Optimal. Leaf size=212 \[ -\frac{\sqrt{d-e x} \sqrt{d+e x} \left (5 a e^4+6 b d^2 e^2+8 c d^4\right )}{16 d^6 x^2}-\frac{e^2 \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \left (5 a e^4+6 b d^2 e^2+8 c d^4\right )}{16 d^7 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\sqrt{d-e x} \sqrt{d+e x} \left (5 a e^2+6 b d^2\right )}{24 d^4 x^4}-\frac{a \sqrt{d-e x} \sqrt{d+e x}}{6 d^2 x^6} \]
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Rubi [A] time = 0.372665, antiderivative size = 248, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 7, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {520, 1251, 897, 1157, 385, 199, 208} \[ -\frac{\left (d^2-e^2 x^2\right ) \left (5 a e^4+6 b d^2 e^2+8 c d^4\right )}{16 d^6 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{e^2 \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \left (5 a e^4+6 b d^2 e^2+8 c d^4\right )}{16 d^7 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (5 a e^2+6 b d^2\right )}{24 d^4 x^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{a \left (d^2-e^2 x^2\right )}{6 d^2 x^6 \sqrt{d-e x} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 520
Rule 1251
Rule 897
Rule 1157
Rule 385
Rule 199
Rule 208
Rubi steps
\begin{align*} \int \frac{a+b x^2+c x^4}{x^7 \sqrt{d-e x} \sqrt{d+e x}} \, dx &=\frac{\sqrt{d^2-e^2 x^2} \int \frac{a+b x^2+c x^4}{x^7 \sqrt{d^2-e^2 x^2}} \, dx}{\sqrt{d-e x} \sqrt{d+e x}}\\ &=\frac{\sqrt{d^2-e^2 x^2} \operatorname{Subst}\left (\int \frac{a+b x+c x^2}{x^4 \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )}{2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{\sqrt{d^2-e^2 x^2} \operatorname{Subst}\left (\int \frac{\frac{c d^4+b d^2 e^2+a e^4}{e^4}-\frac{\left (2 c d^2+b e^2\right ) x^2}{e^4}+\frac{c x^4}{e^4}}{\left (\frac{d^2}{e^2}-\frac{x^2}{e^2}\right )^4} \, dx,x,\sqrt{d^2-e^2 x^2}\right )}{e^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{a \left (d^2-e^2 x^2\right )}{6 d^2 x^6 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\sqrt{d^2-e^2 x^2} \operatorname{Subst}\left (\int \frac{-5 a-\frac{6 \left (c d^4+b d^2 e^2\right )}{e^4}+\frac{6 c d^2 x^2}{e^4}}{\left (\frac{d^2}{e^2}-\frac{x^2}{e^2}\right )^3} \, dx,x,\sqrt{d^2-e^2 x^2}\right )}{6 d^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{a \left (d^2-e^2 x^2\right )}{6 d^2 x^6 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (6 b d^2+5 a e^2\right ) \left (d^2-e^2 x^2\right )}{24 d^4 x^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (\left (6 b+\frac{8 c d^2}{e^2}+\frac{5 a e^2}{d^2}\right ) \sqrt{d^2-e^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\frac{d^2}{e^2}-\frac{x^2}{e^2}\right )^2} \, dx,x,\sqrt{d^2-e^2 x^2}\right )}{8 d^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{a \left (d^2-e^2 x^2\right )}{6 d^2 x^6 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (6 b d^2+5 a e^2\right ) \left (d^2-e^2 x^2\right )}{24 d^4 x^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (8 c d^4+6 b d^2 e^2+5 a e^4\right ) \left (d^2-e^2 x^2\right )}{16 d^6 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (e^2 \left (6 b+\frac{8 c d^2}{e^2}+\frac{5 a e^2}{d^2}\right ) \sqrt{d^2-e^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{d^2}{e^2}-\frac{x^2}{e^2}} \, dx,x,\sqrt{d^2-e^2 x^2}\right )}{16 d^4 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{a \left (d^2-e^2 x^2\right )}{6 d^2 x^6 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (6 b d^2+5 a e^2\right ) \left (d^2-e^2 x^2\right )}{24 d^4 x^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (8 c d^4+6 b d^2 e^2+5 a e^4\right ) \left (d^2-e^2 x^2\right )}{16 d^6 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{e^2 \left (8 c d^4+6 b d^2 e^2+5 a e^4\right ) \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{16 d^7 \sqrt{d-e x} \sqrt{d+e x}}\\ \end{align*}
Mathematica [A] time = 0.20004, size = 173, normalized size = 0.82 \[ \frac{-d \left (d^2-e^2 x^2\right ) \left (a \left (10 d^2 e^2 x^2+8 d^4+15 e^4 x^4\right )+6 \left (3 b d^2 e^2 x^4+2 b d^4 x^2+4 c d^4 x^4\right )\right )-3 e^2 x^6 \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \left (5 a e^4+6 b d^2 e^2+8 c d^4\right )}{48 d^7 x^6 \sqrt{d-e x} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.033, size = 306, normalized size = 1.4 \begin{align*} -{\frac{{\it csgn} \left ( d \right ) }{48\,{d}^{7}{x}^{6}}\sqrt{-ex+d}\sqrt{ex+d} \left ( 15\,\ln \left ( 2\,{\frac{d \left ( \sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{\it csgn} \left ( d \right ) +d \right ) }{x}} \right ){x}^{6}a{e}^{6}+18\,\ln \left ( 2\,{\frac{d \left ( \sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{\it csgn} \left ( d \right ) +d \right ) }{x}} \right ){x}^{6}b{d}^{2}{e}^{4}+24\,\ln \left ( 2\,{\frac{d \left ( \sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{\it csgn} \left ( d \right ) +d \right ) }{x}} \right ){x}^{6}c{d}^{4}{e}^{2}+15\,{\it csgn} \left ( d \right ) d\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{x}^{4}a{e}^{4}+18\,{\it csgn} \left ( d \right ){d}^{3}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{x}^{4}b{e}^{2}+24\,{\it csgn} \left ( d \right ){d}^{5}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{x}^{4}c+10\,{\it csgn} \left ( d \right ){x}^{2}a{d}^{3}{e}^{2}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}+12\,{\it csgn} \left ( d \right ){x}^{2}b{d}^{5}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}+8\,{\it csgn} \left ( d \right ) a{d}^{5}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ){\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44854, size = 298, normalized size = 1.41 \begin{align*} \frac{3 \,{\left (8 \, c d^{4} e^{2} + 6 \, b d^{2} e^{4} + 5 \, a e^{6}\right )} x^{6} \log \left (\frac{\sqrt{e x + d} \sqrt{-e x + d} - d}{x}\right ) -{\left (8 \, a d^{5} + 3 \,{\left (8 \, c d^{5} + 6 \, b d^{3} e^{2} + 5 \, a d e^{4}\right )} x^{4} + 2 \,{\left (6 \, b d^{5} + 5 \, a d^{3} e^{2}\right )} x^{2}\right )} \sqrt{e x + d} \sqrt{-e x + d}}{48 \, d^{7} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: MellinTransformStripError} \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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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