Optimal. Leaf size=141 \[ \frac{64 \left (a+b x^2\right )^{3/4} (12 b c-7 a d)}{105 a^4 e^3 (e x)^{3/2}}-\frac{16 (12 b c-7 a d)}{35 a^3 e^3 (e x)^{3/2} \sqrt [4]{a+b x^2}}-\frac{2 (12 b c-7 a d)}{35 a^2 e^3 (e x)^{3/2} \left (a+b x^2\right )^{5/4}}-\frac{2 c}{7 a e (e x)^{7/2} \left (a+b x^2\right )^{5/4}} \]
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Rubi [A] time = 0.06919, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {453, 273, 264} \[ \frac{64 \left (a+b x^2\right )^{3/4} (12 b c-7 a d)}{105 a^4 e^3 (e x)^{3/2}}-\frac{16 (12 b c-7 a d)}{35 a^3 e^3 (e x)^{3/2} \sqrt [4]{a+b x^2}}-\frac{2 (12 b c-7 a d)}{35 a^2 e^3 (e x)^{3/2} \left (a+b x^2\right )^{5/4}}-\frac{2 c}{7 a e (e x)^{7/2} \left (a+b x^2\right )^{5/4}} \]
Antiderivative was successfully verified.
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Rule 453
Rule 273
Rule 264
Rubi steps
\begin{align*} \int \frac{c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{9/4}} \, dx &=-\frac{2 c}{7 a e (e x)^{7/2} \left (a+b x^2\right )^{5/4}}-\frac{(12 b c-7 a d) \int \frac{1}{(e x)^{5/2} \left (a+b x^2\right )^{9/4}} \, dx}{7 a e^2}\\ &=-\frac{2 c}{7 a e (e x)^{7/2} \left (a+b x^2\right )^{5/4}}-\frac{2 (12 b c-7 a d)}{35 a^2 e^3 (e x)^{3/2} \left (a+b x^2\right )^{5/4}}-\frac{(8 (12 b c-7 a d)) \int \frac{1}{(e x)^{5/2} \left (a+b x^2\right )^{5/4}} \, dx}{35 a^2 e^2}\\ &=-\frac{2 c}{7 a e (e x)^{7/2} \left (a+b x^2\right )^{5/4}}-\frac{2 (12 b c-7 a d)}{35 a^2 e^3 (e x)^{3/2} \left (a+b x^2\right )^{5/4}}-\frac{16 (12 b c-7 a d)}{35 a^3 e^3 (e x)^{3/2} \sqrt [4]{a+b x^2}}-\frac{(32 (12 b c-7 a d)) \int \frac{1}{(e x)^{5/2} \sqrt [4]{a+b x^2}} \, dx}{35 a^3 e^2}\\ &=-\frac{2 c}{7 a e (e x)^{7/2} \left (a+b x^2\right )^{5/4}}-\frac{2 (12 b c-7 a d)}{35 a^2 e^3 (e x)^{3/2} \left (a+b x^2\right )^{5/4}}-\frac{16 (12 b c-7 a d)}{35 a^3 e^3 (e x)^{3/2} \sqrt [4]{a+b x^2}}+\frac{64 (12 b c-7 a d) \left (a+b x^2\right )^{3/4}}{105 a^4 e^3 (e x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0483768, size = 94, normalized size = 0.67 \[ \frac{\sqrt{e x} \left (40 a^2 b x^2 \left (3 c-14 d x^2\right )-10 a^3 \left (3 c+7 d x^2\right )+64 a b^2 x^4 \left (15 c-7 d x^2\right )+768 b^3 c x^6\right )}{105 a^4 e^5 x^4 \left (a+b x^2\right )^{5/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 86, normalized size = 0.6 \begin{align*} -{\frac{2\,x \left ( 224\,a{b}^{2}d{x}^{6}-384\,{b}^{3}c{x}^{6}+280\,{a}^{2}bd{x}^{4}-480\,a{b}^{2}c{x}^{4}+35\,{a}^{3}d{x}^{2}-60\,{a}^{2}bc{x}^{2}+15\,c{a}^{3} \right ) }{105\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}} \left ( ex \right ) ^{-{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{9}{4}} \left (e x\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58664, size = 261, normalized size = 1.85 \begin{align*} \frac{2 \,{\left (32 \,{\left (12 \, b^{3} c - 7 \, a b^{2} d\right )} x^{6} + 40 \,{\left (12 \, a b^{2} c - 7 \, a^{2} b d\right )} x^{4} - 15 \, a^{3} c + 5 \,{\left (12 \, a^{2} b c - 7 \, a^{3} d\right )} x^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{e x}}{105 \,{\left (a^{4} b^{2} e^{5} x^{8} + 2 \, a^{5} b e^{5} x^{6} + a^{6} e^{5} x^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{9}{4}} \left (e x\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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