3.1449 \(\int \frac{3+5 x}{(1-2 x) (2+3 x)^7} \, dx\)

Optimal. Leaf size=87 \[ -\frac{176}{117649 (3 x+2)}-\frac{44}{16807 (3 x+2)^2}-\frac{44}{7203 (3 x+2)^3}-\frac{11}{686 (3 x+2)^4}-\frac{11}{245 (3 x+2)^5}+\frac{1}{126 (3 x+2)^6}-\frac{352 \log (1-2 x)}{823543}+\frac{352 \log (3 x+2)}{823543} \]

[Out]

1/(126*(2 + 3*x)^6) - 11/(245*(2 + 3*x)^5) - 11/(686*(2 + 3*x)^4) - 44/(7203*(2 + 3*x)^3) - 44/(16807*(2 + 3*x
)^2) - 176/(117649*(2 + 3*x)) - (352*Log[1 - 2*x])/823543 + (352*Log[2 + 3*x])/823543

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Rubi [A]  time = 0.0318751, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ -\frac{176}{117649 (3 x+2)}-\frac{44}{16807 (3 x+2)^2}-\frac{44}{7203 (3 x+2)^3}-\frac{11}{686 (3 x+2)^4}-\frac{11}{245 (3 x+2)^5}+\frac{1}{126 (3 x+2)^6}-\frac{352 \log (1-2 x)}{823543}+\frac{352 \log (3 x+2)}{823543} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1/(126*(2 + 3*x)^6) - 11/(245*(2 + 3*x)^5) - 11/(686*(2 + 3*x)^4) - 44/(7203*(2 + 3*x)^3) - 44/(16807*(2 + 3*x
)^2) - 176/(117649*(2 + 3*x)) - (352*Log[1 - 2*x])/823543 + (352*Log[2 + 3*x])/823543

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{3+5 x}{(1-2 x) (2+3 x)^7} \, dx &=\int \left (-\frac{704}{823543 (-1+2 x)}-\frac{1}{7 (2+3 x)^7}+\frac{33}{49 (2+3 x)^6}+\frac{66}{343 (2+3 x)^5}+\frac{132}{2401 (2+3 x)^4}+\frac{264}{16807 (2+3 x)^3}+\frac{528}{117649 (2+3 x)^2}+\frac{1056}{823543 (2+3 x)}\right ) \, dx\\ &=\frac{1}{126 (2+3 x)^6}-\frac{11}{245 (2+3 x)^5}-\frac{11}{686 (2+3 x)^4}-\frac{44}{7203 (2+3 x)^3}-\frac{44}{16807 (2+3 x)^2}-\frac{176}{117649 (2+3 x)}-\frac{352 \log (1-2 x)}{823543}+\frac{352 \log (2+3 x)}{823543}\\ \end{align*}

Mathematica [A]  time = 0.036287, size = 55, normalized size = 0.63 \[ \frac{-\frac{7 \left (3849120 x^5+15075720 x^4+24841080 x^3+22413105 x^2+12254814 x+3013741\right )}{(3 x+2)^6}-31680 \log (3-6 x)+31680 \log (3 x+2)}{74118870} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-7*(3013741 + 12254814*x + 22413105*x^2 + 24841080*x^3 + 15075720*x^4 + 3849120*x^5))/(2 + 3*x)^6 - 31680*Lo
g[3 - 6*x] + 31680*Log[2 + 3*x])/74118870

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Maple [A]  time = 0.008, size = 72, normalized size = 0.8 \begin{align*} -{\frac{352\,\ln \left ( 2\,x-1 \right ) }{823543}}+{\frac{1}{126\, \left ( 2+3\,x \right ) ^{6}}}-{\frac{11}{245\, \left ( 2+3\,x \right ) ^{5}}}-{\frac{11}{686\, \left ( 2+3\,x \right ) ^{4}}}-{\frac{44}{7203\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{44}{16807\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{176}{235298+352947\,x}}+{\frac{352\,\ln \left ( 2+3\,x \right ) }{823543}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-352/823543*ln(2*x-1)+1/126/(2+3*x)^6-11/245/(2+3*x)^5-11/686/(2+3*x)^4-44/7203/(2+3*x)^3-44/16807/(2+3*x)^2-1
76/117649/(2+3*x)+352/823543*ln(2+3*x)

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Maxima [A]  time = 1.03245, size = 103, normalized size = 1.18 \begin{align*} -\frac{3849120 \, x^{5} + 15075720 \, x^{4} + 24841080 \, x^{3} + 22413105 \, x^{2} + 12254814 \, x + 3013741}{10588410 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac{352}{823543} \, \log \left (3 \, x + 2\right ) - \frac{352}{823543} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10588410*(3849120*x^5 + 15075720*x^4 + 24841080*x^3 + 22413105*x^2 + 12254814*x + 3013741)/(729*x^6 + 2916*
x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 352/823543*log(3*x + 2) - 352/823543*log(2*x - 1)

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Fricas [A]  time = 1.1169, size = 463, normalized size = 5.32 \begin{align*} -\frac{26943840 \, x^{5} + 105530040 \, x^{4} + 173887560 \, x^{3} + 156891735 \, x^{2} - 31680 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (3 \, x + 2\right ) + 31680 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (2 \, x - 1\right ) + 85783698 \, x + 21096187}{74118870 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/74118870*(26943840*x^5 + 105530040*x^4 + 173887560*x^3 + 156891735*x^2 - 31680*(729*x^6 + 2916*x^5 + 4860*x
^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*log(3*x + 2) + 31680*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x
^2 + 576*x + 64)*log(2*x - 1) + 85783698*x + 21096187)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 +
576*x + 64)

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Sympy [A]  time = 0.196314, size = 75, normalized size = 0.86 \begin{align*} - \frac{3849120 x^{5} + 15075720 x^{4} + 24841080 x^{3} + 22413105 x^{2} + 12254814 x + 3013741}{7718950890 x^{6} + 30875803560 x^{5} + 51459672600 x^{4} + 45741931200 x^{3} + 22870965600 x^{2} + 6098924160 x + 677658240} - \frac{352 \log{\left (x - \frac{1}{2} \right )}}{823543} + \frac{352 \log{\left (x + \frac{2}{3} \right )}}{823543} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3849120*x**5 + 15075720*x**4 + 24841080*x**3 + 22413105*x**2 + 12254814*x + 3013741)/(7718950890*x**6 + 3087
5803560*x**5 + 51459672600*x**4 + 45741931200*x**3 + 22870965600*x**2 + 6098924160*x + 677658240) - 352*log(x
- 1/2)/823543 + 352*log(x + 2/3)/823543

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Giac [A]  time = 2.88297, size = 72, normalized size = 0.83 \begin{align*} -\frac{3849120 \, x^{5} + 15075720 \, x^{4} + 24841080 \, x^{3} + 22413105 \, x^{2} + 12254814 \, x + 3013741}{10588410 \,{\left (3 \, x + 2\right )}^{6}} + \frac{352}{823543} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{352}{823543} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10588410*(3849120*x^5 + 15075720*x^4 + 24841080*x^3 + 22413105*x^2 + 12254814*x + 3013741)/(3*x + 2)^6 + 35
2/823543*log(abs(3*x + 2)) - 352/823543*log(abs(2*x - 1))