Optimal. Leaf size=177 \[ -\frac {a \sqrt {a^2+2 a b x^2+b^2 x^4} \left (c+d x^2\right )^{3/2}}{2 c x^2 \left (a+b x^2\right )}+\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \sqrt {c+d x^2} (a d+2 b c)}{2 c \left (a+b x^2\right )}-\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} (a d+2 b c) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 \sqrt {c} \left (a+b x^2\right )} \]
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Rubi [A] time = 0.12, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {1250, 446, 78, 50, 63, 208} \[ -\frac {a \sqrt {a^2+2 a b x^2+b^2 x^4} \left (c+d x^2\right )^{3/2}}{2 c x^2 \left (a+b x^2\right )}+\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \sqrt {c+d x^2} (a d+2 b c)}{2 c \left (a+b x^2\right )}-\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} (a d+2 b c) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 \sqrt {c} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 78
Rule 208
Rule 446
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^3} \, 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^3} \, dx}{a b+b^2 x^2}\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \operatorname {Subst}\left (\int \frac {\left (a b+b^2 x\right ) \sqrt {c+d x}}{x^2} \, dx,x,x^2\right )}{2 \left (a b+b^2 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}}{2 c x^2 \left (a+b x^2\right )}+\frac {\left (\left (b^2 c+\frac {a b d}{2}\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c+d x}}{x} \, dx,x,x^2\right )}{2 c \left (a b+b^2 x^2\right )}\\ &=\frac {(2 b c+a d) \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}}{2 c x^2 \left (a+b x^2\right )}+\frac {\left (\left (b^2 c+\frac {a b d}{2}\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right )}{2 \left (a b+b^2 x^2\right )}\\ &=\frac {(2 b c+a d) \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}}{2 c x^2 \left (a+b x^2\right )}+\frac {\left (\left (b^2 c+\frac {a b d}{2}\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{d \left (a b+b^2 x^2\right )}\\ &=\frac {(2 b c+a d) \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}}{2 c x^2 \left (a+b x^2\right )}-\frac {(2 b c+a d) \sqrt {a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 \sqrt {c} \left (a+b x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 90, normalized size = 0.51 \[ -\frac {\sqrt {\left (a+b x^2\right )^2} \left (\sqrt {c} \left (a-2 b x^2\right ) \sqrt {c+d x^2}+x^2 (a d+2 b c) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )\right )}{2 \sqrt {c} x^2 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 141, normalized size = 0.80 \[ \left [\frac {{\left (2 \, b c + a d\right )} \sqrt {c} x^{2} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 2 \, {\left (2 \, b c x^{2} - a c\right )} \sqrt {d x^{2} + c}}{4 \, c x^{2}}, \frac {{\left (2 \, b c + a d\right )} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + {\left (2 \, b c x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, c x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.44, size = 100, normalized size = 0.56 \[ \frac {2 \, \sqrt {d x^{2} + c} b d \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {{\left (2 \, b c d \mathrm {sgn}\left (b x^{2} + a\right ) + a d^{2} \mathrm {sgn}\left (b x^{2} + a\right )\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} - \frac {\sqrt {d x^{2} + c} a d \mathrm {sgn}\left (b x^{2} + a\right )}{x^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 133, normalized size = 0.75 \[ -\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (a \sqrt {c}\, d \,x^{2} \ln \left (\frac {2 c +2 \sqrt {d \,x^{2}+c}\, \sqrt {c}}{x}\right )+2 b \,c^{\frac {3}{2}} x^{2} \ln \left (\frac {2 c +2 \sqrt {d \,x^{2}+c}\, \sqrt {c}}{x}\right )-\sqrt {d \,x^{2}+c}\, a d \,x^{2}-2 \sqrt {d \,x^{2}+c}\, b c \,x^{2}+\left (d \,x^{2}+c \right )^{\frac {3}{2}} a \right )}{2 \left (b \,x^{2}+a \right ) c \,x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.23, size = 83, normalized size = 0.47 \[ -b \sqrt {c} \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right ) - \frac {a d \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right )}{2 \, \sqrt {c}} + \sqrt {d x^{2} + c} b + \frac {\sqrt {d x^{2} + c} a d}{2 \, c} - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a}{2 \, c x^{2}} \]
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 \[ \int \frac {\sqrt {d\,x^2+c}\,\sqrt {{\left (b\,x^2+a\right )}^2}}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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