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

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

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Rubi [A]  time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {5}{4} (1-2 x)^{5/2}+\frac {55}{6} (1-2 x)^{3/2}-\frac {121}{4} \sqrt {1-2 x} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 0.58 \begin {gather*} -\frac {1}{3} \sqrt {1-2 x} \left (15 x^2+40 x+67\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.02, size = 31, normalized size = 0.78 \begin {gather*} -\frac {1}{12} \left (15 (1-2 x)^2-110 (1-2 x)+363\right ) \sqrt {1-2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/12*((363 - 110*(1 - 2*x) + 15*(1 - 2*x)^2)*Sqrt[1 - 2*x])

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fricas [A]  time = 1.50, size = 19, normalized size = 0.48 \begin {gather*} -\frac {1}{3} \, {\left (15 \, x^{2} + 40 \, x + 67\right )} \sqrt {-2 \, x + 1} \end {gather*}

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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giac [A]  time = 1.02, size = 35, normalized size = 0.88 \begin {gather*} -\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 {gather*}

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)

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maple [A]  time = 0.00, size = 20, normalized size = 0.50 \begin {gather*} -\frac {\left (15 x^{2}+40 x +67\right ) \sqrt {-2 x +1}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.64, size = 28, normalized size = 0.70 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 0.03, size = 23, normalized size = 0.58 \begin {gather*} -\frac {\sqrt {1-2\,x}\,\left (220\,x+15\,{\left (2\,x-1\right )}^2+253\right )}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + 3)^2/(1 - 2*x)^(1/2),x)

[Out]

-((1 - 2*x)^(1/2)*(220*x + 15*(2*x - 1)^2 + 253))/12

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sympy [A]  time = 1.39, size = 134, normalized size = 3.35 \begin {gather*} \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 {gather*}

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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