Optimal. Leaf size=110 \[ \frac {\sqrt {a+b x^2} (2 b c-3 a d)}{3 a^2 x}+\frac {(2 b e-a f) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2}}-\frac {c \sqrt {a+b x^2}}{3 a x^3}+\frac {f x \sqrt {a+b x^2}}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 110, 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 = {1807, 1585, 1265, 388, 217, 206} \[ \frac {\sqrt {a+b x^2} (2 b c-3 a d)}{3 a^2 x}+\frac {(2 b e-a f) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2}}-\frac {c \sqrt {a+b x^2}}{3 a x^3}+\frac {f x \sqrt {a+b x^2}}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 217
Rule 388
Rule 1265
Rule 1585
Rule 1807
Rubi steps
\begin {align*} \int \frac {c+d x^2+e x^4+f x^6}{x^4 \sqrt {a+b x^2}} \, dx &=-\frac {c \sqrt {a+b x^2}}{3 a x^3}-\frac {\int \frac {(2 b c-3 a d) x-3 a e x^3-3 a f x^5}{x^3 \sqrt {a+b x^2}} \, dx}{3 a}\\ &=-\frac {c \sqrt {a+b x^2}}{3 a x^3}-\frac {\int \frac {2 b c-3 a d-3 a e x^2-3 a f x^4}{x^2 \sqrt {a+b x^2}} \, dx}{3 a}\\ &=-\frac {c \sqrt {a+b x^2}}{3 a x^3}+\frac {(2 b c-3 a d) \sqrt {a+b x^2}}{3 a^2 x}+\frac {\int \frac {3 a^2 e+3 a^2 f x^2}{\sqrt {a+b x^2}} \, dx}{3 a^2}\\ &=-\frac {c \sqrt {a+b x^2}}{3 a x^3}+\frac {(2 b c-3 a d) \sqrt {a+b x^2}}{3 a^2 x}+\frac {f x \sqrt {a+b x^2}}{2 b}+\frac {(2 b e-a f) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{2 b}\\ &=-\frac {c \sqrt {a+b x^2}}{3 a x^3}+\frac {(2 b c-3 a d) \sqrt {a+b x^2}}{3 a^2 x}+\frac {f x \sqrt {a+b x^2}}{2 b}+\frac {(2 b e-a f) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{2 b}\\ &=-\frac {c \sqrt {a+b x^2}}{3 a x^3}+\frac {(2 b c-3 a d) \sqrt {a+b x^2}}{3 a^2 x}+\frac {f x \sqrt {a+b x^2}}{2 b}+\frac {(2 b e-a f) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 93, normalized size = 0.85 \[ \frac {\sqrt {a+b x^2} \left (3 a^2 f x^4-2 a b \left (c+3 d x^2\right )+4 b^2 c x^2\right )}{6 a^2 b x^3}+\frac {(2 b e-a f) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.79, size = 210, normalized size = 1.91 \[ \left [-\frac {3 \, {\left (2 \, a^{2} b e - a^{3} f\right )} \sqrt {b} x^{3} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (3 \, a^{2} b f x^{4} - 2 \, a b^{2} c + 2 \, {\left (2 \, b^{3} c - 3 \, a b^{2} d\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{12 \, a^{2} b^{2} x^{3}}, -\frac {3 \, {\left (2 \, a^{2} b e - a^{3} f\right )} \sqrt {-b} x^{3} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (3 \, a^{2} b f x^{4} - 2 \, a b^{2} c + 2 \, {\left (2 \, b^{3} c - 3 \, a b^{2} d\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{6 \, a^{2} b^{2} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.56, size = 176, normalized size = 1.60 \[ \frac {\sqrt {b x^{2} + a} f x}{2 \, b} + \frac {{\left (a \sqrt {b} f - 2 \, b^{\frac {3}{2}} e\right )} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right )}{4 \, b^{2}} + \frac {2 \, {\left (3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} \sqrt {b} d + 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{\frac {3}{2}} c - 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a \sqrt {b} d - 2 \, a b^{\frac {3}{2}} c + 3 \, a^{2} \sqrt {b} d\right )}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 117, normalized size = 1.06 \[ -\frac {a f \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}+\frac {e \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}+\frac {\sqrt {b \,x^{2}+a}\, f x}{2 b}-\frac {\sqrt {b \,x^{2}+a}\, d}{a x}+\frac {2 \sqrt {b \,x^{2}+a}\, b c}{3 a^{2} x}-\frac {\sqrt {b \,x^{2}+a}\, c}{3 a \,x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.33, size = 102, normalized size = 0.93 \[ \frac {\sqrt {b x^{2} + a} f x}{2 \, b} + \frac {e \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b}} - \frac {a f \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {3}{2}}} + \frac {2 \, \sqrt {b x^{2} + a} b c}{3 \, a^{2} x} - \frac {\sqrt {b x^{2} + a} d}{a x} - \frac {\sqrt {b x^{2} + a} c}{3 \, a x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.20, size = 143, normalized size = 1.30 \[ \left \{\begin {array}{cl} -\frac {-f\,x^6-3\,e\,x^4+3\,d\,x^2+c}{3\,\sqrt {a}\,x^3} & \text {\ if\ \ }b=0\\ \frac {e\,\ln \left (\sqrt {b}\,x+\sqrt {b\,x^2+a}\right )}{\sqrt {b}}-\frac {d\,\sqrt {b\,x^2+a}}{a\,x}-\frac {a\,f\,\ln \left (2\,\sqrt {b}\,x+2\,\sqrt {b\,x^2+a}\right )}{2\,b^{3/2}}+\frac {f\,x\,\sqrt {b\,x^2+a}}{2\,b}-\frac {c\,\sqrt {b\,x^2+a}\,\left (a-2\,b\,x^2\right )}{3\,a^2\,x^3} & \text {\ if\ \ }b\neq 0 \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 4.73, size = 197, normalized size = 1.79 \[ \frac {\sqrt {a} f x \sqrt {1 + \frac {b x^{2}}{a}}}{2 b} - \frac {a f \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{2 b^{\frac {3}{2}}} + e \left (\begin {cases} \frac {\sqrt {- \frac {a}{b}} \operatorname {asin}{\left (x \sqrt {- \frac {b}{a}} \right )}}{\sqrt {a}} & \text {for}\: a > 0 \wedge b < 0 \\\frac {\sqrt {\frac {a}{b}} \operatorname {asinh}{\left (x \sqrt {\frac {b}{a}} \right )}}{\sqrt {a}} & \text {for}\: a > 0 \wedge b > 0 \\\frac {\sqrt {- \frac {a}{b}} \operatorname {acosh}{\left (x \sqrt {- \frac {b}{a}} \right )}}{\sqrt {- a}} & \text {for}\: b > 0 \wedge a < 0 \end {cases}\right ) - \frac {\sqrt {b} c \sqrt {\frac {a}{b x^{2}} + 1}}{3 a x^{2}} - \frac {\sqrt {b} d \sqrt {\frac {a}{b x^{2}} + 1}}{a} + \frac {2 b^{\frac {3}{2}} c \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________