\(\int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)} \, dx\) [1304]

Optimal result

Integrand size = 22, antiderivative size = 92 \[ \int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)} \, dx=\frac {7}{9 (2+3 x)^7}+\frac {217}{54 (2+3 x)^6}+\frac {121}{5 (2+3 x)^5}+\frac {605}{4 (2+3 x)^4}+\frac {3025}{3 (2+3 x)^3}+\frac {15125}{2 (2+3 x)^2}+\frac {75625}{2+3 x}-378125 \log (2+3 x)+378125 \log (3+5 x) \]

[Out]

7/9/(2+3*x)^7+217/54/(2+3*x)^6+121/5/(2+3*x)^5+605/4/(2+3*x)^4+3025/3/(2+3*x)^3+15125/2/(2+3*x)^2+75625/(2+3*x
)-378125*ln(2+3*x)+378125*ln(3+5*x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {90} \[ \int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)} \, dx=\frac {75625}{3 x+2}+\frac {15125}{2 (3 x+2)^2}+\frac {3025}{3 (3 x+2)^3}+\frac {605}{4 (3 x+2)^4}+\frac {121}{5 (3 x+2)^5}+\frac {217}{54 (3 x+2)^6}+\frac {7}{9 (3 x+2)^7}-378125 \log (3 x+2)+378125 \log (5 x+3) \]

[In]

Int[(1 - 2*x)^2/((2 + 3*x)^8*(3 + 5*x)),x]

[Out]

7/(9*(2 + 3*x)^7) + 217/(54*(2 + 3*x)^6) + 121/(5*(2 + 3*x)^5) + 605/(4*(2 + 3*x)^4) + 3025/(3*(2 + 3*x)^3) +
15125/(2*(2 + 3*x)^2) + 75625/(2 + 3*x) - 378125*Log[2 + 3*x] + 378125*Log[3 + 5*x]

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*} \text {integral}& = \int \left (-\frac {49}{3 (2+3 x)^8}-\frac {217}{3 (2+3 x)^7}-\frac {363}{(2+3 x)^6}-\frac {1815}{(2+3 x)^5}-\frac {9075}{(2+3 x)^4}-\frac {45375}{(2+3 x)^3}-\frac {226875}{(2+3 x)^2}-\frac {1134375}{2+3 x}+\frac {1890625}{3+5 x}\right ) \, dx \\ & = \frac {7}{9 (2+3 x)^7}+\frac {217}{54 (2+3 x)^6}+\frac {121}{5 (2+3 x)^5}+\frac {605}{4 (2+3 x)^4}+\frac {3025}{3 (2+3 x)^3}+\frac {15125}{2 (2+3 x)^2}+\frac {75625}{2+3 x}-378125 \log (2+3 x)+378125 \log (3+5 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.91 \[ \int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)} \, dx=\frac {420+2170 (2+3 x)+13068 (2+3 x)^2+81675 (2+3 x)^3+544500 (2+3 x)^4+4083750 (2+3 x)^5+40837500 (2+3 x)^6}{540 (2+3 x)^7}-378125 \log (5 (2+3 x))+378125 \log (3+5 x) \]

[In]

Integrate[(1 - 2*x)^2/((2 + 3*x)^8*(3 + 5*x)),x]

[Out]

(420 + 2170*(2 + 3*x) + 13068*(2 + 3*x)^2 + 81675*(2 + 3*x)^3 + 544500*(2 + 3*x)^4 + 4083750*(2 + 3*x)^5 + 408
37500*(2 + 3*x)^6)/(540*(2 + 3*x)^7) - 378125*Log[5*(2 + 3*x)] + 378125*Log[3 + 5*x]

Maple [A] (verified)

Time = 2.34 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.61

method	result	size
norman	\(\frac {55130625 x^{6}[phone] x^{4}+\frac {444720375}{2} x^{5}+\frac {1340357535}{4} x^{3}+\frac {1690211853}{10} x^{2}+\frac {4092979271}{90} x +\frac {688425608}{135}}{\left (2+3 x \right )^{7}}-378125 \ln \left (2+3 x \right )+378125 \ln \left (3+5 x \right )\)	\(56\)
risch	\(\frac {55130625 x^{6}[phone] x^{4}+\frac {444720375}{2} x^{5}+\frac {1340357535}{4} x^{3}+\frac {1690211853}{10} x^{2}+\frac {4092979271}{90} x +\frac {688425608}{135}}{\left (2+3 x \right )^{7}}-378125 \ln \left (2+3 x \right )+378125 \ln \left (3+5 x \right )\)	\(57\)
default	\(\frac {7}{9 \left (2+3 x \right )^{7}}+\frac {217}{54 \left (2+3 x \right )^{6}}+\frac {121}{5 \left (2+3 x \right )^{5}}+\frac {605}{4 \left (2+3 x \right )^{4}}+\frac {3025}{3 \left (2+3 x \right )^{3}}+\frac {15125}{2 \left (2+3 x \right )^{2}}+\frac {75625}{2+3 x}-378125 \ln \left (2+3 x \right )+378125 \ln \left (3+5 x \right )\)	\(81\)
parallelrisch	\(-\frac {5162666560 x -1463616000000 \ln \left (x +\frac {3}{5}\right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x^{3}-325248000000 \ln \left (x +\frac {3}{5}\right ) x [phone] \ln \left (\frac {2}{3}+x \right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x [phone] x^{5}[phone] x^{6}[phone] x^{7}[phone] x^{3}[phone] x^{4}[phone] x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right )[phone] \ln \left (\frac {2}{3}+x \right ) x^{7}-529254000000 \ln \left (x +\frac {3}{5}\right ) x^{7}-30976000000 \ln \left (x +\frac {3}{5}\right )[phone] \ln \left (\frac {2}{3}+x \right ) x^{5}-3659040000000 \ln \left (x +\frac {3}{5}\right ) x^{3}-4939704000000 \ln \left (x +\frac {3}{5}\right ) x^{5}-5488560000000 \ln \left (x +\frac {3}{5}\right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right ) x^{6}-2469852000000 \ln \left (x +\frac {3}{5}\right ) x^{6}}{640 \left (2+3 x \right )^{7}}\)	\(178\)

