Optimal. Leaf size=71 \[ -\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{3 x^3}-\frac {b \left (c x^2\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{\sqrt {a}}\right )}{3 \sqrt {a} x^3} \]
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Rubi [A] time = 0.06, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {368, 266, 47, 63, 208} \[ -\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{3 x^3}-\frac {b \left (c x^2\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{\sqrt {a}}\right )}{3 \sqrt {a} x^3} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rule 266
Rule 368
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x^4} \, dx &=\frac {\left (c x^2\right )^{3/2} \operatorname {Subst}\left (\int \frac {\sqrt {a+b x^3}}{x^4} \, dx,x,\sqrt {c x^2}\right )}{x^3}\\ &=\frac {\left (c x^2\right )^{3/2} \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2} \, dx,x,\left (c x^2\right )^{3/2}\right )}{3 x^3}\\ &=-\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{3 x^3}+\frac {\left (b \left (c x^2\right )^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\left (c x^2\right )^{3/2}\right )}{6 x^3}\\ &=-\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{3 x^3}+\frac {\left (c x^2\right )^{3/2} \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \left (c x^2\right )^{3/2}}\right )}{3 x^3}\\ &=-\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{3 x^3}-\frac {b \left (c x^2\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{\sqrt {a}}\right )}{3 \sqrt {a} x^3}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 93, normalized size = 1.31 \[ \frac {-b \left (c x^2\right )^{3/2} \sqrt {\frac {b \left (c x^2\right )^{3/2}}{a}+1} \tanh ^{-1}\left (\sqrt {\frac {b \left (c x^2\right )^{3/2}}{a}+1}\right )-a-b \left (c x^2\right )^{3/2}}{3 x^3 \sqrt {a+b \left (c x^2\right )^{3/2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 206, normalized size = 2.90 \[ \left [\frac {b c x^{3} \sqrt {\frac {c}{a}} \log \left (\frac {b c^{2} x^{4} - 2 \, \sqrt {\sqrt {c x^{2}} b c x^{2} + a} a x \sqrt {\frac {c}{a}} + 2 \, \sqrt {c x^{2}} a}{x^{4}}\right ) - 2 \, \sqrt {\sqrt {c x^{2}} b c x^{2} + a}}{6 \, x^{3}}, -\frac {b c x^{3} \sqrt {-\frac {c}{a}} \arctan \left (-\frac {{\left (a b c^{2} x^{4} \sqrt {-\frac {c}{a}} - \sqrt {c x^{2}} a^{2} \sqrt {-\frac {c}{a}}\right )} \sqrt {\sqrt {c x^{2}} b c x^{2} + a}}{b^{2} c^{4} x^{7} - a^{2} c x}\right ) + \sqrt {\sqrt {c x^{2}} b c x^{2} + a}}{3 \, x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 61, normalized size = 0.86 \[ \frac {\frac {b^{2} c^{3} \arctan \left (\frac {\sqrt {b c^{\frac {3}{2}} x^{3} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {\sqrt {b c^{\frac {3}{2}} x^{3} + a} b c^{\frac {3}{2}}}{x^{3}}}{3 \, b c^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a +\left (c \,x^{2}\right )^{\frac {3}{2}} b}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\left (c x^{2}\right )^{\frac {3}{2}} b + a}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {a+b\,{\left (c\,x^2\right )}^{3/2}}}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a + b \left (c x^{2}\right )^{\frac {3}{2}}}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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