3.1800 \(\int \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^3 \, dx\)

Optimal. Leaf size=79 \[ \frac{1125}{416} (1-2 x)^{13/2}-\frac{12675}{352} (1-2 x)^{11/2}+\frac{28555}{144} (1-2 x)^{9/2}-\frac{64317}{112} (1-2 x)^{7/2}+\frac{144837}{160} (1-2 x)^{5/2}-\frac{65219}{96} (1-2 x)^{3/2} \]

[Out]

(-65219*(1 - 2*x)^(3/2))/96 + (144837*(1 - 2*x)^(5/2))/160 - (64317*(1 - 2*x)^(7
/2))/112 + (28555*(1 - 2*x)^(9/2))/144 - (12675*(1 - 2*x)^(11/2))/352 + (1125*(1
- 2*x)^(13/2))/416

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Rubi [A]  time = 0.0624584, 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 \[ \frac{1125}{416} (1-2 x)^{13/2}-\frac{12675}{352} (1-2 x)^{11/2}+\frac{28555}{144} (1-2 x)^{9/2}-\frac{64317}{112} (1-2 x)^{7/2}+\frac{144837}{160} (1-2 x)^{5/2}-\frac{65219}{96} (1-2 x)^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

(-65219*(1 - 2*x)^(3/2))/96 + (144837*(1 - 2*x)^(5/2))/160 - (64317*(1 - 2*x)^(7
/2))/112 + (28555*(1 - 2*x)^(9/2))/144 - (12675*(1 - 2*x)^(11/2))/352 + (1125*(1
- 2*x)^(13/2))/416

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Rubi in Sympy [A]  time = 9.10429, size = 70, normalized size = 0.89 \[ \frac{1125 \left (- 2 x + 1\right )^{\frac{13}{2}}}{416} - \frac{12675 \left (- 2 x + 1\right )^{\frac{11}{2}}}{352} + \frac{28555 \left (- 2 x + 1\right )^{\frac{9}{2}}}{144} - \frac{64317 \left (- 2 x + 1\right )^{\frac{7}{2}}}{112} + \frac{144837 \left (- 2 x + 1\right )^{\frac{5}{2}}}{160} - \frac{65219 \left (- 2 x + 1\right )^{\frac{3}{2}}}{96} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1125*(-2*x + 1)**(13/2)/416 - 12675*(-2*x + 1)**(11/2)/352 + 28555*(-2*x + 1)**(
9/2)/144 - 64317*(-2*x + 1)**(7/2)/112 + 144837*(-2*x + 1)**(5/2)/160 - 65219*(-
2*x + 1)**(3/2)/96

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Mathematica [A]  time = 0.0533175, size = 38, normalized size = 0.48 \[ -\frac{(1-2 x)^{3/2} \left (3898125 x^5+16206750 x^4+29300075 x^3+30337080 x^2+19918608 x+8261156\right )}{45045} \]

Antiderivative was successfully verified.

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

[Out]

-((1 - 2*x)^(3/2)*(8261156 + 19918608*x + 30337080*x^2 + 29300075*x^3 + 16206750
*x^4 + 3898125*x^5))/45045

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Maple [A]  time = 0.006, size = 35, normalized size = 0.4 \[ -{\frac{3898125\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+8261156}{45045} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/45045*(3898125*x^5+16206750*x^4+29300075*x^3+30337080*x^2+19918608*x+8261156)
*(1-2*x)^(3/2)

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Maxima [A]  time = 1.33592, size = 74, normalized size = 0.94 \[ \frac{1125}{416} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} - \frac{12675}{352} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{28555}{144} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{64317}{112} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{144837}{160} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{65219}{96} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1125/416*(-2*x + 1)^(13/2) - 12675/352*(-2*x + 1)^(11/2) + 28555/144*(-2*x + 1)^
(9/2) - 64317/112*(-2*x + 1)^(7/2) + 144837/160*(-2*x + 1)^(5/2) - 65219/96*(-2*
x + 1)^(3/2)

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Fricas [A]  time = 0.207803, size = 53, normalized size = 0.67 \[ \frac{1}{45045} \,{\left (7796250 \, x^{6} + 28515375 \, x^{5} + 42393400 \, x^{4} + 31374085 \, x^{3} + 9500136 \, x^{2} - 3396296 \, x - 8261156\right )} \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/45045*(7796250*x^6 + 28515375*x^5 + 42393400*x^4 + 31374085*x^3 + 9500136*x^2
- 3396296*x - 8261156)*sqrt(-2*x + 1)

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Sympy [A]  time = 3.13149, size = 70, normalized size = 0.89 \[ \frac{1125 \left (- 2 x + 1\right )^{\frac{13}{2}}}{416} - \frac{12675 \left (- 2 x + 1\right )^{\frac{11}{2}}}{352} + \frac{28555 \left (- 2 x + 1\right )^{\frac{9}{2}}}{144} - \frac{64317 \left (- 2 x + 1\right )^{\frac{7}{2}}}{112} + \frac{144837 \left (- 2 x + 1\right )^{\frac{5}{2}}}{160} - \frac{65219 \left (- 2 x + 1\right )^{\frac{3}{2}}}{96} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1125*(-2*x + 1)**(13/2)/416 - 12675*(-2*x + 1)**(11/2)/352 + 28555*(-2*x + 1)**(
9/2)/144 - 64317*(-2*x + 1)**(7/2)/112 + 144837*(-2*x + 1)**(5/2)/160 - 65219*(-
2*x + 1)**(3/2)/96

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GIAC/XCAS [A]  time = 0.213566, size = 122, normalized size = 1.54 \[ \frac{1125}{416} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} + \frac{12675}{352} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{28555}{144} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{64317}{112} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{144837}{160} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{65219}{96} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1125/416*(2*x - 1)^6*sqrt(-2*x + 1) + 12675/352*(2*x - 1)^5*sqrt(-2*x + 1) + 285
55/144*(2*x - 1)^4*sqrt(-2*x + 1) + 64317/112*(2*x - 1)^3*sqrt(-2*x + 1) + 14483
7/160*(2*x - 1)^2*sqrt(-2*x + 1) - 65219/96*(-2*x + 1)^(3/2)