Optimal. Leaf size=97 \[ \frac {\sqrt {b x^2+c x^4} (2 A c+b B)}{2 b}+\frac {(2 A c+b B) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{2 \sqrt {c}}-\frac {A \left (b x^2+c x^4\right )^{3/2}}{b x^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.21, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2034, 792, 664, 620, 206} \[ \frac {\sqrt {b x^2+c x^4} (2 A c+b B)}{2 b}+\frac {(2 A c+b B) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{2 \sqrt {c}}-\frac {A \left (b x^2+c x^4\right )^{3/2}}{b x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 620
Rule 664
Rule 792
Rule 2034
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \sqrt {b x^2+c x^4}}{x^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(A+B x) \sqrt {b x+c x^2}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {A \left (b x^2+c x^4\right )^{3/2}}{b x^4}+\frac {\left (-2 (-b B+A c)+\frac {3}{2} (-b B+2 A c)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b x+c x^2}}{x} \, dx,x,x^2\right )}{b}\\ &=\frac {(b B+2 A c) \sqrt {b x^2+c x^4}}{2 b}-\frac {A \left (b x^2+c x^4\right )^{3/2}}{b x^4}+\frac {1}{4} (b B+2 A c) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+c x^2}} \, dx,x,x^2\right )\\ &=\frac {(b B+2 A c) \sqrt {b x^2+c x^4}}{2 b}-\frac {A \left (b x^2+c x^4\right )^{3/2}}{b x^4}+\frac {1}{2} (b B+2 A c) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x^2}{\sqrt {b x^2+c x^4}}\right )\\ &=\frac {(b B+2 A c) \sqrt {b x^2+c x^4}}{2 b}-\frac {A \left (b x^2+c x^4\right )^{3/2}}{b x^4}+\frac {(b B+2 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{2 \sqrt {c}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.15, size = 78, normalized size = 0.80 \[ \frac {\sqrt {x^2 \left (b+c x^2\right )} \left (\frac {x (2 A c+b B) \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {c} \sqrt {\frac {c x^2}{b}+1}}-2 A+B x^2\right )}{2 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.98, size = 161, normalized size = 1.66 \[ \left [\frac {{\left (B b + 2 \, A c\right )} \sqrt {c} x^{2} \log \left (-2 \, c x^{2} - b - 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right ) + 2 \, \sqrt {c x^{4} + b x^{2}} {\left (B c x^{2} - 2 \, A c\right )}}{4 \, c x^{2}}, -\frac {{\left (B b + 2 \, A c\right )} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-c}}{c x^{2} + b}\right ) - \sqrt {c x^{4} + b x^{2}} {\left (B c x^{2} - 2 \, A c\right )}}{2 \, c x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.27, size = 92, normalized size = 0.95 \[ \frac {1}{2} \, \sqrt {c x^{2} + b} B x \mathrm {sgn}\relax (x) + \frac {2 \, A b \sqrt {c} \mathrm {sgn}\relax (x)}{{\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b} - \frac {{\left (B b \sqrt {c} \mathrm {sgn}\relax (x) + 2 \, A c^{\frac {3}{2}} \mathrm {sgn}\relax (x)\right )} \log \left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2}\right )}{4 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 130, normalized size = 1.34 \[ \frac {\sqrt {c \,x^{4}+b \,x^{2}}\, \left (2 A b c x \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right )+B \,b^{2} x \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right )+2 \sqrt {c \,x^{2}+b}\, A \,c^{\frac {3}{2}} x^{2}+\sqrt {c \,x^{2}+b}\, B b \sqrt {c}\, x^{2}-2 \left (c \,x^{2}+b \right )^{\frac {3}{2}} A \sqrt {c}\right )}{2 \sqrt {c \,x^{2}+b}\, b \sqrt {c}\, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.48, size = 105, normalized size = 1.08 \[ \frac {1}{2} \, {\left (\sqrt {c} \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right ) - \frac {2 \, \sqrt {c x^{4} + b x^{2}}}{x^{2}}\right )} A + \frac {1}{4} \, {\left (\frac {b \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right )}{\sqrt {c}} + 2 \, \sqrt {c x^{4} + b x^{2}}\right )} B \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (B\,x^2+A\right )\,\sqrt {c\,x^4+b\,x^2}}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x^{2} \left (b + c x^{2}\right )} \left (A + B x^{2}\right )}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________