3.2097 \(\int \frac{(2+3 x) (3+5 x)^3}{(1-2 x)^{3/2}} \, dx\)

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 + (335*(1 - 2*x)^(5/2))/8 - (375*(1
- 2*x)^(7/2))/112

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Rubi [A]  time = 0.0129449, 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} \[ -\frac{375}{112} (1-2 x)^{7/2}+\frac{335}{8} (1-2 x)^{5/2}-\frac{935}{4} (1-2 x)^{3/2}+\frac{8349}{8} \sqrt{1-2 x}+\frac{9317}{16 \sqrt{1-2 x}} \]

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 + (335*(1 - 2*x)^(5/2))/8 - (375*(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+5 x)^3}{(1-2 x)^{3/2}} \, dx &=\int \left (\frac{9317}{16 (1-2 x)^{3/2}}-\frac{8349}{8 \sqrt{1-2 x}}+\frac{2805}{4} \sqrt{1-2 x}-\frac{1675}{8} (1-2 x)^{3/2}+\frac{375}{16} (1-2 x)^{5/2}\right ) \, dx\\ &=\frac{9317}{16 \sqrt{1-2 x}}+\frac{8349}{8} \sqrt{1-2 x}-\frac{935}{4} (1-2 x)^{3/2}+\frac{335}{8} (1-2 x)^{5/2}-\frac{375}{112} (1-2 x)^{7/2}\\ \end{align*}

Mathematica [A]  time = 0.0133034, size = 33, normalized size = 0.5 \[ \frac{-375 x^4-1595 x^3-3590 x^2-9637 x+10015}{7 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 30, normalized size = 0.5 \begin{align*} -{\frac{375\,{x}^{4}+1595\,{x}^{3}+3590\,{x}^{2}+9637\,x-10015}{7}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}

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 = 1.3267, size = 62, normalized size = 0.94 \begin{align*} -\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}} \end{align*}

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, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.59216, size = 107, normalized size = 1.62 \begin{align*} \frac{{\left (375 \, x^{4} + 1595 \, x^{3} + 3590 \, x^{2} + 9637 \, x - 10015\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)*(3+5*x)^3/(1-2*x)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 16.3288, size = 58, normalized size = 0.88 \begin{align*} - \frac{375 \left (1 - 2 x\right )^{\frac{7}{2}}}{112} + \frac{335 \left (1 - 2 x\right )^{\frac{5}{2}}}{8} - \frac{935 \left (1 - 2 x\right )^{\frac{3}{2}}}{4} + \frac{8349 \sqrt{1 - 2 x}}{8} + \frac{9317}{16 \sqrt{1 - 2 x}} \end{align*}

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]

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

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Giac [A]  time = 1.69326, size = 81, normalized size = 1.23 \begin{align*} \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}} \end{align*}

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, 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)