Optimal. Leaf size=178 \[ -\frac{256 \left (a+b x^2\right )^{7/4} (16 b c-11 a d)}{385 a^5 e^3 (e x)^{7/2}}+\frac{64 \left (a+b x^2\right )^{3/4} (16 b c-11 a d)}{55 a^4 e^3 (e x)^{7/2}}-\frac{24 (16 b c-11 a d)}{55 a^3 e^3 (e x)^{7/2} \sqrt [4]{a+b x^2}}-\frac{2 (16 b c-11 a d)}{55 a^2 e^3 (e x)^{7/2} \left (a+b x^2\right )^{5/4}}-\frac{2 c}{11 a e (e x)^{11/2} \left (a+b x^2\right )^{5/4}} \]
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Rubi [A] time = 0.0903243, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {453, 273, 264} \[ -\frac{256 \left (a+b x^2\right )^{7/4} (16 b c-11 a d)}{385 a^5 e^3 (e x)^{7/2}}+\frac{64 \left (a+b x^2\right )^{3/4} (16 b c-11 a d)}{55 a^4 e^3 (e x)^{7/2}}-\frac{24 (16 b c-11 a d)}{55 a^3 e^3 (e x)^{7/2} \sqrt [4]{a+b x^2}}-\frac{2 (16 b c-11 a d)}{55 a^2 e^3 (e x)^{7/2} \left (a+b x^2\right )^{5/4}}-\frac{2 c}{11 a e (e x)^{11/2} \left (a+b x^2\right )^{5/4}} \]
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)^{13/2} \left (a+b x^2\right )^{9/4}} \, dx &=-\frac{2 c}{11 a e (e x)^{11/2} \left (a+b x^2\right )^{5/4}}-\frac{(16 b c-11 a d) \int \frac{1}{(e x)^{9/2} \left (a+b x^2\right )^{9/4}} \, dx}{11 a e^2}\\ &=-\frac{2 c}{11 a e (e x)^{11/2} \left (a+b x^2\right )^{5/4}}-\frac{2 (16 b c-11 a d)}{55 a^2 e^3 (e x)^{7/2} \left (a+b x^2\right )^{5/4}}-\frac{(12 (16 b c-11 a d)) \int \frac{1}{(e x)^{9/2} \left (a+b x^2\right )^{5/4}} \, dx}{55 a^2 e^2}\\ &=-\frac{2 c}{11 a e (e x)^{11/2} \left (a+b x^2\right )^{5/4}}-\frac{2 (16 b c-11 a d)}{55 a^2 e^3 (e x)^{7/2} \left (a+b x^2\right )^{5/4}}-\frac{24 (16 b c-11 a d)}{55 a^3 e^3 (e x)^{7/2} \sqrt [4]{a+b x^2}}-\frac{(96 (16 b c-11 a d)) \int \frac{1}{(e x)^{9/2} \sqrt [4]{a+b x^2}} \, dx}{55 a^3 e^2}\\ &=-\frac{2 c}{11 a e (e x)^{11/2} \left (a+b x^2\right )^{5/4}}-\frac{2 (16 b c-11 a d)}{55 a^2 e^3 (e x)^{7/2} \left (a+b x^2\right )^{5/4}}-\frac{24 (16 b c-11 a d)}{55 a^3 e^3 (e x)^{7/2} \sqrt [4]{a+b x^2}}+\frac{64 (16 b c-11 a d) \left (a+b x^2\right )^{3/4}}{55 a^4 e^3 (e x)^{7/2}}+\frac{(128 (16 b c-11 a d)) \int \frac{\left (a+b x^2\right )^{3/4}}{(e x)^{9/2}} \, dx}{55 a^4 e^2}\\ &=-\frac{2 c}{11 a e (e x)^{11/2} \left (a+b x^2\right )^{5/4}}-\frac{2 (16 b c-11 a d)}{55 a^2 e^3 (e x)^{7/2} \left (a+b x^2\right )^{5/4}}-\frac{24 (16 b c-11 a d)}{55 a^3 e^3 (e x)^{7/2} \sqrt [4]{a+b x^2}}+\frac{64 (16 b c-11 a d) \left (a+b x^2\right )^{3/4}}{55 a^4 e^3 (e x)^{7/2}}-\frac{256 (16 b c-11 a d) \left (a+b x^2\right )^{7/4}}{385 a^5 e^3 (e x)^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.054169, size = 79, normalized size = 0.44 \[ \frac{2 x \left (a x^2 \left (20 a^2 b x^2-5 a^3+160 a b^2 x^4+128 b^3 x^6\right ) (11 a d-16 b c)-35 a^5 c\right )}{385 a^6 (e x)^{13/2} \left (a+b x^2\right )^{5/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 110, normalized size = 0.6 \begin{align*} -{\frac{2\,x \left ( -1408\,a{b}^{3}d{x}^{8}+2048\,{b}^{4}c{x}^{8}-1760\,{a}^{2}{b}^{2}d{x}^{6}+2560\,a{b}^{3}c{x}^{6}-220\,{a}^{3}bd{x}^{4}+320\,{a}^{2}{b}^{2}c{x}^{4}+55\,{a}^{4}d{x}^{2}-80\,{a}^{3}bc{x}^{2}+35\,c{a}^{4} \right ) }{385\,{a}^{5}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}} \left ( ex \right ) ^{-{\frac{13}{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{13}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68508, size = 323, normalized size = 1.81 \begin{align*} -\frac{2 \,{\left (128 \,{\left (16 \, b^{4} c - 11 \, a b^{3} d\right )} x^{8} + 160 \,{\left (16 \, a b^{3} c - 11 \, a^{2} b^{2} d\right )} x^{6} + 35 \, a^{4} c + 20 \,{\left (16 \, a^{2} b^{2} c - 11 \, a^{3} b d\right )} x^{4} - 5 \,{\left (16 \, a^{3} b c - 11 \, a^{4} d\right )} x^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{e x}}{385 \,{\left (a^{5} b^{2} e^{7} x^{10} + 2 \, a^{6} b e^{7} x^{8} + a^{7} e^{7} x^{6}\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{13}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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