Optimal. Leaf size=85 \[ -\frac {64 \left (a+b x^2\right )^{11/4}}{231 a^3 c (c x)^{11/2}}+\frac {16 \left (a+b x^2\right )^{7/4}}{21 a^2 c (c x)^{11/2}}-\frac {2 \left (a+b x^2\right )^{3/4}}{3 a c (c x)^{11/2}} \]
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Rubi [A] time = 0.03, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {273, 264} \[ -\frac {64 \left (a+b x^2\right )^{11/4}}{231 a^3 c (c x)^{11/2}}+\frac {16 \left (a+b x^2\right )^{7/4}}{21 a^2 c (c x)^{11/2}}-\frac {2 \left (a+b x^2\right )^{3/4}}{3 a c (c x)^{11/2}} \]
Antiderivative was successfully verified.
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Rule 264
Rule 273
Rubi steps
\begin {align*} \int \frac {1}{(c x)^{13/2} \sqrt [4]{a+b x^2}} \, dx &=-\frac {2 \left (a+b x^2\right )^{3/4}}{3 a c (c x)^{11/2}}-\frac {8 \int \frac {\left (a+b x^2\right )^{3/4}}{(c x)^{13/2}} \, dx}{3 a}\\ &=-\frac {2 \left (a+b x^2\right )^{3/4}}{3 a c (c x)^{11/2}}+\frac {16 \left (a+b x^2\right )^{7/4}}{21 a^2 c (c x)^{11/2}}+\frac {32 \int \frac {\left (a+b x^2\right )^{7/4}}{(c x)^{13/2}} \, dx}{21 a^2}\\ &=-\frac {2 \left (a+b x^2\right )^{3/4}}{3 a c (c x)^{11/2}}+\frac {16 \left (a+b x^2\right )^{7/4}}{21 a^2 c (c x)^{11/2}}-\frac {64 \left (a+b x^2\right )^{11/4}}{231 a^3 c (c x)^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 52, normalized size = 0.61 \[ -\frac {2 \sqrt {c x} \left (a+b x^2\right )^{3/4} \left (21 a^2-24 a b x^2+32 b^2 x^4\right )}{231 a^3 c^7 x^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 46, normalized size = 0.54 \[ -\frac {2 \, {\left (32 \, b^{2} x^{4} - 24 \, a b x^{2} + 21 \, a^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {c x}}{231 \, a^{3} c^{7} x^{6}} \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} \left (c x\right )^{\frac {13}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 42, normalized size = 0.49 \[ -\frac {2 \left (b \,x^{2}+a \right )^{\frac {3}{4}} \left (32 b^{2} x^{4}-24 a b \,x^{2}+21 a^{2}\right ) x}{231 \left (c x \right )^{\frac {13}{2}} a^{3}} \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} \left (c x\right )^{\frac {13}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.00, size = 54, normalized size = 0.64 \[ -\frac {{\left (b\,x^2+a\right )}^{3/4}\,\left (\frac {2}{11\,a\,c^6}-\frac {16\,b\,x^2}{77\,a^2\,c^6}+\frac {64\,b^2\,x^4}{231\,a^3\,c^6}\right )}{x^5\,\sqrt {c\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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