3.163 \(\int \frac{(b x^2+c x^4)^3}{x^7} \, dx\)

Optimal. Leaf size=39 \[ \frac{3}{2} b^2 c x^2+b^3 \log (x)+\frac{3}{4} b c^2 x^4+\frac{c^3 x^6}{6} \]

[Out]

(3*b^2*c*x^2)/2 + (3*b*c^2*x^4)/4 + (c^3*x^6)/6 + b^3*Log[x]

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Rubi [A]  time = 0.0259172, antiderivative size = 39, 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} \[ \frac{3}{2} b^2 c x^2+b^3 \log (x)+\frac{3}{4} b c^2 x^4+\frac{c^3 x^6}{6} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^3/x^7,x]

[Out]

(3*b^2*c*x^2)/2 + (3*b*c^2*x^4)/4 + (c^3*x^6)/6 + b^3*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 )^3}{x^7} \, dx &=\int \frac{\left (b+c x^2\right )^3}{x} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(b+c x)^3}{x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (3 b^2 c+\frac{b^3}{x}+3 b c^2 x+c^3 x^2\right ) \, dx,x,x^2\right )\\ &=\frac{3}{2} b^2 c x^2+\frac{3}{4} b c^2 x^4+\frac{c^3 x^6}{6}+b^3 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0044091, size = 39, normalized size = 1. \[ \frac{3}{2} b^2 c x^2+b^3 \log (x)+\frac{3}{4} b c^2 x^4+\frac{c^3 x^6}{6} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^3/x^7,x]

[Out]

(3*b^2*c*x^2)/2 + (3*b*c^2*x^4)/4 + (c^3*x^6)/6 + b^3*Log[x]

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Maple [A]  time = 0.043, size = 34, normalized size = 0.9 \begin{align*}{\frac{3\,{b}^{2}c{x}^{2}}{2}}+{\frac{3\,b{c}^{2}{x}^{4}}{4}}+{\frac{{c}^{3}{x}^{6}}{6}}+{b}^{3}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)^3/x^7,x)

[Out]

3/2*b^2*c*x^2+3/4*b*c^2*x^4+1/6*c^3*x^6+b^3*ln(x)

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Maxima [A]  time = 0.98711, size = 49, normalized size = 1.26 \begin{align*} \frac{1}{6} \, c^{3} x^{6} + \frac{3}{4} \, b c^{2} x^{4} + \frac{3}{2} \, b^{2} c x^{2} + \frac{1}{2} \, b^{3} \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^7,x, algorithm="maxima")

[Out]

1/6*c^3*x^6 + 3/4*b*c^2*x^4 + 3/2*b^2*c*x^2 + 1/2*b^3*log(x^2)

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Fricas [A]  time = 1.43224, size = 78, normalized size = 2. \begin{align*} \frac{1}{6} \, c^{3} x^{6} + \frac{3}{4} \, b c^{2} x^{4} + \frac{3}{2} \, b^{2} c x^{2} + b^{3} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^7,x, algorithm="fricas")

[Out]

1/6*c^3*x^6 + 3/4*b*c^2*x^4 + 3/2*b^2*c*x^2 + b^3*log(x)

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Sympy [A]  time = 0.298766, size = 37, normalized size = 0.95 \begin{align*} b^{3} \log{\left (x \right )} + \frac{3 b^{2} c x^{2}}{2} + \frac{3 b c^{2} x^{4}}{4} + \frac{c^{3} x^{6}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)**3/x**7,x)

[Out]

b**3*log(x) + 3*b**2*c*x**2/2 + 3*b*c**2*x**4/4 + c**3*x**6/6

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Giac [A]  time = 1.28543, size = 49, normalized size = 1.26 \begin{align*} \frac{1}{6} \, c^{3} x^{6} + \frac{3}{4} \, b c^{2} x^{4} + \frac{3}{2} \, b^{2} c x^{2} + \frac{1}{2} \, b^{3} \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^7,x, algorithm="giac")

[Out]

1/6*c^3*x^6 + 3/4*b*c^2*x^4 + 3/2*b^2*c*x^2 + 1/2*b^3*log(x^2)