\(\int \frac {(a+b x) (A+B x)}{(d+e x)^4} \, dx\) [1019]

Optimal result

Integrand size = 18, antiderivative size = 75 \[ \int \frac {(a+b x) (A+B x)}{(d+e x)^4} \, dx=-\frac {(b d-a e) (B d-A e)}{3 e^3 (d+e x)^3}+\frac {2 b B d-A b e-a B e}{2 e^3 (d+e x)^2}-\frac {b B}{e^3 (d+e x)} \]

[Out]

-1/3*(-a*e+b*d)*(-A*e+B*d)/e^3/(e*x+d)^3+1/2*(-A*b*e-B*a*e+2*B*b*d)/e^3/(e*x+d)^2-b*B/e^3/(e*x+d)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {78} \[ \int \frac {(a+b x) (A+B x)}{(d+e x)^4} \, dx=\frac {-a B e-A b e+2 b B d}{2 e^3 (d+e x)^2}-\frac {(b d-a e) (B d-A e)}{3 e^3 (d+e x)^3}-\frac {b B}{e^3 (d+e x)} \]

[In]

Int[((a + b*x)*(A + B*x))/(d + e*x)^4,x]

[Out]

-1/3*((b*d - a*e)*(B*d - A*e))/(e^3*(d + e*x)^3) + (2*b*B*d - A*b*e - a*B*e)/(2*e^3*(d + e*x)^2) - (b*B)/(e^3*
(d + e*x))

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b d+a e) (-B d+A e)}{e^2 (d+e x)^4}+\frac {-2 b B d+A b e+a B e}{e^2 (d+e x)^3}+\frac {b B}{e^2 (d+e x)^2}\right ) \, dx \\ & = -\frac {(b d-a e) (B d-A e)}{3 e^3 (d+e x)^3}+\frac {2 b B d-A b e-a B e}{2 e^3 (d+e x)^2}-\frac {b B}{e^3 (d+e x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b x) (A+B x)}{(d+e x)^4} \, dx=-\frac {a e (2 A e+B (d+3 e x))+b \left (A e (d+3 e x)+2 B \left (d^2+3 d e x+3 e^2 x^2\right )\right )}{6 e^3 (d+e x)^3} \]

[In]

Integrate[((a + b*x)*(A + B*x))/(d + e*x)^4,x]

[Out]

-1/6*(a*e*(2*A*e + B*(d + 3*e*x)) + b*(A*e*(d + 3*e*x) + 2*B*(d^2 + 3*d*e*x + 3*e^2*x^2)))/(e^3*(d + e*x)^3)

Maple [A] (verified)

Time = 2.01 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.93

method	result	size
norman	\(\frac {-\frac {b B \,x^{2}}{e}-\frac {\left (A b e +B a e +2 B b d \right ) x}{2 e^{2}}-\frac {2 A a \,e^{2}+A b d e +B a d e +2 b B \,d^{2}}{6 e^{3}}}{\left (e x +d \right )^{3}}\)	\(70\)
risch	\(\frac {-\frac {b B \,x^{2}}{e}-\frac {\left (A b e +B a e +2 B b d \right ) x}{2 e^{2}}-\frac {2 A a \,e^{2}+A b d e +B a d e +2 b B \,d^{2}}{6 e^{3}}}{\left (e x +d \right )^{3}}\)	\(70\)
gosper	\(-\frac {6 b B \,x^{2} e^{2}+3 A x b \,e^{2}+3 B x a \,e^{2}+6 B x b d e +2 A a \,e^{2}+A b d e +B a d e +2 b B \,d^{2}}{6 \left (e x +d \right )^{3} e^{3}}\)	\(71\)
parallelrisch	\(-\frac {6 b B \,x^{2} e^{2}+3 A x b \,e^{2}+3 B x a \,e^{2}+6 B x b d e +2 A a \,e^{2}+A b d e +B a d e +2 b B \,d^{2}}{6 \left (e x +d \right )^{3} e^{3}}\)	\(71\)
default	\(-\frac {b B}{e^{3} \left (e x +d \right )}-\frac {A a \,e^{2}-A b d e -B a d e +b B \,d^{2}}{3 e^{3} \left (e x +d \right )^{3}}-\frac {A b e +B a e -2 B b d}{2 e^{3} \left (e x +d \right )^{2}}\)	\(79\)

[In]

int((b*x+a)*(B*x+A)/(e*x+d)^4,x,method=_RETURNVERBOSE)

[Out]

