3.2017 \(\int \frac{(3+5 x)^2}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=40 \[ -\frac{5}{4} (1-2 x)^{5/2}+\frac{55}{6} (1-2 x)^{3/2}-\frac{121}{4} \sqrt{1-2 x} \]

[Out]

(-121*Sqrt[1 - 2*x])/4 + (55*(1 - 2*x)^(3/2))/6 - (5*(1 - 2*x)^(5/2))/4

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Rubi [A]  time = 0.0075216, 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} \[ -\frac{5}{4} (1-2 x)^{5/2}+\frac{55}{6} (1-2 x)^{3/2}-\frac{121}{4} \sqrt{1-2 x} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^2/Sqrt[1 - 2*x],x]

[Out]

(-121*Sqrt[1 - 2*x])/4 + (55*(1 - 2*x)^(3/2))/6 - (5*(1 - 2*x)^(5/2))/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}{\sqrt{1-2 x}} \, dx &=\int \left (\frac{121}{4 \sqrt{1-2 x}}-\frac{55}{2} \sqrt{1-2 x}+\frac{25}{4} (1-2 x)^{3/2}\right ) \, dx\\ &=-\frac{121}{4} \sqrt{1-2 x}+\frac{55}{6} (1-2 x)^{3/2}-\frac{5}{4} (1-2 x)^{5/2}\\ \end{align*}

Mathematica [A]  time = 0.0072686, size = 23, normalized size = 0.57 \[ -\frac{1}{3} \sqrt{1-2 x} \left (15 x^2+40 x+67\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - 2*x]*(67 + 40*x + 15*x^2))/3

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Maple [A]  time = 0.002, size = 20, normalized size = 0.5 \begin{align*} -{\frac{15\,{x}^{2}+40\,x+67}{3}\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)^(1/2),x)

[Out]

-1/3*(15*x^2+40*x+67)*(1-2*x)^(1/2)

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Maxima [A]  time = 2.14306, size = 38, normalized size = 0.95 \begin{align*} -\frac{5}{4} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{55}{6} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \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)^(1/2),x, algorithm="maxima")

[Out]

-5/4*(-2*x + 1)^(5/2) + 55/6*(-2*x + 1)^(3/2) - 121/4*sqrt(-2*x + 1)

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Fricas [A]  time = 1.53232, size = 57, normalized size = 1.42 \begin{align*} -\frac{1}{3} \,{\left (15 \, x^{2} + 40 \, x + 67\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)^(1/2),x, algorithm="fricas")

[Out]

-1/3*(15*x^2 + 40*x + 67)*sqrt(-2*x + 1)

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Sympy [A]  time = 1.2059, size = 134, normalized size = 3.35 \begin{align*} \begin{cases} - \sqrt{5} i \left (x + \frac{3}{5}\right )^{2} \sqrt{10 x - 5} - \frac{22 \sqrt{5} i \left (x + \frac{3}{5}\right ) \sqrt{10 x - 5}}{15} - \frac{242 \sqrt{5} i \sqrt{10 x - 5}}{75} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\- \sqrt{5} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )^{2} - \frac{22 \sqrt{5} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )}{15} - \frac{242 \sqrt{5} \sqrt{5 - 10 x}}{75} & \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)**(1/2),x)

[Out]

Piecewise((-sqrt(5)*I*(x + 3/5)**2*sqrt(10*x - 5) - 22*sqrt(5)*I*(x + 3/5)*sqrt(10*x - 5)/15 - 242*sqrt(5)*I*s
qrt(10*x - 5)/75, 10*Abs(x + 3/5)/11 > 1), (-sqrt(5)*sqrt(5 - 10*x)*(x + 3/5)**2 - 22*sqrt(5)*sqrt(5 - 10*x)*(
x + 3/5)/15 - 242*sqrt(5)*sqrt(5 - 10*x)/75, True))

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Giac [A]  time = 1.83239, size = 47, normalized size = 1.18 \begin{align*} -\frac{5}{4} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{55}{6} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \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)^(1/2),x, algorithm="giac")

[Out]

-5/4*(2*x - 1)^2*sqrt(-2*x + 1) + 55/6*(-2*x + 1)^(3/2) - 121/4*sqrt(-2*x + 1)