3.1776 \(\int \sqrt{1-2 x} (2+3 x)^3 (3+5 x) \, dx\)

Optimal. Leaf size=66 \[ -\frac{135}{176} (1-2 x)^{11/2}+\frac{69}{8} (1-2 x)^{9/2}-\frac{153}{4} (1-2 x)^{7/2}+\frac{3283}{40} (1-2 x)^{5/2}-\frac{3773}{48} (1-2 x)^{3/2} \]

[Out]

(-3773*(1 - 2*x)^(3/2))/48 + (3283*(1 - 2*x)^(5/2))/40 - (153*(1 - 2*x)^(7/2))/4
+ (69*(1 - 2*x)^(9/2))/8 - (135*(1 - 2*x)^(11/2))/176

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Rubi [A]  time = 0.0509167, antiderivative size = 66, 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 \[ -\frac{135}{176} (1-2 x)^{11/2}+\frac{69}{8} (1-2 x)^{9/2}-\frac{153}{4} (1-2 x)^{7/2}+\frac{3283}{40} (1-2 x)^{5/2}-\frac{3773}{48} (1-2 x)^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

(-3773*(1 - 2*x)^(3/2))/48 + (3283*(1 - 2*x)^(5/2))/40 - (153*(1 - 2*x)^(7/2))/4
+ (69*(1 - 2*x)^(9/2))/8 - (135*(1 - 2*x)^(11/2))/176

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Rubi in Sympy [A]  time = 7.84628, size = 58, normalized size = 0.88 \[ - \frac{135 \left (- 2 x + 1\right )^{\frac{11}{2}}}{176} + \frac{69 \left (- 2 x + 1\right )^{\frac{9}{2}}}{8} - \frac{153 \left (- 2 x + 1\right )^{\frac{7}{2}}}{4} + \frac{3283 \left (- 2 x + 1\right )^{\frac{5}{2}}}{40} - \frac{3773 \left (- 2 x + 1\right )^{\frac{3}{2}}}{48} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-135*(-2*x + 1)**(11/2)/176 + 69*(-2*x + 1)**(9/2)/8 - 153*(-2*x + 1)**(7/2)/4 +
3283*(-2*x + 1)**(5/2)/40 - 3773*(-2*x + 1)**(3/2)/48

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Mathematica [A]  time = 0.0303094, size = 38, normalized size = 0.58 \[ \frac{1}{165} \sqrt{1-2 x} \left (4050 x^5+12645 x^4+15075 x^3+7527 x^2-482 x-4442\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(-4442 - 482*x + 7527*x^2 + 15075*x^3 + 12645*x^4 + 4050*x^5))/16
5

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Maple [A]  time = 0.006, size = 30, normalized size = 0.5 \[ -{\frac{2025\,{x}^{4}+7335\,{x}^{3}+11205\,{x}^{2}+9366\,x+4442}{165} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/165*(2025*x^4+7335*x^3+11205*x^2+9366*x+4442)*(1-2*x)^(3/2)

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Maxima [A]  time = 1.354, size = 62, normalized size = 0.94 \[ -\frac{135}{176} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{69}{8} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{153}{4} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{3283}{40} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{3773}{48} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-135/176*(-2*x + 1)^(11/2) + 69/8*(-2*x + 1)^(9/2) - 153/4*(-2*x + 1)^(7/2) + 32
83/40*(-2*x + 1)^(5/2) - 3773/48*(-2*x + 1)^(3/2)

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Fricas [A]  time = 0.206714, size = 46, normalized size = 0.7 \[ \frac{1}{165} \,{\left (4050 \, x^{5} + 12645 \, x^{4} + 15075 \, x^{3} + 7527 \, x^{2} - 482 \, x - 4442\right )} \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/165*(4050*x^5 + 12645*x^4 + 15075*x^3 + 7527*x^2 - 482*x - 4442)*sqrt(-2*x + 1
)

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Sympy [A]  time = 3.207, size = 58, normalized size = 0.88 \[ - \frac{135 \left (- 2 x + 1\right )^{\frac{11}{2}}}{176} + \frac{69 \left (- 2 x + 1\right )^{\frac{9}{2}}}{8} - \frac{153 \left (- 2 x + 1\right )^{\frac{7}{2}}}{4} + \frac{3283 \left (- 2 x + 1\right )^{\frac{5}{2}}}{40} - \frac{3773 \left (- 2 x + 1\right )^{\frac{3}{2}}}{48} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-135*(-2*x + 1)**(11/2)/176 + 69*(-2*x + 1)**(9/2)/8 - 153*(-2*x + 1)**(7/2)/4 +
3283*(-2*x + 1)**(5/2)/40 - 3773*(-2*x + 1)**(3/2)/48

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GIAC/XCAS [A]  time = 0.214987, size = 100, normalized size = 1.52 \[ \frac{135}{176} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{69}{8} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{153}{4} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{3283}{40} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{3773}{48} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

135/176*(2*x - 1)^5*sqrt(-2*x + 1) + 69/8*(2*x - 1)^4*sqrt(-2*x + 1) + 153/4*(2*
x - 1)^3*sqrt(-2*x + 1) + 3283/40*(2*x - 1)^2*sqrt(-2*x + 1) - 3773/48*(-2*x + 1
)^(3/2)