3.13.25 \(\int \frac {(1-2 x) (2+3 x)^4}{(3+5 x)^3} \, dx\) [1225]

Optimal. Leaf size=52 \[ \frac {1647 x}{3125}-\frac {297 x^2}{1250}-\frac {54 x^3}{125}-\frac {11}{31250 (3+5 x)^2}-\frac {26}{3125 (3+5 x)}+\frac {114 \log (3+5 x)}{3125} \]

[Out]

1647/3125*x-297/1250*x^2-54/125*x^3-11/31250/(3+5*x)^2-26/3125/(3+5*x)+114/3125*ln(3+5*x)

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Rubi [A]
time = 0.02, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \begin {gather*} -\frac {54 x^3}{125}-\frac {297 x^2}{1250}+\frac {1647 x}{3125}-\frac {26}{3125 (5 x+3)}-\frac {11}{31250 (5 x+3)^2}+\frac {114 \log (5 x+3)}{3125} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1647*x)/3125 - (297*x^2)/1250 - (54*x^3)/125 - 11/(31250*(3 + 5*x)^2) - 26/(3125*(3 + 5*x)) + (114*Log[3 + 5*
x])/3125

Rule 78

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)^4}{(3+5 x)^3} \, dx &=\int \left (\frac {1647}{3125}-\frac {297 x}{625}-\frac {162 x^2}{125}+\frac {11}{3125 (3+5 x)^3}+\frac {26}{625 (3+5 x)^2}+\frac {114}{625 (3+5 x)}\right ) \, dx\\ &=\frac {1647 x}{3125}-\frac {297 x^2}{1250}-\frac {54 x^3}{125}-\frac {11}{31250 (3+5 x)^2}-\frac {26}{3125 (3+5 x)}+\frac {114 \log (3+5 x)}{3125}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 51, normalized size = 0.98 \begin {gather*} \frac {17192+87220 x+133650 x^2+13500 x^3-118125 x^4-67500 x^5+228 (3+5 x)^2 \log (3+5 x)}{6250 (3+5 x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(17192 + 87220*x + 133650*x^2 + 13500*x^3 - 118125*x^4 - 67500*x^5 + 228*(3 + 5*x)^2*Log[3 + 5*x])/(6250*(3 +
5*x)^2)

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Maple [A]
time = 0.11, size = 41, normalized size = 0.79

method	result	size
risch	\(-\frac {54 x^{3}}{125}-\frac {297 x^{2}}{1250}+\frac {1647 x}{3125}+\frac {-\frac {26 x}{625}-\frac {791}{31250}}{\left (3+5 x \right )^{2}}+\frac {114 \ln \left (3+5 x \right )}{3125}\)	\(37\)
default	\(\frac {1647 x}{3125}-\frac {297 x^{2}}{1250}-\frac {54 x^{3}}{125}-\frac {11}{31250 \left (3+5 x \right )^{2}}-\frac {26}{3125 \left (3+5 x \right )}+\frac {114 \ln \left (3+5 x \right )}{3125}\)	\(41\)
norman	\(\frac {\frac {8974}{1875} x +\frac {15461}{1125} x^{2}+\frac {54}{25} x^{3}-\frac {189}{10} x^{4}-\frac {54}{5} x^{5}}{\left (3+5 x \right )^{2}}+\frac {114 \ln \left (3+5 x \right )}{3125}\)	\(42\)
meijerg	\(\frac {8 x \left (\frac {5 x}{3}+2\right )}{27 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {32 x^{2}}{27 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {4 x \left (15 x +6\right )}{75 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {114 \ln \left (1+\frac {5 x}{3}\right )}{3125}-\frac {54 x \left (\frac {100}{9} x^{2}+30 x +12\right )}{125 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {1053 x \left (-\frac {625}{27} x^{3}+\frac {500}{9} x^{2}+150 x +60\right )}{6250 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {243 x \left (\frac {1250}{81} x^{4}-\frac {625}{27} x^{3}+\frac {500}{9} x^{2}+150 x +60\right )}{3125 \left (1+\frac {5 x}{3}\right )^{2}}\)	\(127\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1647/3125*x-297/1250*x^2-54/125*x^3-11/31250/(3+5*x)^2-26/3125/(3+5*x)+114/3125*ln(3+5*x)

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Maxima [A]
time = 0.30, size = 41, normalized size = 0.79 \begin {gather*} -\frac {54}{125} \, x^{3} - \frac {297}{1250} \, x^{2} + \frac {1647}{3125} \, x - \frac {1300 \, x + 791}{31250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {114}{3125} \, \log \left (5 \, x + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-54/125*x^3 - 297/1250*x^2 + 1647/3125*x - 1/31250*(1300*x + 791)/(25*x^2 + 30*x + 9) + 114/3125*log(5*x + 3)

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Fricas [A]
time = 1.26, size = 57, normalized size = 1.10 \begin {gather*} -\frac {337500 \, x^{5} + 590625 \, x^{4} - 67500 \, x^{3} - 427275 \, x^{2} - 1140 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 146930 \, x + 791}{31250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/31250*(337500*x^5 + 590625*x^4 - 67500*x^3 - 427275*x^2 - 1140*(25*x^2 + 30*x + 9)*log(5*x + 3) - 146930*x
+ 791)/(25*x^2 + 30*x + 9)

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Sympy [A]
time = 0.04, size = 42, normalized size = 0.81 \begin {gather*} - \frac {54 x^{3}}{125} - \frac {297 x^{2}}{1250} + \frac {1647 x}{3125} - \frac {1300 x + 791}{781250 x^{2} + 937500 x + 281250} + \frac {114 \log {\left (5 x + 3 \right )}}{3125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-54*x**3/125 - 297*x**2/1250 + 1647*x/3125 - (1300*x + 791)/(781250*x**2 + 937500*x + 281250) + 114*log(5*x +
3)/3125

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Giac [A]
time = 1.30, size = 37, normalized size = 0.71 \begin {gather*} -\frac {54}{125} \, x^{3} - \frac {297}{1250} \, x^{2} + \frac {1647}{3125} \, x - \frac {1300 \, x + 791}{31250 \, {\left (5 \, x + 3\right )}^{2}} + \frac {114}{3125} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-54/125*x^3 - 297/1250*x^2 + 1647/3125*x - 1/31250*(1300*x + 791)/(5*x + 3)^2 + 114/3125*log(abs(5*x + 3))

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Mupad [B]
time = 0.03, size = 37, normalized size = 0.71 \begin {gather*} \frac {1647\,x}{3125}+\frac {114\,\ln \left (x+\frac {3}{5}\right )}{3125}-\frac {\frac {26\,x}{15625}+\frac {791}{781250}}{x^2+\frac {6\,x}{5}+\frac {9}{25}}-\frac {297\,x^2}{1250}-\frac {54\,x^3}{125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1647*x)/3125 + (114*log(x + 3/5))/3125 - ((26*x)/15625 + 791/781250)/((6*x)/5 + x^2 + 9/25) - (297*x^2)/1250
- (54*x^3)/125

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