3.2146 \(\int \frac{(2+3 x)^5 (3+5 x)^2}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=105 \[ \frac{6075 (1-2 x)^{11/2}}{1408}-\frac{10845}{128} (1-2 x)^{9/2}+\frac{672003}{896} (1-2 x)^{7/2}-\frac{514017}{128} (1-2 x)^{5/2}+\frac{1965635}{128} (1-2 x)^{3/2}-\frac{8117095}{128} \sqrt{1-2 x}-\frac{6206585}{128 \sqrt{1-2 x}}+\frac{2033647}{384 (1-2 x)^{3/2}} \]

[Out]

2033647/(384*(1 - 2*x)^(3/2)) - 6206585/(128*Sqrt[1 - 2*x]) - (8117095*Sqrt[1 - 2*x])/128 + (1965635*(1 - 2*x)
^(3/2))/128 - (514017*(1 - 2*x)^(5/2))/128 + (672003*(1 - 2*x)^(7/2))/896 - (10845*(1 - 2*x)^(9/2))/128 + (607
5*(1 - 2*x)^(11/2))/1408

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Rubi [A]  time = 0.0196394, antiderivative size = 105, 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} \[ \frac{6075 (1-2 x)^{11/2}}{1408}-\frac{10845}{128} (1-2 x)^{9/2}+\frac{672003}{896} (1-2 x)^{7/2}-\frac{514017}{128} (1-2 x)^{5/2}+\frac{1965635}{128} (1-2 x)^{3/2}-\frac{8117095}{128} \sqrt{1-2 x}-\frac{6206585}{128 \sqrt{1-2 x}}+\frac{2033647}{384 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2033647/(384*(1 - 2*x)^(3/2)) - 6206585/(128*Sqrt[1 - 2*x]) - (8117095*Sqrt[1 - 2*x])/128 + (1965635*(1 - 2*x)
^(3/2))/128 - (514017*(1 - 2*x)^(5/2))/128 + (672003*(1 - 2*x)^(7/2))/896 - (10845*(1 - 2*x)^(9/2))/128 + (607
5*(1 - 2*x)^(11/2))/1408

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)^5 (3+5 x)^2}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac{2033647}{128 (1-2 x)^{5/2}}-\frac{6206585}{128 (1-2 x)^{3/2}}+\frac{8117095}{128 \sqrt{1-2 x}}-\frac{5896905}{128} \sqrt{1-2 x}+\frac{2570085}{128} (1-2 x)^{3/2}-\frac{672003}{128} (1-2 x)^{5/2}+\frac{97605}{128} (1-2 x)^{7/2}-\frac{6075}{128} (1-2 x)^{9/2}\right ) \, dx\\ &=\frac{2033647}{384 (1-2 x)^{3/2}}-\frac{6206585}{128 \sqrt{1-2 x}}-\frac{8117095}{128} \sqrt{1-2 x}+\frac{1965635}{128} (1-2 x)^{3/2}-\frac{514017}{128} (1-2 x)^{5/2}+\frac{672003}{896} (1-2 x)^{7/2}-\frac{10845}{128} (1-2 x)^{9/2}+\frac{6075 (1-2 x)^{11/2}}{1408}\\ \end{align*}

Mathematica [A]  time = 0.0205438, size = 48, normalized size = 0.46 \[ -\frac{127575 x^7+806085 x^6+2456001 x^5+5121279 x^4+9702012 x^3+32450916 x^2-65622552 x+21852008}{231 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(21852008 - 65622552*x + 32450916*x^2 + 9702012*x^3 + 5121279*x^4 + 2456001*x^5 + 806085*x^6 + 127575*x^7)/(2
31*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.003, size = 45, normalized size = 0.4 \begin{align*} -{\frac{127575\,{x}^{7}+806085\,{x}^{6}+2456001\,{x}^{5}+5121279\,{x}^{4}+9702012\,{x}^{3}[phone]\,{x}^{2}-65622552\,x+21852008}{231} \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)^5*(3+5*x)^2/(1-2*x)^(5/2),x)

