Optimal. Leaf size=144 \[ -\frac {\sqrt {a+b \sqrt {c x^2}}}{3 x^3}+\frac {b^2 c \sqrt {a+b \sqrt {c x^2}}}{8 a^2 x}-\frac {b \left (c x^2\right )^{3/2} \sqrt {a+b \sqrt {c x^2}}}{12 a c x^5}-\frac {b^3 \left (c x^2\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{8 a^{5/2} x^3} \]
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Rubi [A]
time = 0.04, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {375, 43, 44, 65,
214} \begin {gather*} -\frac {b^3 \left (c x^2\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{8 a^{5/2} x^3}+\frac {b^2 c \sqrt {a+b \sqrt {c x^2}}}{8 a^2 x}-\frac {b \left (c x^2\right )^{3/2} \sqrt {a+b \sqrt {c x^2}}}{12 a c x^5}-\frac {\sqrt {a+b \sqrt {c x^2}}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 44
Rule 65
Rule 214
Rule 375
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^4} \, dx &=\frac {\left (c x^2\right )^{3/2} \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x^4} \, dx,x,\sqrt {c x^2}\right )}{x^3}\\ &=-\frac {\sqrt {a+b \sqrt {c x^2}}}{3 x^3}+\frac {\left (b \left (c x^2\right )^{3/2}\right ) \text {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x}} \, dx,x,\sqrt {c x^2}\right )}{6 x^3}\\ &=-\frac {\sqrt {a+b \sqrt {c x^2}}}{3 x^3}-\frac {b \left (c x^2\right )^{3/2} \sqrt {a+b \sqrt {c x^2}}}{12 a c x^5}-\frac {\left (b^2 \left (c x^2\right )^{3/2}\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,\sqrt {c x^2}\right )}{8 a x^3}\\ &=-\frac {\sqrt {a+b \sqrt {c x^2}}}{3 x^3}+\frac {b^2 c \sqrt {a+b \sqrt {c x^2}}}{8 a^2 x}-\frac {b \left (c x^2\right )^{3/2} \sqrt {a+b \sqrt {c x^2}}}{12 a c x^5}+\frac {\left (b^3 \left (c x^2\right )^{3/2}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sqrt {c x^2}\right )}{16 a^2 x^3}\\ &=-\frac {\sqrt {a+b \sqrt {c x^2}}}{3 x^3}+\frac {b^2 c \sqrt {a+b \sqrt {c x^2}}}{8 a^2 x}-\frac {b \left (c x^2\right )^{3/2} \sqrt {a+b \sqrt {c x^2}}}{12 a c x^5}+\frac {\left (b^2 \left (c x^2\right )^{3/2}\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sqrt {c x^2}}\right )}{8 a^2 x^3}\\ &=-\frac {\sqrt {a+b \sqrt {c x^2}}}{3 x^3}+\frac {b^2 c \sqrt {a+b \sqrt {c x^2}}}{8 a^2 x}-\frac {b \left (c x^2\right )^{3/2} \sqrt {a+b \sqrt {c x^2}}}{12 a c x^5}-\frac {b^3 \left (c x^2\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{8 a^{5/2} x^3}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.96, size = 63, normalized size = 0.44 \begin {gather*} \frac {2 b^3 \left (c x^2\right )^{3/2} \left (a+b \sqrt {c x^2}\right )^{3/2} \, _2F_1\left (\frac {3}{2},4;\frac {5}{2};1+\frac {b \sqrt {c x^2}}{a}\right )}{3 a^4 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 97, normalized size = 0.67
method | result | size |
default | \(-\frac {3 a^{\frac {9}{2}} \sqrt {a +b \sqrt {c \,x^{2}}}+8 a^{\frac {7}{2}} \left (a +b \sqrt {c \,x^{2}}\right )^{\frac {3}{2}}-3 a^{\frac {5}{2}} \left (a +b \sqrt {c \,x^{2}}\right )^{\frac {5}{2}}+3 \arctanh \left (\frac {\sqrt {a +b \sqrt {c \,x^{2}}}}{\sqrt {a}}\right ) a^{2} b^{3} \left (c \,x^{2}\right )^{\frac {3}{2}}}{24 x^{3} a^{\frac {9}{2}}}\) | \(97\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 252, normalized size = 1.75 \begin {gather*} \left [\frac {3 \, b^{3} c x^{3} \sqrt {\frac {c}{a}} \log \left (\frac {b c x^{2} - 2 \, \sqrt {\sqrt {c x^{2}} b + a} a x \sqrt {\frac {c}{a}} + 2 \, \sqrt {c x^{2}} a}{x^{2}}\right ) + 2 \, {\left (3 \, b^{2} c x^{2} - 2 \, \sqrt {c x^{2}} a b - 8 \, a^{2}\right )} \sqrt {\sqrt {c x^{2}} b + a}}{48 \, a^{2} x^{3}}, -\frac {3 \, b^{3} c x^{3} \sqrt {-\frac {c}{a}} \arctan \left (-\frac {{\left (a b c x^{2} \sqrt {-\frac {c}{a}} - \sqrt {c x^{2}} a^{2} \sqrt {-\frac {c}{a}}\right )} \sqrt {\sqrt {c x^{2}} b + a}}{b^{2} c^{2} x^{3} - a^{2} c x}\right ) - {\left (3 \, b^{2} c x^{2} - 2 \, \sqrt {c x^{2}} a b - 8 \, a^{2}\right )} \sqrt {\sqrt {c x^{2}} b + a}}{24 \, a^{2} x^{3}}\right ] \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {a + b \sqrt {c x^{2}}}}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.22, size = 114, normalized size = 0.79 \begin {gather*} \frac {\frac {3 \, b^{4} c^{2} \arctan \left (\frac {\sqrt {b \sqrt {c} x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {3 \, {\left (b \sqrt {c} x + a\right )}^{\frac {5}{2}} b^{4} c^{2} - 8 \, {\left (b \sqrt {c} x + a\right )}^{\frac {3}{2}} a b^{4} c^{2} - 3 \, \sqrt {b \sqrt {c} x + a} a^{2} b^{4} c^{2}}{a^{2} b^{3} c^{\frac {3}{2}} x^{3}}}{24 \, b \sqrt {c}} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {a+b\,\sqrt {c\,x^2}}}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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