Optimal. Leaf size=85 \[ -\frac {64 \left (a+b x^2\right )^{13/4}}{585 a^3 c (c x)^{13/2}}+\frac {16 \left (a+b x^2\right )^{9/4}}{45 a^2 c (c x)^{13/2}}-\frac {2 \left (a+b x^2\right )^{5/4}}{5 a c (c x)^{13/2}} \]
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Rubi [A] time = 0.02, 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} \[ -\frac {64 \left (a+b x^2\right )^{13/4}}{585 a^3 c (c x)^{13/2}}+\frac {16 \left (a+b x^2\right )^{9/4}}{45 a^2 c (c x)^{13/2}}-\frac {2 \left (a+b x^2\right )^{5/4}}{5 a c (c x)^{13/2}} \]
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 = 52, normalized size = 0.61 \[ -\frac {2 \sqrt {c x} \left (a+b x^2\right )^{5/4} \left (45 a^2-40 a b x^2+32 b^2 x^4\right )}{585 a^3 c^8 x^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.04, size = 57, normalized size = 0.67 \[ -\frac {2 \, {\left (32 \, b^{3} x^{6} - 8 \, a b^{2} x^{4} + 5 \, a^{2} b x^{2} + 45 \, a^{3}\right )} {\left (b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {c x}}{585 \, a^{3} c^{8} x^{7}} \]
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 = 0.01, size = 42, 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 = 5.00, size = 65, normalized size = 0.76 \[ -\frac {{\left (b\,x^2+a\right )}^{1/4}\,\left (\frac {2}{13\,c^7}+\frac {2\,b\,x^2}{117\,a\,c^7}-\frac {16\,b^2\,x^4}{585\,a^2\,c^7}+\frac {64\,b^3\,x^6}{585\,a^3\,c^7}\right )}{x^6\,\sqrt {c\,x}} \]
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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