Optimal. Leaf size=133 \[ -\frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{2 x^2}+\frac {3 b \left (c \sqrt {a+b x^2}\right )^{3/2} \tan ^{-1}\left (\sqrt [4]{1+\frac {b x^2}{a}}\right )}{4 a \left (1+\frac {b x^2}{a}\right )^{3/4}}-\frac {3 b \left (c \sqrt {a+b x^2}\right )^{3/2} \tanh ^{-1}\left (\sqrt [4]{1+\frac {b x^2}{a}}\right )}{4 a \left (1+\frac {b x^2}{a}\right )^{3/4}} \]
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Rubi [A]
time = 0.05, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1973, 272, 43,
65, 304, 209, 212} \begin {gather*} \frac {3 b \text {ArcTan}\left (\sqrt [4]{\frac {b x^2}{a}+1}\right ) \left (c \sqrt {a+b x^2}\right )^{3/2}}{4 a \left (\frac {b x^2}{a}+1\right )^{3/4}}-\frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{2 x^2}-\frac {3 b \left (c \sqrt {a+b x^2}\right )^{3/2} \tanh ^{-1}\left (\sqrt [4]{\frac {b x^2}{a}+1}\right )}{4 a \left (\frac {b x^2}{a}+1\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 65
Rule 209
Rule 212
Rule 272
Rule 304
Rule 1973
Rubi steps
\begin {align*} \int \frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x^3} \, dx &=\frac {\left (c \sqrt {c \sqrt {a+b x^2}}\right ) \int \frac {\left (a+b x^2\right )^{3/4}}{x^3} \, dx}{\sqrt [4]{a+b x^2}}\\ &=\frac {\left (c \sqrt {c \sqrt {a+b x^2}}\right ) \text {Subst}\left (\int \frac {(a+b x)^{3/4}}{x^2} \, dx,x,x^2\right )}{2 \sqrt [4]{a+b x^2}}\\ &=-\frac {c \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}}{2 x^2}+\frac {\left (3 b c \sqrt {c \sqrt {a+b x^2}}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt [4]{a+b x}} \, dx,x,x^2\right )}{8 \sqrt [4]{a+b x^2}}\\ &=-\frac {c \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}}{2 x^2}+\frac {\left (3 c \sqrt {c \sqrt {a+b x^2}}\right ) \text {Subst}\left (\int \frac {x^2}{-\frac {a}{b}+\frac {x^4}{b}} \, dx,x,\sqrt [4]{a+b x^2}\right )}{2 \sqrt [4]{a+b x^2}}\\ &=-\frac {c \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}}{2 x^2}-\frac {\left (3 b c \sqrt {c \sqrt {a+b x^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a}-x^2} \, dx,x,\sqrt [4]{a+b x^2}\right )}{4 \sqrt [4]{a+b x^2}}+\frac {\left (3 b c \sqrt {c \sqrt {a+b x^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a}+x^2} \, dx,x,\sqrt [4]{a+b x^2}\right )}{4 \sqrt [4]{a+b x^2}}\\ &=-\frac {c \sqrt {c \sqrt {a+b x^2}} \sqrt {a+b x^2}}{2 x^2}+\frac {3 b c \sqrt {c \sqrt {a+b x^2}} \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )}{4 \sqrt [4]{a} \sqrt [4]{a+b x^2}}-\frac {3 b c \sqrt {c \sqrt {a+b x^2}} \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )}{4 \sqrt [4]{a} \sqrt [4]{a+b x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 107, normalized size = 0.80 \begin {gather*} -\frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \left (2 \sqrt [4]{a} \left (a+b x^2\right )^{3/4}-3 b x^2 \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )+3 b x^2 \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )\right )}{4 \sqrt [4]{a} x^2 \left (a+b x^2\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (c \sqrt {b \,x^{2}+a}\right )^{\frac {3}{2}}}{x^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 138, normalized size = 1.04 \begin {gather*} \frac {{\left (3 \, c^{4} {\left (\frac {2 \, \arctan \left (\frac {\sqrt {\sqrt {b x^{2} + a} c}}{\left (a c^{2}\right )^{\frac {1}{4}}}\right )}{\left (a c^{2}\right )^{\frac {1}{4}}} + \frac {\log \left (\frac {\sqrt {\sqrt {b x^{2} + a} c} - \left (a c^{2}\right )^{\frac {1}{4}}}{\sqrt {\sqrt {b x^{2} + a} c} + \left (a c^{2}\right )^{\frac {1}{4}}}\right )}{\left (a c^{2}\right )^{\frac {1}{4}}}\right )} - \frac {4 \, \left (\sqrt {b x^{2} + a} c\right )^{\frac {3}{2}} c^{4}}{{\left (b x^{2} + a\right )} c^{2} - a c^{2}}\right )} b}{8 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {\left (c \sqrt {a + b x^{2}}\right )^{\frac {3}{2}}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.13, size = 212, normalized size = 1.59 \begin {gather*} \frac {{\left (\frac {6 \, \sqrt {2} b^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (b x^{2} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {1}{4}}} + \frac {6 \, \sqrt {2} b^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (b x^{2} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {1}{4}}} + \frac {3 \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} b^{2} \log \left (\sqrt {2} {\left (b x^{2} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{2} + a} + \sqrt {-a}\right )}{a} + \frac {3 \, \sqrt {2} b^{2} \log \left (-\sqrt {2} {\left (b x^{2} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{2} + a} + \sqrt {-a}\right )}{\left (-a\right )^{\frac {1}{4}}} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {3}{4}} b}{x^{2}}\right )} c^{\frac {3}{2}}}{16 \, b} \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 {{\left (c\,\sqrt {b\,x^2+a}\right )}^{3/2}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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