3.17.88 \(\int \frac {(2+3 x)^4}{(1-2 x)^3 (3+5 x)^2} \, dx\) [1688]

Optimal. Leaf size=54 \[ \frac {2401}{1936 (1-2 x)^2}-\frac {10633}{5324 (1-2 x)}-\frac {1}{33275 (3+5 x)}-\frac {47481 \log (1-2 x)}{117128}+\frac {138 \log (3+5 x)}{366025} \]

[Out]

2401/1936/(1-2*x)^2-10633/5324/(1-2*x)-1/33275/(3+5*x)-47481/117128*ln(1-2*x)+138/366025*ln(3+5*x)

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Rubi [A]
time = 0.02, antiderivative size = 54, 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 = {90} \begin {gather*} -\frac {10633}{5324 (1-2 x)}-\frac {1}{33275 (5 x+3)}+\frac {2401}{1936 (1-2 x)^2}-\frac {47481 \log (1-2 x)}{117128}+\frac {138 \log (5 x+3)}{366025} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^4/((1 - 2*x)^3*(3 + 5*x)^2),x]

[Out]

2401/(1936*(1 - 2*x)^2) - 10633/(5324*(1 - 2*x)) - 1/(33275*(3 + 5*x)) - (47481*Log[1 - 2*x])/117128 + (138*Lo
g[3 + 5*x])/366025

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin {align*} \int \frac {(2+3 x)^4}{(1-2 x)^3 (3+5 x)^2} \, dx &=\int \left (-\frac {2401}{484 (-1+2 x)^3}-\frac {10633}{2662 (-1+2 x)^2}-\frac {47481}{58564 (-1+2 x)}+\frac {1}{6655 (3+5 x)^2}+\frac {138}{73205 (3+5 x)}\right ) \, dx\\ &=\frac {2401}{1936 (1-2 x)^2}-\frac {10633}{5324 (1-2 x)}-\frac {1}{33275 (3+5 x)}-\frac {47481 \log (1-2 x)}{117128}+\frac {138 \log (3+5 x)}{366025}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 48, normalized size = 0.89 \begin {gather*} \frac {\frac {7263025}{(1-2 x)^2}+\frac {11696300}{-1+2 x}-\frac {176}{3+5 x}-2374050 \log (1-2 x)+2208 \log (6+10 x)}{5856400} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^4/((1 - 2*x)^3*(3 + 5*x)^2),x]

[Out]

(7263025/(1 - 2*x)^2 + 11696300/(-1 + 2*x) - 176/(3 + 5*x) - 2374050*Log[1 - 2*x] + 2208*Log[6 + 10*x])/585640
0

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Maple [A]
time = 0.10, size = 45, normalized size = 0.83

method	result	size
risch	\(\frac {\frac {1329117}{66550} x^{2}+\frac {4364739}{532400} x -\frac {1209091}{532400}}{\left (-1+2 x \right )^{2} \left (3+5 x \right )}-\frac {47481 \ln \left (-1+2 x \right )}{117128}+\frac {138 \ln \left (3+5 x \right )}{366025}\)	\(44\)
default	\(\frac {2401}{1936 \left (-1+2 x \right )^{2}}+\frac {10633}{5324 \left (-1+2 x \right )}-\frac {47481 \ln \left (-1+2 x \right )}{117128}-\frac {1}{33275 \left (3+5 x \right )}+\frac {138 \ln \left (3+5 x \right )}{366025}\)	\(45\)
norman	\(\frac {\frac {277826}{19965} x^{2}+\frac {231529}{79860} x +\frac {1209091}{79860} x^{3}}{\left (-1+2 x \right )^{2} \left (3+5 x \right )}-\frac {47481 \ln \left (-1+2 x \right )}{117128}+\frac {138 \ln \left (3+5 x \right )}{366025}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^4/(1-2*x)^3/(3+5*x)^2,x,method=_RETURNVERBOSE)

[Out]

