3.2070 \(\int \frac{(2+3 x)^6 (3+5 x)}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=105 \[ \frac{3645 (1-2 x)^{13/2}}{1664}-\frac{59049 (1-2 x)^{11/2}}{1408}+\frac{45549}{128} (1-2 x)^{9/2}-\frac{225855}{128} (1-2 x)^{7/2}+\frac{731619}{128} (1-2 x)^{5/2}-\frac{1692705}{128} (1-2 x)^{3/2}+\frac{3916031}{128} \sqrt{1-2 x}+\frac{1294139}{128 \sqrt{1-2 x}} \]

[Out]

1294139/(128*Sqrt[1 - 2*x]) + (3916031*Sqrt[1 - 2*x])/128 - (1692705*(1 - 2*x)^(3/2))/128 + (731619*(1 - 2*x)^
(5/2))/128 - (225855*(1 - 2*x)^(7/2))/128 + (45549*(1 - 2*x)^(9/2))/128 - (59049*(1 - 2*x)^(11/2))/1408 + (364
5*(1 - 2*x)^(13/2))/1664

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Rubi [A]  time = 0.0180748, antiderivative size = 105, 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} \[ \frac{3645 (1-2 x)^{13/2}}{1664}-\frac{59049 (1-2 x)^{11/2}}{1408}+\frac{45549}{128} (1-2 x)^{9/2}-\frac{225855}{128} (1-2 x)^{7/2}+\frac{731619}{128} (1-2 x)^{5/2}-\frac{1692705}{128} (1-2 x)^{3/2}+\frac{3916031}{128} \sqrt{1-2 x}+\frac{1294139}{128 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

Int[((2 + 3*x)^6*(3 + 5*x))/(1 - 2*x)^(3/2),x]

[Out]

1294139/(128*Sqrt[1 - 2*x]) + (3916031*Sqrt[1 - 2*x])/128 - (1692705*(1 - 2*x)^(3/2))/128 + (731619*(1 - 2*x)^
(5/2))/128 - (225855*(1 - 2*x)^(7/2))/128 + (45549*(1 - 2*x)^(9/2))/128 - (59049*(1 - 2*x)^(11/2))/1408 + (364
5*(1 - 2*x)^(13/2))/1664

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)^6 (3+5 x)}{(1-2 x)^{3/2}} \, dx &=\int \left (\frac{1294139}{128 (1-2 x)^{3/2}}-\frac{3916031}{128 \sqrt{1-2 x}}+\frac{5078115}{128} \sqrt{1-2 x}-\frac{3658095}{128} (1-2 x)^{3/2}+\frac{1580985}{128} (1-2 x)^{5/2}-\frac{409941}{128} (1-2 x)^{7/2}+\frac{59049}{128} (1-2 x)^{9/2}-\frac{3645}{128} (1-2 x)^{11/2}\right ) \, dx\\ &=\frac{1294139}{128 \sqrt{1-2 x}}+\frac{3916031}{128} \sqrt{1-2 x}-\frac{1692705}{128} (1-2 x)^{3/2}+\frac{731619}{128} (1-2 x)^{5/2}-\frac{225855}{128} (1-2 x)^{7/2}+\frac{45549}{128} (1-2 x)^{9/2}-\frac{59049 (1-2 x)^{11/2}}{1408}+\frac{3645 (1-2 x)^{13/2}}{1664}\\ \end{align*}

Mathematica [A]  time = 0.0183422, size = 48, normalized size = 0.46 \[ \frac{-40095 x^7-243486 x^6-687420 x^5-1230120 x^4-1663632 x^3-2109792 x^2-4512448 x+4539904}{143 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

Integrate[((2 + 3*x)^6*(3 + 5*x))/(1 - 2*x)^(3/2),x]

[Out]

(4539904 - 4512448*x - 2109792*x^2 - 1663632*x^3 - 1230120*x^4 - 687420*x^5 - 243486*x^6 - 40095*x^7)/(143*Sqr
t[1 - 2*x])

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Maple [A]  time = 0.003, size = 45, normalized size = 0.4 \begin{align*} -{\frac{40095\,{x}^{7}+243486\,{x}^{6}+687420\,{x}^{5}+1230120\,{x}^{4}+1663632\,{x}^{3}+2109792\,{x}^{2}+4512448\,x-4539904}{143}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^6*(3+5*x)/(1-2*x)^(3/2),x)

