Optimal. Leaf size=84 \[ \frac {\sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {b} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \]
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Rubi [A] time = 0.02, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {411} \[ \frac {\sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {b} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \]
Antiderivative was successfully verified.
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Rule 411
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2}} \, dx &=\frac {\sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {b} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}\\ \end {align*}
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Mathematica [C] time = 0.29, size = 133, normalized size = 1.58 \[ \frac {x \left (c+d x^2\right )+\frac {i c \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \left (E\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-\operatorname {EllipticF}\left (i \sinh ^{-1}\left (x \sqrt {\frac {b}{a}}\right ),\frac {a d}{b c}\right )\right )}{\sqrt {\frac {b}{a}}}}{a \sqrt {a+b x^2} \sqrt {c+d x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 181, normalized size = 2.15 \[ \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, \left (\sqrt {-\frac {b}{a}}\, d \,x^{3}+\sqrt {-\frac {b}{a}}\, c x -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, c \EllipticE \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, c \EllipticF \left (\sqrt {-\frac {b}{a}}\, x , \sqrt {\frac {a d}{b c}}\right )\right )}{\left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right ) \sqrt {-\frac {b}{a}}\, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {d\,x^2+c}}{{\left (b\,x^2+a\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\sqrt {c + d x^{2}}}{\left (a + b x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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