3.1206 \(\int \frac{1-2 x}{(2+3 x)^7 (3+5 x)} \, dx\)

Optimal. Leaf size=81 \[ \frac{6875}{3 x+2}+\frac{1375}{2 (3 x+2)^2}+\frac{275}{3 (3 x+2)^3}+\frac{55}{4 (3 x+2)^4}+\frac{11}{5 (3 x+2)^5}+\frac{7}{18 (3 x+2)^6}-34375 \log (3 x+2)+34375 \log (5 x+3) \]

[Out]

7/(18*(2 + 3*x)^6) + 11/(5*(2 + 3*x)^5) + 55/(4*(2 + 3*x)^4) + 275/(3*(2 + 3*x)^3) + 1375/(2*(2 + 3*x)^2) + 68
75/(2 + 3*x) - 34375*Log[2 + 3*x] + 34375*Log[3 + 5*x]

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Rubi [A]  time = 0.0287595, antiderivative size = 81, 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} \[ \frac{6875}{3 x+2}+\frac{1375}{2 (3 x+2)^2}+\frac{275}{3 (3 x+2)^3}+\frac{55}{4 (3 x+2)^4}+\frac{11}{5 (3 x+2)^5}+\frac{7}{18 (3 x+2)^6}-34375 \log (3 x+2)+34375 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

7/(18*(2 + 3*x)^6) + 11/(5*(2 + 3*x)^5) + 55/(4*(2 + 3*x)^4) + 275/(3*(2 + 3*x)^3) + 1375/(2*(2 + 3*x)^2) + 68
75/(2 + 3*x) - 34375*Log[2 + 3*x] + 34375*Log[3 + 5*x]

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{1-2 x}{(2+3 x)^7 (3+5 x)} \, dx &=\int \left (-\frac{7}{(2+3 x)^7}-\frac{33}{(2+3 x)^6}-\frac{165}{(2+3 x)^5}-\frac{825}{(2+3 x)^4}-\frac{4125}{(2+3 x)^3}-\frac{20625}{(2+3 x)^2}-\frac{103125}{2+3 x}+\frac{171875}{3+5 x}\right ) \, dx\\ &=\frac{7}{18 (2+3 x)^6}+\frac{11}{5 (2+3 x)^5}+\frac{55}{4 (2+3 x)^4}+\frac{275}{3 (2+3 x)^3}+\frac{1375}{2 (2+3 x)^2}+\frac{6875}{2+3 x}-34375 \log (2+3 x)+34375 \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0409192, size = 75, normalized size = 0.93 \[ \frac{1237500 (3 x+2)^5+123750 (3 x+2)^4+16500 (3 x+2)^3+2475 (3 x+2)^2+396 (3 x+2)+70}{180 (3 x+2)^6}-34375 \log (3 x+2)+34375 \log (-3 (5 x+3)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(70 + 396*(2 + 3*x) + 2475*(2 + 3*x)^2 + 16500*(2 + 3*x)^3 + 123750*(2 + 3*x)^4 + 1237500*(2 + 3*x)^5)/(180*(2
+ 3*x)^6) - 34375*Log[2 + 3*x] + 34375*Log[-3*(3 + 5*x)]

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Maple [A]  time = 0.006, size = 72, normalized size = 0.9 \begin{align*}{\frac{7}{18\, \left ( 2+3\,x \right ) ^{6}}}+{\frac{11}{5\, \left ( 2+3\,x \right ) ^{5}}}+{\frac{55}{4\, \left ( 2+3\,x \right ) ^{4}}}+{\frac{275}{3\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{1375}{2\, \left ( 2+3\,x \right ) ^{2}}}+6875\, \left ( 2+3\,x \right ) ^{-1}-34375\,\ln \left ( 2+3\,x \right ) +34375\,\ln \left ( 3+5\,x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7/18/(2+3*x)^6+11/5/(2+3*x)^5+55/4/(2+3*x)^4+275/3/(2+3*x)^3+1375/2/(2+3*x)^2+6875/(2+3*x)-34375*ln(2+3*x)+343
75*ln(3+5*x)

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Maxima [A]  time = 1.14679, size = 103, normalized size = 1.27 \begin{align*} \frac{300712500 \, x^{5} + 1012398750 \, x^{4} + 1363675500 \, x^{3} + 918643275 \, x^{2} + 309504888 \, x + 41722762}{180 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + 34375 \, \log \left (5 \, x + 3\right ) - 34375 \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/180*(300712500*x^5 + 1012398750*x^4 + 1363675500*x^3 + 918643275*x^2 + 309504888*x + 41722762)/(729*x^6 + 29
16*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 34375*log(5*x + 3) - 34375*log(3*x + 2)

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Fricas [A]  time = 1.47718, size = 466, normalized size = 5.75 \begin{align*} \frac{300712500 \, x^{5} + 1012398750 \, x^{4} + 1363675500 \, x^{3} + 918643275 \, x^{2} + 6187500 \,{\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 ) - 6187500 \,{\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 ) + 309504888 \, x + 41722762}{180 \,{\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((1-2*x)/(2+3*x)^7/(3+5*x),x, algorithm="fricas")

[Out]

1/180*(300712500*x^5 + 1012398750*x^4 + 1363675500*x^3 + 918643275*x^2 + 6187500*(729*x^6 + 2916*x^5 + 4860*x^
4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*log(5*x + 3) - 6187500*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*
x^2 + 576*x + 64)*log(3*x + 2) + 309504888*x + 41722762)/(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 = 0.184709, size = 71, normalized size = 0.88 \begin{align*} \frac{300712500 x^{5} + 1012398750 x^{4} + 1363675500 x^{3} + 918643275 x^{2} + 309504888 x + 41722762}{131220 x^{6} + 524880 x^{5} + 874800 x^{4} + 777600 x^{3} + 388800 x^{2} + 103680 x + 11520} + 34375 \log{\left (x + \frac{3}{5} \right )} - 34375 \log{\left (x + \frac{2}{3} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(300712500*x**5 + 1012398750*x**4 + 1363675500*x**3 + 918643275*x**2 + 309504888*x + 41722762)/(131220*x**6 +
524880*x**5 + 874800*x**4 + 777600*x**3 + 388800*x**2 + 103680*x + 11520) + 34375*log(x + 3/5) - 34375*log(x +
2/3)

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Giac [A]  time = 2.92259, size = 72, normalized size = 0.89 \begin{align*} \frac{300712500 \, x^{5} + 1012398750 \, x^{4} + 1363675500 \, x^{3} + 918643275 \, x^{2} + 309504888 \, x + 41722762}{180 \,{\left (3 \, x + 2\right )}^{6}} + 34375 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 34375 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/180*(300712500*x^5 + 1012398750*x^4 + 1363675500*x^3 + 918643275*x^2 + 309504888*x + 41722762)/(3*x + 2)^6 +
34375*log(abs(5*x + 3)) - 34375*log(abs(3*x + 2))