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

Optimal. Leaf size=40 \[ -\frac {25}{36} (1-2 x)^{9/2}+\frac {55}{14} (1-2 x)^{7/2}-\frac {121}{20} (1-2 x)^{5/2} \]

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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}{36} (1-2 x)^{9/2}+\frac {55}{14} (1-2 x)^{7/2}-\frac {121}{20} (1-2 x)^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(3/2)*(3 + 5*x)^2,x]

[Out]

(-121*(1 - 2*x)^(5/2))/20 + (55*(1 - 2*x)^(7/2))/14 - (25*(1 - 2*x)^(9/2))/36

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)^{3/2} (3+5 x)^2 \, dx &=\int \left (\frac {121}{4} (1-2 x)^{3/2}-\frac {55}{2} (1-2 x)^{5/2}+\frac {25}{4} (1-2 x)^{7/2}\right ) \, dx\\ &=-\frac {121}{20} (1-2 x)^{5/2}+\frac {55}{14} (1-2 x)^{7/2}-\frac {25}{36} (1-2 x)^{9/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 23, normalized size = 0.58 \begin {gather*} -\frac {1}{315} (1-2 x)^{5/2} \left (875 x^2+1600 x+887\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/315*((1 - 2*x)^(5/2)*(887 + 1600*x + 875*x^2))

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IntegrateAlgebraic [A]  time = 0.02, size = 38, normalized size = 0.95 \begin {gather*} \frac {-875 (1-2 x)^{9/2}+4950 (1-2 x)^{7/2}-7623 (1-2 x)^{5/2}}{1260} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-7623*(1 - 2*x)^(5/2) + 4950*(1 - 2*x)^(7/2) - 875*(1 - 2*x)^(9/2))/1260

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fricas [A]  time = 1.81, size = 29, normalized size = 0.72 \begin {gather*} -\frac {1}{315} \, {\left (3500 \, x^{4} + 2900 \, x^{3} - 1977 \, x^{2} - 1948 \, x + 887\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/315*(3500*x^4 + 2900*x^3 - 1977*x^2 - 1948*x + 887)*sqrt(-2*x + 1)

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giac [A]  time = 1.09, size = 49, normalized size = 1.22 \begin {gather*} -\frac {25}{36} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {55}{14} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {121}{20} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/36*(2*x - 1)^4*sqrt(-2*x + 1) - 55/14*(2*x - 1)^3*sqrt(-2*x + 1) - 121/20*(2*x - 1)^2*sqrt(-2*x + 1)

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maple [A]  time = 0.00, size = 20, normalized size = 0.50 \begin {gather*} -\frac {\left (875 x^{2}+1600 x +887\right ) \left (-2 x +1\right )^{\frac {5}{2}}}{315} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/315*(875*x^2+1600*x+887)*(-2*x+1)^(5/2)

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maxima [A]  time = 0.49, size = 28, normalized size = 0.70 \begin {gather*} -\frac {25}{36} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {55}{14} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {121}{20} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/36*(-2*x + 1)^(9/2) + 55/14*(-2*x + 1)^(7/2) - 121/20*(-2*x + 1)^(5/2)

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mupad [B]  time = 0.03, size = 23, normalized size = 0.58 \begin {gather*} -\frac {{\left (1-2\,x\right )}^{5/2}\,\left (9900\,x+875\,{\left (2\,x-1\right )}^2+2673\right )}{1260} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(3/2)*(5*x + 3)^2,x)

[Out]

-((1 - 2*x)^(5/2)*(9900*x + 875*(2*x - 1)^2 + 2673))/1260

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sympy [B]  time = 1.56, size = 236, normalized size = 5.90 \begin {gather*} \begin {cases} - \frac {20 \sqrt {5} i \left (x + \frac {3}{5}\right )^{4} \sqrt {10 x - 5}}{9} + \frac {220 \sqrt {5} i \left (x + \frac {3}{5}\right )^{3} \sqrt {10 x - 5}}{63} - \frac {121 \sqrt {5} i \left (x + \frac {3}{5}\right )^{2} \sqrt {10 x - 5}}{525} - \frac {2662 \sqrt {5} i \left (x + \frac {3}{5}\right ) \sqrt {10 x - 5}}{7875} - \frac {29282 \sqrt {5} i \sqrt {10 x - 5}}{39375} & \text {for}\: \frac {10 \left |{x + \frac {3}{5}}\right |}{11} > 1 \\- \frac {20 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{4}}{9} + \frac {220 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{3}}{63} - \frac {121 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{2}}{525} - \frac {2662 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )}{7875} - \frac {29282 \sqrt {5} \sqrt {5 - 10 x}}{39375} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-20*sqrt(5)*I*(x + 3/5)**4*sqrt(10*x - 5)/9 + 220*sqrt(5)*I*(x + 3/5)**3*sqrt(10*x - 5)/63 - 121*sq
rt(5)*I*(x + 3/5)**2*sqrt(10*x - 5)/525 - 2662*sqrt(5)*I*(x + 3/5)*sqrt(10*x - 5)/7875 - 29282*sqrt(5)*I*sqrt(
10*x - 5)/39375, 10*Abs(x + 3/5)/11 > 1), (-20*sqrt(5)*sqrt(5 - 10*x)*(x + 3/5)**4/9 + 220*sqrt(5)*sqrt(5 - 10
*x)*(x + 3/5)**3/63 - 121*sqrt(5)*sqrt(5 - 10*x)*(x + 3/5)**2/525 - 2662*sqrt(5)*sqrt(5 - 10*x)*(x + 3/5)/7875
- 29282*sqrt(5)*sqrt(5 - 10*x)/39375, True))

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