3.5.3 \(\int \frac {(a+b x^2)^{5/2}}{x^8} \, dx\) [403]

Optimal. Leaf size=21 \[ -\frac {\left (a+b x^2\right )^{7/2}}{7 a x^7} \]

[Out]

-1/7*(b*x^2+a)^(7/2)/a/x^7

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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 )^{7/2}}{7 a x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(5/2)/x^8,x]

[Out]

-1/7*(a + b*x^2)^(7/2)/(a*x^7)

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 )^{5/2}}{x^8} \, dx &=-\frac {\left (a+b x^2\right )^{7/2}}{7 a x^7}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 21, normalized size = 1.00 \begin {gather*} -\frac {\left (a+b x^2\right )^{7/2}}{7 a x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(5/2)/x^8,x]

[Out]

-1/7*(a + b*x^2)^(7/2)/(a*x^7)

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Maple [A]
time = 0.07, size = 18, normalized size = 0.86

method	result	size
gosper	\(-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{7 a \,x^{7}}\)	\(18\)
default	\(-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{7 a \,x^{7}}\)	\(18\)
trager	\(-\frac {\left (b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}\right ) \sqrt {b \,x^{2}+a}}{7 a \,x^{7}}\)	\(47\)
risch	\(-\frac {\left (b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}\right ) \sqrt {b \,x^{2}+a}}{7 a \,x^{7}}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(5/2)/x^8,x,method=_RETURNVERBOSE)

[Out]

-1/7*(b*x^2+a)^(7/2)/a/x^7

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Maxima [A]
time = 0.28, size = 17, normalized size = 0.81 \begin {gather*} -\frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}}}{7 \, a x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)/x^8,x, algorithm="maxima")

[Out]

-1/7*(b*x^2 + a)^(7/2)/(a*x^7)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (17) = 34\).
time = 0.97, size = 46, normalized size = 2.19 \begin {gather*} -\frac {{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \sqrt {b x^{2} + a}}{7 \, a x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)/x^8,x, algorithm="fricas")

[Out]

-1/7*(b^3*x^6 + 3*a*b^2*x^4 + 3*a^2*b*x^2 + a^3)*sqrt(b*x^2 + a)/(a*x^7)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (17) = 34\).
time = 0.62, size = 95, normalized size = 4.52 \begin {gather*} - \frac {a^{2} \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{7 x^{6}} - \frac {3 a b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{7 x^{4}} - \frac {3 b^{\frac {5}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{7 x^{2}} - \frac {b^{\frac {7}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{7 a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(5/2)/x**8,x)

[Out]

-a**2*sqrt(b)*sqrt(a/(b*x**2) + 1)/(7*x**6) - 3*a*b**(3/2)*sqrt(a/(b*x**2) + 1)/(7*x**4) - 3*b**(5/2)*sqrt(a/(
b*x**2) + 1)/(7*x**2) - b**(7/2)*sqrt(a/(b*x**2) + 1)/(7*a)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (17) = 34\).
time = 1.38, size = 113, normalized size = 5.38 \begin {gather*} \frac {2 \, {\left (7 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} b^{\frac {7}{2}} + 35 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{2} b^{\frac {7}{2}} + 21 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{4} b^{\frac {7}{2}} + a^{6} b^{\frac {7}{2}}\right )}}{7 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)/x^8,x, algorithm="giac")

[Out]

2/7*(7*(sqrt(b)*x - sqrt(b*x^2 + a))^12*b^(7/2) + 35*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^2*b^(7/2) + 21*(sqrt(b)
*x - sqrt(b*x^2 + a))^4*a^4*b^(7/2) + a^6*b^(7/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^7

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Mupad [B]
time = 5.07, size = 71, normalized size = 3.38 \begin {gather*} -\frac {a^2\,\sqrt {b\,x^2+a}}{7\,x^7}-\frac {3\,b^2\,\sqrt {b\,x^2+a}}{7\,x^3}-\frac {b^3\,\sqrt {b\,x^2+a}}{7\,a\,x}-\frac {3\,a\,b\,\sqrt {b\,x^2+a}}{7\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^(5/2)/x^8,x)

[Out]

- (a^2*(a + b*x^2)^(1/2))/(7*x^7) - (3*b^2*(a + b*x^2)^(1/2))/(7*x^3) - (b^3*(a + b*x^2)^(1/2))/(7*a*x) - (3*a
*b*(a + b*x^2)^(1/2))/(7*x^5)

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