3.3.26 \(\int \frac {\sqrt {b x^2+c x^4}}{x^5} \, dx\) [226]

Optimal. Leaf size=25 \[ -\frac {\left (b x^2+c x^4\right )^{3/2}}{3 b x^6} \]

[Out]

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

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Rubi [A]
time = 0.02, 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 )^{3/2}}{3 b x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*x^2 + c*x^4]/x^5,x]

[Out]

-1/3*(b*x^2 + c*x^4)^(3/2)/(b*x^6)

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 {\sqrt {b x^2+c x^4}}{x^5} \, dx &=-\frac {\left (b x^2+c x^4\right )^{3/2}}{3 b x^6}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 25, normalized size = 1.00 \begin {gather*} -\frac {\left (x^2 \left (b+c x^2\right )\right )^{3/2}}{3 b x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*x^2 + c*x^4]/x^5,x]

[Out]

-1/3*(x^2*(b + c*x^2))^(3/2)/(b*x^6)

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Maple [A]
time = 0.09, size = 29, normalized size = 1.16

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)^(1/2)/x^5,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 41, normalized size = 1.64 \begin {gather*} -\frac {\sqrt {c x^{4} + b x^{2}} c}{3 \, b x^{2}} - \frac {\sqrt {c x^{4} + b x^{2}}}{3 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^(1/2)/x^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.32, size = 28, normalized size = 1.12 \begin {gather*} -\frac {\sqrt {c x^{4} + b x^{2}} {\left (c x^{2} + b\right )}}{3 \, b x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^(1/2)/x^5,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2} \left (b + c x^{2}\right )}}{x^{5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)**(1/2)/x**5,x)

[Out]

Integral(sqrt(x**2*(b + c*x**2))/x**5, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (21) = 42\).
time = 20.00, size = 63, normalized size = 2.52 \begin {gather*} \frac {2 \, {\left (3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} c^{\frac {3}{2}} \mathrm {sgn}\left (x\right ) + b^{2} c^{\frac {3}{2}} \mathrm {sgn}\left (x\right )\right )}}{3 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^(1/2)/x^5,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 4.15, size = 28, normalized size = 1.12 \begin {gather*} -\frac {\left (c\,x^2+b\right )\,\sqrt {c\,x^4+b\,x^2}}{3\,b\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2 + c*x^4)^(1/2)/x^5,x)

[Out]

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

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