3.1805 \(\int \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^2 \, dx\)

Optimal. Leaf size=66 \[ -\frac{225}{176} (1-2 x)^{11/2}+\frac{85}{6} (1-2 x)^{9/2}-\frac{3467}{56} (1-2 x)^{7/2}+\frac{1309}{10} (1-2 x)^{5/2}-\frac{5929}{48} (1-2 x)^{3/2} \]

[Out]

(-5929*(1 - 2*x)^(3/2))/48 + (1309*(1 - 2*x)^(5/2))/10 - (3467*(1 - 2*x)^(7/2))/56 + (85*(1 - 2*x)^(9/2))/6 -
(225*(1 - 2*x)^(11/2))/176

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Rubi [A]  time = 0.01326, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {88} \[ -\frac{225}{176} (1-2 x)^{11/2}+\frac{85}{6} (1-2 x)^{9/2}-\frac{3467}{56} (1-2 x)^{7/2}+\frac{1309}{10} (1-2 x)^{5/2}-\frac{5929}{48} (1-2 x)^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5929*(1 - 2*x)^(3/2))/48 + (1309*(1 - 2*x)^(5/2))/10 - (3467*(1 - 2*x)^(7/2))/56 + (85*(1 - 2*x)^(9/2))/6 -
(225*(1 - 2*x)^(11/2))/176

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^2 \, dx &=\int \left (\frac{5929}{16} \sqrt{1-2 x}-\frac{1309}{2} (1-2 x)^{3/2}+\frac{3467}{8} (1-2 x)^{5/2}-\frac{255}{2} (1-2 x)^{7/2}+\frac{225}{16} (1-2 x)^{9/2}\right ) \, dx\\ &=-\frac{5929}{48} (1-2 x)^{3/2}+\frac{1309}{10} (1-2 x)^{5/2}-\frac{3467}{56} (1-2 x)^{7/2}+\frac{85}{6} (1-2 x)^{9/2}-\frac{225}{176} (1-2 x)^{11/2}\\ \end{align*}

Mathematica [A]  time = 0.0142711, size = 33, normalized size = 0.5 \[ -\frac{(1-2 x)^{3/2} \left (23625 x^4+83650 x^3+125115 x^2+102714 x+48098\right )}{1155} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 - 2*x)^(3/2)*(48098 + 102714*x + 125115*x^2 + 83650*x^3 + 23625*x^4))/1155

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Maple [A]  time = 0.003, size = 30, normalized size = 0.5 \begin{align*} -{\frac{23625\,{x}^{4}+83650\,{x}^{3}+125115\,{x}^{2}+102714\,x+48098}{1155} \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)^2*(3+5*x)^2*(1-2*x)^(1/2),x)

[Out]

-1/1155*(23625*x^4+83650*x^3+125115*x^2+102714*x+48098)*(1-2*x)^(3/2)

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Maxima [A]  time = 1.01038, size = 62, normalized size = 0.94 \begin{align*} -\frac{225}{176} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{85}{6} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{3467}{56} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{1309}{10} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{5929}{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)^2*(3+5*x)^2*(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

-225/176*(-2*x + 1)^(11/2) + 85/6*(-2*x + 1)^(9/2) - 3467/56*(-2*x + 1)^(7/2) + 1309/10*(-2*x + 1)^(5/2) - 592
9/48*(-2*x + 1)^(3/2)

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Fricas [A]  time = 1.60726, size = 122, normalized size = 1.85 \begin{align*} \frac{1}{1155} \,{\left (47250 \, x^{5} + 143675 \, x^{4} + 166580 \, x^{3} + 80313 \, x^{2} - 6518 \, x - 48098\right )} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1155*(47250*x^5 + 143675*x^4 + 166580*x^3 + 80313*x^2 - 6518*x - 48098)*sqrt(-2*x + 1)

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Sympy [A]  time = 1.92181, size = 58, normalized size = 0.88 \begin{align*} - \frac{225 \left (1 - 2 x\right )^{\frac{11}{2}}}{176} + \frac{85 \left (1 - 2 x\right )^{\frac{9}{2}}}{6} - \frac{3467 \left (1 - 2 x\right )^{\frac{7}{2}}}{56} + \frac{1309 \left (1 - 2 x\right )^{\frac{5}{2}}}{10} - \frac{5929 \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)**2*(3+5*x)**2*(1-2*x)**(1/2),x)

[Out]

-225*(1 - 2*x)**(11/2)/176 + 85*(1 - 2*x)**(9/2)/6 - 3467*(1 - 2*x)**(7/2)/56 + 1309*(1 - 2*x)**(5/2)/10 - 592
9*(1 - 2*x)**(3/2)/48

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Giac [A]  time = 1.39782, size = 100, normalized size = 1.52 \begin{align*} \frac{225}{176} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{85}{6} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{3467}{56} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{1309}{10} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{5929}{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)^2*(3+5*x)^2*(1-2*x)^(1/2),x, algorithm="giac")

[Out]

225/176*(2*x - 1)^5*sqrt(-2*x + 1) + 85/6*(2*x - 1)^4*sqrt(-2*x + 1) + 3467/56*(2*x - 1)^3*sqrt(-2*x + 1) + 13
09/10*(2*x - 1)^2*sqrt(-2*x + 1) - 5929/48*(-2*x + 1)^(3/2)