3.1531 \(\int \frac{(c+d x)^3}{(a-b x) (a+b x)} \, dx\)

Optimal. Leaf size=76 \[ -\frac{(a d+b c)^3 \log (a-b x)}{2 a b^4}+\frac{(b c-a d)^3 \log (a+b x)}{2 a b^4}-\frac{3 c d^2 x}{b^2}-\frac{d^3 x^2}{2 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0622332, antiderivative size = 76, 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} \[ -\frac{(a d+b c)^3 \log (a-b x)}{2 a b^4}+\frac{(b c-a d)^3 \log (a+b x)}{2 a b^4}-\frac{3 c d^2 x}{b^2}-\frac{d^3 x^2}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/((a - b*x)*(a + b*x)),x]

[Out]

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

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

Mathematica [A]  time = 0.0360463, size = 62, normalized size = 0.82 \[ -\frac{a b^2 d^2 x (6 c+d x)+(b c-a d)^3 (-\log (a+b x))+(a d+b c)^3 \log (a-b x)}{2 a b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^3/((a - b*x)*(a + b*x)),x]

[Out]

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

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Maple [B]  time = 0.006, size = 161, normalized size = 2.1 \begin{align*} -{\frac{{d}^{3}{x}^{2}}{2\,{b}^{2}}}-3\,{\frac{{d}^{2}xc}{{b}^{2}}}-{\frac{{a}^{2}\ln \left ( bx+a \right ){d}^{3}}{2\,{b}^{4}}}+{\frac{3\,a\ln \left ( bx+a \right ) c{d}^{2}}{2\,{b}^{3}}}-{\frac{3\,\ln \left ( bx+a \right ){c}^{2}d}{2\,{b}^{2}}}+{\frac{\ln \left ( bx+a \right ){c}^{3}}{2\,ab}}-{\frac{{a}^{2}\ln \left ( bx-a \right ){d}^{3}}{2\,{b}^{4}}}-{\frac{3\,a\ln \left ( bx-a \right ) c{d}^{2}}{2\,{b}^{3}}}-{\frac{3\,\ln \left ( bx-a \right ){c}^{2}d}{2\,{b}^{2}}}-{\frac{\ln \left ( bx-a \right ){c}^{3}}{2\,ab}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3/(-b*x+a)/(b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.25307, size = 165, normalized size = 2.17 \begin{align*} -\frac{d^{3} x^{2} + 6 \, c d^{2} x}{2 \, b^{2}} + \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{2 \, a b^{4}} - \frac{{\left (b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left (b x - a\right )}{2 \, a b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(-b*x+a)/(b*x+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.28588, size = 246, normalized size = 3.24 \begin{align*} -\frac{a b^{2} d^{3} x^{2} + 6 \, a b^{2} c d^{2} x -{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right ) +{\left (b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left (b x - a\right )}{2 \, a b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(-b*x+a)/(b*x+a),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 1.02269, size = 163, normalized size = 2.14 \begin{align*} - \frac{3 c d^{2} x}{b^{2}} - \frac{d^{3} x^{2}}{2 b^{2}} - \frac{\left (a d - b c\right )^{3} \log{\left (x + \frac{a^{4} d^{3} + 3 a^{2} b^{2} c^{2} d - a \left (a d - b c\right )^{3}}{3 a^{2} b^{2} c d^{2} + b^{4} c^{3}} \right )}}{2 a b^{4}} - \frac{\left (a d + b c\right )^{3} \log{\left (x + \frac{a^{4} d^{3} + 3 a^{2} b^{2} c^{2} d - a \left (a d + b c\right )^{3}}{3 a^{2} b^{2} c d^{2} + b^{4} c^{3}} \right )}}{2 a b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**3/(-b*x+a)/(b*x+a),x)

[Out]

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

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Giac [A]  time = 1.77862, size = 176, normalized size = 2.32 \begin{align*} -\frac{b^{2} d^{3} x^{2} + 6 \, b^{2} c d^{2} x}{2 \, b^{4}} + \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{2 \, a b^{4}} - \frac{{\left (b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left ({\left | b x - a \right |}\right )}{2 \, a b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(-b*x+a)/(b*x+a),x, algorithm="giac")

[Out]

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