Optimal. Leaf size=83 \[ -\frac {64 \left (a+b x^2\right )^{9/4}}{45 a^3 c (c x)^{9/2}}+\frac {16 \left (a+b x^2\right )^{5/4}}{5 a^2 c (c x)^{9/2}}-\frac {2 \sqrt [4]{a+b x^2}}{a c (c x)^{9/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 83, 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} \[ -\frac {64 \left (a+b x^2\right )^{9/4}}{45 a^3 c (c x)^{9/2}}+\frac {16 \left (a+b x^2\right )^{5/4}}{5 a^2 c (c x)^{9/2}}-\frac {2 \sqrt [4]{a+b x^2}}{a c (c x)^{9/2}} \]
Antiderivative was successfully verified.
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Rule 264
Rule 273
Rubi steps
\begin {align*} \int \frac {1}{(c x)^{11/2} \left (a+b x^2\right )^{3/4}} \, dx &=-\frac {2 \sqrt [4]{a+b x^2}}{a c (c x)^{9/2}}-\frac {8 \int \frac {\sqrt [4]{a+b x^2}}{(c x)^{11/2}} \, dx}{a}\\ &=-\frac {2 \sqrt [4]{a+b x^2}}{a c (c x)^{9/2}}+\frac {16 \left (a+b x^2\right )^{5/4}}{5 a^2 c (c x)^{9/2}}+\frac {32 \int \frac {\left (a+b x^2\right )^{5/4}}{(c x)^{11/2}} \, dx}{5 a^2}\\ &=-\frac {2 \sqrt [4]{a+b x^2}}{a c (c x)^{9/2}}+\frac {16 \left (a+b x^2\right )^{5/4}}{5 a^2 c (c x)^{9/2}}-\frac {64 \left (a+b x^2\right )^{9/4}}{45 a^3 c (c x)^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 52, normalized size = 0.63 \[ -\frac {2 \sqrt {c x} \sqrt [4]{a+b x^2} \left (5 a^2-8 a b x^2+32 b^2 x^4\right )}{45 a^3 c^6 x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 46, normalized size = 0.55 \[ -\frac {2 \, {\left (32 \, b^{2} x^{4} - 8 \, a b x^{2} + 5 \, a^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {c x}}{45 \, a^{3} c^{6} x^{5}} \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (c x\right )^{\frac {11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 42, normalized size = 0.51 \[ -\frac {2 \left (b \,x^{2}+a \right )^{\frac {1}{4}} \left (32 b^{2} x^{4}-8 a b \,x^{2}+5 a^{2}\right ) x}{45 \left (c x \right )^{\frac {11}{2}} a^{3}} \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (c x\right )^{\frac {11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.17, size = 54, normalized size = 0.65 \[ -\frac {{\left (b\,x^2+a\right )}^{1/4}\,\left (\frac {2}{9\,a\,c^5}-\frac {16\,b\,x^2}{45\,a^2\,c^5}+\frac {64\,b^2\,x^4}{45\,a^3\,c^5}\right )}{x^4\,\sqrt {c\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 172.06, size = 483, normalized size = 5.82 \[ \frac {5 a^{4} b^{\frac {17}{4}} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {9}{4}\right )}{32 a^{5} b^{4} c^{\frac {11}{2}} x^{4} \Gamma \left (\frac {3}{4}\right ) + 64 a^{4} b^{5} c^{\frac {11}{2}} x^{6} \Gamma \left (\frac {3}{4}\right ) + 32 a^{3} b^{6} c^{\frac {11}{2}} x^{8} \Gamma \left (\frac {3}{4}\right )} + \frac {2 a^{3} b^{\frac {21}{4}} x^{2} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {9}{4}\right )}{32 a^{5} b^{4} c^{\frac {11}{2}} x^{4} \Gamma \left (\frac {3}{4}\right ) + 64 a^{4} b^{5} c^{\frac {11}{2}} x^{6} \Gamma \left (\frac {3}{4}\right ) + 32 a^{3} b^{6} c^{\frac {11}{2}} x^{8} \Gamma \left (\frac {3}{4}\right )} + \frac {21 a^{2} b^{\frac {25}{4}} x^{4} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {9}{4}\right )}{32 a^{5} b^{4} c^{\frac {11}{2}} x^{4} \Gamma \left (\frac {3}{4}\right ) + 64 a^{4} b^{5} c^{\frac {11}{2}} x^{6} \Gamma \left (\frac {3}{4}\right ) + 32 a^{3} b^{6} c^{\frac {11}{2}} x^{8} \Gamma \left (\frac {3}{4}\right )} + \frac {56 a b^{\frac {29}{4}} x^{6} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {9}{4}\right )}{32 a^{5} b^{4} c^{\frac {11}{2}} x^{4} \Gamma \left (\frac {3}{4}\right ) + 64 a^{4} b^{5} c^{\frac {11}{2}} x^{6} \Gamma \left (\frac {3}{4}\right ) + 32 a^{3} b^{6} c^{\frac {11}{2}} x^{8} \Gamma \left (\frac {3}{4}\right )} + \frac {32 b^{\frac {33}{4}} x^{8} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {9}{4}\right )}{32 a^{5} b^{4} c^{\frac {11}{2}} x^{4} \Gamma \left (\frac {3}{4}\right ) + 64 a^{4} b^{5} c^{\frac {11}{2}} x^{6} \Gamma \left (\frac {3}{4}\right ) + 32 a^{3} b^{6} c^{\frac {11}{2}} x^{8} \Gamma \left (\frac {3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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