[In]

int((1-2*x)^2/(2+3*x)^8/(3+5*x),x,method=_RETURNVERBOSE)

[Out]

(55130625*x^6+373744800*x^4+444720375/2*x^5+1340357535/4*x^3+1690211853/10*x^2+4092979271/90*x+688425608/135)/
(2+3*x)^7-378125*ln(2+3*x)+378125*ln(3+5*x)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.68 \[ \int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)} \, dx=\frac {29770537500 \, x^{6} + 120074501250 \, x^{5} + 201822192000 \, x^{4} + 180948267225 \, x^{3} + 91271440062 \, x^{2} + 204187500 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (5 \, x + 3\right ) - 204187500 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (3 \, x + 2\right ) + 24557875626 \, x + 2753702432}{540 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]

[In]

integrate((1-2*x)^2/(2+3*x)^8/(3+5*x),x, algorithm="fricas")

[Out]

1/540*(29770537500*x^6 + 120074501250*x^5 + 201822192000*x^4 + 180948267225*x^3 + 91271440062*x^2 + 204187500*
(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)*log(5*x + 3) - 204187500*
(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)*log(3*x + 2) + 2455787562
6*x + 2753702432)/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.89 \[ \int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)} \, dx=\frac {29770537500 x^{6} + 120074501250 x^{5} + 201822192000 x^{4} + 180948267225 x^{3} + 91271440062 x^{2} + 24557875626 x + 2753702432}{1180980 x^{7} + 5511240 x^{6} + 11022480 x^{5} + 12247200 x^{4} + 8164800 x^{3} + 3265920 x^{2} + 725760 x + 69120} + 378125 \log {\left (x + \frac {3}{5} \right )} - 378125 \log {\left (x + \frac {2}{3} \right )} \]

[In]

integrate((1-2*x)**2/(2+3*x)**8/(3+5*x),x)

[Out]

(29770537500*x**6 + 120074501250*x**5 + 201822192000*x**4 + 180948267225*x**3 + 91271440062*x**2 + 24557875626
*x + 2753702432)/(1180980*x**7 + 5511240*x**6 + 11022480*x**5 + 12247200*x**4 + 8164800*x**3 + 3265920*x**2 +
725760*x + 69120) + 378125*log(x + 3/5) - 378125*log(x + 2/3)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.93 \[ \int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)} \, dx=\frac {29770537500 \, x^{6} + 120074501250 \, x^{5} + 201822192000 \, x^{4} + 180948267225 \, x^{3} + 91271440062 \, x^{2} + 24557875626 \, x + 2753702432}{540 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} + 378125 \, \log \left (5 \, x + 3\right ) - 378125 \, \log \left (3 \, x + 2\right ) \]

[In]

integrate((1-2*x)^2/(2+3*x)^8/(3+5*x),x, algorithm="maxima")

[Out]

1/540*(29770537500*x^6 + 120074501250*x^5 + 201822192000*x^4 + 180948267225*x^3 + 91271440062*x^2 + 2455787562
6*x + 2753702432)/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128) + 37812
5*log(5*x + 3) - 378125*log(3*x + 2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.63 \[ \int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)} \, dx=\frac {29770537500 \, x^{6} + 120074501250 \, x^{5} + 201822192000 \, x^{4} + 180948267225 \, x^{3} + 91271440062 \, x^{2} + 24557875626 \, x + 2753702432}{540 \, {\left (3 \, x + 2\right )}^{7}} + 378125 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 378125 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \]

[In]

integrate((1-2*x)^2/(2+3*x)^8/(3+5*x),x, algorithm="giac")

[Out]

1/540*(29770537500*x^6 + 120074501250*x^5 + 201822192000*x^4 + 180948267225*x^3 + 91271440062*x^2 + 2455787562
6*x + 2753702432)/(3*x + 2)^7 + 378125*log(abs(5*x + 3)) - 378125*log(abs(3*x + 2))

Mupad [B] (verification not implemented)

Time = 1.24 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.82 \[ \int \frac {(1-2 x)^2}{(2+3 x)^8 (3+5 x)} \, dx=\frac {\frac {75625\,x^6}{3}+\frac {1830125\,x^5}{18}+\frac {13842400\,x^4}{81}+\frac {148928615\,x^3}{972}+\frac {20866813\,x^2}{270}+\frac {4092979271\,x}{196830}+\frac {688425608}{295245}}{x^7+\frac {14\,x^6}{3}+\frac {28\,x^5}{3}+\frac {280\,x^4}{27}+\frac {560\,x^3}{81}+\frac {224\,x^2}{81}+\frac {448\,x}{729}+\frac {128}{2187}}-756250\,\mathrm {atanh}\left (30\,x+19\right ) \]

[In]

int((2*x - 1)^2/((3*x + 2)^8*(5*x + 3)),x)

[Out]

((4092979271*x)/196830 + (20866813*x^2)/270 + (148928615*x^3)/972 + (13842400*x^4)/81 + (1830125*x^5)/18 + (75
625*x^6)/3 + 688425608/295245)/((448*x)/729 + (224*x^2)/81 + (560*x^3)/81 + (280*x^4)/27 + (28*x^5)/3 + (14*x^
6)/3 + x^7 + 128/2187) - 756250*atanh(30*x + 19)