3.2056 \(\int \frac{(2+3 x)^3 (3+5 x)}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=66 \[ -\frac{135}{112} (1-2 x)^{7/2}+\frac{621}{40} (1-2 x)^{5/2}-\frac{357}{4} (1-2 x)^{3/2}+\frac{3283}{8} \sqrt{1-2 x}+\frac{3773}{16 \sqrt{1-2 x}} \]

[Out]

3773/(16*Sqrt[1 - 2*x]) + (3283*Sqrt[1 - 2*x])/8 - (357*(1 - 2*x)^(3/2))/4 + (62
1*(1 - 2*x)^(5/2))/40 - (135*(1 - 2*x)^(7/2))/112

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Rubi [A]  time = 0.0590695, 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 \[ -\frac{135}{112} (1-2 x)^{7/2}+\frac{621}{40} (1-2 x)^{5/2}-\frac{357}{4} (1-2 x)^{3/2}+\frac{3283}{8} \sqrt{1-2 x}+\frac{3773}{16 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

3773/(16*Sqrt[1 - 2*x]) + (3283*Sqrt[1 - 2*x])/8 - (357*(1 - 2*x)^(3/2))/4 + (62
1*(1 - 2*x)^(5/2))/40 - (135*(1 - 2*x)^(7/2))/112

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Rubi in Sympy [A]  time = 8.20089, size = 58, normalized size = 0.88 \[ - \frac{135 \left (- 2 x + 1\right )^{\frac{7}{2}}}{112} + \frac{621 \left (- 2 x + 1\right )^{\frac{5}{2}}}{40} - \frac{357 \left (- 2 x + 1\right )^{\frac{3}{2}}}{4} + \frac{3283 \sqrt{- 2 x + 1}}{8} + \frac{3773}{16 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-135*(-2*x + 1)**(7/2)/112 + 621*(-2*x + 1)**(5/2)/40 - 357*(-2*x + 1)**(3/2)/4
+ 3283*sqrt(-2*x + 1)/8 + 3773/(16*sqrt(-2*x + 1))

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Mathematica [A]  time = 0.038557, size = 37, normalized size = 0.56 \[ \frac{\sqrt{1-2 x} \left (675 x^4+2997 x^3+6987 x^2+19154 x-19994\right )}{70 x-35} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(-19994 + 19154*x + 6987*x^2 + 2997*x^3 + 675*x^4))/(-35 + 70*x)

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Maple [A]  time = 0.006, size = 30, normalized size = 0.5 \[ -{\frac{675\,{x}^{4}+2997\,{x}^{3}+6987\,{x}^{2}+19154\,x-19994}{35}{\frac{1}{\sqrt{1-2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/35*(675*x^4+2997*x^3+6987*x^2+19154*x-19994)/(1-2*x)^(1/2)

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Maxima [A]  time = 1.3357, size = 62, normalized size = 0.94 \[ -\frac{135}{112} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{621}{40} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{357}{4} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{3283}{8} \, \sqrt{-2 \, x + 1} + \frac{3773}{16 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-135/112*(-2*x + 1)^(7/2) + 621/40*(-2*x + 1)^(5/2) - 357/4*(-2*x + 1)^(3/2) + 3
283/8*sqrt(-2*x + 1) + 3773/16/sqrt(-2*x + 1)

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Fricas [A]  time = 0.211663, size = 39, normalized size = 0.59 \[ -\frac{675 \, x^{4} + 2997 \, x^{3} + 6987 \, x^{2} + 19154 \, x - 19994}{35 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/35*(675*x^4 + 2997*x^3 + 6987*x^2 + 19154*x - 19994)/sqrt(-2*x + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.210035, size = 81, normalized size = 1.23 \[ \frac{135}{112} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{621}{40} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{357}{4} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{3283}{8} \, \sqrt{-2 \, x + 1} + \frac{3773}{16 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

135/112*(2*x - 1)^3*sqrt(-2*x + 1) + 621/40*(2*x - 1)^2*sqrt(-2*x + 1) - 357/4*(
-2*x + 1)^(3/2) + 3283/8*sqrt(-2*x + 1) + 3773/16/sqrt(-2*x + 1)