3.1269 \(\int \frac {(1-2 x)^2 (3+5 x)^2}{(2+3 x)^8} \, dx\)

Optimal. Leaf size=56 \[ -\frac {100}{729 (3 x+2)^3}+\frac {185}{243 (3 x+2)^4}-\frac {503}{405 (3 x+2)^5}+\frac {259}{729 (3 x+2)^6}-\frac {7}{243 (3 x+2)^7} \]

[Out]

-7/243/(2+3*x)^7+259/729/(2+3*x)^6-503/405/(2+3*x)^5+185/243/(2+3*x)^4-100/729/(2+3*x)^3

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, 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 = {88} \[ -\frac {100}{729 (3 x+2)^3}+\frac {185}{243 (3 x+2)^4}-\frac {503}{405 (3 x+2)^5}+\frac {259}{729 (3 x+2)^6}-\frac {7}{243 (3 x+2)^7} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^2*(3 + 5*x)^2)/(2 + 3*x)^8,x]

[Out]

-7/(243*(2 + 3*x)^7) + 259/(729*(2 + 3*x)^6) - 503/(405*(2 + 3*x)^5) + 185/(243*(2 + 3*x)^4) - 100/(729*(2 + 3
*x)^3)

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 {(1-2 x)^2 (3+5 x)^2}{(2+3 x)^8} \, dx &=\int \left (\frac {49}{81 (2+3 x)^8}-\frac {518}{81 (2+3 x)^7}+\frac {503}{27 (2+3 x)^6}-\frac {740}{81 (2+3 x)^5}+\frac {100}{81 (2+3 x)^4}\right ) \, dx\\ &=-\frac {7}{243 (2+3 x)^7}+\frac {259}{729 (2+3 x)^6}-\frac {503}{405 (2+3 x)^5}+\frac {185}{243 (2+3 x)^4}-\frac {100}{729 (2+3 x)^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 31, normalized size = 0.55 \[ \frac {-40500 x^4-33075 x^3+1107 x^2+1461 x-1423}{3645 (3 x+2)^7} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^2*(3 + 5*x)^2)/(2 + 3*x)^8,x]

[Out]

(-1423 + 1461*x + 1107*x^2 - 33075*x^3 - 40500*x^4)/(3645*(2 + 3*x)^7)

________________________________________________________________________________________

fricas [A]  time = 0.70, size = 59, normalized size = 1.05 \[ -\frac {40500 \, x^{4} + 33075 \, x^{3} - 1107 \, x^{2} - 1461 \, x + 1423}{3645 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2*(3+5*x)^2/(2+3*x)^8,x, algorithm="fricas")

[Out]

-1/3645*(40500*x^4 + 33075*x^3 - 1107*x^2 - 1461*x + 1423)/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 151
20*x^3 + 6048*x^2 + 1344*x + 128)

________________________________________________________________________________________

giac [A]  time = 1.13, size = 29, normalized size = 0.52 \[ -\frac {40500 \, x^{4} + 33075 \, x^{3} - 1107 \, x^{2} - 1461 \, x + 1423}{3645 \, {\left (3 \, x + 2\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2*(3+5*x)^2/(2+3*x)^8,x, algorithm="giac")

[Out]

-1/3645*(40500*x^4 + 33075*x^3 - 1107*x^2 - 1461*x + 1423)/(3*x + 2)^7

________________________________________________________________________________________

maple [A]  time = 0.00, size = 47, normalized size = 0.84 \[ -\frac {7}{243 \left (3 x +2\right )^{7}}+\frac {259}{729 \left (3 x +2\right )^{6}}-\frac {503}{405 \left (3 x +2\right )^{5}}+\frac {185}{243 \left (3 x +2\right )^{4}}-\frac {100}{729 \left (3 x +2\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^2*(5*x+3)^2/(3*x+2)^8,x)

[Out]

-7/243/(3*x+2)^7+259/729/(3*x+2)^6-503/405/(3*x+2)^5+185/243/(3*x+2)^4-100/729/(3*x+2)^3

________________________________________________________________________________________

maxima [A]  time = 0.50, size = 59, normalized size = 1.05 \[ -\frac {40500 \, x^{4} + 33075 \, x^{3} - 1107 \, x^{2} - 1461 \, x + 1423}{3645 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2*(3+5*x)^2/(2+3*x)^8,x, algorithm="maxima")

[Out]

-1/3645*(40500*x^4 + 33075*x^3 - 1107*x^2 - 1461*x + 1423)/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 151
20*x^3 + 6048*x^2 + 1344*x + 128)

________________________________________________________________________________________

mupad [B]  time = 1.09, size = 46, normalized size = 0.82 \[ \frac {185}{243\,{\left (3\,x+2\right )}^4}-\frac {100}{729\,{\left (3\,x+2\right )}^3}-\frac {503}{405\,{\left (3\,x+2\right )}^5}+\frac {259}{729\,{\left (3\,x+2\right )}^6}-\frac {7}{243\,{\left (3\,x+2\right )}^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x - 1)^2*(5*x + 3)^2)/(3*x + 2)^8,x)

[Out]

185/(243*(3*x + 2)^4) - 100/(729*(3*x + 2)^3) - 503/(405*(3*x + 2)^5) + 259/(729*(3*x + 2)^6) - 7/(243*(3*x +
2)^7)

________________________________________________________________________________________

sympy [A]  time = 0.19, size = 54, normalized size = 0.96 \[ \frac {- 40500 x^{4} - 33075 x^{3} + 1107 x^{2} + 1461 x - 1423}{7971615 x^{7} + 37200870 x^{6} + 74401740 x^{5} + 82668600 x^{4} + 55112400 x^{3} + 22044960 x^{2} + 4898880 x + 466560} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**2*(3+5*x)**2/(2+3*x)**8,x)

[Out]

(-40500*x**4 - 33075*x**3 + 1107*x**2 + 1461*x - 1423)/(7971615*x**7 + 37200870*x**6 + 74401740*x**5 + 8266860
0*x**4 + 55112400*x**3 + 22044960*x**2 + 4898880*x + 466560)

________________________________________________________________________________________