Optimal. Leaf size=281 \[ \frac {x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (A c+b B) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{b^{3/4} c^{3/4} \sqrt {b x^2+c x^4}}-\frac {2 x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (A c+b B) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{b^{3/4} c^{3/4} \sqrt {b x^2+c x^4}}+\frac {2 x^{3/2} \left (b+c x^2\right ) (A c+b B)}{b \sqrt {c} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {2 A \sqrt {b x^2+c x^4}}{b x^{3/2}} \]
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Rubi [A] time = 0.31, antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2038, 2032, 329, 305, 220, 1196} \[ \frac {x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (A c+b B) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{b^{3/4} c^{3/4} \sqrt {b x^2+c x^4}}-\frac {2 x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (A c+b B) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{b^{3/4} c^{3/4} \sqrt {b x^2+c x^4}}+\frac {2 x^{3/2} \left (b+c x^2\right ) (A c+b B)}{b \sqrt {c} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {2 A \sqrt {b x^2+c x^4}}{b x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 305
Rule 329
Rule 1196
Rule 2032
Rule 2038
Rubi steps
\begin {align*} \int \frac {A+B x^2}{\sqrt {x} \sqrt {b x^2+c x^4}} \, dx &=-\frac {2 A \sqrt {b x^2+c x^4}}{b x^{3/2}}+\frac {(b B+A c) \int \frac {x^{3/2}}{\sqrt {b x^2+c x^4}} \, dx}{b}\\ &=-\frac {2 A \sqrt {b x^2+c x^4}}{b x^{3/2}}+\frac {\left ((b B+A c) x \sqrt {b+c x^2}\right ) \int \frac {\sqrt {x}}{\sqrt {b+c x^2}} \, dx}{b \sqrt {b x^2+c x^4}}\\ &=-\frac {2 A \sqrt {b x^2+c x^4}}{b x^{3/2}}+\frac {\left (2 (b B+A c) x \sqrt {b+c x^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{b \sqrt {b x^2+c x^4}}\\ &=-\frac {2 A \sqrt {b x^2+c x^4}}{b x^{3/2}}+\frac {\left (2 (b B+A c) x \sqrt {b+c x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {b} \sqrt {c} \sqrt {b x^2+c x^4}}-\frac {\left (2 (b B+A c) x \sqrt {b+c x^2}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {b}}}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {b} \sqrt {c} \sqrt {b x^2+c x^4}}\\ &=\frac {2 (b B+A c) x^{3/2} \left (b+c x^2\right )}{b \sqrt {c} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {2 A \sqrt {b x^2+c x^4}}{b x^{3/2}}-\frac {2 (b B+A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{b^{3/4} c^{3/4} \sqrt {b x^2+c x^4}}+\frac {(b B+A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{b^{3/4} c^{3/4} \sqrt {b x^2+c x^4}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 82, normalized size = 0.29 \[ \frac {2 \sqrt {x} \left (x^2 \sqrt {\frac {c x^2}{b}+1} (A c+b B) \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {c x^2}{b}\right )-3 A \left (b+c x^2\right )\right )}{3 b \sqrt {x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{4} + b x^{2}} {\left (B x^{2} + A\right )} \sqrt {x}}{c x^{5} + b x^{3}}, x\right ) \]
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 \[ \int \frac {B x^{2} + A}{\sqrt {c x^{4} + b x^{2}} \sqrt {x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 377, normalized size = 1.34 \[ \frac {\left (-2 A \,c^{2} x^{2}+2 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, A b c \EllipticE \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, A b c \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )+2 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, B \,b^{2} \EllipticE \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, B \,b^{2} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-2 A b c \right ) \sqrt {x}}{\sqrt {c \,x^{4}+b \,x^{2}}\, b c} \]
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 \[ \int \frac {B x^{2} + A}{\sqrt {c x^{4} + b x^{2}} \sqrt {x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {B\,x^2+A}{\sqrt {x}\,\sqrt {c\,x^4+b\,x^2}} \,d x \]
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 \[ \int \frac {A + B x^{2}}{\sqrt {x} \sqrt {x^{2} \left (b + c x^{2}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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