Optimal. Leaf size=167 \[ \frac {x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (3 b B-5 A c) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{6 b^{9/4} \sqrt [4]{c} \sqrt {b x^2+c x^4}}+\frac {x^{3/2} (3 b B-5 A c)}{3 b^2 \sqrt {b x^2+c x^4}}-\frac {2 A}{3 b \sqrt {x} \sqrt {b x^2+c x^4}} \]
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Rubi [A] time = 0.27, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2038, 2023, 2032, 329, 220} \[ \frac {x^{3/2} (3 b B-5 A c)}{3 b^2 \sqrt {b x^2+c x^4}}+\frac {x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (3 b B-5 A c) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{6 b^{9/4} \sqrt [4]{c} \sqrt {b x^2+c x^4}}-\frac {2 A}{3 b \sqrt {x} \sqrt {b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 329
Rule 2023
Rule 2032
Rule 2038
Rubi steps
\begin {align*} \int \frac {\sqrt {x} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx &=-\frac {2 A}{3 b \sqrt {x} \sqrt {b x^2+c x^4}}-\frac {\left (2 \left (-\frac {3 b B}{2}+\frac {5 A c}{2}\right )\right ) \int \frac {x^{5/2}}{\left (b x^2+c x^4\right )^{3/2}} \, dx}{3 b}\\ &=-\frac {2 A}{3 b \sqrt {x} \sqrt {b x^2+c x^4}}+\frac {(3 b B-5 A c) x^{3/2}}{3 b^2 \sqrt {b x^2+c x^4}}+\frac {(3 b B-5 A c) \int \frac {\sqrt {x}}{\sqrt {b x^2+c x^4}} \, dx}{6 b^2}\\ &=-\frac {2 A}{3 b \sqrt {x} \sqrt {b x^2+c x^4}}+\frac {(3 b B-5 A c) x^{3/2}}{3 b^2 \sqrt {b x^2+c x^4}}+\frac {\left ((3 b B-5 A c) x \sqrt {b+c x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x^2}} \, dx}{6 b^2 \sqrt {b x^2+c x^4}}\\ &=-\frac {2 A}{3 b \sqrt {x} \sqrt {b x^2+c x^4}}+\frac {(3 b B-5 A c) x^{3/2}}{3 b^2 \sqrt {b x^2+c x^4}}+\frac {\left ((3 b B-5 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 )}{3 b^2 \sqrt {b x^2+c x^4}}\\ &=-\frac {2 A}{3 b \sqrt {x} \sqrt {b x^2+c x^4}}+\frac {(3 b B-5 A c) x^{3/2}}{3 b^2 \sqrt {b x^2+c x^4}}+\frac {(3 b B-5 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 )}{6 b^{9/4} \sqrt [4]{c} \sqrt {b x^2+c x^4}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 92, normalized size = 0.55 \[ \frac {x^2 \sqrt {\frac {c x^2}{b}+1} (3 b B-5 A c) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {c x^2}{b}\right )-2 A b-5 A c x^2+3 b B x^2}{3 b^2 \sqrt {x} \sqrt {x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.33, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{4} + b x^{2}} {\left (B x^{2} + A\right )} \sqrt {x}}{c^{2} x^{8} + 2 \, b c x^{6} + b^{2} x^{4}}, x\right ) \]
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 \[ \int \frac {{\left (B x^{2} + A\right )} \sqrt {x}}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 235, normalized size = 1.41 \[ -\frac {\left (c \,x^{2}+b \right ) \left (10 A \,c^{2} x^{2}-6 B b c \,x^{2}+5 \sqrt {-b c}\, \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, A c x \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {-b c}\, \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, B b x \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )+4 A b c \right ) x^{\frac {3}{2}}}{6 \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} b^{2} c} \]
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 \[ \int \frac {{\left (B x^{2} + A\right )} \sqrt {x}}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {x}\,\left (B\,x^2+A\right )}{{\left (c\,x^4+b\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\sqrt {x} \left (A + B x^{2}\right )}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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