Optimal. Leaf size=152 \[ \frac {b \sqrt {a^2+2 a b x^2+b^2 x^4} \left (c+d x^2\right )^{3/2}}{3 d \left (a+b x^2\right )}+\frac {a \sqrt {a^2+2 a b x^2+b^2 x^4} \sqrt {c+d x^2}}{a+b x^2}-\frac {a \sqrt {c} \sqrt {a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a+b x^2} \]
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Rubi [A] time = 0.09, antiderivative size = 152, 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, 80, 50, 63, 208} \[ \frac {b \sqrt {a^2+2 a b x^2+b^2 x^4} \left (c+d x^2\right )^{3/2}}{3 d \left (a+b x^2\right )}+\frac {a \sqrt {a^2+2 a b x^2+b^2 x^4} \sqrt {c+d x^2}}{a+b x^2}-\frac {a \sqrt {c} \sqrt {a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a+b x^2} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
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} \, 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} \, 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} \, dx,x,x^2\right )}{2 \left (a b+b^2 x^2\right )}\\ &=\frac {b \left (c+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d \left (a+b x^2\right )}+\frac {\left (a b \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 \left (a b+b^2 x^2\right )}\\ &=\frac {a \sqrt {c+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac {b \left (c+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d \left (a+b x^2\right )}+\frac {\left (a b c \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 {a \sqrt {c+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac {b \left (c+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d \left (a+b x^2\right )}+\frac {\left (a b c \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 {a \sqrt {c+d x^2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac {b \left (c+d x^2\right )^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d \left (a+b x^2\right )}-\frac {a \sqrt {c} \sqrt {a^2+2 a b x^2+b^2 x^4} \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a+b x^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 83, normalized size = 0.55 \[ \frac {\sqrt {\left (a+b x^2\right )^2} \left (\sqrt {c+d x^2} \left (3 a d+b \left (c+d x^2\right )\right )-3 a \sqrt {c} d \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )\right )}{3 d \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 123, normalized size = 0.81 \[ \left [\frac {3 \, a \sqrt {c} d \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 2 \, {\left (b d x^{2} + b c + 3 \, a d\right )} \sqrt {d x^{2} + c}}{6 \, d}, \frac {3 \, a \sqrt {-c} d \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + {\left (b d x^{2} + b c + 3 \, a d\right )} \sqrt {d x^{2} + c}}{3 \, d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 84, normalized size = 0.55 \[ \frac {a c \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right ) \mathrm {sgn}\left (b x^{2} + a\right )}{\sqrt {-c}} + \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} b d^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 3 \, \sqrt {d x^{2} + c} a d^{3} \mathrm {sgn}\left (b x^{2} + a\right )}{3 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 80, normalized size = 0.53 \[ -\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (3 a \sqrt {c}\, d \ln \left (\frac {2 c +2 \sqrt {d \,x^{2}+c}\, \sqrt {c}}{x}\right )-3 \sqrt {d \,x^{2}+c}\, a d -\left (d \,x^{2}+c \right )^{\frac {3}{2}} b \right )}{3 \left (b \,x^{2}+a \right ) d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.45, size = 45, normalized size = 0.30 \[ -a \sqrt {c} \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right ) + \sqrt {d x^{2} + c} a + \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} b}{3 \, d} \]
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 \[ \int \frac {\sqrt {d\,x^2+c}\,\sqrt {{\left (b\,x^2+a\right )}^2}}{x} \,d x \]
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 \[ \int \frac {\sqrt {c + d x^{2}} \sqrt {\left (a + b x^{2}\right )^{2}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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