Optimal. Leaf size=121 \[ \frac{\sqrt{a+b x^2} \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^4}+\frac{\left (a+b x^2\right )^{3/2} \left (3 a^2 f-2 a b e+b^2 d\right )}{3 b^4}+\frac{\left (a+b x^2\right )^{5/2} (b e-3 a f)}{5 b^4}+\frac{f \left (a+b x^2\right )^{7/2}}{7 b^4} \]
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Rubi [A] time = 0.150072, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {1811, 1799, 1850} \[ \frac{\sqrt{a+b x^2} \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^4}+\frac{\left (a+b x^2\right )^{3/2} \left (3 a^2 f-2 a b e+b^2 d\right )}{3 b^4}+\frac{\left (a+b x^2\right )^{5/2} (b e-3 a f)}{5 b^4}+\frac{f \left (a+b x^2\right )^{7/2}}{7 b^4} \]
Antiderivative was successfully verified.
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Rule 1811
Rule 1799
Rule 1850
Rubi steps
\begin{align*} \int \frac{c x+d x^3+e x^5+f x^7}{\sqrt{a+b x^2}} \, dx &=\int \frac{x \left (c+d x^2+e x^4+f x^6\right )}{\sqrt{a+b x^2}} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{c+d x+e x^2+f x^3}{\sqrt{a+b x}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{b^3 c-a b^2 d+a^2 b e-a^3 f}{b^3 \sqrt{a+b x}}+\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) \sqrt{a+b x}}{b^3}+\frac{(b e-3 a f) (a+b x)^{3/2}}{b^3}+\frac{f (a+b x)^{5/2}}{b^3}\right ) \, dx,x,x^2\right )\\ &=\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \sqrt{a+b x^2}}{b^4}+\frac{\left (b^2 d-2 a b e+3 a^2 f\right ) \left (a+b x^2\right )^{3/2}}{3 b^4}+\frac{(b e-3 a f) \left (a+b x^2\right )^{5/2}}{5 b^4}+\frac{f \left (a+b x^2\right )^{7/2}}{7 b^4}\\ \end{align*}
Mathematica [A] time = 0.0880218, size = 89, normalized size = 0.74 \[ \frac{\sqrt{a+b x^2} \left (8 a^2 b \left (7 e+3 f x^2\right )-48 a^3 f-2 a b^2 \left (35 d+14 e x^2+9 f x^4\right )+b^3 \left (105 c+35 d x^2+21 e x^4+15 f x^6\right )\right )}{105 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 99, normalized size = 0.8 \begin{align*} -{\frac{-15\,f{x}^{6}{b}^{3}+18\,a{b}^{2}f{x}^{4}-21\,{b}^{3}e{x}^{4}-24\,{a}^{2}bf{x}^{2}+28\,a{b}^{2}e{x}^{2}-35\,{b}^{3}d{x}^{2}+48\,{a}^{3}f-56\,{a}^{2}be+70\,a{b}^{2}d-105\,{b}^{3}c}{105\,{b}^{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 = 1.36614, size = 221, normalized size = 1.83 \begin{align*} \frac{{\left (15 \, b^{3} f x^{6} + 3 \,{\left (7 \, b^{3} e - 6 \, a b^{2} f\right )} x^{4} + 105 \, b^{3} c - 70 \, a b^{2} d + 56 \, a^{2} b e - 48 \, a^{3} f +{\left (35 \, b^{3} d - 28 \, a b^{2} e + 24 \, a^{2} b f\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{105 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.61059, size = 238, normalized size = 1.97 \begin{align*} \begin{cases} - \frac{16 a^{3} f \sqrt{a + b x^{2}}}{35 b^{4}} + \frac{8 a^{2} e \sqrt{a + b x^{2}}}{15 b^{3}} + \frac{8 a^{2} f x^{2} \sqrt{a + b x^{2}}}{35 b^{3}} - \frac{2 a d \sqrt{a + b x^{2}}}{3 b^{2}} - \frac{4 a e x^{2} \sqrt{a + b x^{2}}}{15 b^{2}} - \frac{6 a f x^{4} \sqrt{a + b x^{2}}}{35 b^{2}} + \frac{c \sqrt{a + b x^{2}}}{b} + \frac{d x^{2} \sqrt{a + b x^{2}}}{3 b} + \frac{e x^{4} \sqrt{a + b x^{2}}}{5 b} + \frac{f x^{6} \sqrt{a + b x^{2}}}{7 b} & \text{for}\: b \neq 0 \\\frac{\frac{c x^{2}}{2} + \frac{d x^{4}}{4} + \frac{e x^{6}}{6} + \frac{f x^{8}}{8}}{\sqrt{a}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16157, size = 207, normalized size = 1.71 \begin{align*} \frac{105 \, \sqrt{b x^{2} + a} b^{3} c + 35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} b^{2} d - 105 \, \sqrt{b x^{2} + a} a b^{2} d + 15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} f - 63 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a f + 105 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2} f - 105 \, \sqrt{b x^{2} + a} a^{3} f + 21 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} b e - 70 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a b e + 105 \, \sqrt{b x^{2} + a} a^{2} b e}{105 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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