3.13.14 \(\int \frac {(1-2 x)^3 (3+5 x)}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=55 \[ -\frac {103 (1-2 x)^4}{30870 (3 x+2)^4}-\frac {103 (1-2 x)^4}{2205 (3 x+2)^5}+\frac {(1-2 x)^4}{126 (3 x+2)^6} \]

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Rubi [A]  time = 0.01, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {78, 45, 37} \begin {gather*} -\frac {103 (1-2 x)^4}{30870 (3 x+2)^4}-\frac {103 (1-2 x)^4}{2205 (3 x+2)^5}+\frac {(1-2 x)^4}{126 (3 x+2)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^3*(3 + 5*x))/(2 + 3*x)^7,x]

[Out]

(1 - 2*x)^4/(126*(2 + 3*x)^6) - (103*(1 - 2*x)^4)/(2205*(2 + 3*x)^5) - (103*(1 - 2*x)^4)/(30870*(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 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 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 (3+5 x)}{(2+3 x)^7} \, dx &=\frac {(1-2 x)^4}{126 (2+3 x)^6}+\frac {103}{63} \int \frac {(1-2 x)^3}{(2+3 x)^6} \, dx\\ &=\frac {(1-2 x)^4}{126 (2+3 x)^6}-\frac {103 (1-2 x)^4}{2205 (2+3 x)^5}+\frac {206 \int \frac {(1-2 x)^3}{(2+3 x)^5} \, dx}{2205}\\ &=\frac {(1-2 x)^4}{126 (2+3 x)^6}-\frac {103 (1-2 x)^4}{2205 (2+3 x)^5}-\frac {103 (1-2 x)^4}{30870 (2+3 x)^4}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 31, normalized size = 0.56 \begin {gather*} \frac {48600 x^4+14040 x^3+3375 x^2+7218 x-413}{7290 (3 x+2)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^3*(3 + 5*x))/(2 + 3*x)^7,x]

[Out]

(-413 + 7218*x + 3375*x^2 + 14040*x^3 + 48600*x^4)/(7290*(2 + 3*x)^6)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(1-2 x)^3 (3+5 x)}{(2+3 x)^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((1 - 2*x)^3*(3 + 5*x))/(2 + 3*x)^7,x]

[Out]

IntegrateAlgebraic[((1 - 2*x)^3*(3 + 5*x))/(2 + 3*x)^7, x]

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fricas [A]  time = 1.19, size = 54, normalized size = 0.98 \begin {gather*} \frac {48600 \, x^{4} + 14040 \, x^{3} + 3375 \, x^{2} + 7218 \, x - 413}{7290 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3*(3+5*x)/(2+3*x)^7,x, algorithm="fricas")

[Out]

1/7290*(48600*x^4 + 14040*x^3 + 3375*x^2 + 7218*x - 413)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2
+ 576*x + 64)

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giac [A]  time = 0.85, size = 29, normalized size = 0.53 \begin {gather*} \frac {48600 \, x^{4} + 14040 \, x^{3} + 3375 \, x^{2} + 7218 \, x - 413}{7290 \, {\left (3 \, x + 2\right )}^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3*(3+5*x)/(2+3*x)^7,x, algorithm="giac")

[Out]

1/7290*(48600*x^4 + 14040*x^3 + 3375*x^2 + 7218*x - 413)/(3*x + 2)^6

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maple [A]  time = 0.00, size = 47, normalized size = 0.85 \begin {gather*} \frac {343}{1458 \left (3 x +2\right )^{6}}-\frac {428}{729 \left (3 x +2\right )^{3}}+\frac {259}{162 \left (3 x +2\right )^{4}}+\frac {20}{243 \left (3 x +2\right )^{2}}-\frac {2009}{1215 \left (3 x +2\right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^3*(5*x+3)/(3*x+2)^7,x)

[Out]

343/1458/(3*x+2)^6-428/729/(3*x+2)^3+259/162/(3*x+2)^4+20/243/(3*x+2)^2-2009/1215/(3*x+2)^5

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maxima [A]  time = 0.50, size = 54, normalized size = 0.98 \begin {gather*} \frac {48600 \, x^{4} + 14040 \, x^{3} + 3375 \, x^{2} + 7218 \, x - 413}{7290 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3*(3+5*x)/(2+3*x)^7,x, algorithm="maxima")

[Out]

1/7290*(48600*x^4 + 14040*x^3 + 3375*x^2 + 7218*x - 413)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2
+ 576*x + 64)

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mupad [B]  time = 0.03, size = 46, normalized size = 0.84 \begin {gather*} \frac {20}{243\,{\left (3\,x+2\right )}^2}-\frac {428}{729\,{\left (3\,x+2\right )}^3}+\frac {259}{162\,{\left (3\,x+2\right )}^4}-\frac {2009}{1215\,{\left (3\,x+2\right )}^5}+\frac {343}{1458\,{\left (3\,x+2\right )}^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((2*x - 1)^3*(5*x + 3))/(3*x + 2)^7,x)

[Out]

20/(243*(3*x + 2)^2) - 428/(729*(3*x + 2)^3) + 259/(162*(3*x + 2)^4) - 2009/(1215*(3*x + 2)^5) + 343/(1458*(3*
x + 2)^6)

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sympy [A]  time = 0.17, size = 51, normalized size = 0.93 \begin {gather*} - \frac {- 48600 x^{4} - 14040 x^{3} - 3375 x^{2} - 7218 x + 413}{5314410 x^{6} + 21257640 x^{5} + 35429400 x^{4} + 31492800 x^{3} + 15746400 x^{2} + 4199040 x + 466560} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**3*(3+5*x)/(2+3*x)**7,x)

[Out]

-(-48600*x**4 - 14040*x**3 - 3375*x**2 - 7218*x + 413)/(5314410*x**6 + 21257640*x**5 + 35429400*x**4 + 3149280
0*x**3 + 15746400*x**2 + 4199040*x + 466560)

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