[Out]

-1/143*(40095*x^7+243486*x^6+687420*x^5+1230120*x^4+1663632*x^3+2109792*x^2+4512448*x-4539904)/(1-2*x)^(1/2)

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Maxima [A]  time = 1.02744, size = 99, normalized size = 0.94 \begin{align*} \frac{3645}{1664} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} - \frac{59049}{1408} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{45549}{128} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{225855}{128} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{731619}{128} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{1692705}{128} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{3916031}{128} \, \sqrt{-2 \, x + 1} + \frac{1294139}{128 \, \sqrt{-2 \, x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6*(3+5*x)/(1-2*x)^(3/2),x, algorithm="maxima")

[Out]

3645/1664*(-2*x + 1)^(13/2) - 59049/1408*(-2*x + 1)^(11/2) + 45549/128*(-2*x + 1)^(9/2) - 225855/128*(-2*x + 1
)^(7/2) + 731619/128*(-2*x + 1)^(5/2) - 1692705/128*(-2*x + 1)^(3/2) + 3916031/128*sqrt(-2*x + 1) + 1294139/12
8/sqrt(-2*x + 1)

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Fricas [A]  time = 1.56603, size = 181, normalized size = 1.72 \begin{align*} \frac{{\left (40095 \, x^{7} + 243486 \, x^{6} + 687420 \, x^{5} + 1230120 \, x^{4} + 1663632 \, x^{3} + 2109792 \, x^{2} + 4512448 \, x - 4539904\right )} \sqrt{-2 \, x + 1}}{143 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6*(3+5*x)/(1-2*x)^(3/2),x, algorithm="fricas")

[Out]

1/143*(40095*x^7 + 243486*x^6 + 687420*x^5 + 1230120*x^4 + 1663632*x^3 + 2109792*x^2 + 4512448*x - 4539904)*sq
rt(-2*x + 1)/(2*x - 1)

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Sympy [A]  time = 36.4772, size = 94, normalized size = 0.9 \begin{align*} \frac{3645 \left (1 - 2 x\right )^{\frac{13}{2}}}{1664} - \frac{59049 \left (1 - 2 x\right )^{\frac{11}{2}}}{1408} + \frac{45549 \left (1 - 2 x\right )^{\frac{9}{2}}}{128} - \frac{225855 \left (1 - 2 x\right )^{\frac{7}{2}}}{128} + \frac{731619 \left (1 - 2 x\right )^{\frac{5}{2}}}{128} - \frac{1692705 \left (1 - 2 x\right )^{\frac{3}{2}}}{128} + \frac{3916031 \sqrt{1 - 2 x}}{128} + \frac{1294139}{128 \sqrt{1 - 2 x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**6*(3+5*x)/(1-2*x)**(3/2),x)

[Out]

3645*(1 - 2*x)**(13/2)/1664 - 59049*(1 - 2*x)**(11/2)/1408 + 45549*(1 - 2*x)**(9/2)/128 - 225855*(1 - 2*x)**(7
/2)/128 + 731619*(1 - 2*x)**(5/2)/128 - 1692705*(1 - 2*x)**(3/2)/128 + 3916031*sqrt(1 - 2*x)/128 + 1294139/(12
8*sqrt(1 - 2*x))

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Giac [A]  time = 2.03666, size = 146, normalized size = 1.39 \begin{align*} \frac{3645}{1664} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} + \frac{59049}{1408} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{45549}{128} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{225855}{128} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{731619}{128} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{1692705}{128} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{3916031}{128} \, \sqrt{-2 \, x + 1} + \frac{1294139}{128 \, \sqrt{-2 \, x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6*(3+5*x)/(1-2*x)^(3/2),x, algorithm="giac")

[Out]

3645/1664*(2*x - 1)^6*sqrt(-2*x + 1) + 59049/1408*(2*x - 1)^5*sqrt(-2*x + 1) + 45549/128*(2*x - 1)^4*sqrt(-2*x
+ 1) + 225855/128*(2*x - 1)^3*sqrt(-2*x + 1) + 731619/128*(2*x - 1)^2*sqrt(-2*x + 1) - 1692705/128*(-2*x + 1)
^(3/2) + 3916031/128*sqrt(-2*x + 1) + 1294139/128/sqrt(-2*x + 1)