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

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

[Out]

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

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Rubi [A]  time = 0.0211402, antiderivative size = 27, normalized size of antiderivative = 1., 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} \[ -\frac{b^2}{2 x^2}+2 b c \log (x)+\frac{c^2 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^2/x^7,x]

[Out]

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

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]

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 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])

Rubi steps

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

Mathematica [A]  time = 0.0009015, size = 27, normalized size = 1. \[ -\frac{b^2}{2 x^2}+2 b c \log (x)+\frac{c^2 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^2/x^7,x]

[Out]

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

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Maple [A]  time = 0.048, size = 24, normalized size = 0.9 \begin{align*} -{\frac{{b}^{2}}{2\,{x}^{2}}}+{\frac{{c}^{2}{x}^{2}}{2}}+2\,bc\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.992043, size = 32, normalized size = 1.19 \begin{align*} \frac{1}{2} \, c^{2} x^{2} + b c \log \left (x^{2}\right ) - \frac{b^{2}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.24135, size = 59, normalized size = 2.19 \begin{align*} \frac{c^{2} x^{4} + 4 \, b c x^{2} \log \left (x\right ) - b^{2}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.321896, size = 24, normalized size = 0.89 \begin{align*} - \frac{b^{2}}{2 x^{2}} + 2 b c \log{\left (x \right )} + \frac{c^{2} x^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.23333, size = 43, normalized size = 1.59 \begin{align*} \frac{1}{2} \, c^{2} x^{2} + b c \log \left (x^{2}\right ) - \frac{2 \, b c x^{2} + b^{2}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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