Optimal. Leaf size=140 \[ \frac{\sqrt{a+b x^2} \left (70 a^2 b e-105 a^3 f-56 a b^2 d+48 b^3 c\right )}{105 a^4 x}-\frac{\sqrt{a+b x^2} \left (35 a^2 e-28 a b d+24 b^2 c\right )}{105 a^3 x^3}+\frac{\sqrt{a+b x^2} (6 b c-7 a d)}{35 a^2 x^5}-\frac{c \sqrt{a+b x^2}}{7 a x^7} \]
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Rubi [A] time = 0.184285, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094, Rules used = {1803, 12, 264} \[ \frac{\sqrt{a+b x^2} \left (70 a^2 b e-105 a^3 f-56 a b^2 d+48 b^3 c\right )}{105 a^4 x}-\frac{\sqrt{a+b x^2} \left (35 a^2 e-28 a b d+24 b^2 c\right )}{105 a^3 x^3}+\frac{\sqrt{a+b x^2} (6 b c-7 a d)}{35 a^2 x^5}-\frac{c \sqrt{a+b x^2}}{7 a x^7} \]
Antiderivative was successfully verified.
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Rule 1803
Rule 12
Rule 264
Rubi steps
\begin{align*} \int \frac{c+d x^2+e x^4+f x^6}{x^8 \sqrt{a+b x^2}} \, dx &=-\frac{c \sqrt{a+b x^2}}{7 a x^7}-\frac{\int \frac{6 b c-7 a \left (d+e x^2+f x^4\right )}{x^6 \sqrt{a+b x^2}} \, dx}{7 a}\\ &=-\frac{c \sqrt{a+b x^2}}{7 a x^7}+\frac{(6 b c-7 a d) \sqrt{a+b x^2}}{35 a^2 x^5}+\frac{\int \frac{4 b (6 b c-7 a d)-5 a \left (-7 a e-7 a f x^2\right )}{x^4 \sqrt{a+b x^2}} \, dx}{35 a^2}\\ &=-\frac{c \sqrt{a+b x^2}}{7 a x^7}+\frac{(6 b c-7 a d) \sqrt{a+b x^2}}{35 a^2 x^5}-\frac{\left (24 b^2 c-28 a b d+35 a^2 e\right ) \sqrt{a+b x^2}}{105 a^3 x^3}-\frac{\int \frac{2 b \left (24 b^2 c-28 a b d+35 a^2 e\right )-105 a^3 f}{x^2 \sqrt{a+b x^2}} \, dx}{105 a^3}\\ &=-\frac{c \sqrt{a+b x^2}}{7 a x^7}+\frac{(6 b c-7 a d) \sqrt{a+b x^2}}{35 a^2 x^5}-\frac{\left (24 b^2 c-28 a b d+35 a^2 e\right ) \sqrt{a+b x^2}}{105 a^3 x^3}-\frac{\left (48 b^3 c-56 a b^2 d+70 a^2 b e-105 a^3 f\right ) \int \frac{1}{x^2 \sqrt{a+b x^2}} \, dx}{105 a^3}\\ &=-\frac{c \sqrt{a+b x^2}}{7 a x^7}+\frac{(6 b c-7 a d) \sqrt{a+b x^2}}{35 a^2 x^5}-\frac{\left (24 b^2 c-28 a b d+35 a^2 e\right ) \sqrt{a+b x^2}}{105 a^3 x^3}+\frac{\left (48 b^3 c-56 a b^2 d+70 a^2 b e-105 a^3 f\right ) \sqrt{a+b x^2}}{105 a^4 x}\\ \end{align*}
Mathematica [A] time = 0.0782557, size = 103, normalized size = 0.74 \[ \frac{\sqrt{a+b x^2} \left (2 a^2 b x^2 \left (9 c+14 d x^2+35 e x^4\right )-a^3 \left (15 c+21 d x^2+35 x^4 \left (e+3 f x^2\right )\right )-8 a b^2 x^4 \left (3 c+7 d x^2\right )+48 b^3 c x^6\right )}{105 a^4 x^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 111, normalized size = 0.8 \begin{align*} -{\frac{105\,{a}^{3}f{x}^{6}-70\,{a}^{2}be{x}^{6}+56\,a{b}^{2}d{x}^{6}-48\,{b}^{3}c{x}^{6}+35\,{a}^{3}e{x}^{4}-28\,{a}^{2}bd{x}^{4}+24\,a{b}^{2}c{x}^{4}+21\,{a}^{3}d{x}^{2}-18\,{a}^{2}bc{x}^{2}+15\,c{a}^{3}}{105\,{x}^{7}{a}^{4}}\sqrt{b{x}^{2}+a}} \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 = 1.66397, size = 232, normalized size = 1.66 \begin{align*} \frac{{\left ({\left (48 \, b^{3} c - 56 \, a b^{2} d + 70 \, a^{2} b e - 105 \, a^{3} f\right )} x^{6} -{\left (24 \, a b^{2} c - 28 \, a^{2} b d + 35 \, a^{3} e\right )} x^{4} - 15 \, a^{3} c + 3 \,{\left (6 \, a^{2} b c - 7 \, a^{3} d\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{105 \, a^{4} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 4.74422, size = 891, normalized size = 6.36 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22979, size = 748, normalized size = 5.34 \begin{align*} \frac{2 \,{\left (105 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} \sqrt{b} f - 630 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} a \sqrt{b} f + 210 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} b^{\frac{3}{2}} e + 560 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} b^{\frac{5}{2}} d + 1575 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} a^{2} \sqrt{b} f - 910 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} a b^{\frac{3}{2}} e + 1680 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} b^{\frac{7}{2}} c - 1400 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a b^{\frac{5}{2}} d - 2100 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a^{3} \sqrt{b} f + 1540 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a^{2} b^{\frac{3}{2}} e - 1008 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a b^{\frac{7}{2}} c + 1176 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{2} b^{\frac{5}{2}} d + 1575 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{4} \sqrt{b} f - 1260 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{3} b^{\frac{3}{2}} e + 336 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{2} b^{\frac{7}{2}} c - 392 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{3} b^{\frac{5}{2}} d - 630 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{5} \sqrt{b} f + 490 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{4} b^{\frac{3}{2}} e - 48 \, a^{3} b^{\frac{7}{2}} c + 56 \, a^{4} b^{\frac{5}{2}} d + 105 \, a^{6} \sqrt{b} f - 70 \, a^{5} b^{\frac{3}{2}} e\right )}}{105 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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