\(\int \frac {(b x^2+c x^4)^3}{x^{15}} \, dx\) [171]

Optimal result

Integrand size = 17, antiderivative size = 19 \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^{15}} \, dx=-\frac {\left (b+c x^2\right )^4}{8 b x^8} \]

[Out]

-1/8*(c*x^2+b)^4/b/x^8

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1598, 270} \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^{15}} \, dx=-\frac {\left (b+c x^2\right )^4}{8 b x^8} \]

[In]

Int[(b*x^2 + c*x^4)^3/x^15,x]

[Out]

-1/8*(b + c*x^2)^4/(b*x^8)

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 1598

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*} \text {integral}& = \int \frac {\left (b+c x^2\right )^3}{x^9} \, dx \\ & = -\frac {\left (b+c x^2\right )^4}{8 b x^8} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(43\) vs. \(2(19)=38\).

Time = 0.01 (sec) , antiderivative size = 43, normalized size of antiderivative = 2.26 \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^{15}} \, dx=-\frac {b^3}{8 x^8}-\frac {b^2 c}{2 x^6}-\frac {3 b c^2}{4 x^4}-\frac {c^3}{2 x^2} \]

[In]

Integrate[(b*x^2 + c*x^4)^3/x^15,x]

[Out]

-1/8*b^3/x^8 - (b^2*c)/(2*x^6) - (3*b*c^2)/(4*x^4) - c^3/(2*x^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(35\) vs. \(2(17)=34\).

Time = 0.04 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89

method	result	size
gosper	\(-\frac {4 c^{3} x^{6}+6 b \,c^{2} x^{4}+4 b^{2} c \,x^{2}+b^{3}}{8 x^{8}}\)	\(36\)
default	\(-\frac {c^{3}}{2 x^{2}}-\frac {b^{3}}{8 x^{8}}-\frac {b^{2} c}{2 x^{6}}-\frac {3 b \,c^{2}}{4 x^{4}}\)	\(36\)
risch	\(\frac {-\frac {1}{2} c^{3} x^{6}-\frac {3}{4} b \,c^{2} x^{4}-\frac {1}{2} b^{2} c \,x^{2}-\frac {1}{8} b^{3}}{x^{8}}\)	\(37\)
parallelrisch	\(\frac {-4 c^{3} x^{6}-6 b \,c^{2} x^{4}-4 b^{2} c \,x^{2}-b^{3}}{8 x^{8}}\)	\(38\)
norman	\(\frac {-\frac {1}{8} b^{3} x^{6}-\frac {1}{2} c^{3} x^{12}-\frac {3}{4} b \,c^{2} x^{10}-\frac {1}{2} b^{2} c \,x^{8}}{x^{14}}\)	\(40\)

[In]

int((c*x^4+b*x^2)^3/x^15,x,method=_RETURNVERBOSE)

[Out]

-1/8/x^8*(4*c^3*x^6+6*b*c^2*x^4+4*b^2*c*x^2+b^3)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (17) = 34\).

Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.84 \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^{15}} \, dx=-\frac {4 \, c^{3} x^{6} + 6 \, b c^{2} x^{4} + 4 \, b^{2} c x^{2} + b^{3}}{8 \, x^{8}} \]

[In]

integrate((c*x^4+b*x^2)^3/x^15,x, algorithm="fricas")

[Out]

-1/8*(4*c^3*x^6 + 6*b*c^2*x^4 + 4*b^2*c*x^2 + b^3)/x^8

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (15) = 30\).

Time = 0.13 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.95 \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^{15}} \, dx=\frac {- b^{3} - 4 b^{2} c x^{2} - 6 b c^{2} x^{4} - 4 c^{3} x^{6}}{8 x^{8}} \]

[In]

integrate((c*x**4+b*x**2)**3/x**15,x)

[Out]

(-b**3 - 4*b**2*c*x**2 - 6*b*c**2*x**4 - 4*c**3*x**6)/(8*x**8)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (17) = 34\).

Time = 0.20 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.84 \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^{15}} \, dx=-\frac {4 \, c^{3} x^{6} + 6 \, b c^{2} x^{4} + 4 \, b^{2} c x^{2} + b^{3}}{8 \, x^{8}} \]

[In]

integrate((c*x^4+b*x^2)^3/x^15,x, algorithm="maxima")

[Out]

-1/8*(4*c^3*x^6 + 6*b*c^2*x^4 + 4*b^2*c*x^2 + b^3)/x^8

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (17) = 34\).

Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.84 \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^{15}} \, dx=-\frac {4 \, c^{3} x^{6} + 6 \, b c^{2} x^{4} + 4 \, b^{2} c x^{2} + b^{3}}{8 \, x^{8}} \]

[In]

integrate((c*x^4+b*x^2)^3/x^15,x, algorithm="giac")

[Out]

-1/8*(4*c^3*x^6 + 6*b*c^2*x^4 + 4*b^2*c*x^2 + b^3)/x^8

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.95 \[ \int \frac {\left (b x^2+c x^4\right )^3}{x^{15}} \, dx=-\frac {\frac {b^3}{8}+\frac {b^2\,c\,x^2}{2}+\frac {3\,b\,c^2\,x^4}{4}+\frac {c^3\,x^6}{2}}{x^8} \]

[In]

int((b*x^2 + c*x^4)^3/x^15,x)

[Out]

-(b^3/8 + (c^3*x^6)/2 + (b^2*c*x^2)/2 + (3*b*c^2*x^4)/4)/x^8