3.2073 \(\int \frac{(2+3 x)^3 (3+5 x)}{(1-2 x)^{3/2}} \, dx\)

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 + (621*(1 - 2*x)^(5/2))/40 - (135*(
1 - 2*x)^(7/2))/112

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Rubi [A]  time = 0.0119846, 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} \[ -\frac{135}{112} (1-2 x)^{7/2}+\frac{621}{40} (1-2 x)^{5/2}-\frac{357}{4} (1-2 x)^{3/2}+\frac{3283}{8} \sqrt{1-2 x}+\frac{3773}{16 \sqrt{1-2 x}} \]

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 + (621*(1 - 2*x)^(5/2))/40 - (135*(
1 - 2*x)^(7/2))/112

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)^{3/2}} \, dx &=\int \left (\frac{3773}{16 (1-2 x)^{3/2}}-\frac{3283}{8 \sqrt{1-2 x}}+\frac{1071}{4} \sqrt{1-2 x}-\frac{621}{8} (1-2 x)^{3/2}+\frac{135}{16} (1-2 x)^{5/2}\right ) \, dx\\ &=\frac{3773}{16 \sqrt{1-2 x}}+\frac{3283}{8} \sqrt{1-2 x}-\frac{357}{4} (1-2 x)^{3/2}+\frac{621}{40} (1-2 x)^{5/2}-\frac{135}{112} (1-2 x)^{7/2}\\ \end{align*}

Mathematica [A]  time = 0.0127104, size = 33, normalized size = 0.5 \[ \frac{-675 x^4-2997 x^3-6987 x^2-19154 x+19994}{35 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.002, size = 30, normalized size = 0.5 \begin{align*} -{\frac{675\,{x}^{4}+2997\,{x}^{3}+6987\,{x}^{2}+19154\,x-19994}{35}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}

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 = 3.14171, size = 62, normalized size = 0.94 \begin{align*} -\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}} \end{align*}

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, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.67488, size = 109, normalized size = 1.65 \begin{align*} \frac{{\left (675 \, x^{4} + 2997 \, x^{3} + 6987 \, x^{2} + 19154 \, x - 19994\right )} \sqrt{-2 \, x + 1}}{35 \,{\left (2 \, 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)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 16.2832, size = 58, normalized size = 0.88 \begin{align*} - \frac{135 \left (1 - 2 x\right )^{\frac{7}{2}}}{112} + \frac{621 \left (1 - 2 x\right )^{\frac{5}{2}}}{40} - \frac{357 \left (1 - 2 x\right )^{\frac{3}{2}}}{4} + \frac{3283 \sqrt{1 - 2 x}}{8} + \frac{3773}{16 \sqrt{1 - 2 x}} \end{align*}

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]

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

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Giac [A]  time = 2.23297, size = 81, normalized size = 1.23 \begin{align*} \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}} \end{align*}

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, 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)