3.2088 \(\int \frac{(3+5 x)^2}{(1-2 x)^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0073006, 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} \[ -\frac{25}{12} (1-2 x)^{3/2}+\frac{55}{2} \sqrt{1-2 x}+\frac{121}{4 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^2/(1 - 2*x)^(3/2),x]

[Out]

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

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

Mathematica [A]  time = 0.0077595, size = 23, normalized size = 0.57 \[ \frac{-25 x^2-140 x+167}{3 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(167 - 140*x - 25*x^2)/(3*Sqrt[1 - 2*x])

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Maple [A]  time = 0.002, size = 20, normalized size = 0.5 \begin{align*} -{\frac{25\,{x}^{2}+140\,x-167}{3}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(25*x^2+140*x-167)/(1-2*x)^(1/2)

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Maxima [A]  time = 1.14202, size = 38, normalized size = 0.95 \begin{align*} -\frac{25}{12} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{55}{2} \, \sqrt{-2 \, x + 1} + \frac{121}{4 \, \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)^(3/2),x, algorithm="maxima")

[Out]

-25/12*(-2*x + 1)^(3/2) + 55/2*sqrt(-2*x + 1) + 121/4/sqrt(-2*x + 1)

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Fricas [A]  time = 1.53854, size = 72, normalized size = 1.8 \begin{align*} \frac{{\left (25 \, x^{2} + 140 \, x - 167\right )} \sqrt{-2 \, x + 1}}{3 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(25*x^2 + 140*x - 167)*sqrt(-2*x + 1)/(2*x - 1)

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Sympy [B]  time = 1.2354, size = 352, normalized size = 8.8 \begin{align*} \begin{cases} \frac{25 \sqrt{55} i \left (x + \frac{3}{5}\right )^{2} \sqrt{10 x - 5}}{30 \sqrt{11} \left (x + \frac{3}{5}\right ) - 33 \sqrt{11}} + \frac{110 \sqrt{55} i \left (x + \frac{3}{5}\right ) \sqrt{10 x - 5}}{30 \sqrt{11} \left (x + \frac{3}{5}\right ) - 33 \sqrt{11}} - \frac{2420 \sqrt{5} \left (x + \frac{3}{5}\right )}{30 \sqrt{11} \left (x + \frac{3}{5}\right ) - 33 \sqrt{11}} - \frac{242 \sqrt{55} i \sqrt{10 x - 5}}{30 \sqrt{11} \left (x + \frac{3}{5}\right ) - 33 \sqrt{11}} + \frac{2662 \sqrt{5}}{30 \sqrt{11} \left (x + \frac{3}{5}\right ) - 33 \sqrt{11}} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{25 \sqrt{55} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )^{2}}{30 \sqrt{11} \left (x + \frac{3}{5}\right ) - 33 \sqrt{11}} + \frac{110 \sqrt{55} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )}{30 \sqrt{11} \left (x + \frac{3}{5}\right ) - 33 \sqrt{11}} - \frac{242 \sqrt{55} \sqrt{5 - 10 x}}{30 \sqrt{11} \left (x + \frac{3}{5}\right ) - 33 \sqrt{11}} - \frac{2420 \sqrt{5} \left (x + \frac{3}{5}\right )}{30 \sqrt{11} \left (x + \frac{3}{5}\right ) - 33 \sqrt{11}} + \frac{2662 \sqrt{5}}{30 \sqrt{11} \left (x + \frac{3}{5}\right ) - 33 \sqrt{11}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((25*sqrt(55)*I*(x + 3/5)**2*sqrt(10*x - 5)/(30*sqrt(11)*(x + 3/5) - 33*sqrt(11)) + 110*sqrt(55)*I*(x
+ 3/5)*sqrt(10*x - 5)/(30*sqrt(11)*(x + 3/5) - 33*sqrt(11)) - 2420*sqrt(5)*(x + 3/5)/(30*sqrt(11)*(x + 3/5) -
33*sqrt(11)) - 242*sqrt(55)*I*sqrt(10*x - 5)/(30*sqrt(11)*(x + 3/5) - 33*sqrt(11)) + 2662*sqrt(5)/(30*sqrt(11
)*(x + 3/5) - 33*sqrt(11)), 10*Abs(x + 3/5)/11 > 1), (25*sqrt(55)*sqrt(5 - 10*x)*(x + 3/5)**2/(30*sqrt(11)*(x
+ 3/5) - 33*sqrt(11)) + 110*sqrt(55)*sqrt(5 - 10*x)*(x + 3/5)/(30*sqrt(11)*(x + 3/5) - 33*sqrt(11)) - 242*sqrt
(55)*sqrt(5 - 10*x)/(30*sqrt(11)*(x + 3/5) - 33*sqrt(11)) - 2420*sqrt(5)*(x + 3/5)/(30*sqrt(11)*(x + 3/5) - 33
*sqrt(11)) + 2662*sqrt(5)/(30*sqrt(11)*(x + 3/5) - 33*sqrt(11)), True))

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Giac [A]  time = 2.73506, size = 38, normalized size = 0.95 \begin{align*} -\frac{25}{12} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{55}{2} \, \sqrt{-2 \, x + 1} + \frac{121}{4 \, \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)^(3/2),x, algorithm="giac")

[Out]

-25/12*(-2*x + 1)^(3/2) + 55/2*sqrt(-2*x + 1) + 121/4/sqrt(-2*x + 1)