Optimal. Leaf size=117 \[ -\frac {c \sqrt {a+b x^2}}{a x}+\frac {(4 b e-3 a f) x \sqrt {a+b x^2}}{8 b^2}+\frac {f x^3 \sqrt {a+b x^2}}{4 b}+\frac {\left (8 b^2 d-4 a b e+3 a^2 f\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{5/2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1821, 1599,
1173, 396, 223, 212} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 f-4 a b e+8 b^2 d\right )}{8 b^{5/2}}+\frac {x \sqrt {a+b x^2} (4 b e-3 a f)}{8 b^2}-\frac {c \sqrt {a+b x^2}}{a x}+\frac {f x^3 \sqrt {a+b x^2}}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 396
Rule 1173
Rule 1599
Rule 1821
Rubi steps
\begin {align*} \int \frac {c+d x^2+e x^4+f x^6}{x^2 \sqrt {a+b x^2}} \, dx &=-\frac {c \sqrt {a+b x^2}}{a x}-\frac {\int \frac {-a d x-a e x^3-a f x^5}{x \sqrt {a+b x^2}} \, dx}{a}\\ &=-\frac {c \sqrt {a+b x^2}}{a x}-\frac {\int \frac {-a d-a e x^2-a f x^4}{\sqrt {a+b x^2}} \, dx}{a}\\ &=-\frac {c \sqrt {a+b x^2}}{a x}+\frac {f x^3 \sqrt {a+b x^2}}{4 b}-\frac {\int \frac {-4 a b d-a (4 b e-3 a f) x^2}{\sqrt {a+b x^2}} \, dx}{4 a b}\\ &=-\frac {c \sqrt {a+b x^2}}{a x}+\frac {(4 b e-3 a f) x \sqrt {a+b x^2}}{8 b^2}+\frac {f x^3 \sqrt {a+b x^2}}{4 b}+\frac {\left (8 a b^2 d-a^2 (4 b e-3 a f)\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{8 a b^2}\\ &=-\frac {c \sqrt {a+b x^2}}{a x}+\frac {(4 b e-3 a f) x \sqrt {a+b x^2}}{8 b^2}+\frac {f x^3 \sqrt {a+b x^2}}{4 b}+\frac {\left (8 a b^2 d-a^2 (4 b e-3 a f)\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{8 a b^2}\\ &=-\frac {c \sqrt {a+b x^2}}{a x}+\frac {(4 b e-3 a f) x \sqrt {a+b x^2}}{8 b^2}+\frac {f x^3 \sqrt {a+b x^2}}{4 b}+\frac {\left (8 b^2 d-4 a b e+3 a^2 f\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 105, normalized size = 0.90 \begin {gather*} \frac {\sqrt {a+b x^2} \left (-8 b^2 c+4 a b e x^2-3 a^2 f x^2+2 a b f x^4\right )}{8 a b^2 x}+\frac {\left (-8 b^2 d+4 a b e-3 a^2 f\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{8 b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 145, normalized size = 1.24
method | result | size |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (-2 a b f \,x^{4}+3 a^{2} f \,x^{2}-4 a b e \,x^{2}+8 b^{2} c \right )}{8 b^{2} a x}+\frac {3 \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) a^{2} f}{8 b^{\frac {5}{2}}}-\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) a e}{2 b^{\frac {3}{2}}}+\frac {d \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}\) | \(123\) |
default | \(f \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )+e \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )+\frac {d \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}-\frac {c \sqrt {b \,x^{2}+a}}{a x}\) | \(145\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 120, normalized size = 1.03 \begin {gather*} \frac {\sqrt {b x^{2} + a} f x^{3}}{4 \, b} - \frac {3 \, \sqrt {b x^{2} + a} a f x}{8 \, b^{2}} + \frac {d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b}} + \frac {3 \, a^{2} f \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {5}{2}}} + \frac {\sqrt {b x^{2} + a} x e}{2 \, b} - \frac {a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) e}{2 \, b^{\frac {3}{2}}} - \frac {\sqrt {b x^{2} + a} c}{a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.30, size = 229, normalized size = 1.96 \begin {gather*} \left [\frac {{\left (4 \, a^{2} b x e - {\left (8 \, a b^{2} d + 3 \, a^{3} f\right )} x\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (2 \, a b^{2} f x^{4} - 3 \, a^{2} b f x^{2} + 4 \, a b^{2} x^{2} e - 8 \, b^{3} c\right )} \sqrt {b x^{2} + a}}{16 \, a b^{3} x}, \frac {{\left (4 \, a^{2} b x e - {\left (8 \, a b^{2} d + 3 \, a^{3} f\right )} x\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (2 \, a b^{2} f x^{4} - 3 \, a^{2} b f x^{2} + 4 \, a b^{2} x^{2} e - 8 \, b^{3} c\right )} \sqrt {b x^{2} + a}}{8 \, a b^{3} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 4.23, size = 250, normalized size = 2.14 \begin {gather*} - \frac {3 a^{\frac {3}{2}} f x}{8 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {\sqrt {a} e x \sqrt {1 + \frac {b x^{2}}{a}}}{2 b} - \frac {\sqrt {a} f x^{3}}{8 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 a^{2} f \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{8 b^{\frac {5}{2}}} - \frac {a e \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{2 b^{\frac {3}{2}}} + d \left (\begin {cases} \frac {\sqrt {- \frac {a}{b}} \operatorname {asin}{\left (x \sqrt {- \frac {b}{a}} \right )}}{\sqrt {a}} & \text {for}\: a > 0 \wedge b < 0 \\\frac {\sqrt {\frac {a}{b}} \operatorname {asinh}{\left (x \sqrt {\frac {b}{a}} \right )}}{\sqrt {a}} & \text {for}\: a > 0 \wedge b > 0 \\\frac {\sqrt {- \frac {a}{b}} \operatorname {acosh}{\left (x \sqrt {- \frac {b}{a}} \right )}}{\sqrt {- a}} & \text {for}\: b > 0 \wedge a < 0 \end {cases}\right ) - \frac {\sqrt {b} c \sqrt {\frac {a}{b x^{2}} + 1}}{a} + \frac {f x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.27, size = 121, normalized size = 1.03 \begin {gather*} \frac {1}{8} \, \sqrt {b x^{2} + a} {\left (\frac {2 \, f x^{2}}{b} - \frac {3 \, a b f - 4 \, b^{2} e}{b^{3}}\right )} x + \frac {2 \, \sqrt {b} c}{{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a} - \frac {{\left (8 \, b^{\frac {5}{2}} d + 3 \, a^{2} \sqrt {b} f - 4 \, a b^{\frac {3}{2}} e\right )} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right )}{16 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {f\,x^6+e\,x^4+d\,x^2+c}{x^2\,\sqrt {b\,x^2+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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