Optimal. Leaf size=144 \[ -\frac {b^3 \left (c x^2\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{8 a^{5/2} x^3}+\frac {b^2 c \sqrt {a+b \sqrt {c x^2}}}{8 a^2 x}-\frac {b \left (c x^2\right )^{3/2} \sqrt {a+b \sqrt {c x^2}}}{12 a c x^5}-\frac {\sqrt {a+b \sqrt {c x^2}}}{3 x^3} \]
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Rubi [A] time = 0.06, 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 = {368, 47, 51, 63, 208} \[ \frac {b^2 c \sqrt {a+b \sqrt {c x^2}}}{8 a^2 x}-\frac {b^3 \left (c x^2\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{8 a^{5/2} x^3}-\frac {b \left (c x^2\right )^{3/2} \sqrt {a+b \sqrt {c x^2}}}{12 a c x^5}-\frac {\sqrt {a+b \sqrt {c x^2}}}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 208
Rule 368
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^4} \, dx &=\frac {\left (c x^2\right )^{3/2} \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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] time = 0.01, size = 63, normalized size = 0.44 \[ \frac {2 b^3 \left (c x^2\right )^{3/2} \left (a+b \sqrt {c x^2}\right )^{3/2} \, _2F_1\left (\frac {3}{2},4;\frac {5}{2};\frac {\sqrt {c x^2} b}{a}+1\right )}{3 a^4 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 252, normalized size = 1.75 \[ \left [\frac {3 \, b^{3} c x^{3} \sqrt {\frac {c}{a}} \log \left (\frac {b c x^{2} - 2 \, \sqrt {\sqrt {c x^{2}} b + a} a x \sqrt {\frac {c}{a}} + 2 \, \sqrt {c x^{2}} a}{x^{2}}\right ) + 2 \, {\left (3 \, b^{2} c x^{2} - 2 \, \sqrt {c x^{2}} a b - 8 \, a^{2}\right )} \sqrt {\sqrt {c x^{2}} b + a}}{48 \, a^{2} x^{3}}, -\frac {3 \, b^{3} c x^{3} \sqrt {-\frac {c}{a}} \arctan \left (-\frac {{\left (a b c x^{2} \sqrt {-\frac {c}{a}} - \sqrt {c x^{2}} a^{2} \sqrt {-\frac {c}{a}}\right )} \sqrt {\sqrt {c x^{2}} b + a}}{b^{2} c^{2} x^{3} - a^{2} c x}\right ) - {\left (3 \, b^{2} c x^{2} - 2 \, \sqrt {c x^{2}} a b - 8 \, a^{2}\right )} \sqrt {\sqrt {c x^{2}} b + a}}{24 \, a^{2} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 114, normalized size = 0.79 \[ \frac {\frac {3 \, b^{4} c^{2} \arctan \left (\frac {\sqrt {b \sqrt {c} x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {3 \, {\left (b \sqrt {c} x + a\right )}^{\frac {5}{2}} b^{4} c^{2} - 8 \, {\left (b \sqrt {c} x + a\right )}^{\frac {3}{2}} a b^{4} c^{2} - 3 \, \sqrt {b \sqrt {c} x + a} a^{2} b^{4} c^{2}}{a^{2} b^{3} c^{\frac {3}{2}} x^{3}}}{24 \, b \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 97, normalized size = 0.67 \[ -\frac {3 \left (c \,x^{2}\right )^{\frac {3}{2}} a^{2} b^{3} \arctanh \left (\frac {\sqrt {a +\sqrt {c \,x^{2}}\, b}}{\sqrt {a}}\right )+3 \sqrt {a +\sqrt {c \,x^{2}}\, b}\, a^{\frac {9}{2}}+8 \left (a +\sqrt {c \,x^{2}}\, b \right )^{\frac {3}{2}} a^{\frac {7}{2}}-3 \left (a +\sqrt {c \,x^{2}}\, b \right )^{\frac {5}{2}} a^{\frac {5}{2}}}{24 a^{\frac {9}{2}} x^{3}} \]
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 \[ \int \frac {\sqrt {\sqrt {c x^{2}} b + a}}{x^{4}}\,{d x} \]
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 \[ \int \frac {\sqrt {a+b\,\sqrt {c\,x^2}}}{x^4} \,d x \]
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 \[ \int \frac {\sqrt {a + b \sqrt {c x^{2}}}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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