) + 5272
5715/384*(-2*x + 1)^(3/2) - 60160485/128*sqrt(-2*x + 1) + 290521/768*(1620*x - 7
33)/(-2*x + 1)^(3/2)

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Fricas [A]  time = 0.211623, size = 76, normalized size = 0.64 \[ \frac{7016625 \, x^{8} + 47670525 \, x^{7} + 153878760 \, x^{6} + 324478899 \, x^{5} + 540496701 \, x^{4} + 905206628 \, x^{3} + 2892917004 \, x^{2} - 5818266408 \, x + 1938557272}{3003 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3003*(7016625*x^8 + 47670525*x^7 + 153878760*x^6 + 324478899*x^5 + 540496701*x
^4 + 905206628*x^3 + 2892917004*x^2 - 5818266408*x + 1938557272)/((2*x - 1)*sqrt
(-2*x + 1))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (3 x + 2\right )^{5} \left (5 x + 3\right )^{3}}{\left (- 2 x + 1\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((3*x + 2)**5*(5*x + 3)**3/(-2*x + 1)**(5/2), x)

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GIAC/XCAS [A]  time = 0.216024, size = 162, normalized size = 1.37 \[ -\frac{30375}{3328} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} - \frac{277425}{1408} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - \frac{246315}{128} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{10121229}{896} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{2887773}{64} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{52725715}{384} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{60160485}{128} \, \sqrt{-2 \, x + 1} - \frac{290521 \,{\left (1620 \, x - 733\right )}}{768 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-30375/3328*(2*x - 1)^6*sqrt(-2*x + 1) - 277425/1408*(2*x - 1)^5*sqrt(-2*x + 1)
- 246315/128*(2*x - 1)^4*sqrt(-2*x + 1) - 10121229/896*(2*x - 1)^3*sqrt(-2*x + 1
) - 2887773/64*(2*x - 1)^2*sqrt(-2*x + 1) + 52725715/384*(-2*x + 1)^(3/2) - 6016
0485/128*sqrt(-2*x + 1) - 290521/768*(1620*x - 733)/((2*x - 1)*sqrt(-2*x + 1))