3.2004 \(\int \frac{(2+3 x)^2 (3+5 x)}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=53 \[ \frac{45}{56} (1-2 x)^{7/2}-\frac{309}{40} (1-2 x)^{5/2}+\frac{707}{24} (1-2 x)^{3/2}-\frac{539}{8} \sqrt{1-2 x} \]

[Out]

(-539*Sqrt[1 - 2*x])/8 + (707*(1 - 2*x)^(3/2))/24 - (309*(1 - 2*x)^(5/2))/40 + (45*(1 - 2*x)^(7/2))/56

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Rubi [A]  time = 0.0096609, 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} \[ \frac{45}{56} (1-2 x)^{7/2}-\frac{309}{40} (1-2 x)^{5/2}+\frac{707}{24} (1-2 x)^{3/2}-\frac{539}{8} \sqrt{1-2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-539*Sqrt[1 - 2*x])/8 + (707*(1 - 2*x)^(3/2))/24 - (309*(1 - 2*x)^(5/2))/40 + (45*(1 - 2*x)^(7/2))/56

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 \frac{(2+3 x)^2 (3+5 x)}{\sqrt{1-2 x}} \, dx &=\int \left (\frac{539}{8 \sqrt{1-2 x}}-\frac{707}{8} \sqrt{1-2 x}+\frac{309}{8} (1-2 x)^{3/2}-\frac{45}{8} (1-2 x)^{5/2}\right ) \, dx\\ &=-\frac{539}{8} \sqrt{1-2 x}+\frac{707}{24} (1-2 x)^{3/2}-\frac{309}{40} (1-2 x)^{5/2}+\frac{45}{56} (1-2 x)^{7/2}\\ \end{align*}

Mathematica [A]  time = 0.0098676, size = 28, normalized size = 0.53 \[ -\frac{1}{105} \sqrt{1-2 x} \left (675 x^3+2232 x^2+3448 x+4708\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - 2*x]*(4708 + 3448*x + 2232*x^2 + 675*x^3))/105

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Maple [A]  time = 0.003, size = 25, normalized size = 0.5 \begin{align*} -{\frac{675\,{x}^{3}+2232\,{x}^{2}+3448\,x+4708}{105}\sqrt{1-2\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(675*x^3+2232*x^2+3448*x+4708)*(1-2*x)^(1/2)

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Maxima [A]  time = 2.03735, size = 50, normalized size = 0.94 \begin{align*} \frac{45}{56} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{309}{40} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{707}{24} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{539}{8} \, \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)/(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

45/56*(-2*x + 1)^(7/2) - 309/40*(-2*x + 1)^(5/2) + 707/24*(-2*x + 1)^(3/2) - 539/8*sqrt(-2*x + 1)

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Fricas [A]  time = 1.32821, size = 81, normalized size = 1.53 \begin{align*} -\frac{1}{105} \,{\left (675 \, x^{3} + 2232 \, x^{2} + 3448 \, x + 4708\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)/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/105*(675*x^3 + 2232*x^2 + 3448*x + 4708)*sqrt(-2*x + 1)

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Sympy [A]  time = 20.2747, size = 46, normalized size = 0.87 \begin{align*} \frac{45 \left (1 - 2 x\right )^{\frac{7}{2}}}{56} - \frac{309 \left (1 - 2 x\right )^{\frac{5}{2}}}{40} + \frac{707 \left (1 - 2 x\right )^{\frac{3}{2}}}{24} - \frac{539 \sqrt{1 - 2 x}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

45*(1 - 2*x)**(7/2)/56 - 309*(1 - 2*x)**(5/2)/40 + 707*(1 - 2*x)**(3/2)/24 - 539*sqrt(1 - 2*x)/8

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Giac [A]  time = 1.49669, size = 69, normalized size = 1.3 \begin{align*} -\frac{45}{56} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{309}{40} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{707}{24} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{539}{8} \, \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)/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

-45/56*(2*x - 1)^3*sqrt(-2*x + 1) - 309/40*(2*x - 1)^2*sqrt(-2*x + 1) + 707/24*(-2*x + 1)^(3/2) - 539/8*sqrt(-
2*x + 1)