3.2015 \(\int \frac{(2+3 x)^2 (3+5 x)^2}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=66 \[ -\frac{25}{16} (1-2 x)^{9/2}+\frac{255}{14} (1-2 x)^{7/2}-\frac{3467}{40} (1-2 x)^{5/2}+\frac{1309}{6} (1-2 x)^{3/2}-\frac{5929}{16} \sqrt{1-2 x} \]

[Out]

(-5929*Sqrt[1 - 2*x])/16 + (1309*(1 - 2*x)^(3/2))/6 - (3467*(1 - 2*x)^(5/2))/40 + (255*(1 - 2*x)^(7/2))/14 - (
25*(1 - 2*x)^(9/2))/16

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Rubi [A]  time = 0.012597, 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} \[ -\frac{25}{16} (1-2 x)^{9/2}+\frac{255}{14} (1-2 x)^{7/2}-\frac{3467}{40} (1-2 x)^{5/2}+\frac{1309}{6} (1-2 x)^{3/2}-\frac{5929}{16} \sqrt{1-2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5929*Sqrt[1 - 2*x])/16 + (1309*(1 - 2*x)^(3/2))/6 - (3467*(1 - 2*x)^(5/2))/40 + (255*(1 - 2*x)^(7/2))/14 - (
25*(1 - 2*x)^(9/2))/16

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 \frac{(2+3 x)^2 (3+5 x)^2}{\sqrt{1-2 x}} \, dx &=\int \left (\frac{5929}{16 \sqrt{1-2 x}}-\frac{1309}{2} \sqrt{1-2 x}+\frac{3467}{8} (1-2 x)^{3/2}-\frac{255}{2} (1-2 x)^{5/2}+\frac{225}{16} (1-2 x)^{7/2}\right ) \, dx\\ &=-\frac{5929}{16} \sqrt{1-2 x}+\frac{1309}{6} (1-2 x)^{3/2}-\frac{3467}{40} (1-2 x)^{5/2}+\frac{255}{14} (1-2 x)^{7/2}-\frac{25}{16} (1-2 x)^{9/2}\\ \end{align*}

Mathematica [A]  time = 0.0119313, size = 33, normalized size = 0.5 \[ -\frac{1}{105} \sqrt{1-2 x} \left (2625 x^4+10050 x^3+17391 x^2+19574 x+23354\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - 2*x]*(23354 + 19574*x + 17391*x^2 + 10050*x^3 + 2625*x^4))/105

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Maple [A]  time = 0.004, size = 30, normalized size = 0.5 \begin{align*} -{\frac{2625\,{x}^{4}+10050\,{x}^{3}+17391\,{x}^{2}+19574\,x+23354}{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)^2/(1-2*x)^(1/2),x)

[Out]

-1/105*(2625*x^4+10050*x^3+17391*x^2+19574*x+23354)*(1-2*x)^(1/2)

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Maxima [A]  time = 2.18494, size = 62, normalized size = 0.94 \begin{align*} -\frac{25}{16} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{255}{14} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{3467}{40} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{1309}{6} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{5929}{16} \, \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="maxima")

[Out]

-25/16*(-2*x + 1)^(9/2) + 255/14*(-2*x + 1)^(7/2) - 3467/40*(-2*x + 1)^(5/2) + 1309/6*(-2*x + 1)^(3/2) - 5929/
16*sqrt(-2*x + 1)

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Fricas [A]  time = 1.40504, size = 103, normalized size = 1.56 \begin{align*} -\frac{1}{105} \,{\left (2625 \, x^{4} + 10050 \, x^{3} + 17391 \, x^{2} + 19574 \, x + 23354\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]

-1/105*(2625*x^4 + 10050*x^3 + 17391*x^2 + 19574*x + 23354)*sqrt(-2*x + 1)

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Sympy [A]  time = 30.7771, size = 58, normalized size = 0.88 \begin{align*} - \frac{25 \left (1 - 2 x\right )^{\frac{9}{2}}}{16} + \frac{255 \left (1 - 2 x\right )^{\frac{7}{2}}}{14} - \frac{3467 \left (1 - 2 x\right )^{\frac{5}{2}}}{40} + \frac{1309 \left (1 - 2 x\right )^{\frac{3}{2}}}{6} - \frac{5929 \sqrt{1 - 2 x}}{16} \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]

-25*(1 - 2*x)**(9/2)/16 + 255*(1 - 2*x)**(7/2)/14 - 3467*(1 - 2*x)**(5/2)/40 + 1309*(1 - 2*x)**(3/2)/6 - 5929*
sqrt(1 - 2*x)/16

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Giac [A]  time = 1.54153, size = 90, normalized size = 1.36 \begin{align*} -\frac{25}{16} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{255}{14} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{3467}{40} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{1309}{6} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{5929}{16} \, \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="giac")

[Out]

-25/16*(2*x - 1)^4*sqrt(-2*x + 1) - 255/14*(2*x - 1)^3*sqrt(-2*x + 1) - 3467/40*(2*x - 1)^2*sqrt(-2*x + 1) + 1
309/6*(-2*x + 1)^(3/2) - 5929/16*sqrt(-2*x + 1)