\(\int \frac {(a+b x^3)^2}{x^4} \, dx\) [226]

Optimal result

Integrand size = 13, antiderivative size = 27 \[ \int \frac {\left (a+b x^3\right )^2}{x^4} \, dx=-\frac {a^2}{3 x^3}+\frac {b^2 x^3}{3}+2 a b \log (x) \]

[Out]

-1/3*a^2/x^3+1/3*b^2*x^3+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} \[ \int \frac {\left (a+b x^3\right )^2}{x^4} \, dx=-\frac {a^2}{3 x^3}+2 a b \log (x)+\frac {b^2 x^3}{3} \]

[In]

Int[(a + b*x^3)^2/x^4,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^2}{x^4} \, dx=-\frac {a^2}{3 x^3}+\frac {b^2 x^3}{3}+2 a b \log (x) \]

[In]

Integrate[(a + b*x^3)^2/x^4,x]

[Out]

-1/3*a^2/x^3 + (b^2*x^3)/3 + 2*a*b*Log[x]

Maple [A] (verified)

Time = 3.71 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89

method	result	size
default	\(-\frac {a^{2}}{3 x^{3}}+\frac {b^{2} x^{3}}{3}+2 a b \ln \left (x \right )\)	\(24\)
risch	\(-\frac {a^{2}}{3 x^{3}}+\frac {b^{2} x^{3}}{3}+2 a b \ln \left (x \right )\)	\(24\)
norman	\(\frac {-\frac {a^{2}}{3}+\frac {b^{2} x^{6}}{3}}{x^{3}}+2 a b \ln \left (x \right )\)	\(26\)
parallelrisch	\(\frac {b^{2} x^{6}+6 a b \ln \left (x \right ) x^{3}-a^{2}}{3 x^{3}}\)	\(28\)

[In]

int((b*x^3+a)^2/x^4,x,method=_RETURNVERBOSE)

[Out]

-1/3*a^2/x^3+1/3*b^2*x^3+2*a*b*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^2}{x^4} \, dx=\frac {b^{2} x^{6} + 6 \, a b x^{3} \log \left (x\right ) - a^{2}}{3 \, x^{3}} \]

[In]

integrate((b*x^3+a)^2/x^4,x, algorithm="fricas")

[Out]

1/3*(b^2*x^6 + 6*a*b*x^3*log(x) - a^2)/x^3

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x^3\right )^2}{x^4} \, dx=- \frac {a^{2}}{3 x^{3}} + 2 a b \log {\left (x \right )} + \frac {b^{2} x^{3}}{3} \]

[In]

integrate((b*x**3+a)**2/x**4,x)

[Out]

-a**2/(3*x**3) + 2*a*b*log(x) + b**2*x**3/3

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x^3\right )^2}{x^4} \, dx=\frac {1}{3} \, b^{2} x^{3} + \frac {2}{3} \, a b \log \left (x^{3}\right ) - \frac {a^{2}}{3 \, x^{3}} \]

[In]

integrate((b*x^3+a)^2/x^4,x, algorithm="maxima")

[Out]

1/3*b^2*x^3 + 2/3*a*b*log(x^3) - 1/3*a^2/x^3

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b x^3\right )^2}{x^4} \, dx=\frac {1}{3} \, b^{2} x^{3} + 2 \, a b \log \left ({\left | x \right |}\right ) - \frac {2 \, a b x^{3} + a^{2}}{3 \, x^{3}} \]

[In]

integrate((b*x^3+a)^2/x^4,x, algorithm="giac")

[Out]

1/3*b^2*x^3 + 2*a*b*log(abs(x)) - 1/3*(2*a*b*x^3 + a^2)/x^3

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b x^3\right )^2}{x^4} \, dx=\frac {b^2\,x^3}{3}-\frac {a^2}{3\,x^3}+2\,a\,b\,\ln \left (x\right ) \]

[In]

int((a + b*x^3)^2/x^4,x)

[Out]

(b^2*x^3)/3 - a^2/(3*x^3) + 2*a*b*log(x)