2401/1936/(-1+2*x)^2+10633/5324/(-1+2*x)-47481/117128*ln(-1+2*x)-1/33275/(3+5*x)+138/366025*ln(3+5*x)

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Maxima [A]
time = 0.29, size = 46, normalized size = 0.85 \begin {gather*} \frac {10632936 \, x^{2} + 4364739 \, x - 1209091}{532400 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} + \frac {138}{366025} \, \log \left (5 \, x + 3\right ) - \frac {47481}{117128} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)^3/(3+5*x)^2,x, algorithm="maxima")

[Out]

1/532400*(10632936*x^2 + 4364739*x - 1209091)/(20*x^3 - 8*x^2 - 7*x + 3) + 138/366025*log(5*x + 3) - 47481/117
128*log(2*x - 1)

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Fricas [A]
time = 1.44, size = 75, normalized size = 1.39 \begin {gather*} \frac {116962296 \, x^{2} + 2208 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (5 \, x + 3\right ) - 2374050 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (2 \, x - 1\right ) + 48012129 \, x - 13300001}{5856400 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)^3/(3+5*x)^2,x, algorithm="fricas")

[Out]

1/5856400*(116962296*x^2 + 2208*(20*x^3 - 8*x^2 - 7*x + 3)*log(5*x + 3) - 2374050*(20*x^3 - 8*x^2 - 7*x + 3)*l
og(2*x - 1) + 48012129*x - 13300001)/(20*x^3 - 8*x^2 - 7*x + 3)

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Sympy [A]
time = 0.10, size = 44, normalized size = 0.81 \begin {gather*} - \frac {- 10632936 x^{2} - 4364739 x + 1209091}{10648000 x^{3} - 4259200 x^{2} - 3726800 x + 1597200} - \frac {47481 \log {\left (x - \frac {1}{2} \right )}}{117128} + \frac {138 \log {\left (x + \frac {3}{5} \right )}}{366025} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**4/(1-2*x)**3/(3+5*x)**2,x)

[Out]

-(-10632936*x**2 - 4364739*x + 1209091)/(10648000*x**3 - 4259200*x**2 - 3726800*x + 1597200) - 47481*log(x - 1
/2)/117128 + 138*log(x + 3/5)/366025

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Giac [A]
time = 1.55, size = 69, normalized size = 1.28 \begin {gather*} -\frac {1}{33275 \, {\left (5 \, x + 3\right )}} - \frac {1715 \, {\left (\frac {297}{5 \, x + 3} - 89\right )}}{58564 \, {\left (\frac {11}{5 \, x + 3} - 2\right )}^{2}} + \frac {81}{200} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) - \frac {47481}{117128} \, \log \left ({\left | -\frac {11}{5 \, x + 3} + 2 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)^3/(3+5*x)^2,x, algorithm="giac")

[Out]

-1/33275/(5*x + 3) - 1715/58564*(297/(5*x + 3) - 89)/(11/(5*x + 3) - 2)^2 + 81/200*log(1/5*abs(5*x + 3)/(5*x +
3)^2) - 47481/117128*log(abs(-11/(5*x + 3) + 2))

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Mupad [B]
time = 1.15, size = 42, normalized size = 0.78 \begin {gather*} \frac {138\,\ln \left (x+\frac {3}{5}\right )}{366025}-\frac {47481\,\ln \left (x-\frac {1}{2}\right )}{117128}-\frac {\frac {1329117\,x^2}{1331000}+\frac {4364739\,x}{10648000}-\frac {1209091}{10648000}}{-x^3+\frac {2\,x^2}{5}+\frac {7\,x}{20}-\frac {3}{20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*x + 2)^4/((2*x - 1)^3*(5*x + 3)^2),x)

[Out]

(138*log(x + 3/5))/366025 - (47481*log(x - 1/2))/117128 - ((4364739*x)/10648000 + (1329117*x^2)/1331000 - 1209
091/10648000)/((7*x)/20 + (2*x^2)/5 - x^3 - 3/20)

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