3.2137 \(\int \frac{(2+3 x)^3 (3+5 x)}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=66 \[ -\frac{27}{16} (1-2 x)^{5/2}+\frac{207}{8} (1-2 x)^{3/2}-\frac{1071}{4} \sqrt{1-2 x}-\frac{3283}{8 \sqrt{1-2 x}}+\frac{3773}{48 (1-2 x)^{3/2}} \]

[Out]

3773/(48*(1 - 2*x)^(3/2)) - 3283/(8*Sqrt[1 - 2*x]) - (1071*Sqrt[1 - 2*x])/4 + (207*(1 - 2*x)^(3/2))/8 - (27*(1
- 2*x)^(5/2))/16

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Rubi [A]  time = 0.0128068, 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, Rules used = {77} \[ -\frac{27}{16} (1-2 x)^{5/2}+\frac{207}{8} (1-2 x)^{3/2}-\frac{1071}{4} \sqrt{1-2 x}-\frac{3283}{8 \sqrt{1-2 x}}+\frac{3773}{48 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

3773/(48*(1 - 2*x)^(3/2)) - 3283/(8*Sqrt[1 - 2*x]) - (1071*Sqrt[1 - 2*x])/4 + (207*(1 - 2*x)^(3/2))/8 - (27*(1
- 2*x)^(5/2))/16

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{(2+3 x)^3 (3+5 x)}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac{3773}{16 (1-2 x)^{5/2}}-\frac{3283}{8 (1-2 x)^{3/2}}+\frac{1071}{4 \sqrt{1-2 x}}-\frac{621}{8} \sqrt{1-2 x}+\frac{135}{16} (1-2 x)^{3/2}\right ) \, dx\\ &=\frac{3773}{48 (1-2 x)^{3/2}}-\frac{3283}{8 \sqrt{1-2 x}}-\frac{1071}{4} \sqrt{1-2 x}+\frac{207}{8} (1-2 x)^{3/2}-\frac{27}{16} (1-2 x)^{5/2}\\ \end{align*}

Mathematica [A]  time = 0.0139399, size = 33, normalized size = 0.5 \[ -\frac{81 x^4+459 x^3+2403 x^2-5250 x+1726}{3 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1726 - 5250*x + 2403*x^2 + 459*x^3 + 81*x^4)/(3*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.003, size = 30, normalized size = 0.5 \begin{align*} -{\frac{81\,{x}^{4}+459\,{x}^{3}+2403\,{x}^{2}-5250\,x+1726}{3} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(81*x^4+459*x^3+2403*x^2-5250*x+1726)/(1-2*x)^(3/2)

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Maxima [A]  time = 1.09315, size = 57, normalized size = 0.86 \begin{align*} -\frac{27}{16} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{207}{8} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{1071}{4} \, \sqrt{-2 \, x + 1} + \frac{49 \,{\left (804 \, x - 325\right )}}{48 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/16*(-2*x + 1)^(5/2) + 207/8*(-2*x + 1)^(3/2) - 1071/4*sqrt(-2*x + 1) + 49/48*(804*x - 325)/(-2*x + 1)^(3/2
)

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Fricas [A]  time = 1.55004, size = 115, normalized size = 1.74 \begin{align*} -\frac{{\left (81 \, x^{4} + 459 \, x^{3} + 2403 \, x^{2} - 5250 \, x + 1726\right )} \sqrt{-2 \, x + 1}}{3 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(81*x^4 + 459*x^3 + 2403*x^2 - 5250*x + 1726)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

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Sympy [A]  time = 15.0776, size = 58, normalized size = 0.88 \begin{align*} - \frac{27 \left (1 - 2 x\right )^{\frac{5}{2}}}{16} + \frac{207 \left (1 - 2 x\right )^{\frac{3}{2}}}{8} - \frac{1071 \sqrt{1 - 2 x}}{4} - \frac{3283}{8 \sqrt{1 - 2 x}} + \frac{3773}{48 \left (1 - 2 x\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27*(1 - 2*x)**(5/2)/16 + 207*(1 - 2*x)**(3/2)/8 - 1071*sqrt(1 - 2*x)/4 - 3283/(8*sqrt(1 - 2*x)) + 3773/(48*(1
- 2*x)**(3/2))

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Giac [A]  time = 1.70701, size = 76, normalized size = 1.15 \begin{align*} -\frac{27}{16} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{207}{8} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{1071}{4} \, \sqrt{-2 \, x + 1} - \frac{49 \,{\left (804 \, x - 325\right )}}{48 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/16*(2*x - 1)^2*sqrt(-2*x + 1) + 207/8*(-2*x + 1)^(3/2) - 1071/4*sqrt(-2*x + 1) - 49/48*(804*x - 325)/((2*x
- 1)*sqrt(-2*x + 1))