3.2083 \(\int \frac{(2+3 x)^5 (3+5 x)^2}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=105 \[ \frac{6075 (1-2 x)^{13/2}}{1664}-\frac{97605 (1-2 x)^{11/2}}{1408}+\frac{74667}{128} (1-2 x)^{9/2}-\frac{367155}{128} (1-2 x)^{7/2}+\frac{1179381}{128} (1-2 x)^{5/2}-\frac{8117095}{384} (1-2 x)^{3/2}+\frac{6206585}{128} \sqrt{1-2 x}+\frac{2033647}{128 \sqrt{1-2 x}} \]

[Out]

2033647/(128*Sqrt[1 - 2*x]) + (6206585*Sqrt[1 - 2*x])/128 - (8117095*(1 - 2*x)^(3/2))/384 + (1179381*(1 - 2*x)
^(5/2))/128 - (367155*(1 - 2*x)^(7/2))/128 + (74667*(1 - 2*x)^(9/2))/128 - (97605*(1 - 2*x)^(11/2))/1408 + (60
75*(1 - 2*x)^(13/2))/1664

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Rubi [A]  time = 0.0195816, 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} \[ \frac{6075 (1-2 x)^{13/2}}{1664}-\frac{97605 (1-2 x)^{11/2}}{1408}+\frac{74667}{128} (1-2 x)^{9/2}-\frac{367155}{128} (1-2 x)^{7/2}+\frac{1179381}{128} (1-2 x)^{5/2}-\frac{8117095}{384} (1-2 x)^{3/2}+\frac{6206585}{128} \sqrt{1-2 x}+\frac{2033647}{128 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2033647/(128*Sqrt[1 - 2*x]) + (6206585*Sqrt[1 - 2*x])/128 - (8117095*(1 - 2*x)^(3/2))/384 + (1179381*(1 - 2*x)
^(5/2))/128 - (367155*(1 - 2*x)^(7/2))/128 + (74667*(1 - 2*x)^(9/2))/128 - (97605*(1 - 2*x)^(11/2))/1408 + (60
75*(1 - 2*x)^(13/2))/1664

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)^{3/2}} \, dx &=\int \left (\frac{2033647}{128 (1-2 x)^{3/2}}-\frac{6206585}{128 \sqrt{1-2 x}}+\frac{8117095}{128} \sqrt{1-2 x}-\frac{5896905}{128} (1-2 x)^{3/2}+\frac{2570085}{128} (1-2 x)^{5/2}-\frac{672003}{128} (1-2 x)^{7/2}+\frac{97605}{128} (1-2 x)^{9/2}-\frac{6075}{128} (1-2 x)^{11/2}\right ) \, dx\\ &=\frac{2033647}{128 \sqrt{1-2 x}}+\frac{6206585}{128} \sqrt{1-2 x}-\frac{8117095}{384} (1-2 x)^{3/2}+\frac{1179381}{128} (1-2 x)^{5/2}-\frac{367155}{128} (1-2 x)^{7/2}+\frac{74667}{128} (1-2 x)^{9/2}-\frac{97605 (1-2 x)^{11/2}}{1408}+\frac{6075 (1-2 x)^{13/2}}{1664}\\ \end{align*}

Mathematica [A]  time = 0.019236, size = 48, normalized size = 0.46 \[ \frac{-200475 x^7-1201635 x^6-3350637 x^5-5928885 x^4-7945164 x^3-10015804 x^2-21370088 x+21493640}{429 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(21493640 - 21370088*x - 10015804*x^2 - 7945164*x^3 - 5928885*x^4 - 3350637*x^5 - 1201635*x^6 - 200475*x^7)/(4
29*Sqrt[1 - 2*x])

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Maple [A]  time = 0.003, size = 45, normalized size = 0.4 \begin{align*} -{\frac{200475\,{x}^{7}+1201635\,{x}^{6}+3350637\,{x}^{5}+5928885\,{x}^{4}+7945164\,{x}^{3}[phone]\,{x}^{2}[phone]\,x-21493640}{429}{\frac{1}{\sqrt{1-2\,x}}}} \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)^(3/2),x)

