Optimal. Leaf size=154 \[ -\frac {b^{3/2} \left (c \sqrt {a+b x^2}\right )^{3/2} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 a^{3/2} \left (\frac {b x^2}{a}+1\right )^{3/4}}+\frac {b^2 x \left (c \sqrt {a+b x^2}\right )^{3/2}}{2 a \left (a+b x^2\right )}-\frac {b \left (c \sqrt {a+b x^2}\right )^{3/2}}{2 a x}-\frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{3 x^3} \]
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Rubi [A] time = 0.16, antiderivative size = 193, normalized size of antiderivative = 1.25, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6720, 277, 325, 229, 227, 196} \[ \frac {b^2 c x \sqrt {c \sqrt {a+b x^2}}}{2 a \sqrt {a+b x^2}}-\frac {b^{3/2} c \sqrt [4]{\frac {b x^2}{a}+1} \sqrt {c \sqrt {a+b x^2}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 \sqrt {a} \sqrt {a+b x^2}}-\frac {b c \sqrt {a+b x^2} \sqrt {c \sqrt {a+b x^2}}}{2 a x}-\frac {c \sqrt {a+b x^2} \sqrt {c \sqrt {a+b x^2}}}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 196
Rule 227
Rule 229
Rule 277
Rule 325
Rule 6720
Rubi steps
\begin {align*} \int \frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x^4} \, dx &=\frac {\left (c \sqrt {c \sqrt {a+b x^2}}\right ) \int \frac {\left (a+b x^2\right )^{3/4}}{x^4} \, dx}{\sqrt [4]{a+b x^2}}\\ &=-\frac {c \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}}{3 x^3}+\frac {\left (b c \sqrt {c \sqrt {a+b x^2}}\right ) \int \frac {1}{x^2 \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}}{3 x^3}-\frac {b c \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}}{2 a x}+\frac {\left (b^2 c \sqrt {c \sqrt {a+b x^2}}\right ) \int \frac {1}{\sqrt [4]{a+b x^2}} \, dx}{4 a \sqrt [4]{a+b x^2}}\\ &=-\frac {c \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}}{3 x^3}-\frac {b c \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}}{2 a x}+\frac {\left (b^2 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}{4 a \sqrt {a+b x^2}}\\ &=\frac {b^2 c x \sqrt {c \sqrt {a+b x^2}}}{2 a \sqrt {a+b x^2}}-\frac {c \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}}{3 x^3}-\frac {b c \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}}{2 a x}-\frac {\left (b^2 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}{4 a \sqrt {a+b x^2}}\\ &=\frac {b^2 c x \sqrt {c \sqrt {a+b x^2}}}{2 a \sqrt {a+b x^2}}-\frac {c \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}}{3 x^3}-\frac {b c \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}}{2 a x}-\frac {b^{3/2} 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 )}{2 \sqrt {a} \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 57, normalized size = 0.37 \[ -\frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \, _2F_1\left (-\frac {3}{2},-\frac {3}{4};-\frac {1}{2};-\frac {b x^2}{a}\right )}{3 x^3 \left (\frac {b x^2}{a}+1\right )^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {\left (\sqrt {b x^{2} + a} c\right )^{\frac {3}{2}}}{x^{4}}\,{d x} \]
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 \[ \int \frac {\left (\sqrt {b \,x^{2}+a}\, c \right )^{\frac {3}{2}}}{x^{4}}\, dx \]
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 \[ \int \frac {\left (\sqrt {b x^{2} + a} c\right )^{\frac {3}{2}}}{x^{4}}\,{d x} \]
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 \[ \int \frac {{\left (c\,\sqrt {b\,x^2+a}\right )}^{3/2}}{x^4} \,d x \]
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 \[ \int \frac {\left (c \sqrt {a + b x^{2}}\right )^{\frac {3}{2}}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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