\(\int \frac {(a+b x)^2 (A+B x)}{x^8} \, dx\) [103]

Optimal result

Integrand size = 16, antiderivative size = 55 \[ \int \frac {(a+b x)^2 (A+B x)}{x^8} \, dx=-\frac {a^2 A}{7 x^7}-\frac {a (2 A b+a B)}{6 x^6}-\frac {b (A b+2 a B)}{5 x^5}-\frac {b^2 B}{4 x^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 55, 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} \[ \int \frac {(a+b x)^2 (A+B x)}{x^8} \, dx=-\frac {a^2 A}{7 x^7}-\frac {a (a B+2 A b)}{6 x^6}-\frac {b (2 a B+A b)}{5 x^5}-\frac {b^2 B}{4 x^4} \]

[In]

Int[((a + b*x)^2*(A + B*x))/x^8,x]

[Out]

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

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^2 A}{x^8}+\frac {a (2 A b+a B)}{x^7}+\frac {b (A b+2 a B)}{x^6}+\frac {b^2 B}{x^5}\right ) \, dx \\ & = -\frac {a^2 A}{7 x^7}-\frac {a (2 A b+a B)}{6 x^6}-\frac {b (A b+2 a B)}{5 x^5}-\frac {b^2 B}{4 x^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x)^2 (A+B x)}{x^8} \, dx=-\frac {21 b^2 x^2 (4 A+5 B x)+28 a b x (5 A+6 B x)+10 a^2 (6 A+7 B x)}{420 x^7} \]

[In]

Integrate[((a + b*x)^2*(A + B*x))/x^8,x]

[Out]

-1/420*(21*b^2*x^2*(4*A + 5*B*x) + 28*a*b*x*(5*A + 6*B*x) + 10*a^2*(6*A + 7*B*x))/x^7

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.87

method	result	size
default	\(-\frac {a^{2} A}{7 x^{7}}-\frac {a \left (2 A b +B a \right )}{6 x^{6}}-\frac {b \left (A b +2 B a \right )}{5 x^{5}}-\frac {b^{2} B}{4 x^{4}}\)	\(48\)
norman	\(\frac {-\frac {b^{2} B \,x^{3}}{4}+\left (-\frac {1}{5} b^{2} A -\frac {2}{5} a b B \right ) x^{2}+\left (-\frac {1}{3} a b A -\frac {1}{6} a^{2} B \right ) x -\frac {a^{2} A}{7}}{x^{7}}\)	\(51\)
risch	\(\frac {-\frac {b^{2} B \,x^{3}}{4}+\left (-\frac {1}{5} b^{2} A -\frac {2}{5} a b B \right ) x^{2}+\left (-\frac {1}{3} a b A -\frac {1}{6} a^{2} B \right ) x -\frac {a^{2} A}{7}}{x^{7}}\)	\(51\)
gosper	\(-\frac {105 b^{2} B \,x^{3}+84 A \,b^{2} x^{2}+168 B a b \,x^{2}+140 a A b x +70 a^{2} B x +60 a^{2} A}{420 x^{7}}\)	\(52\)
parallelrisch	\(-\frac {105 b^{2} B \,x^{3}+84 A \,b^{2} x^{2}+168 B a b \,x^{2}+140 a A b x +70 a^{2} B x +60 a^{2} A}{420 x^{7}}\)	\(52\)

[In]

int((b*x+a)^2*(B*x+A)/x^8,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^2 (A+B x)}{x^8} \, dx=-\frac {105 \, B b^{2} x^{3} + 60 \, A a^{2} + 84 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} + 70 \, {\left (B a^{2} + 2 \, A a b\right )} x}{420 \, x^{7}} \]

[In]

integrate((b*x+a)^2*(B*x+A)/x^8,x, algorithm="fricas")

[Out]

-1/420*(105*B*b^2*x^3 + 60*A*a^2 + 84*(2*B*a*b + A*b^2)*x^2 + 70*(B*a^2 + 2*A*a*b)*x)/x^7

Sympy [A] (verification not implemented)

Time = 0.65 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.02 \[ \int \frac {(a+b x)^2 (A+B x)}{x^8} \, dx=\frac {- 60 A a^{2} - 105 B b^{2} x^{3} + x^{2} \left (- 84 A b^{2} - 168 B a b\right ) + x \left (- 140 A a b - 70 B a^{2}\right )}{420 x^{7}} \]

[In]

integrate((b*x+a)**2*(B*x+A)/x**8,x)

[Out]

(-60*A*a**2 - 105*B*b**2*x**3 + x**2*(-84*A*b**2 - 168*B*a*b) + x*(-140*A*a*b - 70*B*a**2))/(420*x**7)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^2 (A+B x)}{x^8} \, dx=-\frac {105 \, B b^{2} x^{3} + 60 \, A a^{2} + 84 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} + 70 \, {\left (B a^{2} + 2 \, A a b\right )} x}{420 \, x^{7}} \]

[In]

integrate((b*x+a)^2*(B*x+A)/x^8,x, algorithm="maxima")

[Out]

-1/420*(105*B*b^2*x^3 + 60*A*a^2 + 84*(2*B*a*b + A*b^2)*x^2 + 70*(B*a^2 + 2*A*a*b)*x)/x^7

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^2 (A+B x)}{x^8} \, dx=-\frac {105 \, B b^{2} x^{3} + 168 \, B a b x^{2} + 84 \, A b^{2} x^{2} + 70 \, B a^{2} x + 140 \, A a b x + 60 \, A a^{2}}{420 \, x^{7}} \]

[In]

integrate((b*x+a)^2*(B*x+A)/x^8,x, algorithm="giac")

[Out]

-1/420*(105*B*b^2*x^3 + 168*B*a*b*x^2 + 84*A*b^2*x^2 + 70*B*a^2*x + 140*A*a*b*x + 60*A*a^2)/x^7

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^2 (A+B x)}{x^8} \, dx=-\frac {x^2\,\left (\frac {A\,b^2}{5}+\frac {2\,B\,a\,b}{5}\right )+\frac {A\,a^2}{7}+x\,\left (\frac {B\,a^2}{6}+\frac {A\,b\,a}{3}\right )+\frac {B\,b^2\,x^3}{4}}{x^7} \]

[In]

int(((A + B*x)*(a + b*x)^2)/x^8,x)

[Out]

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