\(\int \frac {(a+b x)^3 (A+B x)}{x^8} \, dx\) [116]

Optimal result

Integrand size = 16, antiderivative size = 75 \[ \int \frac {(a+b x)^3 (A+B x)}{x^8} \, dx=-\frac {a^3 A}{7 x^7}-\frac {a^2 (3 A b+a B)}{6 x^6}-\frac {3 a b (A b+a B)}{5 x^5}-\frac {b^2 (A b+3 a B)}{4 x^4}-\frac {b^3 B}{3 x^3} \]

[Out]

-1/7*a^3*A/x^7-1/6*a^2*(3*A*b+B*a)/x^6-3/5*a*b*(A*b+B*a)/x^5-1/4*b^2*(A*b+3*B*a)/x^4-1/3*b^3*B/x^3

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77} \[ \int \frac {(a+b x)^3 (A+B x)}{x^8} \, dx=-\frac {a^3 A}{7 x^7}-\frac {a^2 (a B+3 A b)}{6 x^6}-\frac {b^2 (3 a B+A b)}{4 x^4}-\frac {3 a b (a B+A b)}{5 x^5}-\frac {b^3 B}{3 x^3} \]

[In]

Int[((a + b*x)^3*(A + B*x))/x^8,x]

[Out]

-1/7*(a^3*A)/x^7 - (a^2*(3*A*b + a*B))/(6*x^6) - (3*a*b*(A*b + a*B))/(5*x^5) - (b^2*(A*b + 3*a*B))/(4*x^4) - (
b^3*B)/(3*x^3)

Rule 77

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^3 A}{x^8}+\frac {a^2 (3 A b+a B)}{x^7}+\frac {3 a b (A b+a B)}{x^6}+\frac {b^2 (A b+3 a B)}{x^5}+\frac {b^3 B}{x^4}\right ) \, dx \\ & = -\frac {a^3 A}{7 x^7}-\frac {a^2 (3 A b+a B)}{6 x^6}-\frac {3 a b (A b+a B)}{5 x^5}-\frac {b^2 (A b+3 a B)}{4 x^4}-\frac {b^3 B}{3 x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x)^3 (A+B x)}{x^8} \, dx=-\frac {35 b^3 x^3 (3 A+4 B x)+63 a b^2 x^2 (4 A+5 B x)+42 a^2 b x (5 A+6 B x)+10 a^3 (6 A+7 B x)}{420 x^7} \]

[In]

Integrate[((a + b*x)^3*(A + B*x))/x^8,x]

[Out]

-1/420*(35*b^3*x^3*(3*A + 4*B*x) + 63*a*b^2*x^2*(4*A + 5*B*x) + 42*a^2*b*x*(5*A + 6*B*x) + 10*a^3*(6*A + 7*B*x
))/x^7

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.88

method	result	size
default	\(-\frac {a^{3} A}{7 x^{7}}-\frac {a^{2} \left (3 A b +B a \right )}{6 x^{6}}-\frac {3 a b \left (A b +B a \right )}{5 x^{5}}-\frac {b^{2} \left (A b +3 B a \right )}{4 x^{4}}-\frac {b^{3} B}{3 x^{3}}\)	\(66\)
norman	\(\frac {-\frac {b^{3} B \,x^{4}}{3}+\left (-\frac {1}{4} b^{3} A -\frac {3}{4} a \,b^{2} B \right ) x^{3}+\left (-\frac {3}{5} a \,b^{2} A -\frac {3}{5} a^{2} b B \right ) x^{2}+\left (-\frac {1}{2} a^{2} b A -\frac {1}{6} a^{3} B \right ) x -\frac {a^{3} A}{7}}{x^{7}}\)	\(74\)
risch	\(\frac {-\frac {b^{3} B \,x^{4}}{3}+\left (-\frac {1}{4} b^{3} A -\frac {3}{4} a \,b^{2} B \right ) x^{3}+\left (-\frac {3}{5} a \,b^{2} A -\frac {3}{5} a^{2} b B \right ) x^{2}+\left (-\frac {1}{2} a^{2} b A -\frac {1}{6} a^{3} B \right ) x -\frac {a^{3} A}{7}}{x^{7}}\)	\(74\)
gosper	\(-\frac {140 b^{3} B \,x^{4}+105 A \,b^{3} x^{3}+315 B a \,b^{2} x^{3}+252 a A \,b^{2} x^{2}+252 B \,a^{2} b \,x^{2}+210 a^{2} A b x +70 a^{3} B x +60 a^{3} A}{420 x^{7}}\)	\(76\)
parallelrisch	\(-\frac {140 b^{3} B \,x^{4}+105 A \,b^{3} x^{3}+315 B a \,b^{2} x^{3}+252 a A \,b^{2} x^{2}+252 B \,a^{2} b \,x^{2}+210 a^{2} A b x +70 a^{3} B x +60 a^{3} A}{420 x^{7}}\)	\(76\)

