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

Optimal. Leaf size=37 \[ \frac {5 (5 x+3)^4}{12 (3 x+2)^4}+\frac {7 (5 x+3)^4}{15 (3 x+2)^5} \]

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Rubi [A]  time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {78, 37} \begin {gather*} \frac {5 (5 x+3)^4}{12 (3 x+2)^4}+\frac {7 (5 x+3)^4}{15 (3 x+2)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(7*(3 + 5*x)^4)/(15*(2 + 3*x)^5) + (5*(3 + 5*x)^4)/(12*(2 + 3*x)^4)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rubi steps

\begin {align*} \int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^6} \, dx &=\frac {7 (3+5 x)^4}{15 (2+3 x)^5}+\frac {5}{3} \int \frac {(3+5 x)^3}{(2+3 x)^5} \, dx\\ &=\frac {7 (3+5 x)^4}{15 (2+3 x)^5}+\frac {5 (3+5 x)^4}{12 (2+3 x)^4}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 31, normalized size = 0.84 \begin {gather*} \frac {405000 x^4+803250 x^3+559800 x^2+153795 x+11758}{4860 (3 x+2)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(11758 + 153795*x + 559800*x^2 + 803250*x^3 + 405000*x^4)/(4860*(2 + 3*x)^5)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.31, size = 49, normalized size = 1.32 \begin {gather*} \frac {405000 \, x^{4} + 803250 \, x^{3} + 559800 \, x^{2} + 153795 \, x + 11758}{4860 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4860*(405000*x^4 + 803250*x^3 + 559800*x^2 + 153795*x + 11758)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240
*x + 32)

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giac [A]  time = 1.21, size = 29, normalized size = 0.78 \begin {gather*} \frac {405000 \, x^{4} + 803250 \, x^{3} + 559800 \, x^{2} + 153795 \, x + 11758}{4860 \, {\left (3 \, x + 2\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4860*(405000*x^4 + 803250*x^3 + 559800*x^2 + 153795*x + 11758)/(3*x + 2)^5

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maple [A]  time = 0.00, size = 47, normalized size = 1.27 \begin {gather*} \frac {185}{243 \left (3 x +2\right )^{3}}-\frac {107}{972 \left (3 x +2\right )^{4}}-\frac {1025}{486 \left (3 x +2\right )^{2}}+\frac {250}{243 \left (3 x +2\right )}+\frac {7}{1215 \left (3 x +2\right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

185/243/(3*x+2)^3-107/972/(3*x+2)^4-1025/486/(3*x+2)^2+250/243/(3*x+2)+7/1215/(3*x+2)^5

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maxima [A]  time = 0.56, size = 49, normalized size = 1.32 \begin {gather*} \frac {405000 \, x^{4} + 803250 \, x^{3} + 559800 \, x^{2} + 153795 \, x + 11758}{4860 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4860*(405000*x^4 + 803250*x^3 + 559800*x^2 + 153795*x + 11758)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240
*x + 32)

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mupad [B]  time = 1.10, size = 46, normalized size = 1.24 \begin {gather*} \frac {250}{243\,\left (3\,x+2\right )}-\frac {1025}{486\,{\left (3\,x+2\right )}^2}+\frac {185}{243\,{\left (3\,x+2\right )}^3}-\frac {107}{972\,{\left (3\,x+2\right )}^4}+\frac {7}{1215\,{\left (3\,x+2\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

250/(243*(3*x + 2)) - 1025/(486*(3*x + 2)^2) + 185/(243*(3*x + 2)^3) - 107/(972*(3*x + 2)^4) + 7/(1215*(3*x +
2)^5)

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sympy [A]  time = 0.16, size = 48, normalized size = 1.30 \begin {gather*} - \frac {- 405000 x^{4} - 803250 x^{3} - 559800 x^{2} - 153795 x - 11758}{1180980 x^{5} + 3936600 x^{4} + 5248800 x^{3} + 3499200 x^{2} + 1166400 x + 155520} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-405000*x**4 - 803250*x**3 - 559800*x**2 - 153795*x - 11758)/(1180980*x**5 + 3936600*x**4 + 5248800*x**3 + 3
499200*x**2 + 1166400*x + 155520)

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