3.7.45 \(\int \frac {x^7}{a+c x^4} \, dx\) [645]

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

[Out]

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

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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 {x^4}{4 c}-\frac {a \log \left (a+c x^4\right )}{4 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^7/(a + c*x^4),x]

[Out]

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

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 {x^7}{a+c x^4} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {x}{a+c x} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (\frac {1}{c}-\frac {a}{c (a+c x)}\right ) \, dx,x,x^4\right )\\ &=\frac {x^4}{4 c}-\frac {a \log \left (a+c x^4\right )}{4 c^2}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 27, normalized size = 1.00 \begin {gather*} \frac {x^4}{4 c}-\frac {a \log \left (a+c x^4\right )}{4 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7/(a + c*x^4),x]

[Out]

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(c*x^4+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 23, normalized size = 0.85 \begin {gather*} \frac {x^{4}}{4 \, c} - \frac {a \log \left (c x^{4} + a\right )}{4 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(c*x^4+a),x, algorithm="maxima")

[Out]

1/4*x^4/c - 1/4*a*log(c*x^4 + a)/c^2

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Fricas [A]
time = 0.34, size = 22, normalized size = 0.81 \begin {gather*} \frac {c x^{4} - a \log \left (c x^{4} + a\right )}{4 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(c*x^4+a),x, algorithm="fricas")

[Out]

1/4*(c*x^4 - a*log(c*x^4 + a))/c^2

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Sympy [A]
time = 0.08, size = 20, normalized size = 0.74 \begin {gather*} - \frac {a \log {\left (a + c x^{4} \right )}}{4 c^{2}} + \frac {x^{4}}{4 c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(c*x**4+a),x)

[Out]

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

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Giac [A]
time = 0.60, size = 24, normalized size = 0.89 \begin {gather*} \frac {x^{4}}{4 \, c} - \frac {a \log \left ({\left | c x^{4} + a \right |}\right )}{4 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(c*x^4+a),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.04, size = 22, normalized size = 0.81 \begin {gather*} -\frac {a\,\ln \left (c\,x^4+a\right )-c\,x^4}{4\,c^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(a + c*x^4),x)

[Out]

-(a*log(a + c*x^4) - c*x^4)/(4*c^2)

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