3.1.59 \(\int \frac {x^7}{b x^2+c x^4} \, dx\)

Optimal. Leaf size=40 \[ \frac {b^2 \log \left (b+c x^2\right )}{2 c^3}-\frac {b x^2}{2 c^2}+\frac {x^4}{4 c} \]

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Rubi [A]  time = 0.03, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1584, 266, 43} \begin {gather*} \frac {b^2 \log \left (b+c x^2\right )}{2 c^3}-\frac {b x^2}{2 c^2}+\frac {x^4}{4 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^7/(b*x^2 + c*x^4),x]

[Out]

-(b*x^2)/(2*c^2) + x^4/(4*c) + (b^2*Log[b + c*x^2])/(2*c^3)

Rule 43

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 266

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]]

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

\begin {align*} \int \frac {x^7}{b x^2+c x^4} \, dx &=\int \frac {x^5}{b+c x^2} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{b+c x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {b}{c^2}+\frac {x}{c}+\frac {b^2}{c^2 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {b x^2}{2 c^2}+\frac {x^4}{4 c}+\frac {b^2 \log \left (b+c x^2\right )}{2 c^3}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 40, normalized size = 1.00 \begin {gather*} \frac {b^2 \log \left (b+c x^2\right )}{2 c^3}-\frac {b x^2}{2 c^2}+\frac {x^4}{4 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7/(b*x^2 + c*x^4),x]

[Out]

-1/2*(b*x^2)/c^2 + x^4/(4*c) + (b^2*Log[b + c*x^2])/(2*c^3)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^7}{b x^2+c x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^7/(b*x^2 + c*x^4),x]

[Out]

IntegrateAlgebraic[x^7/(b*x^2 + c*x^4), x]

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fricas [A]  time = 0.57, size = 33, normalized size = 0.82 \begin {gather*} \frac {c^{2} x^{4} - 2 \, b c x^{2} + 2 \, b^{2} \log \left (c x^{2} + b\right )}{4 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(c*x^4+b*x^2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.15, size = 35, normalized size = 0.88 \begin {gather*} \frac {b^{2} \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{3}} + \frac {c x^{4} - 2 \, b x^{2}}{4 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(c*x^4+b*x^2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.00, size = 35, normalized size = 0.88 \begin {gather*} \frac {x^{4}}{4 c}-\frac {b \,x^{2}}{2 c^{2}}+\frac {b^{2} \ln \left (c \,x^{2}+b \right )}{2 c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(c*x^4+b*x^2),x)

[Out]

-1/2*b*x^2/c^2+1/4*x^4/c+1/2*b^2*ln(c*x^2+b)/c^3

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maxima [A]  time = 1.33, size = 34, normalized size = 0.85 \begin {gather*} \frac {b^{2} \log \left (c x^{2} + b\right )}{2 \, c^{3}} + \frac {c x^{4} - 2 \, b x^{2}}{4 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(c*x^4+b*x^2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.05, size = 33, normalized size = 0.82 \begin {gather*} \frac {2\,b^2\,\ln \left (c\,x^2+b\right )+c^2\,x^4-2\,b\,c\,x^2}{4\,c^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(b*x^2 + c*x^4),x)

[Out]

(2*b^2*log(b + c*x^2) + c^2*x^4 - 2*b*c*x^2)/(4*c^3)

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sympy [A]  time = 0.17, size = 32, normalized size = 0.80 \begin {gather*} \frac {b^{2} \log {\left (b + c x^{2} \right )}}{2 c^{3}} - \frac {b x^{2}}{2 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+b*x**2),x)

[Out]

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

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