3.2028 \(\int \frac{(3+5 x)^3}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=53 \[ \frac{125}{56} (1-2 x)^{7/2}-\frac{165}{8} (1-2 x)^{5/2}+\frac{605}{8} (1-2 x)^{3/2}-\frac{1331}{8} \sqrt{1-2 x} \]

[Out]

(-1331*Sqrt[1 - 2*x])/8 + (605*(1 - 2*x)^(3/2))/8 - (165*(1 - 2*x)^(5/2))/8 + (125*(1 - 2*x)^(7/2))/56

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Rubi [A]  time = 0.0087757, antiderivative size = 53, 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} \[ \frac{125}{56} (1-2 x)^{7/2}-\frac{165}{8} (1-2 x)^{5/2}+\frac{605}{8} (1-2 x)^{3/2}-\frac{1331}{8} \sqrt{1-2 x} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^3/Sqrt[1 - 2*x],x]

[Out]

(-1331*Sqrt[1 - 2*x])/8 + (605*(1 - 2*x)^(3/2))/8 - (165*(1 - 2*x)^(5/2))/8 + (125*(1 - 2*x)^(7/2))/56

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)^3}{\sqrt{1-2 x}} \, dx &=\int \left (\frac{1331}{8 \sqrt{1-2 x}}-\frac{1815}{8} \sqrt{1-2 x}+\frac{825}{8} (1-2 x)^{3/2}-\frac{125}{8} (1-2 x)^{5/2}\right ) \, dx\\ &=-\frac{1331}{8} \sqrt{1-2 x}+\frac{605}{8} (1-2 x)^{3/2}-\frac{165}{8} (1-2 x)^{5/2}+\frac{125}{56} (1-2 x)^{7/2}\\ \end{align*}

Mathematica [A]  time = 0.0095531, size = 28, normalized size = 0.53 \[ -\frac{1}{7} \sqrt{1-2 x} \left (125 x^3+390 x^2+575 x+764\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - 2*x]*(764 + 575*x + 390*x^2 + 125*x^3))/7

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Maple [A]  time = 0.003, size = 25, normalized size = 0.5 \begin{align*} -{\frac{125\,{x}^{3}+390\,{x}^{2}+575\,x+764}{7}\sqrt{1-2\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*(125*x^3+390*x^2+575*x+764)*(1-2*x)^(1/2)

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Maxima [A]  time = 1.14607, size = 50, normalized size = 0.94 \begin{align*} \frac{125}{56} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{165}{8} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{605}{8} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{1331}{8} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125/56*(-2*x + 1)^(7/2) - 165/8*(-2*x + 1)^(5/2) + 605/8*(-2*x + 1)^(3/2) - 1331/8*sqrt(-2*x + 1)

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Fricas [A]  time = 1.6277, size = 74, normalized size = 1.4 \begin{align*} -\frac{1}{7} \,{\left (125 \, x^{3} + 390 \, x^{2} + 575 \, x + 764\right )} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*(125*x^3 + 390*x^2 + 575*x + 764)*sqrt(-2*x + 1)

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Sympy [A]  time = 1.7952, size = 190, normalized size = 3.58 \begin{align*} \begin{cases} - \frac{25 \sqrt{5} i \left (x + \frac{3}{5}\right )^{3} \sqrt{10 x - 5}}{7} - \frac{33 \sqrt{5} i \left (x + \frac{3}{5}\right )^{2} \sqrt{10 x - 5}}{7} - \frac{242 \sqrt{5} i \left (x + \frac{3}{5}\right ) \sqrt{10 x - 5}}{35} - \frac{2662 \sqrt{5} i \sqrt{10 x - 5}}{175} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\- \frac{25 \sqrt{5} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )^{3}}{7} - \frac{33 \sqrt{5} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )^{2}}{7} - \frac{242 \sqrt{5} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )}{35} - \frac{2662 \sqrt{5} \sqrt{5 - 10 x}}{175} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-25*sqrt(5)*I*(x + 3/5)**3*sqrt(10*x - 5)/7 - 33*sqrt(5)*I*(x + 3/5)**2*sqrt(10*x - 5)/7 - 242*sqrt
(5)*I*(x + 3/5)*sqrt(10*x - 5)/35 - 2662*sqrt(5)*I*sqrt(10*x - 5)/175, 10*Abs(x + 3/5)/11 > 1), (-25*sqrt(5)*s
qrt(5 - 10*x)*(x + 3/5)**3/7 - 33*sqrt(5)*sqrt(5 - 10*x)*(x + 3/5)**2/7 - 242*sqrt(5)*sqrt(5 - 10*x)*(x + 3/5)
/35 - 2662*sqrt(5)*sqrt(5 - 10*x)/175, True))

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Giac [A]  time = 1.94007, size = 69, normalized size = 1.3 \begin{align*} -\frac{125}{56} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{165}{8} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{605}{8} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{1331}{8} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125/56*(2*x - 1)^3*sqrt(-2*x + 1) - 165/8*(2*x - 1)^2*sqrt(-2*x + 1) + 605/8*(-2*x + 1)^(3/2) - 1331/8*sqrt(-
2*x + 1)