Optimal. Leaf size=368 \[ \frac {3 \sqrt [4]{c} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (5 b B-7 A c) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{10 b^{11/4} \sqrt {b x^2+c x^4}}-\frac {3 \sqrt [4]{c} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (5 b B-7 A c) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 b^{11/4} \sqrt {b x^2+c x^4}}+\frac {3 \sqrt {c} x^{3/2} \left (b+c x^2\right ) (5 b B-7 A c)}{5 b^3 \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {3 \sqrt {b x^2+c x^4} (5 b B-7 A c)}{5 b^3 x^{3/2}}+\frac {\sqrt {x} (5 b B-7 A c)}{5 b^2 \sqrt {b x^2+c x^4}}-\frac {2 A}{5 b x^{3/2} \sqrt {b x^2+c x^4}} \]
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Rubi [A] time = 0.45, antiderivative size = 368, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2038, 2023, 2025, 2032, 329, 305, 220, 1196} \[ \frac {3 \sqrt {c} x^{3/2} \left (b+c x^2\right ) (5 b B-7 A c)}{5 b^3 \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}+\frac {\sqrt {x} (5 b B-7 A c)}{5 b^2 \sqrt {b x^2+c x^4}}-\frac {3 \sqrt {b x^2+c x^4} (5 b B-7 A c)}{5 b^3 x^{3/2}}+\frac {3 \sqrt [4]{c} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (5 b B-7 A c) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{10 b^{11/4} \sqrt {b x^2+c x^4}}-\frac {3 \sqrt [4]{c} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (5 b B-7 A c) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 b^{11/4} \sqrt {b x^2+c x^4}}-\frac {2 A}{5 b x^{3/2} \sqrt {b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 305
Rule 329
Rule 1196
Rule 2023
Rule 2025
Rule 2032
Rule 2038
Rubi steps
\begin {align*} \int \frac {A+B x^2}{\sqrt {x} \left (b x^2+c x^4\right )^{3/2}} \, dx &=-\frac {2 A}{5 b x^{3/2} \sqrt {b x^2+c x^4}}-\frac {\left (2 \left (-\frac {5 b B}{2}+\frac {7 A c}{2}\right )\right ) \int \frac {x^{3/2}}{\left (b x^2+c x^4\right )^{3/2}} \, dx}{5 b}\\ &=-\frac {2 A}{5 b x^{3/2} \sqrt {b x^2+c x^4}}+\frac {(5 b B-7 A c) \sqrt {x}}{5 b^2 \sqrt {b x^2+c x^4}}+\frac {(3 (5 b B-7 A c)) \int \frac {1}{\sqrt {x} \sqrt {b x^2+c x^4}} \, dx}{10 b^2}\\ &=-\frac {2 A}{5 b x^{3/2} \sqrt {b x^2+c x^4}}+\frac {(5 b B-7 A c) \sqrt {x}}{5 b^2 \sqrt {b x^2+c x^4}}-\frac {3 (5 b B-7 A c) \sqrt {b x^2+c x^4}}{5 b^3 x^{3/2}}+\frac {(3 c (5 b B-7 A c)) \int \frac {x^{3/2}}{\sqrt {b x^2+c x^4}} \, dx}{10 b^3}\\ &=-\frac {2 A}{5 b x^{3/2} \sqrt {b x^2+c x^4}}+\frac {(5 b B-7 A c) \sqrt {x}}{5 b^2 \sqrt {b x^2+c x^4}}-\frac {3 (5 b B-7 A c) \sqrt {b x^2+c x^4}}{5 b^3 x^{3/2}}+\frac {\left (3 c (5 b B-7 A c) x \sqrt {b+c x^2}\right ) \int \frac {\sqrt {x}}{\sqrt {b+c x^2}} \, dx}{10 b^3 \sqrt {b x^2+c x^4}}\\ &=-\frac {2 A}{5 b x^{3/2} \sqrt {b x^2+c x^4}}+\frac {(5 b B-7 A c) \sqrt {x}}{5 b^2 \sqrt {b x^2+c x^4}}-\frac {3 (5 b B-7 A c) \sqrt {b x^2+c x^4}}{5 b^3 x^{3/2}}+\frac {\left (3 c (5 b B-7 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 )}{5 b^3 \sqrt {b x^2+c x^4}}\\ &=-\frac {2 A}{5 b x^{3/2} \sqrt {b x^2+c x^4}}+\frac {(5 b B-7 A c) \sqrt {x}}{5 b^2 \sqrt {b x^2+c x^4}}-\frac {3 (5 b B-7 A c) \sqrt {b x^2+c x^4}}{5 b^3 x^{3/2}}+\frac {\left (3 \sqrt {c} (5 b B-7 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 )}{5 b^{5/2} \sqrt {b x^2+c x^4}}-\frac {\left (3 \sqrt {c} (5 b B-7 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 )}{5 b^{5/2} \sqrt {b x^2+c x^4}}\\ &=-\frac {2 A}{5 b x^{3/2} \sqrt {b x^2+c x^4}}+\frac {(5 b B-7 A c) \sqrt {x}}{5 b^2 \sqrt {b x^2+c x^4}}+\frac {3 \sqrt {c} (5 b B-7 A c) x^{3/2} \left (b+c x^2\right )}{5 b^3 \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {3 (5 b B-7 A c) \sqrt {b x^2+c x^4}}{5 b^3 x^{3/2}}-\frac {3 \sqrt [4]{c} (5 b B-7 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 )}{5 b^{11/4} \sqrt {b x^2+c x^4}}+\frac {3 \sqrt [4]{c} (5 b B-7 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 )}{10 b^{11/4} \sqrt {b x^2+c x^4}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 79, normalized size = 0.21 \[ \frac {2 x^2 \sqrt {\frac {c x^2}{b}+1} (7 A c-5 b B) \, _2F_1\left (-\frac {1}{4},\frac {3}{2};\frac {3}{4};-\frac {c x^2}{b}\right )-2 A b}{5 b^2 x^{3/2} \sqrt {x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.03, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{4} + b x^{2}} {\left (B x^{2} + A\right )} \sqrt {x}}{c^{2} x^{9} + 2 \, b c x^{7} + b^{2} x^{5}}, x\right ) \]
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 \[ \int \frac {B x^{2} + A}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} \sqrt {x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 420, normalized size = 1.14 \[ -\frac {\left (c \,x^{2}+b \right ) \left (-42 A \,c^{2} x^{4}+30 B b c \,x^{4}+42 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, A b c \,x^{2} \EllipticE \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-21 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, A b c \,x^{2} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-30 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, B \,b^{2} x^{2} \EllipticE \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )+15 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, B \,b^{2} x^{2} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-28 A b c \,x^{2}+20 B \,b^{2} x^{2}+4 A \,b^{2}\right ) \sqrt {x}}{10 \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} b^{3}} \]
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 \[ \int \frac {B x^{2} + A}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} \sqrt {x}}\,{d x} \]
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 \[ \int \frac {B\,x^2+A}{\sqrt {x}\,{\left (c\,x^4+b\,x^2\right )}^{3/2}} \,d x \]
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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