3.3.81 \(\int \frac {x^6}{(b x^2+c x^4)^{3/2}} \, dx\) [281]

Optimal. Leaf size=47 \[ -\frac {x^3}{c \sqrt {b x^2+c x^4}}+\frac {2 \sqrt {b x^2+c x^4}}{c^2 x} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2040, 1602} \begin {gather*} \frac {2 \sqrt {b x^2+c x^4}}{c^2 x}-\frac {x^3}{c \sqrt {b x^2+c x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^6/(b*x^2 + c*x^4)^(3/2),x]

[Out]

-(x^3/(c*Sqrt[b*x^2 + c*x^4])) + (2*Sqrt[b*x^2 + c*x^4])/(c^2*x)

Rule 1602

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[Coeff[Pp, x, p]*x^(p - q +
1)*(Qq^(m + 1)/((p + m*q + 1)*Coeff[Qq, x, q])), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rule 2040

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] + Dist[c^j*((m + n*p + n - j + 1)/(a*(n - j)*(p + 1)))
, Int[(c*x)^(m - j)*(a*x^j + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, j, m, n}, x] &&  !IntegerQ[p] && NeQ[n,
j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && LtQ[p, -1] && (IntegerQ[j] || GtQ[c, 0])

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 29, normalized size = 0.62 \begin {gather*} \frac {x \left (2 b+c x^2\right )}{c^2 \sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.11, size = 37, normalized size = 0.79

method	result	size
gosper	\(\frac {\left (c \,x^{2}+b \right ) \left (c \,x^{2}+2 b \right ) x^{3}}{c^{2} \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}\)	\(37\)
default	\(\frac {\left (c \,x^{2}+b \right ) \left (c \,x^{2}+2 b \right ) x^{3}}{c^{2} \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}\)	\(37\)
trager	\(\frac {\left (c \,x^{2}+2 b \right ) \sqrt {c \,x^{4}+b \,x^{2}}}{\left (c \,x^{2}+b \right ) c^{2} x}\)	\(39\)
risch	\(\frac {\left (c \,x^{2}+b \right ) x}{c^{2} \sqrt {x^{2} \left (c \,x^{2}+b \right )}}+\frac {b x}{c^{2} \sqrt {x^{2} \left (c \,x^{2}+b \right )}}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(c*x^4+b*x^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.30, size = 22, normalized size = 0.47 \begin {gather*} \frac {c x^{2} + 2 \, b}{\sqrt {c x^{2} + b} c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(c*x^4+b*x^2)^(3/2),x, algorithm="maxima")

[Out]

(c*x^2 + 2*b)/(sqrt(c*x^2 + b)*c^2)

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Fricas [A]
time = 0.38, size = 39, normalized size = 0.83 \begin {gather*} \frac {\sqrt {c x^{4} + b x^{2}} {\left (c x^{2} + 2 \, b\right )}}{c^{3} x^{3} + b c^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(c*x^4+b*x^2)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6}}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**6/(x**2*(b + c*x**2))**(3/2), x)

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Giac [A]
time = 4.13, size = 51, normalized size = 1.09 \begin {gather*} \frac {\frac {\sqrt {c x^{2} + b}}{c \mathrm {sgn}\left (x\right )} + \frac {b}{\sqrt {c x^{2} + b} c \mathrm {sgn}\left (x\right )}}{c} - \frac {2 \, \sqrt {b} \mathrm {sgn}\left (x\right )}{c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(c*x^4+b*x^2)^(3/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 4.23, size = 38, normalized size = 0.81 \begin {gather*} \frac {\sqrt {c\,x^4+b\,x^2}\,\left (c\,x^2+2\,b\right )}{c^2\,x\,\left (c\,x^2+b\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(b*x^2 + c*x^4)^(3/2),x)

[Out]

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

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