3.2080 \(\int \frac{(2+3 x) (3+5 x)^3}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=66 \[ -\frac{375}{112} (1-2 x)^{7/2}+\frac{335}{8} (1-2 x)^{5/2}-\frac{935}{4} (1-2 x)^{3/2}+\frac{8349}{8} \sqrt{1-2 x}+\frac{9317}{16 \sqrt{1-2 x}} \]

[Out]

9317/(16*Sqrt[1 - 2*x]) + (8349*Sqrt[1 - 2*x])/8 - (935*(1 - 2*x)^(3/2))/4 + (33
5*(1 - 2*x)^(5/2))/8 - (375*(1 - 2*x)^(7/2))/112

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Rubi [A]  time = 0.0552953, 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 \[ -\frac{375}{112} (1-2 x)^{7/2}+\frac{335}{8} (1-2 x)^{5/2}-\frac{935}{4} (1-2 x)^{3/2}+\frac{8349}{8} \sqrt{1-2 x}+\frac{9317}{16 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

9317/(16*Sqrt[1 - 2*x]) + (8349*Sqrt[1 - 2*x])/8 - (935*(1 - 2*x)^(3/2))/4 + (33
5*(1 - 2*x)^(5/2))/8 - (375*(1 - 2*x)^(7/2))/112

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Rubi in Sympy [A]  time = 7.88011, size = 58, normalized size = 0.88 \[ - \frac{375 \left (- 2 x + 1\right )^{\frac{7}{2}}}{112} + \frac{335 \left (- 2 x + 1\right )^{\frac{5}{2}}}{8} - \frac{935 \left (- 2 x + 1\right )^{\frac{3}{2}}}{4} + \frac{8349 \sqrt{- 2 x + 1}}{8} + \frac{9317}{16 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-375*(-2*x + 1)**(7/2)/112 + 335*(-2*x + 1)**(5/2)/8 - 935*(-2*x + 1)**(3/2)/4 +
8349*sqrt(-2*x + 1)/8 + 9317/(16*sqrt(-2*x + 1))

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Mathematica [A]  time = 0.0380085, size = 37, normalized size = 0.56 \[ \frac{\sqrt{1-2 x} \left (375 x^4+1595 x^3+3590 x^2+9637 x-10015\right )}{14 x-7} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(-10015 + 9637*x + 3590*x^2 + 1595*x^3 + 375*x^4))/(-7 + 14*x)

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Maple [A]  time = 0.006, size = 30, normalized size = 0.5 \[ -{\frac{375\,{x}^{4}+1595\,{x}^{3}+3590\,{x}^{2}+9637\,x-10015}{7}{\frac{1}{\sqrt{1-2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/7*(375*x^4+1595*x^3+3590*x^2+9637*x-10015)/(1-2*x)^(1/2)

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Maxima [A]  time = 1.34358, size = 62, normalized size = 0.94 \[ -\frac{375}{112} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{335}{8} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{935}{4} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{8349}{8} \, \sqrt{-2 \, x + 1} + \frac{9317}{16 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-375/112*(-2*x + 1)^(7/2) + 335/8*(-2*x + 1)^(5/2) - 935/4*(-2*x + 1)^(3/2) + 83
49/8*sqrt(-2*x + 1) + 9317/16/sqrt(-2*x + 1)

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Fricas [A]  time = 0.213207, size = 39, normalized size = 0.59 \[ -\frac{375 \, x^{4} + 1595 \, x^{3} + 3590 \, x^{2} + 9637 \, x - 10015}{7 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/7*(375*x^4 + 1595*x^3 + 3590*x^2 + 9637*x - 10015)/sqrt(-2*x + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.223334, size = 81, normalized size = 1.23 \[ \frac{375}{112} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{335}{8} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{935}{4} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{8349}{8} \, \sqrt{-2 \, x + 1} + \frac{9317}{16 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

375/112*(2*x - 1)^3*sqrt(-2*x + 1) + 335/8*(2*x - 1)^2*sqrt(-2*x + 1) - 935/4*(-
2*x + 1)^(3/2) + 8349/8*sqrt(-2*x + 1) + 9317/16/sqrt(-2*x + 1)