Optimal. Leaf size=113 \[ \frac {256 \left (a+b x^2\right )^{17/4}}{3315 a^4 c (c x)^{17/2}}-\frac {64 \left (a+b x^2\right )^{13/4}}{195 a^3 c (c x)^{17/2}}+\frac {8 \left (a+b x^2\right )^{9/4}}{15 a^2 c (c x)^{17/2}}-\frac {2 \left (a+b x^2\right )^{5/4}}{5 a c (c x)^{17/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, 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 = 63, normalized size = 0.56 \[ \frac {2 \left (a+b x^2\right )^{5/4} \left (-195 a^3+180 a^2 b x^2-160 a b^2 x^4+128 b^3 x^6\right )}{3315 a^4 c^9 x^8 \sqrt {c x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.16, size = 68, normalized size = 0.60 \[ \frac {2 \, {\left (128 \, b^{4} x^{8} - 32 \, a b^{3} x^{6} + 20 \, a^{2} b^{2} x^{4} - 15 \, a^{3} b x^{2} - 195 \, a^{4}\right )} {\left (b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {c x}}{3315 \, a^{4} c^{10} x^{9}} \]
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 = 53, normalized size = 0.47 \[ -\frac {2 \left (b \,x^{2}+a \right )^{\frac {5}{4}} \left (-128 b^{3} x^{6}+160 a \,b^{2} x^{4}-180 a^{2} b \,x^{2}+195 a^{3}\right ) x}{3315 \left (c x \right )^{\frac {19}{2}} a^{4}} \]
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 = 5.02, size = 79, normalized size = 0.70 \[ -\frac {{\left (b\,x^2+a\right )}^{1/4}\,\left (\frac {2}{17\,c^9}+\frac {2\,b\,x^2}{221\,a\,c^9}-\frac {8\,b^2\,x^4}{663\,a^2\,c^9}+\frac {64\,b^3\,x^6}{3315\,a^3\,c^9}-\frac {256\,b^4\,x^8}{3315\,a^4\,c^9}\right )}{x^8\,\sqrt {c\,x}} \]
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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