3.16.21 \(\int \frac {(2+3 x)^3}{(1-2 x) (3+5 x)^3} \, dx\) [1521]

Optimal. Leaf size=43 \[ -\frac {1}{2750 (3+5 x)^2}-\frac {101}{15125 (3+5 x)}-\frac {343 \log (1-2 x)}{2662}+\frac {3469 \log (3+5 x)}{166375} \]

[Out]

-1/2750/(3+5*x)^2-101/15125/(3+5*x)-343/2662*ln(1-2*x)+3469/166375*ln(3+5*x)

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Rubi [A]
time = 0.01, antiderivative size = 43, 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 {101}{15125 (5 x+3)}-\frac {1}{2750 (5 x+3)^2}-\frac {343 \log (1-2 x)}{2662}+\frac {3469 \log (5 x+3)}{166375} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2750*1/(3 + 5*x)^2 - 101/(15125*(3 + 5*x)) - (343*Log[1 - 2*x])/2662 + (3469*Log[3 + 5*x])/166375

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)^3}{(1-2 x) (3+5 x)^3} \, dx &=\int \left (-\frac {343}{1331 (-1+2 x)}+\frac {1}{275 (3+5 x)^3}+\frac {101}{3025 (3+5 x)^2}+\frac {3469}{33275 (3+5 x)}\right ) \, dx\\ &=-\frac {1}{2750 (3+5 x)^2}-\frac {101}{15125 (3+5 x)}-\frac {343 \log (1-2 x)}{2662}+\frac {3469 \log (3+5 x)}{166375}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 35, normalized size = 0.81 \begin {gather*} \frac {-\frac {11 (617+1010 x)}{(3+5 x)^2}-42875 \log (1-2 x)+6938 \log (6+10 x)}{332750} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-11*(617 + 1010*x))/(3 + 5*x)^2 - 42875*Log[1 - 2*x] + 6938*Log[6 + 10*x])/332750

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Maple [A]
time = 0.10, size = 36, normalized size = 0.84

method	result	size
risch	\(\frac {-\frac {101 x}{3025}-\frac {617}{30250}}{\left (3+5 x \right )^{2}}-\frac {343 \ln \left (-1+2 x \right )}{2662}+\frac {3469 \ln \left (3+5 x \right )}{166375}\)	\(32\)
norman	\(\frac {\frac {314}{9075} x +\frac {617}{10890} x^{2}}{\left (3+5 x \right )^{2}}-\frac {343 \ln \left (-1+2 x \right )}{2662}+\frac {3469 \ln \left (3+5 x \right )}{166375}\)	\(35\)
default	\(-\frac {343 \ln \left (-1+2 x \right )}{2662}-\frac {1}{2750 \left (3+5 x \right )^{2}}-\frac {101}{15125 \left (3+5 x \right )}+\frac {3469 \ln \left (3+5 x \right )}{166375}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-343/2662*ln(-1+2*x)-1/2750/(3+5*x)^2-101/15125/(3+5*x)+3469/166375*ln(3+5*x)

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Maxima [A]
time = 0.31, size = 36, normalized size = 0.84 \begin {gather*} -\frac {1010 \, x + 617}{30250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {3469}{166375} \, \log \left (5 \, x + 3\right ) - \frac {343}{2662} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30250*(1010*x + 617)/(25*x^2 + 30*x + 9) + 3469/166375*log(5*x + 3) - 343/2662*log(2*x - 1)

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Fricas [A]
time = 0.77, size = 55, normalized size = 1.28 \begin {gather*} \frac {6938 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 42875 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (2 \, x - 1\right ) - 11110 \, x - 6787}{332750 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/332750*(6938*(25*x^2 + 30*x + 9)*log(5*x + 3) - 42875*(25*x^2 + 30*x + 9)*log(2*x - 1) - 11110*x - 6787)/(25
*x^2 + 30*x + 9)

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Sympy [A]
time = 0.07, size = 34, normalized size = 0.79 \begin {gather*} - \frac {1010 x + 617}{756250 x^{2} + 907500 x + 272250} - \frac {343 \log {\left (x - \frac {1}{2} \right )}}{2662} + \frac {3469 \log {\left (x + \frac {3}{5} \right )}}{166375} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(1010*x + 617)/(756250*x**2 + 907500*x + 272250) - 343*log(x - 1/2)/2662 + 3469*log(x + 3/5)/166375

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Giac [A]
time = 2.34, size = 33, normalized size = 0.77 \begin {gather*} -\frac {1010 \, x + 617}{30250 \, {\left (5 \, x + 3\right )}^{2}} + \frac {3469}{166375} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {343}{2662} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30250*(1010*x + 617)/(5*x + 3)^2 + 3469/166375*log(abs(5*x + 3)) - 343/2662*log(abs(2*x - 1))

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Mupad [B]
time = 1.16, size = 30, normalized size = 0.70 \begin {gather*} \frac {3469\,\ln \left (x+\frac {3}{5}\right )}{166375}-\frac {343\,\ln \left (x-\frac {1}{2}\right )}{2662}-\frac {\frac {101\,x}{75625}+\frac {617}{756250}}{x^2+\frac {6\,x}{5}+\frac {9}{25}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3469*log(x + 3/5))/166375 - (343*log(x - 1/2))/2662 - ((101*x)/75625 + 617/756250)/((6*x)/5 + x^2 + 9/25)

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