3.1500 \(\int \frac{1}{(1-2 x) (2+3 x)^7 (3+5 x)} \, dx\)

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

[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

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Rubi [A]  time = 0.0457392, antiderivative size = 97, normalized size of antiderivative = 1., 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} \[ \frac{167115051}{117649 (3 x+2)}+\frac{4774713}{33614 (3 x+2)^2}+\frac{45473}{2401 (3 x+2)^3}+\frac{3897}{1372 (3 x+2)^4}+\frac{111}{245 (3 x+2)^5}+\frac{1}{14 (3 x+2)^6}-\frac{128 \log (1-2 x)}{9058973}-\frac{5849026977 \log (3 x+2)}{823543}+\frac{78125}{11} \log (5 x+3) \]

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*}

Mathematica [A]  time = 0.0368074, size = 97, normalized size = 1. \[ \frac{167115051}{117649 (3 x+2)}+\frac{4774713}{33614 (3 x+2)^2}+\frac{45473}{2401 (3 x+2)^3}+\frac{3897}{1372 (3 x+2)^4}+\frac{111}{245 (3 x+2)^5}+\frac{1}{14 (3 x+2)^6}-\frac{128 \log (1-2 x)}{9058973}-\frac{5849026977 \log (6 x+4)}{823543}+\frac{78125}{11} \log (10 x+6) \]

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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Maple [A]  time = 0.008, size = 80, normalized size = 0.8 \begin{align*} -{\frac{128\,\ln \left ( 2\,x-1 \right ) }{9058973}}+{\frac{1}{14\, \left ( 2+3\,x \right ) ^{6}}}+{\frac{111}{245\, \left ( 2+3\,x \right ) ^{5}}}+{\frac{3897}{1372\, \left ( 2+3\,x \right ) ^{4}}}+{\frac{45473}{2401\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{4774713}{33614\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{167115051}{235298+352947\,x}}-{\frac{5849026977\,\ln \left ( 2+3\,x \right ) }{823543}}+{\frac{78125\,\ln \left ( 3+5\,x \right ) }{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-128/9058973*ln(2*x-1)+1/14/(2+3*x)^6+111/245/(2+3*x)^5+3897/1372/(2+3*x)^4+45473/2401/(2+3*x)^3+4774713/33614
/(2+3*x)^2+167115051/117649/(2+3*x)-5849026977/823543*ln(2+3*x)+78125/11*ln(3+5*x)

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Maxima [A]  time = 1.0645, size = 113, normalized size = 1.16 \begin{align*} \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{align*}

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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Fricas [B]  time = 1.34894, size = 649, normalized size = 6.69 \begin{align*} \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{align*}

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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Sympy [A]  time = 0.256626, size = 85, normalized size = 0.88 \begin{align*} \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{align*}

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 + 112686853
098)/(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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Giac [A]  time = 1.28583, size = 84, normalized size = 0.87 \begin{align*} \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{align*}

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))