Optimal. Leaf size=55 \[ \frac {8 \left (a+b x^2\right )^{5/4}}{5 a^2 c (c x)^{5/2}}-\frac {2 \sqrt [4]{a+b x^2}}{a c (c x)^{5/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {273, 264} \[ \frac {8 \left (a+b x^2\right )^{5/4}}{5 a^2 c (c x)^{5/2}}-\frac {2 \sqrt [4]{a+b x^2}}{a c (c x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 264
Rule 273
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 = 0.02, size = 34, normalized size = 0.62 \[ -\frac {2 x \left (a-4 b x^2\right ) \sqrt [4]{a+b x^2}}{5 a^2 (c x)^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 35, normalized size = 0.64 \[ \frac {2 \, {\left (4 \, b x^{2} - a\right )} {\left (b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {c x}}{5 \, a^{2} c^{4} x^{3}} \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (c x\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 29, normalized size = 0.53 \[ -\frac {2 \left (b \,x^{2}+a \right )^{\frac {1}{4}} \left (-4 b \,x^{2}+a \right ) x}{5 \left (c x \right )^{\frac {7}{2}} a^{2}} \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (c x\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.98, size = 40, normalized size = 0.73 \[ -\frac {{\left (b\,x^2+a\right )}^{1/4}\,\left (\frac {2}{5\,a\,c^3}-\frac {8\,b\,x^2}{5\,a^2\,c^3}\right )}{x^2\,\sqrt {c\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 24.76, size = 78, normalized size = 1.42 \[ - \frac {\sqrt [4]{b} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {5}{4}\right )}{8 a c^{\frac {7}{2}} x^{2} \Gamma \left (\frac {3}{4}\right )} + \frac {b^{\frac {5}{4}} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {5}{4}\right )}{2 a^{2} c^{\frac {7}{2}} \Gamma \left (\frac {3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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