\(\int \frac {(a+b x^3)^5}{x^7} \, dx\) [266]

Optimal result

Integrand size = 13, antiderivative size = 66 \[ \int \frac {\left (a+b x^3\right )^5}{x^7} \, dx=-\frac {a^5}{6 x^6}-\frac {5 a^4 b}{3 x^3}+\frac {10}{3} a^2 b^3 x^3+\frac {5}{6} a b^4 x^6+\frac {b^5 x^9}{9}+10 a^3 b^2 \log (x) \]

[Out]

-1/6*a^5/x^6-5/3*a^4*b/x^3+10/3*a^2*b^3*x^3+5/6*a*b^4*x^6+1/9*b^5*x^9+10*a^3*b^2*ln(x)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 66, 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} \[ \int \frac {\left (a+b x^3\right )^5}{x^7} \, dx=-\frac {a^5}{6 x^6}-\frac {5 a^4 b}{3 x^3}+10 a^3 b^2 \log (x)+\frac {10}{3} a^2 b^3 x^3+\frac {5}{6} a b^4 x^6+\frac {b^5 x^9}{9} \]

[In]

Int[(a + b*x^3)^5/x^7,x]

[Out]

-1/6*a^5/x^6 - (5*a^4*b)/(3*x^3) + (10*a^2*b^3*x^3)/3 + (5*a*b^4*x^6)/6 + (b^5*x^9)/9 + 10*a^3*b^2*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}{3} \text {Subst}\left (\int \frac {(a+b x)^5}{x^3} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left (10 a^2 b^3+\frac {a^5}{x^3}+\frac {5 a^4 b}{x^2}+\frac {10 a^3 b^2}{x}+5 a b^4 x+b^5 x^2\right ) \, dx,x,x^3\right ) \\ & = -\frac {a^5}{6 x^6}-\frac {5 a^4 b}{3 x^3}+\frac {10}{3} a^2 b^3 x^3+\frac {5}{6} a b^4 x^6+\frac {b^5 x^9}{9}+10 a^3 b^2 \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^5}{x^7} \, dx=-\frac {a^5}{6 x^6}-\frac {5 a^4 b}{3 x^3}+\frac {10}{3} a^2 b^3 x^3+\frac {5}{6} a b^4 x^6+\frac {b^5 x^9}{9}+10 a^3 b^2 \log (x) \]

[In]

Integrate[(a + b*x^3)^5/x^7,x]

[Out]

-1/6*a^5/x^6 - (5*a^4*b)/(3*x^3) + (10*a^2*b^3*x^3)/3 + (5*a*b^4*x^6)/6 + (b^5*x^9)/9 + 10*a^3*b^2*Log[x]

Maple [A] (verified)

Time = 3.62 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.86

method	result	size
default	\(-\frac {a^{5}}{6 x^{6}}-\frac {5 a^{4} b}{3 x^{3}}+\frac {10 a^{2} b^{3} x^{3}}{3}+\frac {5 a \,b^{4} x^{6}}{6}+\frac {b^{5} x^{9}}{9}+10 a^{3} b^{2} \ln \left (x \right )\)	\(57\)
norman	\(\frac {-\frac {1}{6} a^{5}+\frac {1}{9} b^{5} x^{15}+\frac {5}{6} a \,b^{4} x^{12}+\frac {10}{3} a^{2} b^{3} x^{9}-\frac {5}{3} a^{4} b \,x^{3}}{x^{6}}+10 a^{3} b^{2} \ln \left (x \right )\)	\(59\)
risch	\(\frac {b^{5} x^{9}}{9}+\frac {5 a \,b^{4} x^{6}}{6}+\frac {10 a^{2} b^{3} x^{3}}{3}+\frac {-\frac {5}{3} a^{4} b \,x^{3}-\frac {1}{6} a^{5}}{x^{6}}+10 a^{3} b^{2} \ln \left (x \right )\)	\(59\)
parallelrisch	\(\frac {2 b^{5} x^{15}+15 a \,b^{4} x^{12}+60 a^{2} b^{3} x^{9}+180 a^{3} b^{2} \ln \left (x \right ) x^{6}-30 a^{4} b \,x^{3}-3 a^{5}}{18 x^{6}}\)	\(62\)

