Optimal. Leaf size=141 \[ \frac{64 \left (a+b x^2\right )^{9/4} (12 b c-13 a d)}{585 a^4 e^3 (e x)^{9/2}}-\frac{16 \left (a+b x^2\right )^{5/4} (12 b c-13 a d)}{65 a^3 e^3 (e x)^{9/2}}+\frac{2 \sqrt [4]{a+b x^2} (12 b c-13 a d)}{13 a^2 e^3 (e x)^{9/2}}-\frac{2 c \sqrt [4]{a+b x^2}}{13 a e (e x)^{13/2}} \]
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Rubi [A] time = 0.0696162, 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} \[ \frac{64 \left (a+b x^2\right )^{9/4} (12 b c-13 a d)}{585 a^4 e^3 (e x)^{9/2}}-\frac{16 \left (a+b x^2\right )^{5/4} (12 b c-13 a d)}{65 a^3 e^3 (e x)^{9/2}}+\frac{2 \sqrt [4]{a+b x^2} (12 b c-13 a d)}{13 a^2 e^3 (e x)^{9/2}}-\frac{2 c \sqrt [4]{a+b x^2}}{13 a e (e x)^{13/2}} \]
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)^{15/2} \left (a+b x^2\right )^{3/4}} \, dx &=-\frac{2 c \sqrt [4]{a+b x^2}}{13 a e (e x)^{13/2}}-\frac{(12 b c-13 a d) \int \frac{1}{(e x)^{11/2} \left (a+b x^2\right )^{3/4}} \, dx}{13 a e^2}\\ &=-\frac{2 c \sqrt [4]{a+b x^2}}{13 a e (e x)^{13/2}}+\frac{2 (12 b c-13 a d) \sqrt [4]{a+b x^2}}{13 a^2 e^3 (e x)^{9/2}}+\frac{(8 (12 b c-13 a d)) \int \frac{\sqrt [4]{a+b x^2}}{(e x)^{11/2}} \, dx}{13 a^2 e^2}\\ &=-\frac{2 c \sqrt [4]{a+b x^2}}{13 a e (e x)^{13/2}}+\frac{2 (12 b c-13 a d) \sqrt [4]{a+b x^2}}{13 a^2 e^3 (e x)^{9/2}}-\frac{16 (12 b c-13 a d) \left (a+b x^2\right )^{5/4}}{65 a^3 e^3 (e x)^{9/2}}-\frac{(32 (12 b c-13 a d)) \int \frac{\left (a+b x^2\right )^{5/4}}{(e x)^{11/2}} \, dx}{65 a^3 e^2}\\ &=-\frac{2 c \sqrt [4]{a+b x^2}}{13 a e (e x)^{13/2}}+\frac{2 (12 b c-13 a d) \sqrt [4]{a+b x^2}}{13 a^2 e^3 (e x)^{9/2}}-\frac{16 (12 b c-13 a d) \left (a+b x^2\right )^{5/4}}{65 a^3 e^3 (e x)^{9/2}}+\frac{64 (12 b c-13 a d) \left (a+b x^2\right )^{9/4}}{585 a^4 e^3 (e x)^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.0477906, size = 94, normalized size = 0.67 \[ -\frac{2 \sqrt{e x} \sqrt [4]{a+b x^2} \left (-4 a^2 b x^2 \left (15 c+26 d x^2\right )+5 a^3 \left (9 c+13 d x^2\right )+32 a b^2 x^4 \left (3 c+13 d x^2\right )-384 b^3 c x^6\right )}{585 a^4 e^8 x^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 86, normalized size = 0.6 \begin{align*} -{\frac{2\,x \left ( 416\,a{b}^{2}d{x}^{6}-384\,{b}^{3}c{x}^{6}-104\,{a}^{2}bd{x}^{4}+96\,a{b}^{2}c{x}^{4}+65\,{a}^{3}d{x}^{2}-60\,{a}^{2}bc{x}^{2}+45\,c{a}^{3} \right ) }{585\,{a}^{4}}\sqrt [4]{b{x}^{2}+a} \left ( ex \right ) ^{-{\frac{15}{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{3}{4}} \left (e x\right )^{\frac{15}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.91521, size = 215, normalized size = 1.52 \begin{align*} \frac{2 \,{\left (32 \,{\left (12 \, b^{3} c - 13 \, a b^{2} d\right )} x^{6} - 8 \,{\left (12 \, a b^{2} c - 13 \, a^{2} b d\right )} x^{4} - 45 \, a^{3} c + 5 \,{\left (12 \, a^{2} b c - 13 \, a^{3} d\right )} x^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{e x}}{585 \, a^{4} e^{8} x^{7}} \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{3}{4}} \left (e x\right )^{\frac{15}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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