3.3.44 \(\int \frac {(b x^2+c x^4)^{3/2}}{x^9} \, dx\) [244]

Optimal. Leaf size=25 \[ -\frac {\left (b x^2+c x^4\right )^{5/2}}{5 b x^{10}} \]

[Out]

-1/5*(c*x^4+b*x^2)^(5/2)/b/x^10

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2039} \begin {gather*} -\frac {\left (b x^2+c x^4\right )^{5/2}}{5 b x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^(3/2)/x^9,x]

[Out]

-1/5*(b*x^2 + c*x^4)^(5/2)/(b*x^10)

Rule 2039

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

Rubi steps

\begin {align*} \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^9} \, dx &=-\frac {\left (b x^2+c x^4\right )^{5/2}}{5 b x^{10}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.06, size = 25, normalized size = 1.00 \begin {gather*} -\frac {\left (x^2 \left (b+c x^2\right )\right )^{5/2}}{5 b x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^(3/2)/x^9,x]

[Out]

-1/5*(x^2*(b + c*x^2))^(5/2)/(b*x^10)

________________________________________________________________________________________

Maple [A]
time = 0.10, size = 29, normalized size = 1.16

method	result	size
gosper	\(-\frac {\left (c \,x^{2}+b \right ) \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}{5 x^{8} b}\)	\(29\)
default	\(-\frac {\left (c \,x^{2}+b \right ) \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}{5 x^{8} b}\)	\(29\)
trager	\(-\frac {\left (c^{2} x^{4}+2 b c \,x^{2}+b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}}}{5 b \,x^{6}}\)	\(40\)
risch	\(-\frac {\sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \left (c^{2} x^{4}+2 b c \,x^{2}+b^{2}\right )}{5 x^{6} b}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)^(3/2)/x^9,x,method=_RETURNVERBOSE)

[Out]

-1/5/x^8*(c*x^2+b)/b*(c*x^4+b*x^2)^(3/2)

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (21) = 42\).
time = 0.29, size = 81, normalized size = 3.24 \begin {gather*} -\frac {\sqrt {c x^{4} + b x^{2}} c^{2}}{5 \, b x^{2}} + \frac {\sqrt {c x^{4} + b x^{2}} c}{10 \, x^{4}} + \frac {3 \, \sqrt {c x^{4} + b x^{2}} b}{10 \, x^{6}} - \frac {{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}{2 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^(3/2)/x^9,x, algorithm="maxima")

[Out]

-1/5*sqrt(c*x^4 + b*x^2)*c^2/(b*x^2) + 1/10*sqrt(c*x^4 + b*x^2)*c/x^4 + 3/10*sqrt(c*x^4 + b*x^2)*b/x^6 - 1/2*(
c*x^4 + b*x^2)^(3/2)/x^8

________________________________________________________________________________________

Fricas [A]
time = 0.32, size = 39, normalized size = 1.56 \begin {gather*} -\frac {{\left (c^{2} x^{4} + 2 \, b c x^{2} + b^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{5 \, b x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^(3/2)/x^9,x, algorithm="fricas")

[Out]

-1/5*(c^2*x^4 + 2*b*c*x^2 + b^2)*sqrt(c*x^4 + b*x^2)/(b*x^6)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}{x^{9}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)**(3/2)/x**9,x)

[Out]

Integral((x**2*(b + c*x**2))**(3/2)/x**9, x)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (21) = 42\).
time = 4.34, size = 92, normalized size = 3.68 \begin {gather*} \frac {2 \, {\left (5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{8} c^{\frac {5}{2}} \mathrm {sgn}\left (x\right ) + 10 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} b^{2} c^{\frac {5}{2}} \mathrm {sgn}\left (x\right ) + b^{4} c^{\frac {5}{2}} \mathrm {sgn}\left (x\right )\right )}}{5 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^(3/2)/x^9,x, algorithm="giac")

[Out]

2/5*(5*(sqrt(c)*x - sqrt(c*x^2 + b))^8*c^(5/2)*sgn(x) + 10*(sqrt(c)*x - sqrt(c*x^2 + b))^4*b^2*c^(5/2)*sgn(x)
+ b^4*c^(5/2)*sgn(x))/((sqrt(c)*x - sqrt(c*x^2 + b))^2 - b)^5

________________________________________________________________________________________

Mupad [B]
time = 4.38, size = 30, normalized size = 1.20 \begin {gather*} -\frac {{\left (c\,x^2+b\right )}^2\,\sqrt {c\,x^4+b\,x^2}}{5\,b\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2 + c*x^4)^(3/2)/x^9,x)

[Out]

-((b + c*x^2)^2*(b*x^2 + c*x^4)^(1/2))/(5*b*x^6)

________________________________________________________________________________________