3.2139 \(\int \frac{(2+3 x) (3+5 x)}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=38 \[ -\frac{15}{4} \sqrt{1-2 x}-\frac{17}{\sqrt{1-2 x}}+\frac{77}{12 (1-2 x)^{3/2}} \]

[Out]

77/(12*(1 - 2*x)^(3/2)) - 17/Sqrt[1 - 2*x] - (15*Sqrt[1 - 2*x])/4

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Rubi [A]  time = 0.0079842, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ -\frac{15}{4} \sqrt{1-2 x}-\frac{17}{\sqrt{1-2 x}}+\frac{77}{12 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

77/(12*(1 - 2*x)^(3/2)) - 17/Sqrt[1 - 2*x] - (15*Sqrt[1 - 2*x])/4

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)}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac{77}{4 (1-2 x)^{5/2}}-\frac{17}{(1-2 x)^{3/2}}+\frac{15}{4 \sqrt{1-2 x}}\right ) \, dx\\ &=\frac{77}{12 (1-2 x)^{3/2}}-\frac{17}{\sqrt{1-2 x}}-\frac{15}{4} \sqrt{1-2 x}\\ \end{align*}

Mathematica [A]  time = 0.008615, size = 23, normalized size = 0.61 \[ -\frac{45 x^2-147 x+43}{3 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(43 - 147*x + 45*x^2)/(3*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.003, size = 20, normalized size = 0.5 \begin{align*} -{\frac{45\,{x}^{2}-147\,x+43}{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)/(1-2*x)^(5/2),x)

[Out]

-1/3*(45*x^2-147*x+43)/(1-2*x)^(3/2)

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Maxima [A]  time = 1.19001, size = 32, normalized size = 0.84 \begin{align*} -\frac{15}{4} \, \sqrt{-2 \, x + 1} + \frac{408 \, x - 127}{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)/(1-2*x)^(5/2),x, algorithm="maxima")

[Out]

-15/4*sqrt(-2*x + 1) + 1/12*(408*x - 127)/(-2*x + 1)^(3/2)

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Fricas [A]  time = 1.46499, size = 82, normalized size = 2.16 \begin{align*} -\frac{{\left (45 \, x^{2} - 147 \, x + 43\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)/(1-2*x)^(5/2),x, algorithm="fricas")

[Out]

-1/3*(45*x^2 - 147*x + 43)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

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Sympy [B]  time = 0.647696, size = 75, normalized size = 1.97 \begin{align*} \frac{45 x^{2}}{6 x \sqrt{1 - 2 x} - 3 \sqrt{1 - 2 x}} - \frac{147 x}{6 x \sqrt{1 - 2 x} - 3 \sqrt{1 - 2 x}} + \frac{43}{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)/(1-2*x)**(5/2),x)

[Out]

45*x**2/(6*x*sqrt(1 - 2*x) - 3*sqrt(1 - 2*x)) - 147*x/(6*x*sqrt(1 - 2*x) - 3*sqrt(1 - 2*x)) + 43/(6*x*sqrt(1 -
2*x) - 3*sqrt(1 - 2*x))

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Giac [A]  time = 1.2754, size = 42, normalized size = 1.11 \begin{align*} -\frac{15}{4} \, \sqrt{-2 \, x + 1} - \frac{408 \, x - 127}{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)/(1-2*x)^(5/2),x, algorithm="giac")

[Out]

-15/4*sqrt(-2*x + 1) - 1/12*(408*x - 127)/((2*x - 1)*sqrt(-2*x + 1))