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 = 0.20, antiderivative size = 100, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 208
Rule 897
Rule 1153
Rule 1621
Rule 1799
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*}
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Mathematica [A] time = 0.41, size = 131, normalized size = 1.31 \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.19, size = 92, normalized size = 0.92 \begin {gather*} \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} \left (-4 a^2 f x^2+6 a b e x^2+2 a b f x^4-3 b^2 c\right )}{6 a b^2 x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 210, normalized size = 2.10 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.55, size = 114, normalized size = 1.14 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 127, normalized size = 1.27 \begin {gather*} \frac {\sqrt {b \,x^{2}+a}\, f \,x^{2}}{3 b}-\frac {d \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{\sqrt {a}}+\frac {b c \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{2 a^{\frac {3}{2}}}-\frac {2 \sqrt {b \,x^{2}+a}\, a f}{3 b^{2}}+\frac {\sqrt {b \,x^{2}+a}\, e}{b}-\frac {\sqrt {b \,x^{2}+a}\, c}{2 a \,x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 104, normalized size = 1.04 \begin {gather*} \frac {\sqrt {b x^{2} + a} f x^{2}}{3 \, b} + \frac {b c \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, a^{\frac {3}{2}}} - \frac {d \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{\sqrt {a}} + \frac {\sqrt {b x^{2} + a} e}{b} - \frac {2 \, \sqrt {b x^{2} + a} a f}{3 \, b^{2}} - \frac {\sqrt {b x^{2} + a} c}{2 \, a x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.95, size = 99, normalized size = 0.99 \begin {gather*} \frac {e\,\sqrt {b\,x^2+a}}{b}-\frac {d\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {c\,\sqrt {b\,x^2+a}}{2\,a\,x^2}+\frac {b\,c\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2\,a^{3/2}}-\frac {f\,\sqrt {b\,x^2+a}\,\left (2\,a-b\,x^2\right )}{3\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 127.43, size = 138, normalized size = 1.38 \begin {gather*} 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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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