Optimal. Leaf size=177 \[ -\frac {a \sqrt {a^2+2 a b x^2+b^2 x^4} \left (c+d x^2\right )^{3/2}}{c x \left (a+b x^2\right )}+\frac {x \sqrt {a^2+2 a b x^2+b^2 x^4} \sqrt {c+d x^2} (2 a d+b c)}{2 c \left (a+b x^2\right )}+\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} (2 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 \sqrt {d} \left (a+b x^2\right )} \]
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Rubi [A] time = 0.09, 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 = {1250, 453, 195, 217, 206} \[ -\frac {a \sqrt {a^2+2 a b x^2+b^2 x^4} \left (c+d x^2\right )^{3/2}}{c x \left (a+b x^2\right )}+\frac {x \sqrt {a^2+2 a b x^2+b^2 x^4} \sqrt {c+d x^2} (2 a d+b c)}{2 c \left (a+b x^2\right )}+\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} (2 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 \sqrt {d} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 453
Rule 1250
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 ) \operatorname {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*}
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Mathematica [A] time = 0.11, size = 122, normalized size = 0.69 \[ \frac {\sqrt {\left (a+b x^2\right )^2} \sqrt {c+d x^2} \left (\sqrt {c} \sqrt {d} \left (b x^2-2 a\right ) \sqrt {\frac {d x^2}{c}+1}+x (2 a d+b c) \sinh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )\right )}{2 \sqrt {c} \sqrt {d} x \left (a+b x^2\right ) \sqrt {\frac {d x^2}{c}+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 134, normalized size = 0.76 \[ \left [\frac {{\left (b c + 2 \, a d\right )} \sqrt {d} x \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (b d x^{2} - 2 \, a d\right )} \sqrt {d x^{2} + c}}{4 \, d x}, -\frac {{\left (b c + 2 \, a d\right )} \sqrt {-d} x \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - {\left (b d x^{2} - 2 \, a d\right )} \sqrt {d x^{2} + c}}{2 \, d x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.47, size = 116, normalized size = 0.66 \[ \frac {1}{2} \, \sqrt {d x^{2} + c} b x \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {2 \, a c \sqrt {d} \mathrm {sgn}\left (b x^{2} + a\right )}{{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c} - \frac {{\left (b c \sqrt {d} \mathrm {sgn}\left (b x^{2} + a\right ) + 2 \, a d^{\frac {3}{2}} \mathrm {sgn}\left (b x^{2} + a\right )\right )} \log \left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2}\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 128, normalized size = 0.72 \[ \frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (2 a c d x \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )+b \,c^{2} x \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )+2 \sqrt {d \,x^{2}+c}\, a \,d^{\frac {3}{2}} x^{2}+\sqrt {d \,x^{2}+c}\, b c \sqrt {d}\, x^{2}-2 \left (d \,x^{2}+c \right )^{\frac {3}{2}} a \sqrt {d}\right )}{2 \left (b \,x^{2}+a \right ) c \sqrt {d}\, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 59, normalized size = 0.33 \[ \frac {1}{2} \, \sqrt {d x^{2} + c} b x + \frac {b c \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{2 \, \sqrt {d}} + a \sqrt {d} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right ) - \frac {\sqrt {d x^{2} + c} a}{x} \]
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 \[ \int \frac {\sqrt {d\,x^2+c}\,\sqrt {{\left (b\,x^2+a\right )}^2}}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c + d x^{2}} \sqrt {\left (a + b x^{2}\right )^{2}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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