3.3.29 \(\int \frac {(a+b x^3)^2}{x^7} \, dx\) [229]

Optimal. Leaf size=26 \[ -\frac {a^2}{6 x^6}-\frac {2 a b}{3 x^3}+b^2 \log (x) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \begin {gather*} -\frac {a^2}{6 x^6}-\frac {2 a b}{3 x^3}+b^2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^2/x^7,x]

[Out]

-1/6*a^2/x^6 - (2*a*b)/(3*x^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*} \int \frac {\left (a+b x^3\right )^2}{x^7} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {(a+b x)^2}{x^3} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (\frac {a^2}{x^3}+\frac {2 a b}{x^2}+\frac {b^2}{x}\right ) \, dx,x,x^3\right )\\ &=-\frac {a^2}{6 x^6}-\frac {2 a b}{3 x^3}+b^2 \log (x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 26, normalized size = 1.00 \begin {gather*} -\frac {a^2}{6 x^6}-\frac {2 a b}{3 x^3}+b^2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^2/x^7,x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.12, size = 23, normalized size = 0.88

method	result	size
default	\(-\frac {a^{2}}{6 x^{6}}-\frac {2 a b}{3 x^{3}}+b^{2} \ln \left (x \right )\)	\(23\)
norman	\(\frac {-\frac {1}{6} a^{2}-\frac {2}{3} a b \,x^{3}}{x^{6}}+b^{2} \ln \left (x \right )\)	\(25\)
risch	\(\frac {-\frac {1}{6} a^{2}-\frac {2}{3} a b \,x^{3}}{x^{6}}+b^{2} \ln \left (x \right )\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^2/x^7,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.30, size = 26, normalized size = 1.00 \begin {gather*} \frac {1}{3} \, b^{2} \log \left (x^{3}\right ) - \frac {4 \, a b x^{3} + a^{2}}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^7,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 28, normalized size = 1.08 \begin {gather*} \frac {6 \, b^{2} x^{6} \log \left (x\right ) - 4 \, a b x^{3} - a^{2}}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^7,x, algorithm="fricas")

[Out]

1/6*(6*b^2*x^6*log(x) - 4*a*b*x^3 - a^2)/x^6

________________________________________________________________________________________

Sympy [A]
time = 0.07, size = 24, normalized size = 0.92 \begin {gather*} b^{2} \log {\left (x \right )} + \frac {- a^{2} - 4 a b x^{3}}{6 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**2/x**7,x)

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 2.08, size = 32, normalized size = 1.23 \begin {gather*} b^{2} \log \left ({\left | x \right |}\right ) - \frac {3 \, b^{2} x^{6} + 4 \, a b x^{3} + a^{2}}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^7,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.04, size = 25, normalized size = 0.96 \begin {gather*} b^2\,\ln \left (x\right )-\frac {\frac {a^2}{6}+\frac {2\,b\,a\,x^3}{3}}{x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^2/x^7,x)

[Out]

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

________________________________________________________________________________________