[Out]

-1/429*(200475*x^7+1201635*x^6+3350637*x^5+5928885*x^4+7945164*x^3+10015804*x^2+21370088*x-21493640)/(1-2*x)^(
1/2)

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Maxima [A]  time = 1.18423, size = 99, normalized size = 0.94 \begin{align*} \frac{6075}{1664} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} - \frac{97605}{1408} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{74667}{128} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{367155}{128} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{1179381}{128} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{8117095}{384} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{6206585}{128} \, \sqrt{-2 \, x + 1} + \frac{2033647}{128 \, \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)^(3/2),x, algorithm="maxima")

[Out]

6075/1664*(-2*x + 1)^(13/2) - 97605/1408*(-2*x + 1)^(11/2) + 74667/128*(-2*x + 1)^(9/2) - 367155/128*(-2*x + 1
)^(7/2) + 1179381/128*(-2*x + 1)^(5/2) - 8117095/384*(-2*x + 1)^(3/2) + 6206585/128*sqrt(-2*x + 1) + 2033647/1
28/sqrt(-2*x + 1)

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Fricas [A]  time = 1.63464, size = 189, normalized size = 1.8 \begin{align*} \frac{{\left (200475 \, x^{7} + 1201635 \, x^{6} + 3350637 \, x^{5} + 5928885 \, x^{4} + 7945164 \, x^{3} + 10015804 \, x^{2} + 21370088 \, x - 21493640\right )} \sqrt{-2 \, x + 1}}{429 \,{\left (2 \, 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)^(3/2),x, algorithm="fricas")

[Out]

1/429*(200475*x^7 + 1201635*x^6 + 3350637*x^5 + 5928885*x^4 + 7945164*x^3 + 10015804*x^2 + 21370088*x - 214936
40)*sqrt(-2*x + 1)/(2*x - 1)

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Sympy [A]  time = 35.4697, size = 94, normalized size = 0.9 \begin{align*} \frac{6075 \left (1 - 2 x\right )^{\frac{13}{2}}}{1664} - \frac{97605 \left (1 - 2 x\right )^{\frac{11}{2}}}{1408} + \frac{74667 \left (1 - 2 x\right )^{\frac{9}{2}}}{128} - \frac{367155 \left (1 - 2 x\right )^{\frac{7}{2}}}{128} + \frac{1179381 \left (1 - 2 x\right )^{\frac{5}{2}}}{128} - \frac{8117095 \left (1 - 2 x\right )^{\frac{3}{2}}}{384} + \frac{6206585 \sqrt{1 - 2 x}}{128} + \frac{2033647}{128 \sqrt{1 - 2 x}} \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)**(3/2),x)

[Out]

6075*(1 - 2*x)**(13/2)/1664 - 97605*(1 - 2*x)**(11/2)/1408 + 74667*(1 - 2*x)**(9/2)/128 - 367155*(1 - 2*x)**(7
/2)/128 + 1179381*(1 - 2*x)**(5/2)/128 - 8117095*(1 - 2*x)**(3/2)/384 + 6206585*sqrt(1 - 2*x)/128 + 2033647/(1
28*sqrt(1 - 2*x))

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Giac [A]  time = 2.41143, size = 146, normalized size = 1.39 \begin{align*} \frac{6075}{1664} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} + \frac{97605}{1408} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{74667}{128} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{367155}{128} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{1179381}{128} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{8117095}{384} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{6206585}{128} \, \sqrt{-2 \, x + 1} + \frac{2033647}{128 \, \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)^(3/2),x, algorithm="giac")

[Out]

6075/1664*(2*x - 1)^6*sqrt(-2*x + 1) + 97605/1408*(2*x - 1)^5*sqrt(-2*x + 1) + 74667/128*(2*x - 1)^4*sqrt(-2*x
+ 1) + 367155/128*(2*x - 1)^3*sqrt(-2*x + 1) + 1179381/128*(2*x - 1)^2*sqrt(-2*x + 1) - 8117095/384*(-2*x + 1
)^(3/2) + 6206585/128*sqrt(-2*x + 1) + 2033647/128/sqrt(-2*x + 1)