3.1530 \(\int \frac{(c+d x)^4}{(a-b x) (a+b x)} \, dx\)

Optimal. Leaf size=103 \[ -\frac{d^2 x \left (a^2 d^2+6 b^2 c^2\right )}{b^4}-\frac{(a d+b c)^4 \log (a-b x)}{2 a b^5}+\frac{(b c-a d)^4 \log (a+b x)}{2 a b^5}-\frac{2 c d^3 x^2}{b^2}-\frac{d^4 x^3}{3 b^2} \]

[Out]

-((d^2*(6*b^2*c^2 + a^2*d^2)*x)/b^4) - (2*c*d^3*x^2)/b^2 - (d^4*x^3)/(3*b^2) - ((b*c + a*d)^4*Log[a - b*x])/(2
*a*b^5) + ((b*c - a*d)^4*Log[a + b*x])/(2*a*b^5)

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Rubi [A]  time = 0.101209, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {72} \[ -\frac{d^2 x \left (a^2 d^2+6 b^2 c^2\right )}{b^4}-\frac{(a d+b c)^4 \log (a-b x)}{2 a b^5}+\frac{(b c-a d)^4 \log (a+b x)}{2 a b^5}-\frac{2 c d^3 x^2}{b^2}-\frac{d^4 x^3}{3 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^4/((a - b*x)*(a + b*x)),x]

[Out]

-((d^2*(6*b^2*c^2 + a^2*d^2)*x)/b^4) - (2*c*d^3*x^2)/b^2 - (d^4*x^3)/(3*b^2) - ((b*c + a*d)^4*Log[a - b*x])/(2
*a*b^5) + ((b*c - a*d)^4*Log[a + b*x])/(2*a*b^5)

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

\begin{align*} \int \frac{(c+d x)^4}{(a-b x) (a+b x)} \, dx &=\int \left (\frac{-6 b^2 c^2 d^2-a^2 d^4}{b^4}-\frac{4 c d^3 x}{b^2}-\frac{d^4 x^2}{b^2}+\frac{(b c+a d)^4}{2 a b^4 (a-b x)}+\frac{(-b c+a d)^4}{2 a b^4 (a+b x)}\right ) \, dx\\ &=-\frac{d^2 \left (6 b^2 c^2+a^2 d^2\right ) x}{b^4}-\frac{2 c d^3 x^2}{b^2}-\frac{d^4 x^3}{3 b^2}-\frac{(b c+a d)^4 \log (a-b x)}{2 a b^5}+\frac{(b c-a d)^4 \log (a+b x)}{2 a b^5}\\ \end{align*}

Mathematica [A]  time = 0.0485051, size = 86, normalized size = 0.83 \[ \frac{-2 a b d^2 x \left (3 a^2 d^2+b^2 \left (18 c^2+6 c d x+d^2 x^2\right )\right )+3 (b c-a d)^4 \log (a+b x)-3 (a d+b c)^4 \log (a-b x)}{6 a b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^4/((a - b*x)*(a + b*x)),x]

[Out]

(-2*a*b*d^2*x*(3*a^2*d^2 + b^2*(18*c^2 + 6*c*d*x + d^2*x^2)) - 3*(b*c + a*d)^4*Log[a - b*x] + 3*(b*c - a*d)^4*
Log[a + b*x])/(6*a*b^5)

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Maple [B]  time = 0.007, size = 229, normalized size = 2.2 \begin{align*} -{\frac{{d}^{4}{x}^{3}}{3\,{b}^{2}}}-2\,{\frac{c{d}^{3}{x}^{2}}{{b}^{2}}}-{\frac{{d}^{4}{a}^{2}x}{{b}^{4}}}-6\,{\frac{{d}^{2}{c}^{2}x}{{b}^{2}}}+{\frac{{a}^{3}\ln \left ( bx+a \right ){d}^{4}}{2\,{b}^{5}}}-2\,{\frac{{a}^{2}\ln \left ( bx+a \right ) c{d}^{3}}{{b}^{4}}}+3\,{\frac{a\ln \left ( bx+a \right ){c}^{2}{d}^{2}}{{b}^{3}}}-2\,{\frac{\ln \left ( bx+a \right ){c}^{3}d}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ){c}^{4}}{2\,ab}}-{\frac{{a}^{3}\ln \left ( bx-a \right ){d}^{4}}{2\,{b}^{5}}}-2\,{\frac{{a}^{2}\ln \left ( bx-a \right ) c{d}^{3}}{{b}^{4}}}-3\,{\frac{a\ln \left ( bx-a \right ){c}^{2}{d}^{2}}{{b}^{3}}}-2\,{\frac{\ln \left ( bx-a \right ){c}^{3}d}{{b}^{2}}}-{\frac{\ln \left ( bx-a \right ){c}^{4}}{2\,ab}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^4/(-b*x+a)/(b*x+a),x)

[Out]

