3.2096 \(\int \frac{(2+3 x)^2 (3+5 x)^3}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=79 \[ \frac{125}{32} (1-2 x)^{9/2}-\frac{12675}{224} (1-2 x)^{7/2}+\frac{5711}{16} (1-2 x)^{5/2}-\frac{21439}{16} (1-2 x)^{3/2}+\frac{144837}{32} \sqrt{1-2 x}+\frac{65219}{32 \sqrt{1-2 x}} \]

[Out]

65219/(32*Sqrt[1 - 2*x]) + (144837*Sqrt[1 - 2*x])/32 - (21439*(1 - 2*x)^(3/2))/16 + (5711*(1 - 2*x)^(5/2))/16
- (12675*(1 - 2*x)^(7/2))/224 + (125*(1 - 2*x)^(9/2))/32

________________________________________________________________________________________

Rubi [A]  time = 0.0161979, 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, Rules used = {88} \[ \frac{125}{32} (1-2 x)^{9/2}-\frac{12675}{224} (1-2 x)^{7/2}+\frac{5711}{16} (1-2 x)^{5/2}-\frac{21439}{16} (1-2 x)^{3/2}+\frac{144837}{32} \sqrt{1-2 x}+\frac{65219}{32 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

65219/(32*Sqrt[1 - 2*x]) + (144837*Sqrt[1 - 2*x])/32 - (21439*(1 - 2*x)^(3/2))/16 + (5711*(1 - 2*x)^(5/2))/16
- (12675*(1 - 2*x)^(7/2))/224 + (125*(1 - 2*x)^(9/2))/32

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{(2+3 x)^2 (3+5 x)^3}{(1-2 x)^{3/2}} \, dx &=\int \left (\frac{65219}{32 (1-2 x)^{3/2}}-\frac{144837}{32 \sqrt{1-2 x}}+\frac{64317}{16} \sqrt{1-2 x}-\frac{28555}{16} (1-2 x)^{3/2}+\frac{12675}{32} (1-2 x)^{5/2}-\frac{1125}{32} (1-2 x)^{7/2}\right ) \, dx\\ &=\frac{65219}{32 \sqrt{1-2 x}}+\frac{144837}{32} \sqrt{1-2 x}-\frac{21439}{16} (1-2 x)^{3/2}+\frac{5711}{16} (1-2 x)^{5/2}-\frac{12675}{224} (1-2 x)^{7/2}+\frac{125}{32} (1-2 x)^{9/2}\\ \end{align*}

Mathematica [A]  time = 0.0159379, size = 38, normalized size = 0.48 \[ \frac{-875 x^5-4150 x^4-9501 x^3-15948 x^2-37944 x+38700}{7 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(38700 - 37944*x - 15948*x^2 - 9501*x^3 - 4150*x^4 - 875*x^5)/(7*Sqrt[1 - 2*x])

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 35, normalized size = 0.4 \begin{align*} -{\frac{875\,{x}^{5}+4150\,{x}^{4}+9501\,{x}^{3}+15948\,{x}^{2}+37944\,x-38700}{7}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*(875*x^5+4150*x^4+9501*x^3+15948*x^2+37944*x-38700)/(1-2*x)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.05822, size = 74, normalized size = 0.94 \begin{align*} \frac{125}{32} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{12675}{224} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{5711}{16} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{21439}{16} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{144837}{32} \, \sqrt{-2 \, x + 1} + \frac{65219}{32 \, \sqrt{-2 \, x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125/32*(-2*x + 1)^(9/2) - 12675/224*(-2*x + 1)^(7/2) + 5711/16*(-2*x + 1)^(5/2) - 21439/16*(-2*x + 1)^(3/2) +
144837/32*sqrt(-2*x + 1) + 65219/32/sqrt(-2*x + 1)

________________________________________________________________________________________

Fricas [A]  time = 1.58219, size = 124, normalized size = 1.57 \begin{align*} \frac{{\left (875 \, x^{5} + 4150 \, x^{4} + 9501 \, x^{3} + 15948 \, x^{2} + 37944 \, x - 38700\right )} \sqrt{-2 \, x + 1}}{7 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*(875*x^5 + 4150*x^4 + 9501*x^3 + 15948*x^2 + 37944*x - 38700)*sqrt(-2*x + 1)/(2*x - 1)

________________________________________________________________________________________

Sympy [A]  time = 22.1413, size = 70, normalized size = 0.89 \begin{align*} \frac{125 \left (1 - 2 x\right )^{\frac{9}{2}}}{32} - \frac{12675 \left (1 - 2 x\right )^{\frac{7}{2}}}{224} + \frac{5711 \left (1 - 2 x\right )^{\frac{5}{2}}}{16} - \frac{21439 \left (1 - 2 x\right )^{\frac{3}{2}}}{16} + \frac{144837 \sqrt{1 - 2 x}}{32} + \frac{65219}{32 \sqrt{1 - 2 x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125*(1 - 2*x)**(9/2)/32 - 12675*(1 - 2*x)**(7/2)/224 + 5711*(1 - 2*x)**(5/2)/16 - 21439*(1 - 2*x)**(3/2)/16 +
144837*sqrt(1 - 2*x)/32 + 65219/(32*sqrt(1 - 2*x))

________________________________________________________________________________________

Giac [A]  time = 2.41436, size = 103, normalized size = 1.3 \begin{align*} \frac{125}{32} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{12675}{224} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{5711}{16} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{21439}{16} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{144837}{32} \, \sqrt{-2 \, x + 1} + \frac{65219}{32 \, \sqrt{-2 \, x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125/32*(2*x - 1)^4*sqrt(-2*x + 1) + 12675/224*(2*x - 1)^3*sqrt(-2*x + 1) + 5711/16*(2*x - 1)^2*sqrt(-2*x + 1)
- 21439/16*(-2*x + 1)^(3/2) + 144837/32*sqrt(-2*x + 1) + 65219/32/sqrt(-2*x + 1)