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 = 0.0948647, antiderivative size = 152, normalized size of antiderivative = 1., 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 1250
Rule 446
Rule 80
Rule 50
Rule 63
Rule 208
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*}
Mathematica [A] time = 0.0460664, 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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Maple [A] time = 0.01, size = 80, normalized size = 0.5 \begin{align*} -{\frac{1}{ \left ( 3\,b{x}^{2}+3\,a \right ) d}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}} \left ( 3\,\sqrt{c}\ln \left ( 2\,{\frac{\sqrt{c}\sqrt{d{x}^{2}+c}+c}{x}} \right ) ad-b \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}-3\,\sqrt{d{x}^{2}+c}ad \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d x^{2} + c} \sqrt{{\left (b x^{2} + a\right )}^{2}}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83764, size = 300, normalized size = 1.97 \begin{align*} \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 ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c + d x^{2}} \sqrt{\left (a + b x^{2}\right )^{2}}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1103, size = 113, normalized size = 0.74 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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