∫ 1________ (1−2x)(2+3x)7(3+5x) dx

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

Optimal. Leaf size=97 \[ \frac {167115051}{117649 (3 x+2)}+\frac {4774713}{33614 (3 x+2)^2}+\frac {45473}{2401 (3 x+2)^3}+\frac {3897}{1372 (3 x+2)^4}+\frac {111}{245 (3 x+2)^5}+\frac {1}{14 (3 x+2)^6}-\frac {128 \log (1-2 x)}{9058973}-\frac {5849026977 \log (3 x+2)}{823543}+\frac {78125}{11} \log (5 x+3) \]

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Rubi [A] time = 0.05, antiderivative size = 97, 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 = {72} \begin {gather*} \frac {167115051}{117649 (3 x+2)}+\frac {4774713}{33614 (3 x+2)^2}+\frac {45473}{2401 (3 x+2)^3}+\frac {3897}{1372 (3 x+2)^4}+\frac {111}{245 (3 x+2)^5}+\frac {1}{14 (3 x+2)^6}-\frac {128 \log (1-2 x)}{9058973}-\frac {5849026977 \log (3 x+2)}{823543}+\frac {78125}{11} \log (5 x+3) \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

1/(14*(2 + 3*x)^6) + 111/(245*(2 + 3*x)^5) + 3897/(1372*(2 + 3*x)^4) + 45473/(2401*(2 + 3*x)^3) + 4774713/(336

14*(2 + 3*x)^2) + 167115051/(117649*(2 + 3*x)) - (128*Log[1 - 2*x])/9058973 - (5849026977*Log[2 + 3*x])/823543

+ (78125*Log[3 + 5*x])/11

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(

e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {1}{(1-2 x) (2+3 x)^7 (3+5 x)} \, dx &=\int \left (-\frac {256}{9058973 (-1+2 x)}-\frac {9}{7 (2+3 x)^7}-\frac {333}{49 (2+3 x)^6}-\frac {11691}{343 (2+3 x)^5}-\frac {409257}{2401 (2+3 x)^4}-\frac {14324139}{16807 (2+3 x)^3}-\frac {501345153}{117649 (2+3 x)^2}-\frac {17547080931}{823543 (2+3 x)}+\frac {390625}{11 (3+5 x)}\right ) \, dx\\ &=\frac {1}{14 (2+3 x)^6}+\frac {111}{245 (2+3 x)^5}+\frac {3897}{1372 (2+3 x)^4}+\frac {45473}{2401 (2+3 x)^3}+\frac {4774713}{33614 (2+3 x)^2}+\frac {167115051}{117649 (2+3 x)}-\frac {128 \log (1-2 x)}{9058973}-\frac {5849026977 \log (2+3 x)}{823543}+\frac {78125}{11} \log (3+5 x)\\ \end {align*}

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Mathematica [A] time = 0.06, size = 97, normalized size = 1.00 \begin {gather*} \frac {167115051}{117649 (3 x+2)}+\frac {4774713}{33614 (3 x+2)^2}+\frac {45473}{2401 (3 x+2)^3}+\frac {3897}{1372 (3 x+2)^4}+\frac {111}{245 (3 x+2)^5}+\frac {1}{14 (3 x+2)^6}-\frac {128 \log (1-2 x)}{9058973}-\frac {5849026977 \log (6 x+4)}{823543}+\frac {78125}{11} \log (10 x+6) \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

1/(14*(2 + 3*x)^6) + 111/(245*(2 + 3*x)^5) + 3897/(1372*(2 + 3*x)^4) + 45473/(2401*(2 + 3*x)^3) + 4774713/(336

14*(2 + 3*x)^2) + 167115051/(117649*(2 + 3*x)) - (128*Log[1 - 2*x])/9058973 - (5849026977*Log[4 + 6*x])/823543

+ (78125*Log[6 + 10*x])/11

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

Veriﬁcation is not applicable to the result.

[In]

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

[Out]

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

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fricas [B] time = 1.71, size = 173, normalized size = 1.78 \begin {gather*} \frac {62537794385220 \, x^{5} + 210543906566070 \, x^{4} + 283597312285980 \, x^{3} + 191046007176255 \, x^{2} + 1286785937500 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (5 \, x + 3\right ) - 1286785934940 \, {\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 ) - 2560 \, {\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 ) + 64366302153384 \, x + 8676887688546}{181179460 \, {\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/(1-2*x)/(2+3*x)^7/(3+5*x),x, algorithm="fricas")

[Out]

1/181179460*(62537794385220*x^5 + 210543906566070*x^4 + 283597312285980*x^3 + 191046007176255*x^2 + 1286785937

500*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*log(5*x + 3) - 1286785934940*(729*x^6 +

2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*log(3*x + 2) - 2560*(729*x^6 + 2916*x^5 + 4860*x^4 +

