Optimal. Leaf size=177 \[ \frac {(b c+2 a d) x \sqrt {c+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 c \left (a+b x^2\right )}-\frac {a \left (c+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{c x \left (a+b x^2\right )}+\frac {(b c+2 a d) \sqrt {a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 \sqrt {d} \left (a+b x^2\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {1264, 464, 201,
223, 212} \begin {gather*} -\frac {a \sqrt {a^2+2 a b x^2+b^2 x^4} \left (c+d x^2\right )^{3/2}}{c x \left (a+b x^2\right )}+\frac {x \sqrt {a^2+2 a b x^2+b^2 x^4} \sqrt {c+d x^2} (2 a d+b c)}{2 c \left (a+b x^2\right )}+\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} (2 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 \sqrt {d} \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 201
Rule 212
Rule 223
Rule 464
Rule 1264
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{x^2} \, dx &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {\left (a b+b^2 x^2\right ) \sqrt {c+d x^2}}{x^2} \, dx}{a b+b^2 x^2}\\ &=-\frac {a \left (c+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{c x \left (a+b x^2\right )}+-\frac {\left (\left (-b^2 c-2 a b d\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}\right ) \int \sqrt {c+d x^2} \, dx}{c \left (a b+b^2 x^2\right )}\\ &=\frac {(b c+2 a d) x \sqrt {c+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 c \left (a+b x^2\right )}-\frac {a \left (c+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{c x \left (a+b x^2\right )}+-\frac {\left (\left (-b^2 c-2 a b d\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{2 \left (a b+b^2 x^2\right )}\\ &=\frac {(b c+2 a d) x \sqrt {c+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 c \left (a+b x^2\right )}-\frac {a \left (c+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{c x \left (a+b x^2\right )}+-\frac {\left (\left (-b^2 c-2 a b d\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}\right ) \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{2 \left (a b+b^2 x^2\right )}\\ &=\frac {(b c+2 a d) x \sqrt {c+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 c \left (a+b x^2\right )}-\frac {a \left (c+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{c x \left (a+b x^2\right )}+\frac {(b c+2 a d) \sqrt {a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 \sqrt {d} \left (a+b x^2\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.11, size = 93, normalized size = 0.53 \begin {gather*} -\frac {\sqrt {\left (a+b x^2\right )^2} \left (\sqrt {d} \left (2 a-b x^2\right ) \sqrt {c+d x^2}+(b c+2 a d) x \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )\right )}{2 \sqrt {d} x \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.12, size = 130, normalized size = 0.73
method | result | size |
risch | \(-\frac {\sqrt {d \,x^{2}+c}\, \left (-b \,x^{2}+2 a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{2 x \left (b \,x^{2}+a \right )}+\frac {\left (a \sqrt {d}\, \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )+\frac {\ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right ) b c}{2 \sqrt {d}}\right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{b \,x^{2}+a}\) | \(112\) |
default | \(-\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (-2 \sqrt {d \,x^{2}+c}\, d^{\frac {3}{2}} a \,x^{2}-\sqrt {d \,x^{2}+c}\, \sqrt {d}\, b c \,x^{2}+2 \left (d \,x^{2}+c \right )^{\frac {3}{2}} \sqrt {d}\, a -2 \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right ) a c d x -\ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right ) b \,c^{2} x \right )}{2 \left (b \,x^{2}+a \right ) c x \sqrt {d}}\) | \(130\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 59, normalized size = 0.33 \begin {gather*} \frac {1}{2} \, \sqrt {d x^{2} + c} b x + \frac {b c \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{2 \, \sqrt {d}} + a \sqrt {d} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right ) - \frac {\sqrt {d x^{2} + c} a}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 134, normalized size = 0.76 \begin {gather*} \left [\frac {{\left (b c + 2 \, a d\right )} \sqrt {d} x \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (b d x^{2} - 2 \, a d\right )} \sqrt {d x^{2} + c}}{4 \, d x}, -\frac {{\left (b c + 2 \, a d\right )} \sqrt {-d} x \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - {\left (b d x^{2} - 2 \, a d\right )} \sqrt {d x^{2} + c}}{2 \, d x}\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 {\sqrt {c + d x^{2}} \sqrt {\left (a + b x^{2}\right )^{2}}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 3.62, size = 116, normalized size = 0.66 \begin {gather*} \frac {1}{2} \, \sqrt {d x^{2} + c} b x \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {2 \, a c \sqrt {d} \mathrm {sgn}\left (b x^{2} + a\right )}{{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c} - \frac {{\left (b c \sqrt {d} \mathrm {sgn}\left (b x^{2} + a\right ) + 2 \, a d^{\frac {3}{2}} \mathrm {sgn}\left (b x^{2} + a\right )\right )} \log \left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2}\right )}{4 \, d} \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 {\sqrt {d\,x^2+c}\,\sqrt {{\left (b\,x^2+a\right )}^2}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________