3.4.83 \(\int \frac {(a+b x^2)^{3/2}}{x^6} \, dx\) [383]

Optimal. Leaf size=21 \[ -\frac {\left (a+b x^2\right )^{5/2}}{5 a x^5} \]

[Out]

-1/5*(b*x^2+a)^(5/2)/a/x^5

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Rubi [A]
time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {270} \begin {gather*} -\frac {\left (a+b x^2\right )^{5/2}}{5 a x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(3/2)/x^6,x]

[Out]

-1/5*(a + b*x^2)^(5/2)/(a*x^5)

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]

Rubi steps

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

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Mathematica [A]
time = 0.06, size = 21, normalized size = 1.00 \begin {gather*} -\frac {\left (a+b x^2\right )^{5/2}}{5 a x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(3/2)/x^6,x]

[Out]

-1/5*(a + b*x^2)^(5/2)/(a*x^5)

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Maple [A]
time = 0.04, size = 18, normalized size = 0.86

method	result	size
gosper	\(-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5 a \,x^{5}}\)	\(18\)
default	\(-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5 a \,x^{5}}\)	\(18\)
trager	\(-\frac {\left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right ) \sqrt {b \,x^{2}+a}}{5 a \,x^{5}}\)	\(36\)
risch	\(-\frac {\left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right ) \sqrt {b \,x^{2}+a}}{5 a \,x^{5}}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*(b*x^2+a)^(5/2)/a/x^5

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Maxima [A]
time = 0.33, size = 17, normalized size = 0.81 \begin {gather*} -\frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{5 \, a x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*(b*x^2 + a)^(5/2)/(a*x^5)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (17) = 34\).
time = 0.85, size = 35, normalized size = 1.67 \begin {gather*} -\frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt {b x^{2} + a}}{5 \, a x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (17) = 34\).
time = 0.45, size = 68, normalized size = 3.24 \begin {gather*} - \frac {a \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{5 x^{4}} - \frac {2 b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{5 x^{2}} - \frac {b^{\frac {5}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{5 a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*sqrt(b)*sqrt(a/(b*x**2) + 1)/(5*x**4) - 2*b**(3/2)*sqrt(a/(b*x**2) + 1)/(5*x**2) - b**(5/2)*sqrt(a/(b*x**2)
+ 1)/(5*a)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (17) = 34\).
time = 0.71, size = 86, normalized size = 4.10 \begin {gather*} \frac {2 \, {\left (5 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} b^{\frac {5}{2}} + 10 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b^{\frac {5}{2}} + a^{4} b^{\frac {5}{2}}\right )}}{5 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 5.29, size = 17, normalized size = 0.81 \begin {gather*} -\frac {{\left (b\,x^2+a\right )}^{5/2}}{5\,a\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^(3/2)/x^6,x)

[Out]

-(a + b*x^2)^(5/2)/(5*a*x^5)

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