3.1933 \(\int (1-2 x)^{5/2} (2+3 x) (3+5 x) \, dx\)

Optimal. Leaf size=40 \[ -\frac {15}{44} (1-2 x)^{11/2}+\frac {17}{9} (1-2 x)^{9/2}-\frac {11}{4} (1-2 x)^{7/2} \]

[Out]

-11/4*(1-2*x)^(7/2)+17/9*(1-2*x)^(9/2)-15/44*(1-2*x)^(11/2)

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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 = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \[ -\frac {15}{44} (1-2 x)^{11/2}+\frac {17}{9} (1-2 x)^{9/2}-\frac {11}{4} (1-2 x)^{7/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-11*(1 - 2*x)^(7/2))/4 + (17*(1 - 2*x)^(9/2))/9 - (15*(1 - 2*x)^(11/2))/44

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int (1-2 x)^{5/2} (2+3 x) (3+5 x) \, dx &=\int \left (\frac {77}{4} (1-2 x)^{5/2}-17 (1-2 x)^{7/2}+\frac {15}{4} (1-2 x)^{9/2}\right ) \, dx\\ &=-\frac {11}{4} (1-2 x)^{7/2}+\frac {17}{9} (1-2 x)^{9/2}-\frac {15}{44} (1-2 x)^{11/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 23, normalized size = 0.58 \[ -\frac {1}{99} (1-2 x)^{7/2} \left (135 x^2+239 x+119\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/99*((1 - 2*x)^(7/2)*(119 + 239*x + 135*x^2))

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fricas [A]  time = 0.93, size = 34, normalized size = 0.85 \[ \frac {1}{99} \, {\left (1080 \, x^{5} + 292 \, x^{4} - 1106 \, x^{3} - 129 \, x^{2} + 475 \, x - 119\right )} \sqrt {-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/99*(1080*x^5 + 292*x^4 - 1106*x^3 - 129*x^2 + 475*x - 119)*sqrt(-2*x + 1)

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giac [A]  time = 0.92, size = 49, normalized size = 1.22 \[ \frac {15}{44} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {17}{9} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {11}{4} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

15/44*(2*x - 1)^5*sqrt(-2*x + 1) + 17/9*(2*x - 1)^4*sqrt(-2*x + 1) + 11/4*(2*x - 1)^3*sqrt(-2*x + 1)

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maple [A]  time = 0.00, size = 20, normalized size = 0.50 \[ -\frac {\left (135 x^{2}+239 x +119\right ) \left (-2 x +1\right )^{\frac {7}{2}}}{99} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/99*(135*x^2+239*x+119)*(-2*x+1)^(7/2)

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maxima [A]  time = 0.56, size = 28, normalized size = 0.70 \[ -\frac {15}{44} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {17}{9} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {11}{4} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-15/44*(-2*x + 1)^(11/2) + 17/9*(-2*x + 1)^(9/2) - 11/4*(-2*x + 1)^(7/2)

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mupad [B]  time = 0.03, size = 23, normalized size = 0.58 \[ -\frac {{\left (1-2\,x\right )}^{7/2}\,\left (1496\,x+135\,{\left (2\,x-1\right )}^2+341\right )}{396} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((1 - 2*x)^(7/2)*(1496*x + 135*(2*x - 1)^2 + 341))/396

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sympy [B]  time = 1.55, size = 85, normalized size = 2.12 \[ \frac {120 x^{5} \sqrt {1 - 2 x}}{11} + \frac {292 x^{4} \sqrt {1 - 2 x}}{99} - \frac {1106 x^{3} \sqrt {1 - 2 x}}{99} - \frac {43 x^{2} \sqrt {1 - 2 x}}{33} + \frac {475 x \sqrt {1 - 2 x}}{99} - \frac {119 \sqrt {1 - 2 x}}{99} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

120*x**5*sqrt(1 - 2*x)/11 + 292*x**4*sqrt(1 - 2*x)/99 - 1106*x**3*sqrt(1 - 2*x)/99 - 43*x**2*sqrt(1 - 2*x)/33
+ 475*x*sqrt(1 - 2*x)/99 - 119*sqrt(1 - 2*x)/99

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