Optimal. Leaf size=141 \[ -\frac{64 \left (a+b x^2\right )^{7/4} (12 b c-11 a d)}{231 a^4 e^3 (e x)^{7/2}}+\frac{16 \left (a+b x^2\right )^{3/4} (12 b c-11 a d)}{33 a^3 e^3 (e x)^{7/2}}-\frac{2 (12 b c-11 a d)}{11 a^2 e^3 (e x)^{7/2} \sqrt [4]{a+b x^2}}-\frac{2 c}{11 a e (e x)^{11/2} \sqrt [4]{a+b x^2}} \]
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Rubi [A] time = 0.0645813, 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} \[ -\frac{64 \left (a+b x^2\right )^{7/4} (12 b c-11 a d)}{231 a^4 e^3 (e x)^{7/2}}+\frac{16 \left (a+b x^2\right )^{3/4} (12 b c-11 a d)}{33 a^3 e^3 (e x)^{7/2}}-\frac{2 (12 b c-11 a d)}{11 a^2 e^3 (e x)^{7/2} \sqrt [4]{a+b x^2}}-\frac{2 c}{11 a e (e x)^{11/2} \sqrt [4]{a+b x^2}} \]
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 )^{5/4}} \, dx &=-\frac{2 c}{11 a e (e x)^{11/2} \sqrt [4]{a+b x^2}}-\frac{(12 b c-11 a d) \int \frac{1}{(e x)^{9/2} \left (a+b x^2\right )^{5/4}} \, dx}{11 a e^2}\\ &=-\frac{2 c}{11 a e (e x)^{11/2} \sqrt [4]{a+b x^2}}-\frac{2 (12 b c-11 a d)}{11 a^2 e^3 (e x)^{7/2} \sqrt [4]{a+b x^2}}-\frac{(8 (12 b c-11 a d)) \int \frac{1}{(e x)^{9/2} \sqrt [4]{a+b x^2}} \, dx}{11 a^2 e^2}\\ &=-\frac{2 c}{11 a e (e x)^{11/2} \sqrt [4]{a+b x^2}}-\frac{2 (12 b c-11 a d)}{11 a^2 e^3 (e x)^{7/2} \sqrt [4]{a+b x^2}}+\frac{16 (12 b c-11 a d) \left (a+b x^2\right )^{3/4}}{33 a^3 e^3 (e x)^{7/2}}+\frac{(32 (12 b c-11 a d)) \int \frac{\left (a+b x^2\right )^{3/4}}{(e x)^{9/2}} \, dx}{33 a^3 e^2}\\ &=-\frac{2 c}{11 a e (e x)^{11/2} \sqrt [4]{a+b x^2}}-\frac{2 (12 b c-11 a d)}{11 a^2 e^3 (e x)^{7/2} \sqrt [4]{a+b x^2}}+\frac{16 (12 b c-11 a d) \left (a+b x^2\right )^{3/4}}{33 a^3 e^3 (e x)^{7/2}}-\frac{64 (12 b c-11 a d) \left (a+b x^2\right )^{7/4}}{231 a^4 e^3 (e x)^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0397234, size = 68, normalized size = 0.48 \[ \frac{2 x \left (-x^2 \left (-3 a^2+8 a b x^2+32 b^2 x^4\right ) (12 b c-11 a d)-21 a^3 c\right )}{231 a^4 (e x)^{13/2} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 86, normalized size = 0.6 \begin{align*} -{\frac{2\,x \left ( -352\,a{b}^{2}d{x}^{6}+384\,{b}^{3}c{x}^{6}-88\,{a}^{2}bd{x}^{4}+96\,a{b}^{2}c{x}^{4}+33\,{a}^{3}d{x}^{2}-36\,{a}^{2}bc{x}^{2}+21\,c{a}^{3} \right ) }{231\,{a}^{4}}{\frac{1}{\sqrt [4]{b{x}^{2}+a}}} \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{5}{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 = 1.6039, size = 238, normalized size = 1.69 \begin{align*} -\frac{2 \,{\left (32 \,{\left (12 \, b^{3} c - 11 \, a b^{2} d\right )} x^{6} + 8 \,{\left (12 \, a b^{2} c - 11 \, a^{2} b d\right )} x^{4} + 21 \, a^{3} c - 3 \,{\left (12 \, a^{2} b c - 11 \, a^{3} d\right )} x^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{e x}}{231 \,{\left (a^{4} b e^{7} x^{8} + a^{5} 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{5}{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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