4320*x^3 + 2160*x^2 + 576*x + 64)*log(2*x - 1) + 64366302153384*x + 8676887688546)/(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 = 0.86, size = 62, normalized size = 0.64 \begin {gather*} \frac {3 \, {\left (270726382620 \, x^{5} + 911445482970 \, x^{4} + 1227693992580 \, x^{3} + 827038992105 \, x^{2} + 278642000664 \, x + 37562284366\right )}}{2352980 \, {\left (3 \, x + 2\right )}^{6}} + \frac {78125}{11} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {5849026977}{823543} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {128}{9058973} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2352980*(270726382620*x^5 + 911445482970*x^4 + 1227693992580*x^3 + 827038992105*x^2 + 278642000664*x + 37562

284366)/(3*x + 2)^6 + 78125/11*log(abs(5*x + 3)) - 5849026977/823543*log(abs(3*x + 2)) - 128/9058973*log(abs(2

*x - 1))

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maple [A] time = 0.01, size = 80, normalized size = 0.82 \begin {gather*} -\frac {128 \ln \left (2 x -1\right )}{9058973}-\frac {5849026977 \ln \left (3 x +2\right )}{823543}+\frac {78125 \ln \left (5 x +3\right )}{11}+\frac {1}{14 \left (3 x +2\right )^{6}}+\frac {111}{245 \left (3 x +2\right )^{5}}+\frac {3897}{1372 \left (3 x +2\right )^{4}}+\frac {45473}{2401 \left (3 x +2\right )^{3}}+\frac {4774713}{33614 \left (3 x +2\right )^{2}}+\frac {167115051}{117649 \left (3 x +2\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

78125/11*ln(5*x+3)+1/14/(3*x+2)^6+111/245/(3*x+2)^5+3897/1372/(3*x+2)^4+45473/2401/(3*x+2)^3+4774713/33614/(3*

x+2)^2+167115051/117649/(3*x+2)-5849026977/823543*ln(3*x+2)-128/9058973*ln(2*x-1)

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maxima [A] time = 0.65, size = 84, normalized size = 0.87 \begin {gather*} \frac {3 \, {\left (270726382620 \, x^{5} + 911445482970 \, x^{4} + 1227693992580 \, x^{3} + 827038992105 \, x^{2} + 278642000664 \, x + 37562284366\right )}}{2352980 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {78125}{11} \, \log \left (5 \, x + 3\right ) - \frac {5849026977}{823543} \, \log \left (3 \, x + 2\right ) - \frac {128}{9058973} \, \log \left (2 \, x - 1\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2352980*(270726382620*x^5 + 911445482970*x^4 + 1227693992580*x^3 + 827038992105*x^2 + 278642000664*x + 37562

284366)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 78125/11*log(5*x + 3) - 584902697

7/823543*log(3*x + 2) - 128/9058973*log(2*x - 1)

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mupad [B] time = 1.12, size = 75, normalized size = 0.77 \begin {gather*} \frac {78125\,\ln \left (x+\frac {3}{5}\right )}{11}-\frac {5849026977\,\ln \left (x+\frac {2}{3}\right )}{823543}-\frac {128\,\ln \left (x-\frac {1}{2}\right )}{9058973}+\frac {\frac {55705017\,x^5}{117649}+\frac {1125241337\,x^4}{705894}+\frac {6820522181\,x^3}{3176523}+\frac {18378644269\,x^2}{12706092}+\frac {7740055574\,x}{15882615}+\frac {18781142183}{285887070}}{x^6+4\,x^5+\frac {20\,x^4}{3}+\frac {160\,x^3}{27}+\frac {80\,x^2}{27}+\frac {64\,x}{81}+\frac {64}{729}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(78125*log(x + 3/5))/11 - (5849026977*log(x + 2/3))/823543 - (128*log(x - 1/2))/9058973 + ((7740055574*x)/1588

2615 + (18378644269*x^2)/12706092 + (6820522181*x^3)/3176523 + (1125241337*x^4)/705894 + (55705017*x^5)/117649

+ 18781142183/285887070)/((64*x)/81 + (80*x^2)/27 + (160*x^3)/27 + (20*x^4)/3 + 4*x^5 + x^6 + 64/729)

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sympy [A] time = 0.28, size = 87, normalized size = 0.90 \begin {gather*} - \frac {- 812179147860 x^{5} - 2734336448910 x^{4} - 3683081977740 x^{3} - 2481116976315 x^{2} - 835926001992 x - 112686853098}{1715322420 x^{6} + 6861289680 x^{5} + 11435482800 x^{4} + 10164873600 x^{3} + 5082436800 x^{2} + 1355316480 x + 150590720} - \frac {128 \log {\left (x - \frac {1}{2} \right )}}{9058973} + \frac {78125 \log {\left (x + \frac {3}{5} \right )}}{11} - \frac {5849026977 \log {\left (x + \frac {2}{3} \right )}}{823543} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-812179147860*x**5 - 2734336448910*x**4 - 3683081977740*x**3 - 2481116976315*x**2 - 835926001992*x - 1126868

53098)/(1715322420*x**6 + 6861289680*x**5 + 11435482800*x**4 + 10164873600*x**3 + 5082436800*x**2 + 1355316480

*x + 150590720) - 128*log(x - 1/2)/9058973 + 78125*log(x + 3/5)/11 - 5849026977*log(x + 2/3)/823543

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