3.7.39 \(\int \frac {(a+b x^4)^3}{x} \, dx\) [639]

Optimal. Leaf size=39 \[ \frac {3}{4} a^2 b x^4+\frac {3}{8} a b^2 x^8+\frac {b^3 x^{12}}{12}+a^3 \log (x) \]

[Out]

3/4*a^2*b*x^4+3/8*a*b^2*x^8+1/12*b^3*x^12+a^3*ln(x)

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Rubi [A]
time = 0.01, antiderivative size = 39, 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*} a^3 \log (x)+\frac {3}{4} a^2 b x^4+\frac {3}{8} a b^2 x^8+\frac {b^3 x^{12}}{12} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^3/x,x]

[Out]

(3*a^2*b*x^4)/4 + (3*a*b^2*x^8)/8 + (b^3*x^12)/12 + a^3*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 )^3}{x} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {(a+b x)^3}{x} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (3 a^2 b+\frac {a^3}{x}+3 a b^2 x+b^3 x^2\right ) \, dx,x,x^4\right )\\ &=\frac {3}{4} a^2 b x^4+\frac {3}{8} a b^2 x^8+\frac {b^3 x^{12}}{12}+a^3 \log (x)\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 39, normalized size = 1.00 \begin {gather*} \frac {3}{4} a^2 b x^4+\frac {3}{8} a b^2 x^8+\frac {b^3 x^{12}}{12}+a^3 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^3/x,x]

[Out]

(3*a^2*b*x^4)/4 + (3*a*b^2*x^8)/8 + (b^3*x^12)/12 + a^3*Log[x]

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Maple [A]
time = 0.13, size = 34, normalized size = 0.87

method	result	size
default	\(\frac {3 a^{2} b \,x^{4}}{4}+\frac {3 a \,b^{2} x^{8}}{8}+\frac {b^{3} x^{12}}{12}+a^{3} \ln \left (x \right )\)	\(34\)
norman	\(\frac {3 a^{2} b \,x^{4}}{4}+\frac {3 a \,b^{2} x^{8}}{8}+\frac {b^{3} x^{12}}{12}+a^{3} \ln \left (x \right )\)	\(34\)
risch	\(\frac {3 a^{2} b \,x^{4}}{4}+\frac {3 a \,b^{2} x^{8}}{8}+\frac {b^{3} x^{12}}{12}+a^{3} \ln \left (x \right )\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*a^2*b*x^4+3/8*a*b^2*x^8+1/12*b^3*x^12+a^3*ln(x)

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Maxima [A]
time = 0.30, size = 36, normalized size = 0.92 \begin {gather*} \frac {1}{12} \, b^{3} x^{12} + \frac {3}{8} \, a b^{2} x^{8} + \frac {3}{4} \, a^{2} b x^{4} + \frac {1}{4} \, a^{3} \log \left (x^{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*b^3*x^12 + 3/8*a*b^2*x^8 + 3/4*a^2*b*x^4 + 1/4*a^3*log(x^4)

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Fricas [A]
time = 0.36, size = 33, normalized size = 0.85 \begin {gather*} \frac {1}{12} \, b^{3} x^{12} + \frac {3}{8} \, a b^{2} x^{8} + \frac {3}{4} \, a^{2} b x^{4} + a^{3} \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*b^3*x^12 + 3/8*a*b^2*x^8 + 3/4*a^2*b*x^4 + a^3*log(x)

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Sympy [A]
time = 0.03, size = 37, normalized size = 0.95 \begin {gather*} a^{3} \log {\left (x \right )} + \frac {3 a^{2} b x^{4}}{4} + \frac {3 a b^{2} x^{8}}{8} + \frac {b^{3} x^{12}}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*log(x) + 3*a**2*b*x**4/4 + 3*a*b**2*x**8/8 + b**3*x**12/12

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Giac [A]
time = 0.60, size = 36, normalized size = 0.92 \begin {gather*} \frac {1}{12} \, b^{3} x^{12} + \frac {3}{8} \, a b^{2} x^{8} + \frac {3}{4} \, a^{2} b x^{4} + \frac {1}{4} \, a^{3} \log \left (x^{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*b^3*x^12 + 3/8*a*b^2*x^8 + 3/4*a^2*b*x^4 + 1/4*a^3*log(x^4)

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Mupad [B]
time = 0.04, size = 33, normalized size = 0.85 \begin {gather*} a^3\,\ln \left (x\right )+\frac {b^3\,x^{12}}{12}+\frac {3\,a^2\,b\,x^4}{4}+\frac {3\,a\,b^2\,x^8}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^4)^3/x,x)

[Out]

a^3*log(x) + (b^3*x^12)/12 + (3*a^2*b*x^4)/4 + (3*a*b^2*x^8)/8

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