[Out]

-1/231*(127575*x^7+806085*x^6+2456001*x^5+5121279*x^4+9702012*x^3+32450916*x^2-65622552*x+21852008)/(1-2*x)^(3
/2)

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Maxima [A]  time = 1.14561, size = 93, normalized size = 0.89 \begin{align*} \frac{6075}{1408} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - \frac{10845}{128} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{672003}{896} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{514017}{128} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{1965635}{128} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{8117095}{128} \, \sqrt{-2 \, x + 1} + \frac{26411 \,{\left (705 \, x - 314\right )}}{192 \,{\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)^5*(3+5*x)^2/(1-2*x)^(5/2),x, algorithm="maxima")

[Out]

6075/1408*(-2*x + 1)^(11/2) - 10845/128*(-2*x + 1)^(9/2) + 672003/896*(-2*x + 1)^(7/2) - 514017/128*(-2*x + 1)
^(5/2) + 1965635/128*(-2*x + 1)^(3/2) - 8117095/128*sqrt(-2*x + 1) + 26411/192*(705*x - 314)/(-2*x + 1)^(3/2)

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Fricas [A]  time = 1.54635, size = 200, normalized size = 1.9 \begin{align*} -\frac{{\left (127575 \, x^{7} + 806085 \, x^{6} + 2456001 \, x^{5} + 5121279 \, x^{4} + 9702012 \, x^{3} + 32450916 \, x^{2} - 65622552 \, x + 21852008\right )} \sqrt{-2 \, x + 1}}{231 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/231*(127575*x^7 + 806085*x^6 + 2456001*x^5 + 5121279*x^4 + 9702012*x^3 + 32450916*x^2 - 65622552*x + 218520
08)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

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Sympy [A]  time = 30.2473, size = 94, normalized size = 0.9 \begin{align*} \frac{6075 \left (1 - 2 x\right )^{\frac{11}{2}}}{1408} - \frac{10845 \left (1 - 2 x\right )^{\frac{9}{2}}}{128} + \frac{672003 \left (1 - 2 x\right )^{\frac{7}{2}}}{896} - \frac{514017 \left (1 - 2 x\right )^{\frac{5}{2}}}{128} + \frac{1965635 \left (1 - 2 x\right )^{\frac{3}{2}}}{128} - \frac{8117095 \sqrt{1 - 2 x}}{128} - \frac{6206585}{128 \sqrt{1 - 2 x}} + \frac{2033647}{384 \left (1 - 2 x\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6075*(1 - 2*x)**(11/2)/1408 - 10845*(1 - 2*x)**(9/2)/128 + 672003*(1 - 2*x)**(7/2)/896 - 514017*(1 - 2*x)**(5/
2)/128 + 1965635*(1 - 2*x)**(3/2)/128 - 8117095*sqrt(1 - 2*x)/128 - 6206585/(128*sqrt(1 - 2*x)) + 2033647/(384
*(1 - 2*x)**(3/2))

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Giac [A]  time = 1.64435, size = 140, normalized size = 1.33 \begin{align*} -\frac{6075}{1408} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - \frac{10845}{128} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{672003}{896} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{514017}{128} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{1965635}{128} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{8117095}{128} \, \sqrt{-2 \, x + 1} - \frac{26411 \,{\left (705 \, x - 314\right )}}{192 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6075/1408*(2*x - 1)^5*sqrt(-2*x + 1) - 10845/128*(2*x - 1)^4*sqrt(-2*x + 1) - 672003/896*(2*x - 1)^3*sqrt(-2*
x + 1) - 514017/128*(2*x - 1)^2*sqrt(-2*x + 1) + 1965635/128*(-2*x + 1)^(3/2) - 8117095/128*sqrt(-2*x + 1) - 2
6411/192*(705*x - 314)/((2*x - 1)*sqrt(-2*x + 1))