\(\int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^7} \, dx\) [1190]

Optimal result

Integrand size = 20, antiderivative size = 55 \[ \int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^7} \, dx=\frac {7 (3+5 x)^4}{18 (2+3 x)^6}+\frac {29 (3+5 x)^4}{45 (2+3 x)^5}+\frac {29 (3+5 x)^4}{36 (2+3 x)^4} \]

[Out]

7/18*(3+5*x)^4/(2+3*x)^6+29/45*(3+5*x)^4/(2+3*x)^5+29/36*(3+5*x)^4/(2+3*x)^4

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {79, 47, 37} \[ \int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^7} \, dx=\frac {29 (5 x+3)^4}{36 (3 x+2)^4}+\frac {29 (5 x+3)^4}{45 (3 x+2)^5}+\frac {7 (5 x+3)^4}{18 (3 x+2)^6} \]

[In]

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

[Out]

(7*(3 + 5*x)^4)/(18*(2 + 3*x)^6) + (29*(3 + 5*x)^4)/(45*(2 + 3*x)^5) + (29*(3 + 5*x)^4)/(36*(2 + 3*x)^4)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rubi steps \begin{align*} \text {integral}& = \frac {7 (3+5 x)^4}{18 (2+3 x)^6}+\frac {29}{9} \int \frac {(3+5 x)^3}{(2+3 x)^6} \, dx \\ & = \frac {7 (3+5 x)^4}{18 (2+3 x)^6}+\frac {29 (3+5 x)^4}{45 (2+3 x)^5}+\frac {29}{9} \int \frac {(3+5 x)^3}{(2+3 x)^5} \, dx \\ & = \frac {7 (3+5 x)^4}{18 (2+3 x)^6}+\frac {29 (3+5 x)^4}{45 (2+3 x)^5}+\frac {29 (3+5 x)^4}{36 (2+3 x)^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.56 \[ \int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^7} \, dx=\frac {-13198+78048 x+587925 x^2+1066500 x^3+607500 x^4}{14580 (2+3 x)^6} \]

[In]

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

[Out]

(-13198 + 78048*x + 587925*x^2 + 1066500*x^3 + 607500*x^4)/(14580*(2 + 3*x)^6)

Maple [A] (verified)

Time = 2.88 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.53

method	result	size
norman	\(\frac {\frac {125}{3} x^{4}+\frac {1975}{27} x^{3}+\frac {2168}{405} x +\frac {4355}{108} x^{2}-\frac {6599}{7290}}{\left (2+3 x \right )^{6}}\)	\(29\)
gosper	\(\frac {607500 x^{4}+1066500 x^{3}+587925 x^{2}+78048 x -13198}{14580 \left (2+3 x \right )^{6}}\)	\(30\)
risch	\(\frac {\frac {125}{3} x^{4}+\frac {1975}{27} x^{3}+\frac {2168}{405} x +\frac {4355}{108} x^{2}-\frac {6599}{7290}}{\left (2+3 x \right )^{6}}\)	\(30\)
parallelrisch	\(\frac {6599 x^{6}+26396 x^{5}+70660 x^{4}+85920 x^{3}+45360 x^{2}+8640 x}{640 \left (2+3 x \right )^{6}}\)	\(39\)
default	\(\frac {7}{1458 \left (2+3 x \right )^{6}}+\frac {125}{243 \left (2+3 x \right )^{2}}+\frac {185}{324 \left (2+3 x \right )^{4}}-\frac {107}{1215 \left (2+3 x \right )^{5}}-\frac {1025}{729 \left (2+3 x \right )^{3}}\)	\(47\)
meijerg	\(\frac {9 x \left (\frac {243}{32} x^{5}+\frac {243}{8} x^{4}+\frac {405}{8} x^{3}+45 x^{2}+\frac {45}{2} x +6\right )}{256 \left (1+\frac {3 x}{2}\right )^{6}}+\frac {27 x^{2} \left (\frac {81}{16} x^{4}+\frac {81}{4} x^{3}+\frac {135}{4} x^{2}+30 x +15\right )}{1280 \left (1+\frac {3 x}{2}\right )^{6}}-\frac {3 x^{3} \left (\frac {27}{8} x^{3}+\frac {27}{2} x^{2}+\frac {45}{2} x +20\right )}{512 \left (1+\frac {3 x}{2}\right )^{6}}-\frac {65 x^{4} \left (\frac {9}{4} x^{2}+9 x +15\right )}{1536 \left (1+\frac {3 x}{2}\right )^{6}}-\frac {25 x^{5} \left (\frac {3 x}{2}+6\right )}{384 \left (1+\frac {3 x}{2}\right )^{6}}\)	\(135\)

[In]

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

[Out]

(125/3*x^4+1975/27*x^3+2168/405*x+4355/108*x^2-6599/7290)/(2+3*x)^6

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.98 \[ \int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^7} \, dx=\frac {607500 \, x^{4} + 1066500 \, x^{3} + 587925 \, x^{2} + 78048 \, x - 13198}{14580 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

[In]

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

[Out]

1/14580*(607500*x^4 + 1066500*x^3 + 587925*x^2 + 78048*x - 13198)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 +
2160*x^2 + 576*x + 64)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^7} \, dx=- \frac {- 607500 x^{4} - 1066500 x^{3} - 587925 x^{2} - 78048 x + 13198}{10628820 x^{6} + 42515280 x^{5} + 70858800 x^{4} + 62985600 x^{3} + 31492800 x^{2} + 8398080 x + 933120} \]

[In]

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

[Out]

-(-607500*x**4 - 1066500*x**3 - 587925*x**2 - 78048*x + 13198)/(10628820*x**6 + 42515280*x**5 + 70858800*x**4
+ 62985600*x**3 + 31492800*x**2 + 8398080*x + 933120)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.98 \[ \int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^7} \, dx=\frac {607500 \, x^{4} + 1066500 \, x^{3} + 587925 \, x^{2} + 78048 \, x - 13198}{14580 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

[In]

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

[Out]

1/14580*(607500*x^4 + 1066500*x^3 + 587925*x^2 + 78048*x - 13198)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 +
2160*x^2 + 576*x + 64)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.53 \[ \int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^7} \, dx=\frac {607500 \, x^{4} + 1066500 \, x^{3} + 587925 \, x^{2} + 78048 \, x - 13198}{14580 \, {\left (3 \, x + 2\right )}^{6}} \]

[In]

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

[Out]

1/14580*(607500*x^4 + 1066500*x^3 + 587925*x^2 + 78048*x - 13198)/(3*x + 2)^6

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.84 \[ \int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^7} \, dx=\frac {125}{243\,{\left (3\,x+2\right )}^2}-\frac {1025}{729\,{\left (3\,x+2\right )}^3}+\frac {185}{324\,{\left (3\,x+2\right )}^4}-\frac {107}{1215\,{\left (3\,x+2\right )}^5}+\frac {7}{1458\,{\left (3\,x+2\right )}^6} \]

[In]

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

[Out]

125/(243*(3*x + 2)^2) - 1025/(729*(3*x + 2)^3) + 185/(324*(3*x + 2)^4) - 107/(1215*(3*x + 2)^5) + 7/(1458*(3*x
+ 2)^6)