3.2052 \(\int \frac{(2+3 x)^7 (3+5 x)}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=118 \[ -\frac{729}{256} (1-2 x)^{15/2}+\frac{101331 (1-2 x)^{13/2}}{1664}-\frac{821583 (1-2 x)^{11/2}}{1408}+\frac{422919}{128} (1-2 x)^{9/2}-\frac{787185}{64} (1-2 x)^{7/2}+\frac{4084101}{128} (1-2 x)^{5/2}-\frac{7882483}{128} (1-2 x)^{3/2}+\frac{15647317}{128} \sqrt{1-2 x}+\frac{9058973}{256 \sqrt{1-2 x}} \]

[Out]

9058973/(256*Sqrt[1 - 2*x]) + (15647317*Sqrt[1 - 2*x])/128 - (7882483*(1 - 2*x)^
(3/2))/128 + (4084101*(1 - 2*x)^(5/2))/128 - (787185*(1 - 2*x)^(7/2))/64 + (4229
19*(1 - 2*x)^(9/2))/128 - (821583*(1 - 2*x)^(11/2))/1408 + (101331*(1 - 2*x)^(13
/2))/1664 - (729*(1 - 2*x)^(15/2))/256

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Rubi [A]  time = 0.0829428, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -\frac{729}{256} (1-2 x)^{15/2}+\frac{101331 (1-2 x)^{13/2}}{1664}-\frac{821583 (1-2 x)^{11/2}}{1408}+\frac{422919}{128} (1-2 x)^{9/2}-\frac{787185}{64} (1-2 x)^{7/2}+\frac{4084101}{128} (1-2 x)^{5/2}-\frac{7882483}{128} (1-2 x)^{3/2}+\frac{15647317}{128} \sqrt{1-2 x}+\frac{9058973}{256 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]  Int[((2 + 3*x)^7*(3 + 5*x))/(1 - 2*x)^(3/2),x]

[Out]

9058973/(256*Sqrt[1 - 2*x]) + (15647317*Sqrt[1 - 2*x])/128 - (7882483*(1 - 2*x)^
(3/2))/128 + (4084101*(1 - 2*x)^(5/2))/128 - (787185*(1 - 2*x)^(7/2))/64 + (4229
19*(1 - 2*x)^(9/2))/128 - (821583*(1 - 2*x)^(11/2))/1408 + (101331*(1 - 2*x)^(13
/2))/1664 - (729*(1 - 2*x)^(15/2))/256

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Rubi in Sympy [A]  time = 12.5871, size = 105, normalized size = 0.89 \[ - \frac{729 \left (- 2 x + 1\right )^{\frac{15}{2}}}{256} + \frac{101331 \left (- 2 x + 1\right )^{\frac{13}{2}}}{1664} - \frac{821583 \left (- 2 x + 1\right )^{\frac{11}{2}}}{1408} + \frac{422919 \left (- 2 x + 1\right )^{\frac{9}{2}}}{128} - \frac{787185 \left (- 2 x + 1\right )^{\frac{7}{2}}}{64} + \frac{4084101 \left (- 2 x + 1\right )^{\frac{5}{2}}}{128} - \frac{7882483 \left (- 2 x + 1\right )^{\frac{3}{2}}}{128} + \frac{15647317 \sqrt{- 2 x + 1}}{128} + \frac{9058973}{256 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**7*(3+5*x)/(1-2*x)**(3/2),x)

[Out]

-729*(-2*x + 1)**(15/2)/256 + 101331*(-2*x + 1)**(13/2)/1664 - 821583*(-2*x + 1)
**(11/2)/1408 + 422919*(-2*x + 1)**(9/2)/128 - 787185*(-2*x + 1)**(7/2)/64 + 408
4101*(-2*x + 1)**(5/2)/128 - 7882483*(-2*x + 1)**(3/2)/128 + 15647317*sqrt(-2*x
+ 1)/128 + 9058973/(256*sqrt(-2*x + 1))

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Mathematica [A]  time = 0.0579659, size = 53, normalized size = 0.45 \[ -\frac{104247 x^8+697653 x^7+2168775 x^6+4220622 x^5+5949090 x^4+6921432 x^3+8106616 x^2+16881328 x-16936240}{143 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]  Integrate[((2 + 3*x)^7*(3 + 5*x))/(1 - 2*x)^(3/2),x]