[In]

int((b*x+a)^3*(B*x+A)/x^8,x,method=_RETURNVERBOSE)

[Out]

-1/7*a^3*A/x^7-1/6*a^2*(3*A*b+B*a)/x^6-3/5*a*b*(A*b+B*a)/x^5-1/4*b^2*(A*b+3*B*a)/x^4-1/3*b^3*B/x^3

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x)^3 (A+B x)}{x^8} \, dx=-\frac {140 \, B b^{3} x^{4} + 60 \, A a^{3} + 105 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 252 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} + 70 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{420 \, x^{7}} \]

[In]

integrate((b*x+a)^3*(B*x+A)/x^8,x, algorithm="fricas")

[Out]

-1/420*(140*B*b^3*x^4 + 60*A*a^3 + 105*(3*B*a*b^2 + A*b^3)*x^3 + 252*(B*a^2*b + A*a*b^2)*x^2 + 70*(B*a^3 + 3*A
*a^2*b)*x)/x^7

Sympy [A] (verification not implemented)

Time = 1.10 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.09 \[ \int \frac {(a+b x)^3 (A+B x)}{x^8} \, dx=\frac {- 60 A a^{3} - 140 B b^{3} x^{4} + x^{3} \left (- 105 A b^{3} - 315 B a b^{2}\right ) + x^{2} \left (- 252 A a b^{2} - 252 B a^{2} b\right ) + x \left (- 210 A a^{2} b - 70 B a^{3}\right )}{420 x^{7}} \]

[In]

integrate((b*x+a)**3*(B*x+A)/x**8,x)

[Out]

(-60*A*a**3 - 140*B*b**3*x**4 + x**3*(-105*A*b**3 - 315*B*a*b**2) + x**2*(-252*A*a*b**2 - 252*B*a**2*b) + x*(-
210*A*a**2*b - 70*B*a**3))/(420*x**7)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x)^3 (A+B x)}{x^8} \, dx=-\frac {140 \, B b^{3} x^{4} + 60 \, A a^{3} + 105 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 252 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} + 70 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{420 \, x^{7}} \]

[In]

integrate((b*x+a)^3*(B*x+A)/x^8,x, algorithm="maxima")

[Out]

-1/420*(140*B*b^3*x^4 + 60*A*a^3 + 105*(3*B*a*b^2 + A*b^3)*x^3 + 252*(B*a^2*b + A*a*b^2)*x^2 + 70*(B*a^3 + 3*A
*a^2*b)*x)/x^7

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^3 (A+B x)}{x^8} \, dx=-\frac {140 \, B b^{3} x^{4} + 315 \, B a b^{2} x^{3} + 105 \, A b^{3} x^{3} + 252 \, B a^{2} b x^{2} + 252 \, A a b^{2} x^{2} + 70 \, B a^{3} x + 210 \, A a^{2} b x + 60 \, A a^{3}}{420 \, x^{7}} \]

[In]

integrate((b*x+a)^3*(B*x+A)/x^8,x, algorithm="giac")

[Out]

-1/420*(140*B*b^3*x^4 + 315*B*a*b^2*x^3 + 105*A*b^3*x^3 + 252*B*a^2*b*x^2 + 252*A*a*b^2*x^2 + 70*B*a^3*x + 210
*A*a^2*b*x + 60*A*a^3)/x^7

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b x)^3 (A+B x)}{x^8} \, dx=-\frac {x^2\,\left (\frac {3\,B\,a^2\,b}{5}+\frac {3\,A\,a\,b^2}{5}\right )+x\,\left (\frac {B\,a^3}{6}+\frac {A\,b\,a^2}{2}\right )+\frac {A\,a^3}{7}+x^3\,\left (\frac {A\,b^3}{4}+\frac {3\,B\,a\,b^2}{4}\right )+\frac {B\,b^3\,x^4}{3}}{x^7} \]

[In]

int(((A + B*x)*(a + b*x)^3)/x^8,x)

[Out]

-(x^2*((3*A*a*b^2)/5 + (3*B*a^2*b)/5) + x*((B*a^3)/6 + (A*a^2*b)/2) + (A*a^3)/7 + x^3*((A*b^3)/4 + (3*B*a*b^2)
/4) + (B*b^3*x^4)/3)/x^7