3.2150 \(\int \frac{(2+3 x) (3+5 x)^2}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=53 \[ \frac{25}{8} (1-2 x)^{3/2}-\frac{505}{8} \sqrt{1-2 x}-\frac{1133}{8 \sqrt{1-2 x}}+\frac{847}{24 (1-2 x)^{3/2}} \]

[Out]

847/(24*(1 - 2*x)^(3/2)) - 1133/(8*Sqrt[1 - 2*x]) - (505*Sqrt[1 - 2*x])/8 + (25*(1 - 2*x)^(3/2))/8

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Rubi [A]  time = 0.0100809, antiderivative size = 53, 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} \[ \frac{25}{8} (1-2 x)^{3/2}-\frac{505}{8} \sqrt{1-2 x}-\frac{1133}{8 \sqrt{1-2 x}}+\frac{847}{24 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

847/(24*(1 - 2*x)^(3/2)) - 1133/(8*Sqrt[1 - 2*x]) - (505*Sqrt[1 - 2*x])/8 + (25*(1 - 2*x)^(3/2))/8

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)^2}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac{847}{8 (1-2 x)^{5/2}}-\frac{1133}{8 (1-2 x)^{3/2}}+\frac{505}{8 \sqrt{1-2 x}}-\frac{75}{8} \sqrt{1-2 x}\right ) \, dx\\ &=\frac{847}{24 (1-2 x)^{3/2}}-\frac{1133}{8 \sqrt{1-2 x}}-\frac{505}{8} \sqrt{1-2 x}+\frac{25}{8} (1-2 x)^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0110197, size = 28, normalized size = 0.53 \[ -\frac{75 x^3+645 x^2-1551 x+499}{3 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(499 - 1551*x + 645*x^2 + 75*x^3)/(3*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.003, size = 25, normalized size = 0.5 \begin{align*} -{\frac{75\,{x}^{3}+645\,{x}^{2}-1551\,x+499}{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)^2/(1-2*x)^(5/2),x)

[Out]

-1/3*(75*x^3+645*x^2-1551*x+499)/(1-2*x)^(3/2)

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Maxima [A]  time = 2.70761, size = 45, normalized size = 0.85 \begin{align*} \frac{25}{8} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{505}{8} \, \sqrt{-2 \, x + 1} + \frac{11 \,{\left (309 \, x - 116\right )}}{12 \,{\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)^2/(1-2*x)^(5/2),x, algorithm="maxima")

[Out]

25/8*(-2*x + 1)^(3/2) - 505/8*sqrt(-2*x + 1) + 11/12*(309*x - 116)/(-2*x + 1)^(3/2)

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Fricas [A]  time = 1.51311, size = 99, normalized size = 1.87 \begin{align*} -\frac{{\left (75 \, x^{3} + 645 \, x^{2} - 1551 \, x + 499\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)^2/(1-2*x)^(5/2),x, algorithm="fricas")

[Out]

-1/3*(75*x^3 + 645*x^2 - 1551*x + 499)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

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Sympy [B]  time = 0.69489, size = 102, normalized size = 1.92 \begin{align*} \frac{75 x^{3}}{6 x \sqrt{1 - 2 x} - 3 \sqrt{1 - 2 x}} + \frac{645 x^{2}}{6 x \sqrt{1 - 2 x} - 3 \sqrt{1 - 2 x}} - \frac{1551 x}{6 x \sqrt{1 - 2 x} - 3 \sqrt{1 - 2 x}} + \frac{499}{6 x \sqrt{1 - 2 x} - 3 \sqrt{1 - 2 x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

75*x**3/(6*x*sqrt(1 - 2*x) - 3*sqrt(1 - 2*x)) + 645*x**2/(6*x*sqrt(1 - 2*x) - 3*sqrt(1 - 2*x)) - 1551*x/(6*x*s
qrt(1 - 2*x) - 3*sqrt(1 - 2*x)) + 499/(6*x*sqrt(1 - 2*x) - 3*sqrt(1 - 2*x))

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Giac [A]  time = 2.08672, size = 54, normalized size = 1.02 \begin{align*} \frac{25}{8} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{505}{8} \, \sqrt{-2 \, x + 1} - \frac{11 \,{\left (309 \, x - 116\right )}}{12 \,{\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)^2/(1-2*x)^(5/2),x, algorithm="giac")

[Out]

25/8*(-2*x + 1)^(3/2) - 505/8*sqrt(-2*x + 1) - 11/12*(309*x - 116)/((2*x - 1)*sqrt(-2*x + 1))