[Out]

-(-16936240 + 16881328*x + 8106616*x^2 + 6921432*x^3 + 5949090*x^4 + 4220622*x^5
+ 2168775*x^6 + 697653*x^7 + 104247*x^8)/(143*Sqrt[1 - 2*x])

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Maple [A]  time = 0.005, size = 50, normalized size = 0.4 \[ -{\frac{104247\,{x}^{8}+697653\,{x}^{7}+2168775\,{x}^{6}+4220622\,{x}^{5}+5949090\,{x}^{4}+6921432\,{x}^{3}+8106616\,{x}^{2}[phone]\,x-16936240}{143}{\frac{1}{\sqrt{1-2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^7*(3+5*x)/(1-2*x)^(3/2),x)

[Out]

-1/143*(104247*x^8+697653*x^7+2168775*x^6+4220622*x^5+5949090*x^4+6921432*x^3+81
06616*x^2+16881328*x-16936240)/(1-2*x)^(1/2)

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Maxima [A]  time = 1.37361, size = 111, normalized size = 0.94 \[ -\frac{729}{256} \,{\left (-2 \, x + 1\right )}^{\frac{15}{2}} + \frac{101331}{1664} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} - \frac{821583}{1408} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{422919}{128} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{787185}{64} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{4084101}{128} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{7882483}{128} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{15647317}{128} \, \sqrt{-2 \, x + 1} + \frac{9058973}{256 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)*(3*x + 2)^7/(-2*x + 1)^(3/2),x, algorithm="maxima")

[Out]

-729/256*(-2*x + 1)^(15/2) + 101331/1664*(-2*x + 1)^(13/2) - 821583/1408*(-2*x +
1)^(11/2) + 422919/128*(-2*x + 1)^(9/2) - 787185/64*(-2*x + 1)^(7/2) + 4084101/
128*(-2*x + 1)^(5/2) - 7882483/128*(-2*x + 1)^(3/2) + 15647317/128*sqrt(-2*x + 1
) + 9058973/256/sqrt(-2*x + 1)

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Fricas [A]  time = 0.231421, size = 66, normalized size = 0.56 \[ -\frac{104247 \, x^{8} + 697653 \, x^{7} + 2168775 \, x^{6} + 4220622 \, x^{5} + 5949090 \, x^{4} + 6921432 \, x^{3} + 8106616 \, x^{2} + 16881328 \, x - 16936240}{143 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)*(3*x + 2)^7/(-2*x + 1)^(3/2),x, algorithm="fricas")

[Out]

-1/143*(104247*x^8 + 697653*x^7 + 2168775*x^6 + 4220622*x^5 + 5949090*x^4 + 6921
432*x^3 + 8106616*x^2 + 16881328*x - 16936240)/sqrt(-2*x + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (3 x + 2\right )^{7} \left (5 x + 3\right )}{\left (- 2 x + 1\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+3*x)**7*(3+5*x)/(1-2*x)**(3/2),x)

[Out]

Integral((3*x + 2)**7*(5*x + 3)/(-2*x + 1)**(3/2), x)

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GIAC/XCAS [A]  time = 0.21127, size = 167, normalized size = 1.42 \[ \frac{729}{256} \,{\left (2 \, x - 1\right )}^{7} \sqrt{-2 \, x + 1} + \frac{101331}{1664} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} + \frac{821583}{1408} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{422919}{128} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{787185}{64} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{4084101}{128} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{7882483}{128} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{15647317}{128} \, \sqrt{-2 \, x + 1} + \frac{9058973}{256 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)*(3*x + 2)^7/(-2*x + 1)^(3/2),x, algorithm="giac")

[Out]

729/256*(2*x - 1)^7*sqrt(-2*x + 1) + 101331/1664*(2*x - 1)^6*sqrt(-2*x + 1) + 82
1583/1408*(2*x - 1)^5*sqrt(-2*x + 1) + 422919/128*(2*x - 1)^4*sqrt(-2*x + 1) + 7
87185/64*(2*x - 1)^3*sqrt(-2*x + 1) + 4084101/128*(2*x - 1)^2*sqrt(-2*x + 1) - 7
882483/128*(-2*x + 1)^(3/2) + 15647317/128*sqrt(-2*x + 1) + 9058973/256/sqrt(-2*
x + 1)