3.1315 \(\int \frac{(1-2 x)^2}{(2+3 x)^3 (3+5 x)^2} \, dx\)

Optimal. Leaf size=46 \[ -\frac{154}{3 x+2}-\frac{121}{5 x+3}-\frac{49}{6 (3 x+2)^2}+1133 \log (3 x+2)-1133 \log (5 x+3) \]

[Out]

-49/(6*(2 + 3*x)^2) - 154/(2 + 3*x) - 121/(3 + 5*x) + 1133*Log[2 + 3*x] - 1133*Log[3 + 5*x]

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Rubi [A]  time = 0.0217176, antiderivative size = 46, normalized size of antiderivative = 1., 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 = {88} \[ -\frac{154}{3 x+2}-\frac{121}{5 x+3}-\frac{49}{6 (3 x+2)^2}+1133 \log (3 x+2)-1133 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-49/(6*(2 + 3*x)^2) - 154/(2 + 3*x) - 121/(3 + 5*x) + 1133*Log[2 + 3*x] - 1133*Log[3 + 5*x]

Rule 88

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{(1-2 x)^2}{(2+3 x)^3 (3+5 x)^2} \, dx &=\int \left (\frac{49}{(2+3 x)^3}+\frac{462}{(2+3 x)^2}+\frac{3399}{2+3 x}+\frac{605}{(3+5 x)^2}-\frac{5665}{3+5 x}\right ) \, dx\\ &=-\frac{49}{6 (2+3 x)^2}-\frac{154}{2+3 x}-\frac{121}{3+5 x}+1133 \log (2+3 x)-1133 \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0252509, size = 48, normalized size = 1.04 \[ -\frac{154}{3 x+2}-\frac{121}{5 x+3}-\frac{49}{6 (3 x+2)^2}+1133 \log (5 (3 x+2))-1133 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-49/(6*(2 + 3*x)^2) - 154/(2 + 3*x) - 121/(3 + 5*x) + 1133*Log[5*(2 + 3*x)] - 1133*Log[3 + 5*x]

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Maple [A]  time = 0.008, size = 45, normalized size = 1. \begin{align*} -{\frac{49}{6\, \left ( 2+3\,x \right ) ^{2}}}-154\, \left ( 2+3\,x \right ) ^{-1}-121\, \left ( 3+5\,x \right ) ^{-1}+1133\,\ln \left ( 2+3\,x \right ) -1133\,\ln \left ( 3+5\,x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-49/6/(2+3*x)^2-154/(2+3*x)-121/(3+5*x)+1133*ln(2+3*x)-1133*ln(3+5*x)

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Maxima [A]  time = 2.55682, size = 62, normalized size = 1.35 \begin{align*} -\frac{20394 \, x^{2} + 26513 \, x + 8595}{6 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} - 1133 \, \log \left (5 \, x + 3\right ) + 1133 \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(20394*x^2 + 26513*x + 8595)/(45*x^3 + 87*x^2 + 56*x + 12) - 1133*log(5*x + 3) + 1133*log(3*x + 2)

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Fricas [A]  time = 1.3425, size = 223, normalized size = 4.85 \begin{align*} -\frac{20394 \, x^{2} + 6798 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (5 \, x + 3\right ) - 6798 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (3 \, x + 2\right ) + 26513 \, x + 8595}{6 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(20394*x^2 + 6798*(45*x^3 + 87*x^2 + 56*x + 12)*log(5*x + 3) - 6798*(45*x^3 + 87*x^2 + 56*x + 12)*log(3*x
+ 2) + 26513*x + 8595)/(45*x^3 + 87*x^2 + 56*x + 12)

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Sympy [A]  time = 0.151591, size = 41, normalized size = 0.89 \begin{align*} - \frac{20394 x^{2} + 26513 x + 8595}{270 x^{3} + 522 x^{2} + 336 x + 72} - 1133 \log{\left (x + \frac{3}{5} \right )} + 1133 \log{\left (x + \frac{2}{3} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(20394*x**2 + 26513*x + 8595)/(270*x**3 + 522*x**2 + 336*x + 72) - 1133*log(x + 3/5) + 1133*log(x + 2/3)

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Giac [A]  time = 1.71319, size = 66, normalized size = 1.43 \begin{align*} -\frac{121}{5 \, x + 3} + \frac{35 \,{\left (\frac{202}{5 \, x + 3} + 501\right )}}{2 \,{\left (\frac{1}{5 \, x + 3} + 3\right )}^{2}} + 1133 \, \log \left ({\left | -\frac{1}{5 \, x + 3} - 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-121/(5*x + 3) + 35/2*(202/(5*x + 3) + 501)/(1/(5*x + 3) + 3)^2 + 1133*log(abs(-1/(5*x + 3) - 3))