[In]

int((b*x^3+a)^5/x^7,x,method=_RETURNVERBOSE)

[Out]

-1/6*a^5/x^6-5/3*a^4*b/x^3+10/3*a^2*b^3*x^3+5/6*a*b^4*x^6+1/9*b^5*x^9+10*a^3*b^2*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^3\right )^5}{x^7} \, dx=\frac {2 \, b^{5} x^{15} + 15 \, a b^{4} x^{12} + 60 \, a^{2} b^{3} x^{9} + 180 \, a^{3} b^{2} x^{6} \log \left (x\right ) - 30 \, a^{4} b x^{3} - 3 \, a^{5}}{18 \, x^{6}} \]

[In]

integrate((b*x^3+a)^5/x^7,x, algorithm="fricas")

[Out]

1/18*(2*b^5*x^15 + 15*a*b^4*x^12 + 60*a^2*b^3*x^9 + 180*a^3*b^2*x^6*log(x) - 30*a^4*b*x^3 - 3*a^5)/x^6

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b x^3\right )^5}{x^7} \, dx=10 a^{3} b^{2} \log {\left (x \right )} + \frac {10 a^{2} b^{3} x^{3}}{3} + \frac {5 a b^{4} x^{6}}{6} + \frac {b^{5} x^{9}}{9} + \frac {- a^{5} - 10 a^{4} b x^{3}}{6 x^{6}} \]

[In]

integrate((b*x**3+a)**5/x**7,x)

[Out]

10*a**3*b**2*log(x) + 10*a**2*b**3*x**3/3 + 5*a*b**4*x**6/6 + b**5*x**9/9 + (-a**5 - 10*a**4*b*x**3)/(6*x**6)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x^3\right )^5}{x^7} \, dx=\frac {1}{9} \, b^{5} x^{9} + \frac {5}{6} \, a b^{4} x^{6} + \frac {10}{3} \, a^{2} b^{3} x^{3} + \frac {10}{3} \, a^{3} b^{2} \log \left (x^{3}\right ) - \frac {10 \, a^{4} b x^{3} + a^{5}}{6 \, x^{6}} \]

[In]

integrate((b*x^3+a)^5/x^7,x, algorithm="maxima")

[Out]

1/9*b^5*x^9 + 5/6*a*b^4*x^6 + 10/3*a^2*b^3*x^3 + 10/3*a^3*b^2*log(x^3) - 1/6*(10*a^4*b*x^3 + a^5)/x^6

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b x^3\right )^5}{x^7} \, dx=\frac {1}{9} \, b^{5} x^{9} + \frac {5}{6} \, a b^{4} x^{6} + \frac {10}{3} \, a^{2} b^{3} x^{3} + 10 \, a^{3} b^{2} \log \left ({\left | x \right |}\right ) - \frac {30 \, a^{3} b^{2} x^{6} + 10 \, a^{4} b x^{3} + a^{5}}{6 \, x^{6}} \]

[In]

integrate((b*x^3+a)^5/x^7,x, algorithm="giac")

[Out]

1/9*b^5*x^9 + 5/6*a*b^4*x^6 + 10/3*a^2*b^3*x^3 + 10*a^3*b^2*log(abs(x)) - 1/6*(30*a^3*b^2*x^6 + 10*a^4*b*x^3 +
a^5)/x^6

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x^3\right )^5}{x^7} \, dx=\frac {b^5\,x^9}{9}-\frac {\frac {a^5}{6}+\frac {5\,b\,a^4\,x^3}{3}}{x^6}+\frac {5\,a\,b^4\,x^6}{6}+\frac {10\,a^2\,b^3\,x^3}{3}+10\,a^3\,b^2\,\ln \left (x\right ) \]

[In]

int((a + b*x^3)^5/x^7,x)

[Out]

(b^5*x^9)/9 - (a^5/6 + (5*a^4*b*x^3)/3)/x^6 + (5*a*b^4*x^6)/6 + (10*a^2*b^3*x^3)/3 + 10*a^3*b^2*log(x)