3.1999 \(\int \frac{(2+3 x) (3+5 x)^2}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=53 \[ \frac{75}{56} (1-2 x)^{7/2}-\frac{101}{8} (1-2 x)^{5/2}+\frac{1133}{24} (1-2 x)^{3/2}-\frac{847}{8} \sqrt{1-2 x} \]

[Out]

(-847*Sqrt[1 - 2*x])/8 + (1133*(1 - 2*x)^(3/2))/24 - (101*(1 - 2*x)^(5/2))/8 + (
75*(1 - 2*x)^(7/2))/56

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Rubi [A]  time = 0.0500309, antiderivative size = 53, 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 \[ \frac{75}{56} (1-2 x)^{7/2}-\frac{101}{8} (1-2 x)^{5/2}+\frac{1133}{24} (1-2 x)^{3/2}-\frac{847}{8} \sqrt{1-2 x} \]

Antiderivative was successfully verified.

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

[Out]

(-847*Sqrt[1 - 2*x])/8 + (1133*(1 - 2*x)^(3/2))/24 - (101*(1 - 2*x)^(5/2))/8 + (
75*(1 - 2*x)^(7/2))/56

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Rubi in Sympy [A]  time = 7.00804, size = 46, normalized size = 0.87 \[ \frac{75 \left (- 2 x + 1\right )^{\frac{7}{2}}}{56} - \frac{101 \left (- 2 x + 1\right )^{\frac{5}{2}}}{8} + \frac{1133 \left (- 2 x + 1\right )^{\frac{3}{2}}}{24} - \frac{847 \sqrt{- 2 x + 1}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

75*(-2*x + 1)**(7/2)/56 - 101*(-2*x + 1)**(5/2)/8 + 1133*(-2*x + 1)**(3/2)/24 -
847*sqrt(-2*x + 1)/8

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Mathematica [A]  time = 0.030088, size = 28, normalized size = 0.53 \[ -\frac{1}{21} \sqrt{1-2 x} \left (225 x^3+723 x^2+1091 x+1469\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(1469 + 1091*x + 723*x^2 + 225*x^3))/21

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Maple [A]  time = 0.006, size = 25, normalized size = 0.5 \[ -{\frac{225\,{x}^{3}+723\,{x}^{2}+1091\,x+1469}{21}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/21*(225*x^3+723*x^2+1091*x+1469)*(1-2*x)^(1/2)

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Maxima [A]  time = 1.34657, size = 50, normalized size = 0.94 \[ \frac{75}{56} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{101}{8} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{1133}{24} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{847}{8} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

75/56*(-2*x + 1)^(7/2) - 101/8*(-2*x + 1)^(5/2) + 1133/24*(-2*x + 1)^(3/2) - 847
/8*sqrt(-2*x + 1)

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Fricas [A]  time = 0.222704, size = 32, normalized size = 0.6 \[ -\frac{1}{21} \,{\left (225 \, x^{3} + 723 \, x^{2} + 1091 \, x + 1469\right )} \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/21*(225*x^3 + 723*x^2 + 1091*x + 1469)*sqrt(-2*x + 1)

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Sympy [A]  time = 6.61471, size = 46, normalized size = 0.87 \[ \frac{75 \left (- 2 x + 1\right )^{\frac{7}{2}}}{56} - \frac{101 \left (- 2 x + 1\right )^{\frac{5}{2}}}{8} + \frac{1133 \left (- 2 x + 1\right )^{\frac{3}{2}}}{24} - \frac{847 \sqrt{- 2 x + 1}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

75*(-2*x + 1)**(7/2)/56 - 101*(-2*x + 1)**(5/2)/8 + 1133*(-2*x + 1)**(3/2)/24 -
847*sqrt(-2*x + 1)/8

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GIAC/XCAS [A]  time = 0.210893, size = 69, normalized size = 1.3 \[ -\frac{75}{56} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{101}{8} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{1133}{24} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{847}{8} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-75/56*(2*x - 1)^3*sqrt(-2*x + 1) - 101/8*(2*x - 1)^2*sqrt(-2*x + 1) + 1133/24*(
-2*x + 1)^(3/2) - 847/8*sqrt(-2*x + 1)