Optimal. Leaf size=104 \[ -\frac {3 b c \sqrt {\frac {c}{a+b x^2}}}{2 a^2}+\frac {3 b c \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c}{a+b x^2}} \tanh ^{-1}\left (\sqrt {\frac {b x^2}{a}+1}\right )}{2 a^2}-\frac {c \sqrt {\frac {c}{a+b x^2}}}{2 a x^2} \]
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Rubi [A] time = 0.15, antiderivative size = 112, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6720, 266, 51, 63, 208} \[ -\frac {3 c \left (a+b x^2\right ) \sqrt {\frac {c}{a+b x^2}}}{2 a^2 x^2}+\frac {3 b c \sqrt {a+b x^2} \sqrt {\frac {c}{a+b x^2}} \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{5/2}}+\frac {c \sqrt {\frac {c}{a+b x^2}}}{a x^2} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 266
Rule 6720
Rubi steps
\begin {align*} \int \frac {\left (\frac {c}{a+b x^2}\right )^{3/2}}{x^3} \, dx &=\left (c \sqrt {\frac {c}{a+b x^2}} \sqrt {a+b x^2}\right ) \int \frac {1}{x^3 \left (a+b x^2\right )^{3/2}} \, dx\\ &=\frac {1}{2} \left (c \sqrt {\frac {c}{a+b x^2}} \sqrt {a+b x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)^{3/2}} \, dx,x,x^2\right )\\ &=\frac {c \sqrt {\frac {c}{a+b x^2}}}{a x^2}+\frac {\left (3 c \sqrt {\frac {c}{a+b x^2}} \sqrt {a+b x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,x^2\right )}{2 a}\\ &=\frac {c \sqrt {\frac {c}{a+b x^2}}}{a x^2}-\frac {3 c \sqrt {\frac {c}{a+b x^2}} \left (a+b x^2\right )}{2 a^2 x^2}-\frac {\left (3 b c \sqrt {\frac {c}{a+b x^2}} \sqrt {a+b x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{4 a^2}\\ &=\frac {c \sqrt {\frac {c}{a+b x^2}}}{a x^2}-\frac {3 c \sqrt {\frac {c}{a+b x^2}} \left (a+b x^2\right )}{2 a^2 x^2}-\frac {\left (3 c \sqrt {\frac {c}{a+b x^2}} \sqrt {a+b x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{2 a^2}\\ &=\frac {c \sqrt {\frac {c}{a+b x^2}}}{a x^2}-\frac {3 c \sqrt {\frac {c}{a+b x^2}} \left (a+b x^2\right )}{2 a^2 x^2}+\frac {3 b c \sqrt {\frac {c}{a+b x^2}} \sqrt {a+b x^2} \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 40, normalized size = 0.38 \[ -\frac {b c \sqrt {\frac {c}{a+b x^2}} \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};\frac {b x^2}{a}+1\right )}{a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 175, normalized size = 1.68 \[ \left [\frac {3 \, b c x^{2} \sqrt {\frac {c}{a}} \log \left (-\frac {b c x^{2} + 2 \, a c + 2 \, {\left (a b x^{2} + a^{2}\right )} \sqrt {\frac {c}{b x^{2} + a}} \sqrt {\frac {c}{a}}}{x^{2}}\right ) - 2 \, {\left (3 \, b c x^{2} + a c\right )} \sqrt {\frac {c}{b x^{2} + a}}}{4 \, a^{2} x^{2}}, -\frac {3 \, b c x^{2} \sqrt {-\frac {c}{a}} \arctan \left (\frac {a \sqrt {\frac {c}{b x^{2} + a}} \sqrt {-\frac {c}{a}}}{c}\right ) + {\left (3 \, b c x^{2} + a c\right )} \sqrt {\frac {c}{b x^{2} + a}}}{2 \, a^{2} x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 103, normalized size = 0.99 \[ -\frac {1}{2} \, c {\left (\frac {3 \, b c \arctan \left (\frac {\sqrt {b c x^{2} + a c}}{\sqrt {-a c}}\right )}{\sqrt {-a c} a^{2}} + \frac {2 \, a b c^{2} - 3 \, {\left (b c x^{2} + a c\right )} b c}{{\left (\sqrt {b c x^{2} + a c} a c - {\left (b c x^{2} + a c\right )}^{\frac {3}{2}}\right )} a^{2}}\right )} \mathrm {sgn}\left (b x^{2} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 81, normalized size = 0.78 \[ \frac {\left (\frac {c}{b \,x^{2}+a}\right )^{\frac {3}{2}} \left (b \,x^{2}+a \right ) \left (3 \sqrt {b \,x^{2}+a}\, a b \,x^{2} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )-3 a^{\frac {3}{2}} b \,x^{2}-a^{\frac {5}{2}}\right )}{2 a^{\frac {7}{2}} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.99, size = 121, normalized size = 1.16 \[ -\frac {1}{4} \, b c {\left (\frac {2 \, c \sqrt {\frac {c}{b x^{2} + a}}}{a^{2} c - \frac {a^{3} c}{b x^{2} + a}} + \frac {3 \, c \log \left (\frac {a \sqrt {\frac {c}{b x^{2} + a}} - \sqrt {a c}}{a \sqrt {\frac {c}{b x^{2} + a}} + \sqrt {a c}}\right )}{\sqrt {a c} a^{2}} + \frac {4 \, \sqrt {\frac {c}{b x^{2} + a}}}{a^{2}}\right )} \]
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 \[ \int \frac {{\left (\frac {c}{b\,x^2+a}\right )}^{3/2}}{x^3} \,d x \]
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 \[ \int \frac {\left (\frac {c}{a + b x^{2}}\right )^{\frac {3}{2}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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