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

Optimal. Leaf size=53 \[ \frac {125}{88} (1-2 x)^{11/2}-\frac {275}{24} (1-2 x)^{9/2}+\frac {1815}{56} (1-2 x)^{7/2}-\frac {1331}{40} (1-2 x)^{5/2} \]

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Rubi [A]  time = 0.01, antiderivative size = 53, 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 {125}{88} (1-2 x)^{11/2}-\frac {275}{24} (1-2 x)^{9/2}+\frac {1815}{56} (1-2 x)^{7/2}-\frac {1331}{40} (1-2 x)^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(3/2)*(3 + 5*x)^3,x]

[Out]

(-1331*(1 - 2*x)^(5/2))/40 + (1815*(1 - 2*x)^(7/2))/56 - (275*(1 - 2*x)^(9/2))/24 + (125*(1 - 2*x)^(11/2))/88

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)^3 \, dx &=\int \left (\frac {1331}{8} (1-2 x)^{3/2}-\frac {1815}{8} (1-2 x)^{5/2}+\frac {825}{8} (1-2 x)^{7/2}-\frac {125}{8} (1-2 x)^{9/2}\right ) \, dx\\ &=-\frac {1331}{40} (1-2 x)^{5/2}+\frac {1815}{56} (1-2 x)^{7/2}-\frac {275}{24} (1-2 x)^{9/2}+\frac {125}{88} (1-2 x)^{11/2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 28, normalized size = 0.53 \begin {gather*} -\frac {(1-2 x)^{5/2} \left (13125 x^3+33250 x^2+31775 x+12592\right )}{1155} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/1155*((1 - 2*x)^(5/2)*(12592 + 31775*x + 33250*x^2 + 13125*x^3))

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IntegrateAlgebraic [A]  time = 0.02, size = 49, normalized size = 0.92 \begin {gather*} \frac {13125 (1-2 x)^{11/2}-105875 (1-2 x)^{9/2}+299475 (1-2 x)^{7/2}-307461 (1-2 x)^{5/2}}{9240} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-307461*(1 - 2*x)^(5/2) + 299475*(1 - 2*x)^(7/2) - 105875*(1 - 2*x)^(9/2) + 13125*(1 - 2*x)^(11/2))/9240

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fricas [A]  time = 1.48, size = 34, normalized size = 0.64 \begin {gather*} -\frac {1}{1155} \, {\left (52500 \, x^{5} + 80500 \, x^{4} + 7225 \, x^{3} - 43482 \, x^{2} - 18593 \, x + 12592\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)^3,x, algorithm="fricas")

[Out]

-1/1155*(52500*x^5 + 80500*x^4 + 7225*x^3 - 43482*x^2 - 18593*x + 12592)*sqrt(-2*x + 1)

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giac [A]  time = 0.82, size = 65, normalized size = 1.23 \begin {gather*} -\frac {125}{88} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {275}{24} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {1815}{56} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {1331}{40} \, {\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)^3,x, algorithm="giac")

[Out]

-125/88*(2*x - 1)^5*sqrt(-2*x + 1) - 275/24*(2*x - 1)^4*sqrt(-2*x + 1) - 1815/56*(2*x - 1)^3*sqrt(-2*x + 1) -
1331/40*(2*x - 1)^2*sqrt(-2*x + 1)

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maple [A]  time = 0.00, size = 25, normalized size = 0.47 \begin {gather*} -\frac {\left (13125 x^{3}+33250 x^{2}+31775 x +12592\right ) \left (-2 x +1\right )^{\frac {5}{2}}}{1155} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1155*(13125*x^3+33250*x^2+31775*x+12592)*(-2*x+1)^(5/2)

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maxima [A]  time = 0.58, size = 37, normalized size = 0.70 \begin {gather*} \frac {125}{88} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {275}{24} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {1815}{56} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {1331}{40} \, {\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)^3,x, algorithm="maxima")

[Out]

125/88*(-2*x + 1)^(11/2) - 275/24*(-2*x + 1)^(9/2) + 1815/56*(-2*x + 1)^(7/2) - 1331/40*(-2*x + 1)^(5/2)

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mupad [B]  time = 0.04, size = 37, normalized size = 0.70 \begin {gather*} \frac {1815\,{\left (1-2\,x\right )}^{7/2}}{56}-\frac {1331\,{\left (1-2\,x\right )}^{5/2}}{40}-\frac {275\,{\left (1-2\,x\right )}^{9/2}}{24}+\frac {125\,{\left (1-2\,x\right )}^{11/2}}{88} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(3/2)*(5*x + 3)^3,x)

[Out]

(1815*(1 - 2*x)^(7/2))/56 - (1331*(1 - 2*x)^(5/2))/40 - (275*(1 - 2*x)^(9/2))/24 + (125*(1 - 2*x)^(11/2))/88

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sympy [B]  time = 2.20, size = 286, normalized size = 5.40 \begin {gather*} \begin {cases} - \frac {100 \sqrt {5} i \left (x + \frac {3}{5}\right )^{5} \sqrt {10 x - 5}}{11} + \frac {40 \sqrt {5} i \left (x + \frac {3}{5}\right )^{4} \sqrt {10 x - 5}}{3} - \frac {11 \sqrt {5} i \left (x + \frac {3}{5}\right )^{3} \sqrt {10 x - 5}}{21} - \frac {121 \sqrt {5} i \left (x + \frac {3}{5}\right )^{2} \sqrt {10 x - 5}}{175} - \frac {2662 \sqrt {5} i \left (x + \frac {3}{5}\right ) \sqrt {10 x - 5}}{2625} - \frac {29282 \sqrt {5} i \sqrt {10 x - 5}}{13125} & \text {for}\: \frac {10 \left |{x + \frac {3}{5}}\right |}{11} > 1 \\- \frac {100 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{5}}{11} + \frac {40 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{4}}{3} - \frac {11 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{3}}{21} - \frac {121 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{2}}{175} - \frac {2662 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )}{2625} - \frac {29282 \sqrt {5} \sqrt {5 - 10 x}}{13125} & \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)**3,x)

[Out]

Piecewise((-100*sqrt(5)*I*(x + 3/5)**5*sqrt(10*x - 5)/11 + 40*sqrt(5)*I*(x + 3/5)**4*sqrt(10*x - 5)/3 - 11*sqr
t(5)*I*(x + 3/5)**3*sqrt(10*x - 5)/21 - 121*sqrt(5)*I*(x + 3/5)**2*sqrt(10*x - 5)/175 - 2662*sqrt(5)*I*(x + 3/
5)*sqrt(10*x - 5)/2625 - 29282*sqrt(5)*I*sqrt(10*x - 5)/13125, 10*Abs(x + 3/5)/11 > 1), (-100*sqrt(5)*sqrt(5 -
10*x)*(x + 3/5)**5/11 + 40*sqrt(5)*sqrt(5 - 10*x)*(x + 3/5)**4/3 - 11*sqrt(5)*sqrt(5 - 10*x)*(x + 3/5)**3/21
- 121*sqrt(5)*sqrt(5 - 10*x)*(x + 3/5)**2/175 - 2662*sqrt(5)*sqrt(5 - 10*x)*(x + 3/5)/2625 - 29282*sqrt(5)*sqr
t(5 - 10*x)/13125, True))

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