3.1818 \(\int \sqrt{1-2 x} (2+3 x) (3+5 x)^3 \, dx\)

Optimal. Leaf size=66 \[ -\frac{375}{176} (1-2 x)^{11/2}+\frac{1675}{72} (1-2 x)^{9/2}-\frac{2805}{28} (1-2 x)^{7/2}+\frac{8349}{40} (1-2 x)^{5/2}-\frac{9317}{48} (1-2 x)^{3/2} \]

[Out]

(-9317*(1 - 2*x)^(3/2))/48 + (8349*(1 - 2*x)^(5/2))/40 - (2805*(1 - 2*x)^(7/2))/28 + (1675*(1 - 2*x)^(9/2))/72
- (375*(1 - 2*x)^(11/2))/176

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Rubi [A]  time = 0.0116714, antiderivative size = 66, 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} \[ -\frac{375}{176} (1-2 x)^{11/2}+\frac{1675}{72} (1-2 x)^{9/2}-\frac{2805}{28} (1-2 x)^{7/2}+\frac{8349}{40} (1-2 x)^{5/2}-\frac{9317}{48} (1-2 x)^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-9317*(1 - 2*x)^(3/2))/48 + (8349*(1 - 2*x)^(5/2))/40 - (2805*(1 - 2*x)^(7/2))/28 + (1675*(1 - 2*x)^(9/2))/72
- (375*(1 - 2*x)^(11/2))/176

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 \sqrt{1-2 x} (2+3 x) (3+5 x)^3 \, dx &=\int \left (\frac{9317}{16} \sqrt{1-2 x}-\frac{8349}{8} (1-2 x)^{3/2}+\frac{2805}{4} (1-2 x)^{5/2}-\frac{1675}{8} (1-2 x)^{7/2}+\frac{375}{16} (1-2 x)^{9/2}\right ) \, dx\\ &=-\frac{9317}{48} (1-2 x)^{3/2}+\frac{8349}{40} (1-2 x)^{5/2}-\frac{2805}{28} (1-2 x)^{7/2}+\frac{1675}{72} (1-2 x)^{9/2}-\frac{375}{176} (1-2 x)^{11/2}\\ \end{align*}

Mathematica [A]  time = 0.0145903, size = 33, normalized size = 0.5 \[ -\frac{(1-2 x)^{3/2} \left (118125 x^4+408625 x^3+598350 x^2+482583 x+223231\right )}{3465} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 - 2*x)^(3/2)*(223231 + 482583*x + 598350*x^2 + 408625*x^3 + 118125*x^4))/3465

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Maple [A]  time = 0.003, size = 30, normalized size = 0.5 \begin{align*} -{\frac{118125\,{x}^{4}+408625\,{x}^{3}+598350\,{x}^{2}+482583\,x+223231}{3465} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3465*(118125*x^4+408625*x^3+598350*x^2+482583*x+223231)*(1-2*x)^(3/2)

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Maxima [A]  time = 1.75219, size = 62, normalized size = 0.94 \begin{align*} -\frac{375}{176} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{1675}{72} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{2805}{28} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{8349}{40} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{9317}{48} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-375/176*(-2*x + 1)^(11/2) + 1675/72*(-2*x + 1)^(9/2) - 2805/28*(-2*x + 1)^(7/2) + 8349/40*(-2*x + 1)^(5/2) -
9317/48*(-2*x + 1)^(3/2)

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Fricas [A]  time = 1.60495, size = 127, normalized size = 1.92 \begin{align*} \frac{1}{3465} \,{\left (236250 \, x^{5} + 699125 \, x^{4} + 788075 \, x^{3} + 366816 \, x^{2} - 36121 \, x - 223231\right )} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3465*(236250*x^5 + 699125*x^4 + 788075*x^3 + 366816*x^2 - 36121*x - 223231)*sqrt(-2*x + 1)

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Sympy [A]  time = 1.91706, size = 58, normalized size = 0.88 \begin{align*} - \frac{375 \left (1 - 2 x\right )^{\frac{11}{2}}}{176} + \frac{1675 \left (1 - 2 x\right )^{\frac{9}{2}}}{72} - \frac{2805 \left (1 - 2 x\right )^{\frac{7}{2}}}{28} + \frac{8349 \left (1 - 2 x\right )^{\frac{5}{2}}}{40} - \frac{9317 \left (1 - 2 x\right )^{\frac{3}{2}}}{48} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-375*(1 - 2*x)**(11/2)/176 + 1675*(1 - 2*x)**(9/2)/72 - 2805*(1 - 2*x)**(7/2)/28 + 8349*(1 - 2*x)**(5/2)/40 -
9317*(1 - 2*x)**(3/2)/48

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Giac [A]  time = 1.70161, size = 100, normalized size = 1.52 \begin{align*} \frac{375}{176} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{1675}{72} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{2805}{28} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{8349}{40} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{9317}{48} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

375/176*(2*x - 1)^5*sqrt(-2*x + 1) + 1675/72*(2*x - 1)^4*sqrt(-2*x + 1) + 2805/28*(2*x - 1)^3*sqrt(-2*x + 1) +
8349/40*(2*x - 1)^2*sqrt(-2*x + 1) - 9317/48*(-2*x + 1)^(3/2)