3.1861 \(\int (1-2 x)^{3/2} (2+3 x)^2 (3+5 x) \, dx\)

Optimal. Leaf size=53 \[ \frac{45}{88} (1-2 x)^{11/2}-\frac{103}{24} (1-2 x)^{9/2}+\frac{101}{8} (1-2 x)^{7/2}-\frac{539}{40} (1-2 x)^{5/2} \]

[Out]

(-539*(1 - 2*x)^(5/2))/40 + (101*(1 - 2*x)^(7/2))/8 - (103*(1 - 2*x)^(9/2))/24 + (45*(1 - 2*x)^(11/2))/88

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Rubi [A]  time = 0.0099535, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {77} \[ \frac{45}{88} (1-2 x)^{11/2}-\frac{103}{24} (1-2 x)^{9/2}+\frac{101}{8} (1-2 x)^{7/2}-\frac{539}{40} (1-2 x)^{5/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-539*(1 - 2*x)^(5/2))/40 + (101*(1 - 2*x)^(7/2))/8 - (103*(1 - 2*x)^(9/2))/24 + (45*(1 - 2*x)^(11/2))/88

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)^{3/2} (2+3 x)^2 (3+5 x) \, dx &=\int \left (\frac{539}{8} (1-2 x)^{3/2}-\frac{707}{8} (1-2 x)^{5/2}+\frac{309}{8} (1-2 x)^{7/2}-\frac{45}{8} (1-2 x)^{9/2}\right ) \, dx\\ &=-\frac{539}{40} (1-2 x)^{5/2}+\frac{101}{8} (1-2 x)^{7/2}-\frac{103}{24} (1-2 x)^{9/2}+\frac{45}{88} (1-2 x)^{11/2}\\ \end{align*}

Mathematica [A]  time = 0.0127899, size = 28, normalized size = 0.53 \[ -\frac{1}{165} (1-2 x)^{5/2} \left (675 x^3+1820 x^2+1840 x+764\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 - 2*x)^(5/2)*(764 + 1840*x + 1820*x^2 + 675*x^3))/165

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Maple [A]  time = 0.003, size = 25, normalized size = 0.5 \begin{align*} -{\frac{675\,{x}^{3}+1820\,{x}^{2}+1840\,x+764}{165} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/165*(675*x^3+1820*x^2+1840*x+764)*(1-2*x)^(5/2)

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Maxima [A]  time = 1.39208, size = 50, normalized size = 0.94 \begin{align*} \frac{45}{88} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - \frac{103}{24} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{101}{8} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{539}{40} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

45/88*(-2*x + 1)^(11/2) - 103/24*(-2*x + 1)^(9/2) + 101/8*(-2*x + 1)^(7/2) - 539/40*(-2*x + 1)^(5/2)

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Fricas [A]  time = 1.30718, size = 109, normalized size = 2.06 \begin{align*} -\frac{1}{165} \,{\left (2700 \, x^{5} + 4580 \, x^{4} + 755 \, x^{3} - 2484 \, x^{2} - 1216 \, x + 764\right )} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/165*(2700*x^5 + 4580*x^4 + 755*x^3 - 2484*x^2 - 1216*x + 764)*sqrt(-2*x + 1)

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Sympy [A]  time = 7.76258, size = 46, normalized size = 0.87 \begin{align*} \frac{45 \left (1 - 2 x\right )^{\frac{11}{2}}}{88} - \frac{103 \left (1 - 2 x\right )^{\frac{9}{2}}}{24} + \frac{101 \left (1 - 2 x\right )^{\frac{7}{2}}}{8} - \frac{539 \left (1 - 2 x\right )^{\frac{5}{2}}}{40} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

45*(1 - 2*x)**(11/2)/88 - 103*(1 - 2*x)**(9/2)/24 + 101*(1 - 2*x)**(7/2)/8 - 539*(1 - 2*x)**(5/2)/40

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Giac [A]  time = 2.11521, size = 88, normalized size = 1.66 \begin{align*} -\frac{45}{88} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - \frac{103}{24} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{101}{8} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{539}{40} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-45/88*(2*x - 1)^5*sqrt(-2*x + 1) - 103/24*(2*x - 1)^4*sqrt(-2*x + 1) - 101/8*(2*x - 1)^3*sqrt(-2*x + 1) - 539
/40*(2*x - 1)^2*sqrt(-2*x + 1)