-1/3*d^4*x^3/b^2-2*c*d^3*x^2/b^2-d^4/b^4*a^2*x-6*d^2/b^2*c^2*x+1/2/b^5*a^3*ln(b*x+a)*d^4-2/b^4*a^2*ln(b*x+a)*c
*d^3+3/b^3*a*ln(b*x+a)*c^2*d^2-2/b^2*ln(b*x+a)*c^3*d+1/2/b/a*ln(b*x+a)*c^4-1/2/b^5*a^3*ln(b*x-a)*d^4-2/b^4*a^2
*ln(b*x-a)*c*d^3-3/b^3*a*ln(b*x-a)*c^2*d^2-2/b^2*ln(b*x-a)*c^3*d-1/2/b/a*ln(b*x-a)*c^4

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Maxima [A]  time = 1.06548, size = 242, normalized size = 2.35 \begin{align*} -\frac{b^{2} d^{4} x^{3} + 6 \, b^{2} c d^{3} x^{2} + 3 \,{\left (6 \, b^{2} c^{2} d^{2} + a^{2} d^{4}\right )} x}{3 \, b^{4}} + \frac{{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (b x + a\right )}{2 \, a b^{5}} - \frac{{\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (b x - a\right )}{2 \, a b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4/(-b*x+a)/(b*x+a),x, algorithm="maxima")

[Out]

-1/3*(b^2*d^4*x^3 + 6*b^2*c*d^3*x^2 + 3*(6*b^2*c^2*d^2 + a^2*d^4)*x)/b^4 + 1/2*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^
2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*log(b*x + a)/(a*b^5) - 1/2*(b^4*c^4 + 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d
^2 + 4*a^3*b*c*d^3 + a^4*d^4)*log(b*x - a)/(a*b^5)

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Fricas [A]  time = 1.38803, size = 360, normalized size = 3.5 \begin{align*} -\frac{2 \, a b^{3} d^{4} x^{3} + 12 \, a b^{3} c d^{3} x^{2} + 6 \,{\left (6 \, a b^{3} c^{2} d^{2} + a^{3} b d^{4}\right )} x - 3 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (b x + a\right ) + 3 \,{\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (b x - a\right )}{6 \, a b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4/(-b*x+a)/(b*x+a),x, algorithm="fricas")

[Out]

-1/6*(2*a*b^3*d^4*x^3 + 12*a*b^3*c*d^3*x^2 + 6*(6*a*b^3*c^2*d^2 + a^3*b*d^4)*x - 3*(b^4*c^4 - 4*a*b^3*c^3*d +
6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*log(b*x + a) + 3*(b^4*c^4 + 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 4
*a^3*b*c*d^3 + a^4*d^4)*log(b*x - a))/(a*b^5)

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Sympy [B]  time = 1.35187, size = 214, normalized size = 2.08 \begin{align*} - \frac{2 c d^{3} x^{2}}{b^{2}} - \frac{d^{4} x^{3}}{3 b^{2}} - \frac{x \left (a^{2} d^{4} + 6 b^{2} c^{2} d^{2}\right )}{b^{4}} + \frac{\left (a d - b c\right )^{4} \log{\left (x + \frac{4 a^{4} c d^{3} + 4 a^{2} b^{2} c^{3} d + \frac{a \left (a d - b c\right )^{4}}{b}}{a^{4} d^{4} + 6 a^{2} b^{2} c^{2} d^{2} + b^{4} c^{4}} \right )}}{2 a b^{5}} - \frac{\left (a d + b c\right )^{4} \log{\left (x + \frac{4 a^{4} c d^{3} + 4 a^{2} b^{2} c^{3} d - \frac{a \left (a d + b c\right )^{4}}{b}}{a^{4} d^{4} + 6 a^{2} b^{2} c^{2} d^{2} + b^{4} c^{4}} \right )}}{2 a b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**4/(-b*x+a)/(b*x+a),x)

[Out]

-2*c*d**3*x**2/b**2 - d**4*x**3/(3*b**2) - x*(a**2*d**4 + 6*b**2*c**2*d**2)/b**4 + (a*d - b*c)**4*log(x + (4*a
**4*c*d**3 + 4*a**2*b**2*c**3*d + a*(a*d - b*c)**4/b)/(a**4*d**4 + 6*a**2*b**2*c**2*d**2 + b**4*c**4))/(2*a*b*
*5) - (a*d + b*c)**4*log(x + (4*a**4*c*d**3 + 4*a**2*b**2*c**3*d - a*(a*d + b*c)**4/b)/(a**4*d**4 + 6*a**2*b**
2*c**2*d**2 + b**4*c**4))/(2*a*b**5)

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Giac [A]  time = 3.28404, size = 247, normalized size = 2.4 \begin{align*} \frac{{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{2 \, a b^{5}} - \frac{{\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left ({\left | b x - a \right |}\right )}{2 \, a b^{5}} - \frac{b^{4} d^{4} x^{3} + 6 \, b^{4} c d^{3} x^{2} + 18 \, b^{4} c^{2} d^{2} x + 3 \, a^{2} b^{2} d^{4} x}{3 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4/(-b*x+a)/(b*x+a),x, algorithm="giac")

[Out]

1/2*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*log(abs(b*x + a))/(a*b^5) - 1/2*(b
^4*c^4 + 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 + a^4*d^4)*log(abs(b*x - a))/(a*b^5) - 1/3*(b^4*d^4
*x^3 + 6*b^4*c*d^3*x^2 + 18*b^4*c^2*d^2*x + 3*a^2*b^2*d^4*x)/b^6