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

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

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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 {25}{12} (1-2 x)^{3/2}+\frac {55}{2} \sqrt {1-2 x}+\frac {121}{4 \sqrt {1-2 x}} \end {gather*}

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*}

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Mathematica [A]  time = 0.02, size = 23, normalized size = 0.58 \begin {gather*} \frac {-25 x^2-140 x+167}{3 \sqrt {1-2 x}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.02, size = 31, normalized size = 0.78 \begin {gather*} \frac {-25 (1-2 x)^2+330 (1-2 x)+363}{12 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(363 + 330*(1 - 2*x) - 25*(1 - 2*x)^2)/(12*Sqrt[1 - 2*x])

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fricas [A]  time = 1.57, size = 26, normalized size = 0.65 \begin {gather*} \frac {{\left (25 \, x^{2} + 140 \, x - 167\right )} \sqrt {-2 \, x + 1}}{3 \, {\left (2 \, x - 1\right )}} \end {gather*}

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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giac [A]  time = 1.13, size = 28, normalized size = 0.70 \begin {gather*} -\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 {gather*}

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)

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maple [A]  time = 0.00, size = 20, normalized size = 0.50 \begin {gather*} -\frac {25 x^{2}+140 x -167}{3 \sqrt {-2 x +1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.44, size = 28, normalized size = 0.70 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 0.03, size = 23, normalized size = 0.58 \begin {gather*} -\frac {660\,x+25\,{\left (2\,x-1\right )}^2-693}{12\,\sqrt {1-2\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + 3)^2/(1 - 2*x)^(3/2),x)

[Out]

-(660*x + 25*(2*x - 1)^2 - 693)/(12*(1 - 2*x)^(1/2))

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sympy [B]  time = 1.40, size = 352, normalized size = 8.80 \begin {gather*} \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 {gather*}

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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