1/(e*x+d)^3*(-b*B/e*x^2-1/2*(A*b*e+B*a*e+2*B*b*d)/e^2*x-1/6*(2*A*a*e^2+A*b*d*e+B*a*d*e+2*B*b*d^2)/e^3)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.24 \[ \int \frac {(a+b x) (A+B x)}{(d+e x)^4} \, dx=-\frac {6 \, B b e^{2} x^{2} + 2 \, B b d^{2} + 2 \, A a e^{2} + {\left (B a + A b\right )} d e + 3 \, {\left (2 \, B b d e + {\left (B a + A b\right )} e^{2}\right )} x}{6 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]

[In]

integrate((b*x+a)*(B*x+A)/(e*x+d)^4,x, algorithm="fricas")

[Out]

-1/6*(6*B*b*e^2*x^2 + 2*B*b*d^2 + 2*A*a*e^2 + (B*a + A*b)*d*e + 3*(2*B*b*d*e + (B*a + A*b)*e^2)*x)/(e^6*x^3 +
3*d*e^5*x^2 + 3*d^2*e^4*x + d^3*e^3)

Sympy [A] (verification not implemented)

Time = 0.73 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.43 \[ \int \frac {(a+b x) (A+B x)}{(d+e x)^4} \, dx=\frac {- 2 A a e^{2} - A b d e - B a d e - 2 B b d^{2} - 6 B b e^{2} x^{2} + x \left (- 3 A b e^{2} - 3 B a e^{2} - 6 B b d e\right )}{6 d^{3} e^{3} + 18 d^{2} e^{4} x + 18 d e^{5} x^{2} + 6 e^{6} x^{3}} \]

[In]

integrate((b*x+a)*(B*x+A)/(e*x+d)**4,x)

[Out]

(-2*A*a*e**2 - A*b*d*e - B*a*d*e - 2*B*b*d**2 - 6*B*b*e**2*x**2 + x*(-3*A*b*e**2 - 3*B*a*e**2 - 6*B*b*d*e))/(6
*d**3*e**3 + 18*d**2*e**4*x + 18*d*e**5*x**2 + 6*e**6*x**3)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.24 \[ \int \frac {(a+b x) (A+B x)}{(d+e x)^4} \, dx=-\frac {6 \, B b e^{2} x^{2} + 2 \, B b d^{2} + 2 \, A a e^{2} + {\left (B a + A b\right )} d e + 3 \, {\left (2 \, B b d e + {\left (B a + A b\right )} e^{2}\right )} x}{6 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]

[In]

integrate((b*x+a)*(B*x+A)/(e*x+d)^4,x, algorithm="maxima")

[Out]

-1/6*(6*B*b*e^2*x^2 + 2*B*b*d^2 + 2*A*a*e^2 + (B*a + A*b)*d*e + 3*(2*B*b*d*e + (B*a + A*b)*e^2)*x)/(e^6*x^3 +
3*d*e^5*x^2 + 3*d^2*e^4*x + d^3*e^3)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x) (A+B x)}{(d+e x)^4} \, dx=-\frac {6 \, B b e^{2} x^{2} + 6 \, B b d e x + 3 \, B a e^{2} x + 3 \, A b e^{2} x + 2 \, B b d^{2} + B a d e + A b d e + 2 \, A a e^{2}}{6 \, {\left (e x + d\right )}^{3} e^{3}} \]

[In]

integrate((b*x+a)*(B*x+A)/(e*x+d)^4,x, algorithm="giac")

[Out]

-1/6*(6*B*b*e^2*x^2 + 6*B*b*d*e*x + 3*B*a*e^2*x + 3*A*b*e^2*x + 2*B*b*d^2 + B*a*d*e + A*b*d*e + 2*A*a*e^2)/((e
*x + d)^3*e^3)

Mupad [B] (verification not implemented)

Time = 1.16 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.21 \[ \int \frac {(a+b x) (A+B x)}{(d+e x)^4} \, dx=-\frac {\frac {2\,A\,a\,e^2+2\,B\,b\,d^2+A\,b\,d\,e+B\,a\,d\,e}{6\,e^3}+\frac {x\,\left (A\,b\,e+B\,a\,e+2\,B\,b\,d\right )}{2\,e^2}+\frac {B\,b\,x^2}{e}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3} \]

[In]

int(((A + B*x)*(a + b*x))/(d + e*x)^4,x)

[Out]

-((2*A*a*e^2 + 2*B*b*d^2 + A*b*d*e + B*a*d*e)/(6*e^3) + (x*(A*b*e + B*a*e + 2*B*b*d))/(2*e^2) + (B*b*x^2)/e)/(
d^3 + e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x)