3.2161 \(\int \frac{(2+3 x) (3+5 x)^3}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=66 \[ -\frac{75}{16} (1-2 x)^{5/2}+\frac{1675}{24} (1-2 x)^{3/2}-\frac{2805}{4} \sqrt{1-2 x}-\frac{8349}{8 \sqrt{1-2 x}}+\frac{9317}{48 (1-2 x)^{3/2}} \]

[Out]

9317/(48*(1 - 2*x)^(3/2)) - 8349/(8*Sqrt[1 - 2*x]) - (2805*Sqrt[1 - 2*x])/4 + (1675*(1 - 2*x)^(3/2))/24 - (75*
(1 - 2*x)^(5/2))/16

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Rubi [A]  time = 0.0125482, 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} \[ -\frac{75}{16} (1-2 x)^{5/2}+\frac{1675}{24} (1-2 x)^{3/2}-\frac{2805}{4} \sqrt{1-2 x}-\frac{8349}{8 \sqrt{1-2 x}}+\frac{9317}{48 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

9317/(48*(1 - 2*x)^(3/2)) - 8349/(8*Sqrt[1 - 2*x]) - (2805*Sqrt[1 - 2*x])/4 + (1675*(1 - 2*x)^(3/2))/24 - (75*
(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+5 x)^3}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac{9317}{16 (1-2 x)^{5/2}}-\frac{8349}{8 (1-2 x)^{3/2}}+\frac{2805}{4 \sqrt{1-2 x}}-\frac{1675}{8} \sqrt{1-2 x}+\frac{375}{16} (1-2 x)^{3/2}\right ) \, dx\\ &=\frac{9317}{48 (1-2 x)^{3/2}}-\frac{8349}{8 \sqrt{1-2 x}}-\frac{2805}{4} \sqrt{1-2 x}+\frac{1675}{24} (1-2 x)^{3/2}-\frac{75}{16} (1-2 x)^{5/2}\\ \end{align*}

Mathematica [A]  time = 0.0139388, size = 33, normalized size = 0.5 \[ -\frac{225 x^4+1225 x^3+6240 x^2-13533 x+4457}{3 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(4457 - 13533*x + 6240*x^2 + 1225*x^3 + 225*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{225\,{x}^{4}+1225\,{x}^{3}+6240\,{x}^{2}-13533\,x+4457}{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+5*x)^3/(1-2*x)^(5/2),x)

[Out]

-1/3*(225*x^4+1225*x^3+6240*x^2-13533*x+4457)/(1-2*x)^(3/2)

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Maxima [A]  time = 1.96033, size = 57, normalized size = 0.86 \begin{align*} -\frac{75}{16} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{1675}{24} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{2805}{4} \, \sqrt{-2 \, x + 1} + \frac{121 \,{\left (828 \, x - 337\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+5*x)^3/(1-2*x)^(5/2),x, algorithm="maxima")

[Out]

-75/16*(-2*x + 1)^(5/2) + 1675/24*(-2*x + 1)^(3/2) - 2805/4*sqrt(-2*x + 1) + 121/48*(828*x - 337)/(-2*x + 1)^(
3/2)

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Fricas [A]  time = 1.54208, size = 119, normalized size = 1.8 \begin{align*} -\frac{{\left (225 \, x^{4} + 1225 \, x^{3} + 6240 \, x^{2} - 13533 \, x + 4457\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+5*x)^3/(1-2*x)^(5/2),x, algorithm="fricas")

[Out]

-1/3*(225*x^4 + 1225*x^3 + 6240*x^2 - 13533*x + 4457)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

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Sympy [A]  time = 15.0144, size = 58, normalized size = 0.88 \begin{align*} - \frac{75 \left (1 - 2 x\right )^{\frac{5}{2}}}{16} + \frac{1675 \left (1 - 2 x\right )^{\frac{3}{2}}}{24} - \frac{2805 \sqrt{1 - 2 x}}{4} - \frac{8349}{8 \sqrt{1 - 2 x}} + \frac{9317}{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+5*x)**3/(1-2*x)**(5/2),x)

[Out]

-75*(1 - 2*x)**(5/2)/16 + 1675*(1 - 2*x)**(3/2)/24 - 2805*sqrt(1 - 2*x)/4 - 8349/(8*sqrt(1 - 2*x)) + 9317/(48*
(1 - 2*x)**(3/2))

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Giac [A]  time = 2.07256, size = 76, normalized size = 1.15 \begin{align*} -\frac{75}{16} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{1675}{24} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{2805}{4} \, \sqrt{-2 \, x + 1} - \frac{121 \,{\left (828 \, x - 337\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+5*x)^3/(1-2*x)^(5/2),x, algorithm="giac")

[Out]

-75/16*(2*x - 1)^2*sqrt(-2*x + 1) + 1675/24*(-2*x + 1)^(3/2) - 2805/4*sqrt(-2*x + 1) - 121/48*(828*x - 337)/((
2*x - 1)*sqrt(-2*x + 1))