3.2095 \(\int \frac{(2+3 x)^3 (3+5 x)^3}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=92 \[ -\frac{3375}{704} (1-2 x)^{11/2}+\frac{1275}{16} (1-2 x)^{9/2}-\frac{260055}{448} (1-2 x)^{7/2}+\frac{98209}{40} (1-2 x)^{5/2}-\frac{444983}{64} (1-2 x)^{3/2}+\frac{302379}{16} \sqrt{1-2 x}+\frac{456533}{64 \sqrt{1-2 x}} \]

[Out]

456533/(64*Sqrt[1 - 2*x]) + (302379*Sqrt[1 - 2*x])/16 - (444983*(1 - 2*x)^(3/2))/64 + (98209*(1 - 2*x)^(5/2))/
40 - (260055*(1 - 2*x)^(7/2))/448 + (1275*(1 - 2*x)^(9/2))/16 - (3375*(1 - 2*x)^(11/2))/704

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Rubi [A]  time = 0.0192098, antiderivative size = 92, 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} \[ -\frac{3375}{704} (1-2 x)^{11/2}+\frac{1275}{16} (1-2 x)^{9/2}-\frac{260055}{448} (1-2 x)^{7/2}+\frac{98209}{40} (1-2 x)^{5/2}-\frac{444983}{64} (1-2 x)^{3/2}+\frac{302379}{16} \sqrt{1-2 x}+\frac{456533}{64 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

456533/(64*Sqrt[1 - 2*x]) + (302379*Sqrt[1 - 2*x])/16 - (444983*(1 - 2*x)^(3/2))/64 + (98209*(1 - 2*x)^(5/2))/
40 - (260055*(1 - 2*x)^(7/2))/448 + (1275*(1 - 2*x)^(9/2))/16 - (3375*(1 - 2*x)^(11/2))/704

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)^3 (3+5 x)^3}{(1-2 x)^{3/2}} \, dx &=\int \left (\frac{456533}{64 (1-2 x)^{3/2}}-\frac{302379}{16 \sqrt{1-2 x}}+\frac{1334949}{64} \sqrt{1-2 x}-\frac{98209}{8} (1-2 x)^{3/2}+\frac{260055}{64} (1-2 x)^{5/2}-\frac{11475}{16} (1-2 x)^{7/2}+\frac{3375}{64} (1-2 x)^{9/2}\right ) \, dx\\ &=\frac{456533}{64 \sqrt{1-2 x}}+\frac{302379}{16} \sqrt{1-2 x}-\frac{444983}{64} (1-2 x)^{3/2}+\frac{98209}{40} (1-2 x)^{5/2}-\frac{260055}{448} (1-2 x)^{7/2}+\frac{1275}{16} (1-2 x)^{9/2}-\frac{3375}{704} (1-2 x)^{11/2}\\ \end{align*}

Mathematica [A]  time = 0.0188514, size = 43, normalized size = 0.47 \[ \frac{-118125 x^6-627375 x^5-1564350 x^4-2569643 x^3-3611453 x^2-8012926 x+8096086}{385 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8096086 - 8012926*x - 3611453*x^2 - 2569643*x^3 - 1564350*x^4 - 627375*x^5 - 118125*x^6)/(385*Sqrt[1 - 2*x])

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Maple [A]  time = 0.003, size = 40, normalized size = 0.4 \begin{align*} -{\frac{118125\,{x}^{6}+627375\,{x}^{5}+1564350\,{x}^{4}+2569643\,{x}^{3}+3611453\,{x}^{2}+8012926\,x-8096086}{385}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/385*(118125*x^6+627375*x^5+1564350*x^4+2569643*x^3+3611453*x^2+8012926*x-8096086)/(1-2*x)^(1/2)

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Maxima [A]  time = 2.5011, size = 86, normalized size = 0.93 \begin{align*} -\frac{3375}{704} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{1275}{16} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{260055}{448} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{98209}{40} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{444983}{64} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{302379}{16} \, \sqrt{-2 \, x + 1} + \frac{456533}{64 \, \sqrt{-2 \, x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3375/704*(-2*x + 1)^(11/2) + 1275/16*(-2*x + 1)^(9/2) - 260055/448*(-2*x + 1)^(7/2) + 98209/40*(-2*x + 1)^(5/
2) - 444983/64*(-2*x + 1)^(3/2) + 302379/16*sqrt(-2*x + 1) + 456533/64/sqrt(-2*x + 1)

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Fricas [A]  time = 1.52041, size = 165, normalized size = 1.79 \begin{align*} \frac{{\left (118125 \, x^{6} + 627375 \, x^{5} + 1564350 \, x^{4} + 2569643 \, x^{3} + 3611453 \, x^{2} + 8012926 \, x - 8096086\right )} \sqrt{-2 \, x + 1}}{385 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/385*(118125*x^6 + 627375*x^5 + 1564350*x^4 + 2569643*x^3 + 3611453*x^2 + 8012926*x - 8096086)*sqrt(-2*x + 1)
/(2*x - 1)

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Sympy [A]  time = 28.8409, size = 82, normalized size = 0.89 \begin{align*} - \frac{3375 \left (1 - 2 x\right )^{\frac{11}{2}}}{704} + \frac{1275 \left (1 - 2 x\right )^{\frac{9}{2}}}{16} - \frac{260055 \left (1 - 2 x\right )^{\frac{7}{2}}}{448} + \frac{98209 \left (1 - 2 x\right )^{\frac{5}{2}}}{40} - \frac{444983 \left (1 - 2 x\right )^{\frac{3}{2}}}{64} + \frac{302379 \sqrt{1 - 2 x}}{16} + \frac{456533}{64 \sqrt{1 - 2 x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3375*(1 - 2*x)**(11/2)/704 + 1275*(1 - 2*x)**(9/2)/16 - 260055*(1 - 2*x)**(7/2)/448 + 98209*(1 - 2*x)**(5/2)/
40 - 444983*(1 - 2*x)**(3/2)/64 + 302379*sqrt(1 - 2*x)/16 + 456533/(64*sqrt(1 - 2*x))

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Giac [A]  time = 1.94652, size = 124, normalized size = 1.35 \begin{align*} \frac{3375}{704} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{1275}{16} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{260055}{448} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{98209}{40} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{444983}{64} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{302379}{16} \, \sqrt{-2 \, x + 1} + \frac{456533}{64 \, \sqrt{-2 \, x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3375/704*(2*x - 1)^5*sqrt(-2*x + 1) + 1275/16*(2*x - 1)^4*sqrt(-2*x + 1) + 260055/448*(2*x - 1)^3*sqrt(-2*x +
1) + 98209/40*(2*x - 1)^2*sqrt(-2*x + 1) - 444983/64*(-2*x + 1)^(3/2) + 302379/16*sqrt(-2*x + 1) + 456533/64/s
qrt(-2*x + 1)