Optimal. Leaf size=79 \[ \frac {3 c \sqrt {b x^2+c x^4}}{2 x}-\frac {\left (b x^2+c x^4\right )^{3/2}}{2 x^5}-\frac {3}{2} \sqrt {b} c \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2045, 2046,
2033, 212} \begin {gather*} \frac {3 c \sqrt {b x^2+c x^4}}{2 x}-\frac {3}{2} \sqrt {b} c \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )-\frac {\left (b x^2+c x^4\right )^{3/2}}{2 x^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 2033
Rule 2045
Rule 2046
Rubi steps
\begin {align*} \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^6} \, dx &=-\frac {\left (b x^2+c x^4\right )^{3/2}}{2 x^5}+\frac {1}{2} (3 c) \int \frac {\sqrt {b x^2+c x^4}}{x^2} \, dx\\ &=\frac {3 c \sqrt {b x^2+c x^4}}{2 x}-\frac {\left (b x^2+c x^4\right )^{3/2}}{2 x^5}+\frac {1}{2} (3 b c) \int \frac {1}{\sqrt {b x^2+c x^4}} \, dx\\ &=\frac {3 c \sqrt {b x^2+c x^4}}{2 x}-\frac {\left (b x^2+c x^4\right )^{3/2}}{2 x^5}-\frac {1}{2} (3 b c) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {b x^2+c x^4}}\right )\\ &=\frac {3 c \sqrt {b x^2+c x^4}}{2 x}-\frac {\left (b x^2+c x^4\right )^{3/2}}{2 x^5}-\frac {3}{2} \sqrt {b} c \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.08, size = 83, normalized size = 1.05 \begin {gather*} -\frac {\sqrt {x^2 \left (b+c x^2\right )} \left (\left (b-2 c x^2\right ) \sqrt {b+c x^2}+3 \sqrt {b} c x^2 \tanh ^{-1}\left (\frac {\sqrt {b+c x^2}}{\sqrt {b}}\right )\right )}{2 x^3 \sqrt {b+c x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.09, size = 102, normalized size = 1.29
method | result | size |
risch | \(-\frac {b \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{2 x^{3}}+\frac {\left (-\frac {3 \sqrt {b}\, \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {c \,x^{2}+b}}{x}\right ) c}{2}+\sqrt {c \,x^{2}+b}\, c \right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{x \sqrt {c \,x^{2}+b}}\) | \(88\) |
default | \(-\frac {\left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \left (3 b^{\frac {3}{2}} \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {c \,x^{2}+b}}{x}\right ) c \,x^{2}-\left (c \,x^{2}+b \right )^{\frac {3}{2}} c \,x^{2}+\left (c \,x^{2}+b \right )^{\frac {5}{2}}-3 \sqrt {c \,x^{2}+b}\, b c \,x^{2}\right )}{2 x^{5} \left (c \,x^{2}+b \right )^{\frac {3}{2}} b}\) | \(102\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.42, size = 147, normalized size = 1.86 \begin {gather*} \left [\frac {3 \, \sqrt {b} c x^{3} \log \left (-\frac {c x^{3} + 2 \, b x - 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {b}}{x^{3}}\right ) + 2 \, \sqrt {c x^{4} + b x^{2}} {\left (2 \, c x^{2} - b\right )}}{4 \, x^{3}}, \frac {3 \, \sqrt {-b} c x^{3} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-b}}{c x^{3} + b x}\right ) + \sqrt {c x^{4} + b x^{2}} {\left (2 \, c x^{2} - b\right )}}{2 \, x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}{x^{6}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 3.51, size = 69, normalized size = 0.87 \begin {gather*} \frac {\frac {3 \, b c^{2} \arctan \left (\frac {\sqrt {c x^{2} + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-b}} + 2 \, \sqrt {c x^{2} + b} c^{2} \mathrm {sgn}\left (x\right ) - \frac {\sqrt {c x^{2} + b} b c \mathrm {sgn}\left (x\right )}{x^{2}}}{2 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,x^4+b\,x^2\right )}^{3/2}}{x^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________