Optimal. Leaf size=88 \[ -\frac {64 \left (a-b x^2\right )^{13/4}}{585 a^3 c (c x)^{13/2}}+\frac {16 \left (a-b x^2\right )^{9/4}}{45 a^2 c (c x)^{13/2}}-\frac {2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{13/2}} \]
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Rubi [A] time = 0.03, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {273, 264} \[ -\frac {64 \left (a-b x^2\right )^{13/4}}{585 a^3 c (c x)^{13/2}}+\frac {16 \left (a-b x^2\right )^{9/4}}{45 a^2 c (c x)^{13/2}}-\frac {2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{13/2}} \]
Antiderivative was successfully verified.
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Rule 264
Rule 273
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{a-b x^2}}{(c x)^{15/2}} \, dx &=-\frac {2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{13/2}}-\frac {8 \int \frac {\left (a-b x^2\right )^{5/4}}{(c x)^{15/2}} \, dx}{5 a}\\ &=-\frac {2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{13/2}}+\frac {16 \left (a-b x^2\right )^{9/4}}{45 a^2 c (c x)^{13/2}}+\frac {32 \int \frac {\left (a-b x^2\right )^{9/4}}{(c x)^{15/2}} \, dx}{45 a^2}\\ &=-\frac {2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{13/2}}+\frac {16 \left (a-b x^2\right )^{9/4}}{45 a^2 c (c x)^{13/2}}-\frac {64 \left (a-b x^2\right )^{13/4}}{585 a^3 c (c x)^{13/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 53, normalized size = 0.60 \[ -\frac {2 \sqrt {c x} \left (a-b x^2\right )^{5/4} \left (45 a^2+40 a b x^2+32 b^2 x^4\right )}{585 a^3 c^8 x^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 58, normalized size = 0.66 \[ \frac {2 \, {\left (32 \, b^{3} x^{6} + 8 \, a b^{2} x^{4} + 5 \, a^{2} b x^{2} - 45 \, a^{3}\right )} {\left (-b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {c x}}{585 \, a^{3} c^{8} x^{7}} \]
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 \[ \int \frac {{\left (-b x^{2} + a\right )}^{\frac {1}{4}}}{\left (c x\right )^{\frac {15}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 43, normalized size = 0.49 \[ -\frac {2 \left (-b \,x^{2}+a \right )^{\frac {5}{4}} \left (32 b^{2} x^{4}+40 a b \,x^{2}+45 a^{2}\right ) x}{585 \left (c x \right )^{\frac {15}{2}} a^{3}} \]
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 \[ \int \frac {{\left (-b x^{2} + a\right )}^{\frac {1}{4}}}{\left (c x\right )^{\frac {15}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.91, size = 65, normalized size = 0.74 \[ \frac {{\left (a-b\,x^2\right )}^{1/4}\,\left (\frac {2\,b\,x^2}{117\,a\,c^7}-\frac {2}{13\,c^7}+\frac {16\,b^2\,x^4}{585\,a^2\,c^7}+\frac {64\,b^3\,x^6}{585\,a^3\,c^7}\right )}{x^6\,\sqrt {c\,x}} \]
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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