\(\int \frac {(a+b x^7)^2}{x^8} \, dx\) [1440]

Optimal result

Integrand size = 13, antiderivative size = 27 \[ \int \frac {\left (a+b x^7\right )^2}{x^8} \, dx=-\frac {a^2}{7 x^7}+\frac {b^2 x^7}{7}+2 a b \log (x) \]

[Out]

-1/7*a^2/x^7+1/7*b^2*x^7+2*a*b*ln(x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \[ \int \frac {\left (a+b x^7\right )^2}{x^8} \, dx=-\frac {a^2}{7 x^7}+2 a b \log (x)+\frac {b^2 x^7}{7} \]

[In]

Int[(a + b*x^7)^2/x^8,x]

[Out]

-1/7*a^2/x^7 + (b^2*x^7)/7 + 2*a*b*Log[x]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{7} \text {Subst}\left (\int \frac {(a+b x)^2}{x^2} \, dx,x,x^7\right ) \\ & = \frac {1}{7} \text {Subst}\left (\int \left (b^2+\frac {a^2}{x^2}+\frac {2 a b}{x}\right ) \, dx,x,x^7\right ) \\ & = -\frac {a^2}{7 x^7}+\frac {b^2 x^7}{7}+2 a b \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^7\right )^2}{x^8} \, dx=-\frac {a^2}{7 x^7}+\frac {b^2 x^7}{7}+2 a b \log (x) \]

[In]

Integrate[(a + b*x^7)^2/x^8,x]

[Out]

-1/7*a^2/x^7 + (b^2*x^7)/7 + 2*a*b*Log[x]

Maple [A] (verified)

Time = 5.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89

method	result	size
default	\(-\frac {a^{2}}{7 x^{7}}+\frac {b^{2} x^{7}}{7}+2 a b \ln \left (x \right )\)	\(24\)
risch	\(-\frac {a^{2}}{7 x^{7}}+\frac {b^{2} x^{7}}{7}+2 a b \ln \left (x \right )\)	\(24\)
norman	\(\frac {-\frac {a^{2}}{7}+\frac {b^{2} x^{14}}{7}}{x^{7}}+2 a b \ln \left (x \right )\)	\(26\)
parallelrisch	\(\frac {b^{2} x^{14}+14 a b \ln \left (x \right ) x^{7}-a^{2}}{7 x^{7}}\)	\(28\)

[In]

int((b*x^7+a)^2/x^8,x,method=_RETURNVERBOSE)

[Out]

-1/7*a^2/x^7+1/7*b^2*x^7+2*a*b*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^7\right )^2}{x^8} \, dx=\frac {b^{2} x^{14} + 14 \, a b x^{7} \log \left (x\right ) - a^{2}}{7 \, x^{7}} \]

[In]

integrate((b*x^7+a)^2/x^8,x, algorithm="fricas")

[Out]

1/7*(b^2*x^14 + 14*a*b*x^7*log(x) - a^2)/x^7

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x^7\right )^2}{x^8} \, dx=- \frac {a^{2}}{7 x^{7}} + 2 a b \log {\left (x \right )} + \frac {b^{2} x^{7}}{7} \]

[In]

integrate((b*x**7+a)**2/x**8,x)

[Out]

-a**2/(7*x**7) + 2*a*b*log(x) + b**2*x**7/7

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x^7\right )^2}{x^8} \, dx=\frac {1}{7} \, b^{2} x^{7} + \frac {2}{7} \, a b \log \left (x^{7}\right ) - \frac {a^{2}}{7 \, x^{7}} \]

[In]

integrate((b*x^7+a)^2/x^8,x, algorithm="maxima")

[Out]

1/7*b^2*x^7 + 2/7*a*b*log(x^7) - 1/7*a^2/x^7

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b x^7\right )^2}{x^8} \, dx=\frac {1}{7} \, b^{2} x^{7} + 2 \, a b \log \left ({\left | x \right |}\right ) - \frac {2 \, a b x^{7} + a^{2}}{7 \, x^{7}} \]

[In]

integrate((b*x^7+a)^2/x^8,x, algorithm="giac")

[Out]

1/7*b^2*x^7 + 2*a*b*log(abs(x)) - 1/7*(2*a*b*x^7 + a^2)/x^7

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b x^7\right )^2}{x^8} \, dx=\frac {b^2\,x^7}{7}-\frac {a^2}{7\,x^7}+2\,a\,b\,\ln \left (x\right ) \]

[In]

int((a + b*x^7)^2/x^8,x)

[Out]

(b^2*x^7)/7 - a^2/(7*x^7) + 2*a*b*log(x)