\(\int \frac {(a+b x)^3}{(c+d x) (e+f x)} \, dx\) [1484]

Optimal result

Integrand size = 22, antiderivative size = 104 \[ \int \frac {(a+b x)^3}{(c+d x) (e+f x)} \, dx=-\frac {b^2 (b d e+b c f-3 a d f) x}{d^2 f^2}+\frac {b^3 x^2}{2 d f}-\frac {(b c-a d)^3 \log (c+d x)}{d^3 (d e-c f)}+\frac {(b e-a f)^3 \log (e+f x)}{f^3 (d e-c f)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {84} \[ \int \frac {(a+b x)^3}{(c+d x) (e+f x)} \, dx=-\frac {b^2 x (-3 a d f+b c f+b d e)}{d^2 f^2}-\frac {(b c-a d)^3 \log (c+d x)}{d^3 (d e-c f)}+\frac {(b e-a f)^3 \log (e+f x)}{f^3 (d e-c f)}+\frac {b^3 x^2}{2 d f} \]

[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 84

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*} \text {integral}& = \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] (verified)

Time = 0.08 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x)^3}{(c+d x) (e+f x)} \, dx=\frac {b^2 d f (d e-c f) x (6 a d f+b (-2 d e-2 c f+d f x))-2 (b c-a d)^3 f^3 \log (c+d x)+2 d^3 (b e-a f)^3 \log (e+f x)}{2 d^3 f^3 (d e-c f)} \]

[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))

Maple [A] (verified)

Time = 2.60 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.45

method	result	size
default	\(\frac {b^{2} \left (\frac {1}{2} b d f \,x^{2}+3 a d f x -b c f x -b d e x \right )}{d^{2} f^{2}}+\frac {\left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \ln \left (d x +c \right )}{d^{3} \left (c f -d e \right )}+\frac {\left (a^{3} f^{3}-3 a^{2} b e \,f^{2}+3 a \,b^{2} e^{2} f -b^{3} e^{3}\right ) \ln \left (f x +e \right )}{f^{3} \left (c f -d e \right )}\)	\(151\)
norman	\(\frac {b^{2} \left (3 a d f -b c f -b d e \right ) x}{d^{2} f^{2}}+\frac {b^{3} x^{2}}{2 d f}+\frac {\left (a^{3} f^{3}-3 a^{2} b e \,f^{2}+3 a \,b^{2} e^{2} f -b^{3} e^{3}\right ) \ln \left (f x +e \right )}{f^{3} \left (c f -d e \right )}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (d x +c \right )}{\left (c f -d e \right ) d^{3}}\)	\(156\)
parallelrisch	\(-\frac {-x^{2} b^{3} c \,d^{2} f^{3}+x^{2} b^{3} d^{3} e \,f^{2}+2 \ln \left (d x +c \right ) a^{3} d^{3} f^{3}-6 \ln \left (d x +c \right ) a^{2} b c \,d^{2} f^{3}+6 \ln \left (d x +c \right ) a \,b^{2} c^{2} d \,f^{3}-2 \ln \left (d x +c \right ) b^{3} c^{3} f^{3}-2 \ln \left (f x +e \right ) a^{3} d^{3} f^{3}+6 \ln \left (f x +e \right ) a^{2} b \,d^{3} e \,f^{2}-6 \ln \left (f x +e \right ) a \,b^{2} d^{3} e^{2} f +2 \ln \left (f x +e \right ) b^{3} d^{3} e^{3}-6 x a \,b^{2} c \,d^{2} f^{3}+6 x a \,b^{2} d^{3} e \,f^{2}+2 x \,b^{3} c^{2} d \,f^{3}-2 x \,b^{3} d^{3} e^{2} f}{2 d^{3} f^{3} \left (c f -d e \right )}\)	\(247\)
risch	\(\frac {b^{3} x^{2}}{2 d f}+\frac {3 b^{2} a x}{d f}-\frac {b^{3} c x}{d^{2} f}-\frac {b^{3} e x}{d \,f^{2}}-\frac {\ln \left (d x +c \right ) a^{3}}{c f -d e}+\frac {3 \ln \left (d x +c \right ) a^{2} b c}{\left (c f -d e \right ) d}-\frac {3 \ln \left (d x +c \right ) a \,b^{2} c^{2}}{\left (c f -d e \right ) d^{2}}+\frac {\ln \left (d x +c \right ) b^{3} c^{3}}{\left (c f -d e \right ) d^{3}}+\frac {\ln \left (-f x -e \right ) a^{3}}{c f -d e}-\frac {3 \ln \left (-f x -e \right ) a^{2} b e}{f \left (c f -d e \right )}+\frac {3 \ln \left (-f x -e \right ) a \,b^{2} e^{2}}{f^{2} \left (c f -d e \right )}-\frac {\ln \left (-f x -e \right ) b^{3} e^{3}}{f^{3} \left (c f -d e \right )}\)	\(269\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 207 vs. \(2 (102) = 204\).

Time = 0.29 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.99 \[ \int \frac {(a+b x)^3}{(c+d x) (e+f x)} \, dx=-\frac {2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{3} \log \left (d x + c\right ) - {\left (b^{3} d^{3} e f^{2} - b^{3} c d^{2} f^{3}\right )} x^{2} + 2 \, {\left (b^{3} d^{3} e^{2} f - 3 \, a b^{2} d^{3} e f^{2} - {\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2}\right )} f^{3}\right )} x - 2 \, {\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{3} e^{2} f + 3 \, a^{2} b d^{3} e f^{2} - a^{3} d^{3} f^{3}\right )} \log \left (f x + e\right )}{2 \, {\left (d^{4} e f^{3} - c d^{3} f^{4}\right )}} \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^3}{(c+d x) (e+f x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.55 \[ \int \frac {(a+b x)^3}{(c+d x) (e+f x)} \, dx=-\frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (d x + c\right )}{d^{4} e - c d^{3} f} + \frac {{\left (b^{3} e^{3} - 3 \, a b^{2} e^{2} f + 3 \, a^{2} b e f^{2} - a^{3} f^{3}\right )} \log \left (f x + e\right )}{d e f^{3} - c f^{4}} + \frac {b^{3} d f x^{2} - 2 \, {\left (b^{3} d e + {\left (b^{3} c - 3 \, a b^{2} d\right )} f\right )} x}{2 \, d^{2} f^{2}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.57 \[ \int \frac {(a+b x)^3}{(c+d x) (e+f x)} \, dx=-\frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{4} e - c d^{3} f} + \frac {{\left (b^{3} e^{3} - 3 \, a b^{2} e^{2} f + 3 \, a^{2} b e f^{2} - a^{3} f^{3}\right )} \log \left ({\left | f x + e \right |}\right )}{d e f^{3} - c f^{4}} + \frac {b^{3} d f x^{2} - 2 \, b^{3} d e x - 2 \, b^{3} c f x + 6 \, a b^{2} d f x}{2 \, d^{2} f^{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.93 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.05 \[ \int \frac {(a+b x)^3}{(c+d x) (e+f x)} \, dx=x\,\left (\frac {3\,a\,b^2}{d\,f}-\frac {b^3\,\left (c\,f+d\,e\right )}{d^2\,f^2}\right )+\frac {b^3\,x^2}{2\,d\,f}-\frac {\ln \left (c+d\,x\right )\,{\left (a\,d-b\,c\right )}^3}{d^3\,\left (c\,f-d\,e\right )}+\frac {\ln \left (e+f\,x\right )\,{\left (a\,f-b\,e\right )}^3}{f^3\,\left (c\,f-d\,e\right )} \]

[In]

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

[Out]

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