Optimal. Leaf size=115 \[ -\frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x}+\frac {3 b x \left (c \sqrt {a+b x^2}\right )^{3/2}}{a+b x^2}-\frac {3 \sqrt {b} \left (c \sqrt {a+b x^2}\right )^{3/2} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a} \left (\frac {b x^2}{a}+1\right )^{3/4}} \]
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Rubi [A] time = 0.14, antiderivative size = 142, normalized size of antiderivative = 1.23, 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 = {6720, 277, 229, 227, 196} \[ -\frac {c \sqrt {a+b x^2} \sqrt {c \sqrt {a+b x^2}}}{x}+\frac {3 b c x \sqrt {c \sqrt {a+b x^2}}}{\sqrt {a+b x^2}}-\frac {3 \sqrt {a} \sqrt {b} c \sqrt [4]{\frac {b x^2}{a}+1} \sqrt {c \sqrt {a+b x^2}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 196
Rule 227
Rule 229
Rule 277
Rule 6720
Rubi steps
\begin {align*} \int \frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x^2} \, dx &=\frac {\left (c \sqrt {c \sqrt {a+b x^2}}\right ) \int \frac {\left (a+b x^2\right )^{3/4}}{x^2} \, dx}{\sqrt [4]{a+b x^2}}\\ &=-\frac {c \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}}{x}+\frac {\left (3 b c \sqrt {c \sqrt {a+b x^2}}\right ) \int \frac {1}{\sqrt [4]{a+b x^2}} \, dx}{2 \sqrt [4]{a+b x^2}}\\ &=-\frac {c \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}}{x}+\frac {\left (3 b c \sqrt {c \sqrt {a+b x^2}} \sqrt [4]{1+\frac {b x^2}{a}}\right ) \int \frac {1}{\sqrt [4]{1+\frac {b x^2}{a}}} \, dx}{2 \sqrt {a+b x^2}}\\ &=\frac {3 b c x \sqrt {c \sqrt {a+b x^2}}}{\sqrt {a+b x^2}}-\frac {c \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}}{x}-\frac {\left (3 b c \sqrt {c \sqrt {a+b x^2}} \sqrt [4]{1+\frac {b x^2}{a}}\right ) \int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{5/4}} \, dx}{2 \sqrt {a+b x^2}}\\ &=\frac {3 b c x \sqrt {c \sqrt {a+b x^2}}}{\sqrt {a+b x^2}}-\frac {c \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}}{x}-\frac {3 \sqrt {a} \sqrt {b} c \sqrt {c \sqrt {a+b x^2}} \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 55, normalized size = 0.48 \[ -\frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \, _2F_1\left (-\frac {3}{4},-\frac {1}{2};\frac {1}{2};-\frac {b x^2}{a}\right )}{x \left (\frac {b x^2}{a}+1\right )^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.06, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{2} + a} \sqrt {\sqrt {b x^{2} + a} c} c}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\sqrt {b x^{2} + a} c\right )^{\frac {3}{2}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[ \int \frac {\left (\sqrt {b \,x^{2}+a}\, c \right )^{\frac {3}{2}}}{x^{2}}\, dx \]
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 \[ \int \frac {\left (\sqrt {b x^{2} + a} c\right )^{\frac {3}{2}}}{x^{2}}\,{d x} \]
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 \[ \int \frac {{\left (c\,\sqrt {b\,x^2+a}\right )}^{3/2}}{x^2} \,d x \]
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 \[ \int \frac {\left (c \sqrt {a + b x^{2}}\right )^{\frac {3}{2}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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