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

Optimal. Leaf size=40 \[ -\frac {25}{44} (1-2 x)^{11/2}+\frac {55}{18} (1-2 x)^{9/2}-\frac {121}{28} (1-2 x)^{7/2} \]

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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}{44} (1-2 x)^{11/2}+\frac {55}{18} (1-2 x)^{9/2}-\frac {121}{28} (1-2 x)^{7/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(5/2)*(3 + 5*x)^2,x]

[Out]

(-121*(1 - 2*x)^(7/2))/28 + (55*(1 - 2*x)^(9/2))/18 - (25*(1 - 2*x)^(11/2))/44

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 (1-2 x)^{5/2} (3+5 x)^2 \, dx &=\int \left (\frac {121}{4} (1-2 x)^{5/2}-\frac {55}{2} (1-2 x)^{7/2}+\frac {25}{4} (1-2 x)^{9/2}\right ) \, dx\\ &=-\frac {121}{28} (1-2 x)^{7/2}+\frac {55}{18} (1-2 x)^{9/2}-\frac {25}{44} (1-2 x)^{11/2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 0.58 \begin {gather*} -\frac {1}{693} (1-2 x)^{7/2} \left (1575 x^2+2660 x+1271\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/693*((1 - 2*x)^(7/2)*(1271 + 2660*x + 1575*x^2))

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IntegrateAlgebraic [A]  time = 0.02, size = 38, normalized size = 0.95 \begin {gather*} \frac {-1575 (1-2 x)^{11/2}+8470 (1-2 x)^{9/2}-11979 (1-2 x)^{7/2}}{2772} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-11979*(1 - 2*x)^(7/2) + 8470*(1 - 2*x)^(9/2) - 1575*(1 - 2*x)^(11/2))/2772

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fricas [A]  time = 1.50, size = 34, normalized size = 0.85 \begin {gather*} \frac {1}{693} \, {\left (12600 \, x^{5} + 2380 \, x^{4} - 12302 \, x^{3} - 867 \, x^{2} + 4966 \, x - 1271\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/693*(12600*x^5 + 2380*x^4 - 12302*x^3 - 867*x^2 + 4966*x - 1271)*sqrt(-2*x + 1)

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giac [A]  time = 0.95, size = 49, normalized size = 1.22 \begin {gather*} \frac {25}{44} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {55}{18} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {121}{28} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/44*(2*x - 1)^5*sqrt(-2*x + 1) + 55/18*(2*x - 1)^4*sqrt(-2*x + 1) + 121/28*(2*x - 1)^3*sqrt(-2*x + 1)

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maple [A]  time = 0.00, size = 20, normalized size = 0.50 \begin {gather*} -\frac {\left (1575 x^{2}+2660 x +1271\right ) \left (-2 x +1\right )^{\frac {7}{2}}}{693} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/693*(1575*x^2+2660*x+1271)*(-2*x+1)^(7/2)

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maxima [A]  time = 0.60, size = 28, normalized size = 0.70 \begin {gather*} -\frac {25}{44} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {55}{18} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {121}{28} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/44*(-2*x + 1)^(11/2) + 55/18*(-2*x + 1)^(9/2) - 121/28*(-2*x + 1)^(7/2)

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mupad [B]  time = 0.03, size = 23, normalized size = 0.58 \begin {gather*} -\frac {{\left (1-2\,x\right )}^{7/2}\,\left (16940\,x+1575\,{\left (2\,x-1\right )}^2+3509\right )}{2772} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(5/2)*(5*x + 3)^2,x)

[Out]

-((1 - 2*x)^(7/2)*(16940*x + 1575*(2*x - 1)^2 + 3509))/2772

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sympy [B]  time = 1.67, size = 85, normalized size = 2.12 \begin {gather*} \frac {200 x^{5} \sqrt {1 - 2 x}}{11} + \frac {340 x^{4} \sqrt {1 - 2 x}}{99} - \frac {12302 x^{3} \sqrt {1 - 2 x}}{693} - \frac {289 x^{2} \sqrt {1 - 2 x}}{231} + \frac {4966 x \sqrt {1 - 2 x}}{693} - \frac {1271 \sqrt {1 - 2 x}}{693} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

200*x**5*sqrt(1 - 2*x)/11 + 340*x**4*sqrt(1 - 2*x)/99 - 12302*x**3*sqrt(1 - 2*x)/693 - 289*x**2*sqrt(1 - 2*x)/
231 + 4966*x*sqrt(1 - 2*x)/693 - 1271*sqrt(1 - 2*x)/693

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