Optimal. Leaf size=203 \[ -\frac {5 c^{3/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (7 b B-9 A c) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{42 b^{13/4} \sqrt {b x^2+c x^4}}-\frac {5 \sqrt {b x^2+c x^4} (7 b B-9 A c)}{21 b^3 x^{5/2}}+\frac {7 b B-9 A c}{7 b^2 \sqrt {x} \sqrt {b x^2+c x^4}}-\frac {2 A}{7 b x^{5/2} \sqrt {b x^2+c x^4}} \]
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Rubi [A] time = 0.33, antiderivative size = 203, 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, 2023, 2025, 2032, 329, 220} \[ -\frac {5 c^{3/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (7 b B-9 A c) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{42 b^{13/4} \sqrt {b x^2+c x^4}}-\frac {5 \sqrt {b x^2+c x^4} (7 b B-9 A c)}{21 b^3 x^{5/2}}+\frac {7 b B-9 A c}{7 b^2 \sqrt {x} \sqrt {b x^2+c x^4}}-\frac {2 A}{7 b x^{5/2} \sqrt {b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 329
Rule 2023
Rule 2025
Rule 2032
Rule 2038
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x^{3/2} \left (b x^2+c x^4\right )^{3/2}} \, dx &=-\frac {2 A}{7 b x^{5/2} \sqrt {b x^2+c x^4}}-\frac {\left (2 \left (-\frac {7 b B}{2}+\frac {9 A c}{2}\right )\right ) \int \frac {\sqrt {x}}{\left (b x^2+c x^4\right )^{3/2}} \, dx}{7 b}\\ &=-\frac {2 A}{7 b x^{5/2} \sqrt {b x^2+c x^4}}+\frac {7 b B-9 A c}{7 b^2 \sqrt {x} \sqrt {b x^2+c x^4}}+\frac {(5 (7 b B-9 A c)) \int \frac {1}{x^{3/2} \sqrt {b x^2+c x^4}} \, dx}{14 b^2}\\ &=-\frac {2 A}{7 b x^{5/2} \sqrt {b x^2+c x^4}}+\frac {7 b B-9 A c}{7 b^2 \sqrt {x} \sqrt {b x^2+c x^4}}-\frac {5 (7 b B-9 A c) \sqrt {b x^2+c x^4}}{21 b^3 x^{5/2}}-\frac {(5 c (7 b B-9 A c)) \int \frac {\sqrt {x}}{\sqrt {b x^2+c x^4}} \, dx}{42 b^3}\\ &=-\frac {2 A}{7 b x^{5/2} \sqrt {b x^2+c x^4}}+\frac {7 b B-9 A c}{7 b^2 \sqrt {x} \sqrt {b x^2+c x^4}}-\frac {5 (7 b B-9 A c) \sqrt {b x^2+c x^4}}{21 b^3 x^{5/2}}-\frac {\left (5 c (7 b B-9 A c) x \sqrt {b+c x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x^2}} \, dx}{42 b^3 \sqrt {b x^2+c x^4}}\\ &=-\frac {2 A}{7 b x^{5/2} \sqrt {b x^2+c x^4}}+\frac {7 b B-9 A c}{7 b^2 \sqrt {x} \sqrt {b x^2+c x^4}}-\frac {5 (7 b B-9 A c) \sqrt {b x^2+c x^4}}{21 b^3 x^{5/2}}-\frac {\left (5 c (7 b B-9 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 )}{21 b^3 \sqrt {b x^2+c x^4}}\\ &=-\frac {2 A}{7 b x^{5/2} \sqrt {b x^2+c x^4}}+\frac {7 b B-9 A c}{7 b^2 \sqrt {x} \sqrt {b x^2+c x^4}}-\frac {5 (7 b B-9 A c) \sqrt {b x^2+c x^4}}{21 b^3 x^{5/2}}-\frac {5 c^{3/4} (7 b B-9 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 )}{42 b^{13/4} \sqrt {b x^2+c x^4}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 79, normalized size = 0.39 \[ \frac {2 x^2 \sqrt {\frac {c x^2}{b}+1} (9 A c-7 b B) \, _2F_1\left (-\frac {3}{4},\frac {3}{2};\frac {1}{4};-\frac {c x^2}{b}\right )-6 A b}{21 b^2 x^{5/2} \sqrt {x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{4} + b x^{2}} {\left (B x^{2} + A\right )} \sqrt {x}}{c^{2} x^{10} + 2 \, b c x^{8} + b^{2} x^{6}}, x\right ) \]
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 \[ \int \frac {B x^{2} + A}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} x^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 254, normalized size = 1.25 \[ \frac {\left (c \,x^{2}+b \right ) \left (90 A \,c^{2} x^{4}-70 B b c \,x^{4}+45 \sqrt {-b c}\, \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, A c \,x^{3} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-35 \sqrt {-b c}\, \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, B b \,x^{3} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )+36 A b c \,x^{2}-28 B \,b^{2} x^{2}-12 A \,b^{2}\right )}{42 \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} b^{3} \sqrt {x}} \]
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 \[ \int \frac {B x^{2} + A}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} x^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {B\,x^2+A}{x^{3/2}\,{\left (c\,x^4+b\,x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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