Optimal. Leaf size=59 \[ -\frac {2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{9/2}}+\frac {8 \left (a-b x^2\right )^{9/4}}{45 a^2 c (c x)^{9/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 59, 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 )^{9/4}}{45 a^2 c (c x)^{9/2}}-\frac {2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 279
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{a-b x^2}}{(c x)^{11/2}} \, dx &=-\frac {2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{9/2}}-\frac {4 \int \frac {\left (a-b x^2\right )^{5/4}}{(c x)^{11/2}} \, dx}{5 a}\\ &=-\frac {2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{9/2}}+\frac {8 \left (a-b x^2\right )^{9/4}}{45 a^2 c (c x)^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 48, normalized size = 0.81 \begin {gather*} -\frac {2 x \sqrt [4]{a-b x^2} \left (5 a^2-a b x^2-4 b^2 x^4\right )}{45 a^2 (c x)^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 32, normalized size = 0.54
method | result | size |
gosper | \(-\frac {2 x \left (-b \,x^{2}+a \right )^{\frac {5}{4}} \left (4 b \,x^{2}+5 a \right )}{45 a^{2} \left (c x \right )^{\frac {11}{2}}}\) | \(32\) |
risch | \(-\frac {2 \left (-b \,x^{2}+a \right )^{\frac {1}{4}} \left (\left (-b \,x^{2}+a \right )^{3}\right )^{\frac {1}{4}} \left (-4 b^{2} x^{4}-a b \,x^{2}+5 a^{2}\right )}{45 \sqrt {c x}\, \left (-\left (b \,x^{2}-a \right )^{3}\right )^{\frac {1}{4}} c^{5} x^{4} a^{2}}\) | \(75\) |
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 = 1.32, size = 46, normalized size = 0.78 \begin {gather*} \frac {2 \, {\left (4 \, b^{2} x^{4} + a b x^{2} - 5 \, a^{2}\right )} {\left (-b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {c x}}{45 \, a^{2} c^{6} x^{5}} \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 = 55.50, size = 462, normalized size = 7.83 \begin {gather*} \begin {cases} - \frac {5 \sqrt [4]{b} \sqrt [4]{\frac {a}{b x^{2}} - 1} \Gamma \left (- \frac {9}{4}\right )}{8 c^{\frac {11}{2}} x^{4} \Gamma \left (- \frac {1}{4}\right )} + \frac {b^{\frac {5}{4}} \sqrt [4]{\frac {a}{b x^{2}} - 1} \Gamma \left (- \frac {9}{4}\right )}{8 a c^{\frac {11}{2}} x^{2} \Gamma \left (- \frac {1}{4}\right )} + \frac {b^{\frac {9}{4}} \sqrt [4]{\frac {a}{b x^{2}} - 1} \Gamma \left (- \frac {9}{4}\right )}{2 a^{2} c^{\frac {11}{2}} \Gamma \left (- \frac {1}{4}\right )} & \text {for}\: \left |{\frac {a}{b x^{2}}}\right | > 1 \\\frac {5 a^{3} b^{\frac {5}{4}} \sqrt [4]{- \frac {a}{b x^{2}} + 1} e^{\frac {i \pi }{4}} \Gamma \left (- \frac {9}{4}\right )}{x^{2} \left (- 8 a^{3} b c^{\frac {11}{2}} x^{2} \Gamma \left (- \frac {1}{4}\right ) + 8 a^{2} b^{2} c^{\frac {11}{2}} x^{4} \Gamma \left (- \frac {1}{4}\right )\right )} - \frac {6 a^{2} b^{\frac {9}{4}} \sqrt [4]{- \frac {a}{b x^{2}} + 1} e^{\frac {i \pi }{4}} \Gamma \left (- \frac {9}{4}\right )}{- 8 a^{3} b c^{\frac {11}{2}} x^{2} \Gamma \left (- \frac {1}{4}\right ) + 8 a^{2} b^{2} c^{\frac {11}{2}} x^{4} \Gamma \left (- \frac {1}{4}\right )} - \frac {3 a b^{\frac {13}{4}} x^{2} \sqrt [4]{- \frac {a}{b x^{2}} + 1} e^{\frac {i \pi }{4}} \Gamma \left (- \frac {9}{4}\right )}{- 8 a^{3} b c^{\frac {11}{2}} x^{2} \Gamma \left (- \frac {1}{4}\right ) + 8 a^{2} b^{2} c^{\frac {11}{2}} x^{4} \Gamma \left (- \frac {1}{4}\right )} + \frac {4 b^{\frac {17}{4}} x^{4} \sqrt [4]{- \frac {a}{b x^{2}} + 1} e^{\frac {i \pi }{4}} \Gamma \left (- \frac {9}{4}\right )}{- 8 a^{3} b c^{\frac {11}{2}} x^{2} \Gamma \left (- \frac {1}{4}\right ) + 8 a^{2} b^{2} c^{\frac {11}{2}} x^{4} \Gamma \left (- \frac {1}{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 = 4.89, size = 51, normalized size = 0.86 \begin {gather*} \frac {{\left (a-b\,x^2\right )}^{1/4}\,\left (\frac {2\,b\,x^2}{45\,a\,c^5}-\frac {2}{9\,c^5}+\frac {8\,b^2\,x^4}{45\,a^2\,c^5}\right )}{x^4\,\sqrt {c\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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