Optimal. Leaf size=57 \[ -\frac {2 \sqrt [4]{a-b x^2}}{a c (c x)^{5/2}}+\frac {8 \left (a-b x^2\right )^{5/4}}{5 a^2 c (c x)^{5/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {279, 270}
\begin {gather*} \frac {8 \left (a-b x^2\right )^{5/4}}{5 a^2 c (c x)^{5/2}}-\frac {2 \sqrt [4]{a-b x^2}}{a c (c x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 279
Rubi steps
\begin {align*} \int \frac {1}{(c x)^{7/2} \left (a-b x^2\right )^{3/4}} \, dx &=-\frac {2 \sqrt [4]{a-b x^2}}{a c (c x)^{5/2}}-\frac {4 \int \frac {\sqrt [4]{a-b x^2}}{(c x)^{7/2}} \, dx}{a}\\ &=-\frac {2 \sqrt [4]{a-b x^2}}{a c (c x)^{5/2}}+\frac {8 \left (a-b x^2\right )^{5/4}}{5 a^2 c (c x)^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 35, normalized size = 0.61 \begin {gather*} -\frac {2 x \sqrt [4]{a-b x^2} \left (a+4 b x^2\right )}{5 a^2 (c x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 30, normalized size = 0.53
method | result | size |
gosper | \(-\frac {2 x \left (-b \,x^{2}+a \right )^{\frac {1}{4}} \left (4 b \,x^{2}+a \right )}{5 a^{2} \left (c x \right )^{\frac {7}{2}}}\) | \(30\) |
risch | \(-\frac {2 \left (-b \,x^{2}+a \right )^{\frac {1}{4}} \left (\left (-b \,x^{2}+a \right )^{3}\right )^{\frac {1}{4}} \left (4 b \,x^{2}+a \right )}{5 \sqrt {c x}\, \left (-\left (b \,x^{2}-a \right )^{3}\right )^{\frac {1}{4}} c^{3} a^{2} x^{2}}\) | \(62\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.75, size = 34, normalized size = 0.60 \begin {gather*} -\frac {2 \, {\left (4 \, b x^{2} + a\right )} {\left (-b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {c x}}{5 \, a^{2} c^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 15.65, size = 352, normalized size = 6.18 \begin {gather*} \begin {cases} - \frac {\sqrt [4]{b} \sqrt [4]{\frac {a}{b x^{2}} - 1} \Gamma \left (- \frac {5}{4}\right )}{8 a c^{\frac {7}{2}} x^{2} \Gamma \left (\frac {3}{4}\right )} - \frac {b^{\frac {5}{4}} \sqrt [4]{\frac {a}{b x^{2}} - 1} \Gamma \left (- \frac {5}{4}\right )}{2 a^{2} c^{\frac {7}{2}} \Gamma \left (\frac {3}{4}\right )} & \text {for}\: \left |{\frac {a}{b x^{2}}}\right | > 1 \\- \frac {a^{2} b^{\frac {5}{4}} \sqrt [4]{- \frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {5}{4}\right )}{- 8 a^{3} b c^{\frac {7}{2}} x^{2} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right ) + 8 a^{2} b^{2} c^{\frac {7}{2}} x^{4} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right )} - \frac {3 a b^{\frac {9}{4}} x^{2} \sqrt [4]{- \frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {5}{4}\right )}{- 8 a^{3} b c^{\frac {7}{2}} x^{2} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right ) + 8 a^{2} b^{2} c^{\frac {7}{2}} x^{4} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right )} + \frac {4 b^{\frac {13}{4}} x^{4} \sqrt [4]{- \frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {5}{4}\right )}{- 8 a^{3} b c^{\frac {7}{2}} x^{2} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right ) + 8 a^{2} b^{2} c^{\frac {7}{2}} x^{4} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right )} & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.10, size = 41, normalized size = 0.72 \begin {gather*} -\frac {{\left (a-b\,x^2\right )}^{1/4}\,\left (\frac {2}{5\,a\,c^3}+\frac {8\,b\,x^2}{5\,a^2\,c^3}\right )}{x^2\,\sqrt {c\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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