3.2151 \(\int \frac{(3+5 x)^2}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=40 \[ -\frac{25}{4} \sqrt{1-2 x}-\frac{55}{2 \sqrt{1-2 x}}+\frac{121}{12 (1-2 x)^{3/2}} \]

[Out]

121/(12*(1 - 2*x)^(3/2)) - 55/(2*Sqrt[1 - 2*x]) - (25*Sqrt[1 - 2*x])/4

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Rubi [A]  time = 0.0071897, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {43} \[ -\frac{25}{4} \sqrt{1-2 x}-\frac{55}{2 \sqrt{1-2 x}}+\frac{121}{12 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^2/(1 - 2*x)^(5/2),x]

[Out]

121/(12*(1 - 2*x)^(3/2)) - 55/(2*Sqrt[1 - 2*x]) - (25*Sqrt[1 - 2*x])/4

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{(3+5 x)^2}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac{121}{4 (1-2 x)^{5/2}}-\frac{55}{2 (1-2 x)^{3/2}}+\frac{25}{4 \sqrt{1-2 x}}\right ) \, dx\\ &=\frac{121}{12 (1-2 x)^{3/2}}-\frac{55}{2 \sqrt{1-2 x}}-\frac{25}{4} \sqrt{1-2 x}\\ \end{align*}

Mathematica [A]  time = 0.0082924, size = 23, normalized size = 0.57 \[ -\frac{75 x^2-240 x+71}{3 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(71 - 240*x + 75*x^2)/(3*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.002, size = 20, normalized size = 0.5 \begin{align*} -{\frac{75\,{x}^{2}-240\,x+71}{3} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(75*x^2-240*x+71)/(1-2*x)^(3/2)

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Maxima [A]  time = 1.10929, size = 32, normalized size = 0.8 \begin{align*} -\frac{25}{4} \, \sqrt{-2 \, x + 1} + \frac{11 \,{\left (60 \, x - 19\right )}}{12 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2/(1-2*x)^(5/2),x, algorithm="maxima")

[Out]

-25/4*sqrt(-2*x + 1) + 11/12*(60*x - 19)/(-2*x + 1)^(3/2)

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Fricas [A]  time = 1.54897, size = 82, normalized size = 2.05 \begin{align*} -\frac{{\left (75 \, x^{2} - 240 \, x + 71\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((3+5*x)^2/(1-2*x)^(5/2),x, algorithm="fricas")

[Out]

-1/3*(75*x^2 - 240*x + 71)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

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Sympy [B]  time = 0.622128, size = 75, normalized size = 1.88 \begin{align*} \frac{75 x^{2}}{6 x \sqrt{1 - 2 x} - 3 \sqrt{1 - 2 x}} - \frac{240 x}{6 x \sqrt{1 - 2 x} - 3 \sqrt{1 - 2 x}} + \frac{71}{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((3+5*x)**2/(1-2*x)**(5/2),x)

[Out]

75*x**2/(6*x*sqrt(1 - 2*x) - 3*sqrt(1 - 2*x)) - 240*x/(6*x*sqrt(1 - 2*x) - 3*sqrt(1 - 2*x)) + 71/(6*x*sqrt(1 -
2*x) - 3*sqrt(1 - 2*x))

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Giac [A]  time = 2.47437, size = 42, normalized size = 1.05 \begin{align*} -\frac{25}{4} \, \sqrt{-2 \, x + 1} - \frac{11 \,{\left (60 \, x - 19\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((3+5*x)^2/(1-2*x)^(5/2),x, algorithm="giac")

[Out]

-25/4*sqrt(-2*x + 1) - 11/12*(60*x - 19)/((2*x - 1)*sqrt(-2*x + 1))