Optimal. Leaf size=117 \[ \frac {256 \left (a-b x^2\right )^{17/4}}{3315 a^4 c (c x)^{17/2}}-\frac {64 \left (a-b x^2\right )^{13/4}}{195 a^3 c (c x)^{17/2}}+\frac {8 \left (a-b x^2\right )^{9/4}}{15 a^2 c (c x)^{17/2}}-\frac {2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{17/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {273, 264} \[ \frac {256 \left (a-b x^2\right )^{17/4}}{3315 a^4 c (c x)^{17/2}}-\frac {64 \left (a-b x^2\right )^{13/4}}{195 a^3 c (c x)^{17/2}}+\frac {8 \left (a-b x^2\right )^{9/4}}{15 a^2 c (c x)^{17/2}}-\frac {2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{17/2}} \]
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)^{19/2}} \, dx &=-\frac {2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{17/2}}-\frac {12 \int \frac {\left (a-b x^2\right )^{5/4}}{(c x)^{19/2}} \, dx}{5 a}\\ &=-\frac {2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{17/2}}+\frac {8 \left (a-b x^2\right )^{9/4}}{15 a^2 c (c x)^{17/2}}+\frac {32 \int \frac {\left (a-b x^2\right )^{9/4}}{(c x)^{19/2}} \, dx}{15 a^2}\\ &=-\frac {2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{17/2}}+\frac {8 \left (a-b x^2\right )^{9/4}}{15 a^2 c (c x)^{17/2}}-\frac {64 \left (a-b x^2\right )^{13/4}}{195 a^3 c (c x)^{17/2}}-\frac {128 \int \frac {\left (a-b x^2\right )^{13/4}}{(c x)^{19/2}} \, dx}{195 a^3}\\ &=-\frac {2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{17/2}}+\frac {8 \left (a-b x^2\right )^{9/4}}{15 a^2 c (c x)^{17/2}}-\frac {64 \left (a-b x^2\right )^{13/4}}{195 a^3 c (c x)^{17/2}}+\frac {256 \left (a-b x^2\right )^{17/4}}{3315 a^4 c (c x)^{17/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 64, normalized size = 0.55 \[ -\frac {2 \left (a-b x^2\right )^{5/4} \left (195 a^3+180 a^2 b x^2+160 a b^2 x^4+128 b^3 x^6\right )}{3315 a^4 c^9 x^8 \sqrt {c x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.01, size = 69, normalized size = 0.59 \[ \frac {2 \, {\left (128 \, b^{4} x^{8} + 32 \, a b^{3} x^{6} + 20 \, a^{2} b^{2} x^{4} + 15 \, a^{3} b x^{2} - 195 \, a^{4}\right )} {\left (-b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {c x}}{3315 \, a^{4} c^{10} x^{9}} \]
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 \[ \int \frac {{\left (-b x^{2} + a\right )}^{\frac {1}{4}}}{\left (c x\right )^{\frac {19}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 54, normalized size = 0.46 \[ -\frac {2 \left (-b \,x^{2}+a \right )^{\frac {5}{4}} \left (128 b^{3} x^{6}+160 a \,b^{2} x^{4}+180 a^{2} b \,x^{2}+195 a^{3}\right ) x}{3315 \left (c x \right )^{\frac {19}{2}} a^{4}} \]
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 \[ \int \frac {{\left (-b x^{2} + a\right )}^{\frac {1}{4}}}{\left (c x\right )^{\frac {19}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.93, size = 79, normalized size = 0.68 \[ \frac {{\left (a-b\,x^2\right )}^{1/4}\,\left (\frac {2\,b\,x^2}{221\,a\,c^9}-\frac {2}{17\,c^9}+\frac {8\,b^2\,x^4}{663\,a^2\,c^9}+\frac {64\,b^3\,x^6}{3315\,a^3\,c^9}+\frac {256\,b^4\,x^8}{3315\,a^4\,c^9}\right )}{x^8\,\sqrt {c\,x}} \]
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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