3.1484 \(\int \frac{(a+b x)^3}{(c+d x) (e+f x)} \, dx\)

Optimal. Leaf size=104 \[ -\frac{b^2 x (-3 a d f+b c f+b d e)}{d^2 f^2}-\frac{(b c-a d)^3 \log (c+d x)}{d^3 (d e-c f)}+\frac{(b e-a f)^3 \log (e+f x)}{f^3 (d e-c f)}+\frac{b^3 x^2}{2 d f} \]

[Out]

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

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Rubi [A]  time = 0.118409, antiderivative size = 104, 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 = {72} \[ -\frac{b^2 x (-3 a d f+b c f+b d e)}{d^2 f^2}-\frac{(b c-a d)^3 \log (c+d x)}{d^3 (d e-c f)}+\frac{(b e-a f)^3 \log (e+f x)}{f^3 (d e-c f)}+\frac{b^3 x^2}{2 d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0794704, size = 99, normalized size = 0.95 \[ \frac{b^2 d f x (d e-c f) (6 a d f+b (-2 c f-2 d e+d f x))-2 f^3 (b c-a d)^3 \log (c+d x)+2 d^3 (b e-a f)^3 \log (e+f x)}{2 d^3 f^3 (d e-c f)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.01, size = 257, normalized size = 2.5 \begin{align*}{\frac{{b}^{3}{x}^{2}}{2\,df}}+3\,{\frac{a{b}^{2}x}{df}}-{\frac{{b}^{3}cx}{{d}^{2}f}}-{\frac{{b}^{3}ex}{d{f}^{2}}}-{\frac{\ln \left ( dx+c \right ){a}^{3}}{cf-de}}+3\,{\frac{\ln \left ( dx+c \right ){a}^{2}cb}{d \left ( cf-de \right ) }}-3\,{\frac{\ln \left ( dx+c \right ) a{b}^{2}{c}^{2}}{{d}^{2} \left ( cf-de \right ) }}+{\frac{\ln \left ( dx+c \right ){b}^{3}{c}^{3}}{{d}^{3} \left ( cf-de \right ) }}+{\frac{\ln \left ( fx+e \right ){a}^{3}}{cf-de}}-3\,{\frac{\ln \left ( fx+e \right ){a}^{2}be}{f \left ( cf-de \right ) }}+3\,{\frac{\ln \left ( fx+e \right ) a{b}^{2}{e}^{2}}{{f}^{2} \left ( cf-de \right ) }}-{\frac{\ln \left ( fx+e \right ){b}^{3}{e}^{3}}{{f}^{3} \left ( cf-de \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.07692, size = 217, normalized size = 2.09 \begin{align*} -\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 (d x + c\right )}{d^{4} e - c d^{3} f} + \frac{{\left (b^{3} e^{3} - 3 \, a b^{2} e^{2} f + 3 \, a^{2} b e f^{2} - a^{3} f^{3}\right )} \log \left (f x + e\right )}{d e f^{3} - c f^{4}} + \frac{b^{3} d f x^{2} - 2 \,{\left (b^{3} d e +{\left (b^{3} c - 3 \, a b^{2} d\right )} f\right )} x}{2 \, d^{2} f^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.09659, size = 408, normalized size = 3.92 \begin{align*} -\frac{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{3} \log \left (d x + c\right ) -{\left (b^{3} d^{3} e f^{2} - b^{3} c d^{2} f^{3}\right )} x^{2} + 2 \,{\left (b^{3} d^{3} e^{2} f - 3 \, a b^{2} d^{3} e f^{2} -{\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2}\right )} f^{3}\right )} x - 2 \,{\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{3} e^{2} f + 3 \, a^{2} b d^{3} e f^{2} - a^{3} d^{3} f^{3}\right )} \log \left (f x + e\right )}{2 \,{\left (d^{4} e f^{3} - c d^{3} f^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 8.14758, size = 614, normalized size = 5.9 \begin{align*} \frac{b^{3} x^{2}}{2 d f} + \frac{\left (a f - b e\right )^{3} \log{\left (x + \frac{a^{3} c d^{2} f^{3} + a^{3} d^{3} e f^{2} - 6 a^{2} b c d^{2} e f^{2} + 3 a b^{2} c^{2} d e f^{2} + 3 a b^{2} c d^{2} e^{2} f - b^{3} c^{3} e f^{2} - b^{3} c d^{2} e^{3} - \frac{c^{2} d^{2} f \left (a f - b e\right )^{3}}{c f - d e} + \frac{2 c d^{3} e \left (a f - b e\right )^{3}}{c f - d e} - \frac{d^{4} e^{2} \left (a f - b e\right )^{3}}{f \left (c f - d e\right )}}{2 a^{3} d^{3} f^{3} - 3 a^{2} b c d^{2} f^{3} - 3 a^{2} b d^{3} e f^{2} + 3 a b^{2} c^{2} d f^{3} + 3 a b^{2} d^{3} e^{2} f - b^{3} c^{3} f^{3} - b^{3} d^{3} e^{3}} \right )}}{f^{3} \left (c f - d e\right )} + \frac{x \left (3 a b^{2} d f - b^{3} c f - b^{3} d e\right )}{d^{2} f^{2}} - \frac{\left (a d - b c\right )^{3} \log{\left (x + \frac{a^{3} c d^{2} f^{3} + a^{3} d^{3} e f^{2} - 6 a^{2} b c d^{2} e f^{2} + 3 a b^{2} c^{2} d e f^{2} + 3 a b^{2} c d^{2} e^{2} f - b^{3} c^{3} e f^{2} - b^{3} c d^{2} e^{3} + \frac{c^{2} f^{4} \left (a d - b c\right )^{3}}{d \left (c f - d e\right )} - \frac{2 c e f^{3} \left (a d - b c\right )^{3}}{c f - d e} + \frac{d e^{2} f^{2} \left (a d - b c\right )^{3}}{c f - d e}}{2 a^{3} d^{3} f^{3} - 3 a^{2} b c d^{2} f^{3} - 3 a^{2} b d^{3} e f^{2} + 3 a b^{2} c^{2} d f^{3} + 3 a b^{2} d^{3} e^{2} f - b^{3} c^{3} f^{3} - b^{3} d^{3} e^{3}} \right )}}{d^{3} \left (c f - d e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError