Optimal. Leaf size=100 \[ \frac{(b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{3/2}}+\frac{\sqrt{a+b x^2} (b e-a f)}{b^2}+\frac{f \left (a+b x^2\right )^{3/2}}{3 b^2}-\frac{c \sqrt{a+b x^2}}{2 a x^2} \]
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Rubi [A] time = 0.202996, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {1799, 1621, 897, 1153, 208} \[ \frac{(b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{3/2}}+\frac{\sqrt{a+b x^2} (b e-a f)}{b^2}+\frac{f \left (a+b x^2\right )^{3/2}}{3 b^2}-\frac{c \sqrt{a+b x^2}}{2 a x^2} \]
Antiderivative was successfully verified.
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Rule 1799
Rule 1621
Rule 897
Rule 1153
Rule 208
Rubi steps
\begin{align*} \int \frac{c+d x^2+e x^4+f x^6}{x^3 \sqrt{a+b x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{c+d x+e x^2+f x^3}{x^2 \sqrt{a+b x}} \, dx,x,x^2\right )\\ &=-\frac{c \sqrt{a+b x^2}}{2 a x^2}-\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{2} (b c-2 a d)-a e x-a f x^2}{x \sqrt{a+b x}} \, dx,x,x^2\right )}{2 a}\\ &=-\frac{c \sqrt{a+b x^2}}{2 a x^2}-\frac{\operatorname{Subst}\left (\int \frac{\frac{\frac{1}{2} b^2 (b c-2 a d)+a^2 b e-a^3 f}{b^2}-\frac{\left (a b e-2 a^2 f\right ) x^2}{b^2}-\frac{a f x^4}{b^2}}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )}{a b}\\ &=-\frac{c \sqrt{a+b x^2}}{2 a x^2}-\frac{\operatorname{Subst}\left (\int \left (-a \left (e-\frac{a f}{b}\right )-\frac{a f x^2}{b}+\frac{b c-2 a d}{2 \left (-\frac{a}{b}+\frac{x^2}{b}\right )}\right ) \, dx,x,\sqrt{a+b x^2}\right )}{a b}\\ &=\frac{(b e-a f) \sqrt{a+b x^2}}{b^2}-\frac{c \sqrt{a+b x^2}}{2 a x^2}+\frac{f \left (a+b x^2\right )^{3/2}}{3 b^2}-\frac{(b c-2 a d) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )}{2 a b}\\ &=\frac{(b e-a f) \sqrt{a+b x^2}}{b^2}-\frac{c \sqrt{a+b x^2}}{2 a x^2}+\frac{f \left (a+b x^2\right )^{3/2}}{3 b^2}+\frac{(b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.382209, size = 131, normalized size = 1.31 \[ \frac{3 b^3 c x^2 \sqrt{\frac{b x^2}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{b x^2}{a}+1}\right )-\left (a+b x^2\right ) \left (4 a^2 f x^2-2 a b x^2 \left (3 e+f x^2\right )+3 b^2 c\right )}{6 a b^2 x^2 \sqrt{a+b x^2}}-\frac{d \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{\sqrt{a}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 127, normalized size = 1.3 \begin{align*}{\frac{f{x}^{2}}{3\,b}\sqrt{b{x}^{2}+a}}-{\frac{2\,af}{3\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{e}{b}\sqrt{b{x}^{2}+a}}-{d\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}-{\frac{c}{2\,a{x}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{bc}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{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 = 1.46024, size = 482, normalized size = 4.82 \begin{align*} \left [-\frac{3 \,{\left (b^{3} c - 2 \, a b^{2} d\right )} \sqrt{a} x^{2} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) - 2 \,{\left (2 \, a^{2} b f x^{4} - 3 \, a b^{2} c + 2 \,{\left (3 \, a^{2} b e - 2 \, a^{3} f\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{12 \, a^{2} b^{2} x^{2}}, -\frac{3 \,{\left (b^{3} c - 2 \, a b^{2} d\right )} \sqrt{-a} x^{2} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left (2 \, a^{2} b f x^{4} - 3 \, a b^{2} c + 2 \,{\left (3 \, a^{2} b e - 2 \, a^{3} f\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{6 \, a^{2} b^{2} x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 40.1251, size = 138, normalized size = 1.38 \begin{align*} e \left (\begin{cases} \frac{x^{2}}{2 \sqrt{a}} & \text{for}\: b = 0 \\\frac{\sqrt{a + b x^{2}}}{b} & \text{otherwise} \end{cases}\right ) + f \left (\begin{cases} - \frac{2 a \sqrt{a + b x^{2}}}{3 b^{2}} + \frac{x^{2} \sqrt{a + b x^{2}}}{3 b} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 \sqrt{a}} & \text{otherwise} \end{cases}\right ) - \frac{\sqrt{b} c \sqrt{\frac{a}{b x^{2}} + 1}}{2 a x} - \frac{d \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{\sqrt{a}} + \frac{b c \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2 a^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23002, size = 154, normalized size = 1.54 \begin{align*} -\frac{\frac{3 \,{\left (b^{2} c - 2 \, a b d\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{3 \, \sqrt{b x^{2} + a} b c}{a x^{2}} - \frac{2 \,{\left ({\left (b x^{2} + a\right )}^{\frac{3}{2}} b^{2} f - 3 \, \sqrt{b x^{2} + a} a b^{2} f + 3 \, \sqrt{b x^{2} + a} b^{3} e\right )}}{b^{3}}}{6 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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