3.2068 \(\int \frac{(2+3 x)^3 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=79 \[ \frac{75}{32} (1-2 x)^{9/2}-\frac{7695}{224} (1-2 x)^{7/2}+\frac{17541}{80} (1-2 x)^{5/2}-\frac{39977}{48} (1-2 x)^{3/2}+\frac{91091}{32} \sqrt{1-2 x}+\frac{41503}{32 \sqrt{1-2 x}} \]

[Out]

41503/(32*Sqrt[1 - 2*x]) + (91091*Sqrt[1 - 2*x])/32 - (39977*(1 - 2*x)^(3/2))/48
+ (17541*(1 - 2*x)^(5/2))/80 - (7695*(1 - 2*x)^(7/2))/224 + (75*(1 - 2*x)^(9/2)
)/32

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Rubi [A]  time = 0.0708148, antiderivative size = 79, 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 \[ \frac{75}{32} (1-2 x)^{9/2}-\frac{7695}{224} (1-2 x)^{7/2}+\frac{17541}{80} (1-2 x)^{5/2}-\frac{39977}{48} (1-2 x)^{3/2}+\frac{91091}{32} \sqrt{1-2 x}+\frac{41503}{32 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

41503/(32*Sqrt[1 - 2*x]) + (91091*Sqrt[1 - 2*x])/32 - (39977*(1 - 2*x)^(3/2))/48
+ (17541*(1 - 2*x)^(5/2))/80 - (7695*(1 - 2*x)^(7/2))/224 + (75*(1 - 2*x)^(9/2)
)/32

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Rubi in Sympy [A]  time = 10.1163, size = 70, normalized size = 0.89 \[ \frac{75 \left (- 2 x + 1\right )^{\frac{9}{2}}}{32} - \frac{7695 \left (- 2 x + 1\right )^{\frac{7}{2}}}{224} + \frac{17541 \left (- 2 x + 1\right )^{\frac{5}{2}}}{80} - \frac{39977 \left (- 2 x + 1\right )^{\frac{3}{2}}}{48} + \frac{91091 \sqrt{- 2 x + 1}}{32} + \frac{41503}{32 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

75*(-2*x + 1)**(9/2)/32 - 7695*(-2*x + 1)**(7/2)/224 + 17541*(-2*x + 1)**(5/2)/8
0 - 39977*(-2*x + 1)**(3/2)/48 + 91091*sqrt(-2*x + 1)/32 + 41503/(32*sqrt(-2*x +
1))

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Mathematica [A]  time = 0.0504687, size = 38, normalized size = 0.48 \[ \frac{-7875 x^5-38025 x^4-88443 x^3-150253 x^2-359726 x+367286}{105 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(367286 - 359726*x - 150253*x^2 - 88443*x^3 - 38025*x^4 - 7875*x^5)/(105*Sqrt[1
- 2*x])

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Maple [A]  time = 0.006, size = 35, normalized size = 0.4 \[ -{\frac{7875\,{x}^{5}+38025\,{x}^{4}+88443\,{x}^{3}+150253\,{x}^{2}+359726\,x-367286}{105}{\frac{1}{\sqrt{1-2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/105*(7875*x^5+38025*x^4+88443*x^3+150253*x^2+359726*x-367286)/(1-2*x)^(1/2)

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Maxima [A]  time = 1.35021, size = 74, normalized size = 0.94 \[ \frac{75}{32} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{7695}{224} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{17541}{80} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{39977}{48} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{91091}{32} \, \sqrt{-2 \, x + 1} + \frac{41503}{32 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

75/32*(-2*x + 1)^(9/2) - 7695/224*(-2*x + 1)^(7/2) + 17541/80*(-2*x + 1)^(5/2) -
39977/48*(-2*x + 1)^(3/2) + 91091/32*sqrt(-2*x + 1) + 41503/32/sqrt(-2*x + 1)

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Fricas [A]  time = 0.207585, size = 46, normalized size = 0.58 \[ -\frac{7875 \, x^{5} + 38025 \, x^{4} + 88443 \, x^{3} + 150253 \, x^{2} + 359726 \, x - 367286}{105 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/105*(7875*x^5 + 38025*x^4 + 88443*x^3 + 150253*x^2 + 359726*x - 367286)/sqrt(
-2*x + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.21052, size = 103, normalized size = 1.3 \[ \frac{75}{32} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{7695}{224} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{17541}{80} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{39977}{48} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{91091}{32} \, \sqrt{-2 \, x + 1} + \frac{41503}{32 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

75/32*(2*x - 1)^4*sqrt(-2*x + 1) + 7695/224*(2*x - 1)^3*sqrt(-2*x + 1) + 17541/8
0*(2*x - 1)^2*sqrt(-2*x + 1) - 39977/48*(-2*x + 1)^(3/2) + 91091/32*sqrt(-2*x +
1) + 41503/32/sqrt(-2*x + 1)