3.7.31 \(\int \frac {(a+b x^4)^2}{x^5} \, dx\) [631]

Optimal. Leaf size=27 \[ -\frac {a^2}{4 x^4}+\frac {b^2 x^4}{4}+2 a b \log (x) \]

[Out]

-1/4*a^2/x^4+1/4*b^2*x^4+2*a*b*ln(x)

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Rubi [A]
time = 0.01, antiderivative size = 27, 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}{4 x^4}+2 a b \log (x)+\frac {b^2 x^4}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^2/x^5,x]

[Out]

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

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Mathematica [A]
time = 0.00, size = 27, normalized size = 1.00 \begin {gather*} -\frac {a^2}{4 x^4}+\frac {b^2 x^4}{4}+2 a b \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^2/x^5,x]

[Out]

-1/4*a^2/x^4 + (b^2*x^4)/4 + 2*a*b*Log[x]

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Maple [A]
time = 0.14, size = 24, normalized size = 0.89

method	result	size
default	\(-\frac {a^{2}}{4 x^{4}}+\frac {b^{2} x^{4}}{4}+2 a b \ln \left (x \right )\)	\(24\)
risch	\(-\frac {a^{2}}{4 x^{4}}+\frac {b^{2} x^{4}}{4}+2 a b \ln \left (x \right )\)	\(24\)
norman	\(\frac {-\frac {a^{2}}{4}+\frac {b^{2} x^{8}}{4}}{x^{4}}+2 a b \ln \left (x \right )\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)^2/x^5,x,method=_RETURNVERBOSE)

[Out]

-1/4*a^2/x^4+1/4*b^2*x^4+2*a*b*ln(x)

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Maxima [A]
time = 0.29, size = 25, normalized size = 0.93 \begin {gather*} \frac {1}{4} \, b^{2} x^{4} + \frac {1}{2} \, a b \log \left (x^{4}\right ) - \frac {a^{2}}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^2/x^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.36, size = 27, normalized size = 1.00 \begin {gather*} \frac {b^{2} x^{8} + 8 \, a b x^{4} \log \left (x\right ) - a^{2}}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^2/x^5,x, algorithm="fricas")

[Out]

1/4*(b^2*x^8 + 8*a*b*x^4*log(x) - a^2)/x^4

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Sympy [A]
time = 0.05, size = 24, normalized size = 0.89 \begin {gather*} - \frac {a^{2}}{4 x^{4}} + 2 a b \log {\left (x \right )} + \frac {b^{2} x^{4}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)**2/x**5,x)

[Out]

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

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Giac [A]
time = 0.51, size = 33, normalized size = 1.22 \begin {gather*} \frac {1}{4} \, b^{2} x^{4} + \frac {1}{2} \, a b \log \left (x^{4}\right ) - \frac {2 \, a b x^{4} + a^{2}}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^2/x^5,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.03, size = 23, normalized size = 0.85 \begin {gather*} \frac {b^2\,x^4}{4}-\frac {a^2}{4\,x^4}+2\,a\,b\,\ln \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^4)^2/x^5,x)

[Out]

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

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