Optimal. Leaf size=137 \[ \frac {x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (A c+b B) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 b^{5/4} c^{5/4} \sqrt {b x^2+c x^4}}-\frac {x^{3/2} (b B-A c)}{b c \sqrt {b x^2+c x^4}} \]
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Rubi [A] time = 0.22, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2037, 2032, 329, 220} \[ \frac {x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (A c+b B) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 b^{5/4} c^{5/4} \sqrt {b x^2+c x^4}}-\frac {x^{3/2} (b B-A c)}{b c \sqrt {b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 329
Rule 2032
Rule 2037
Rubi steps
\begin {align*} \int \frac {x^{5/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx &=-\frac {(b B-A c) x^{3/2}}{b c \sqrt {b x^2+c x^4}}+\frac {(b B+A c) \int \frac {\sqrt {x}}{\sqrt {b x^2+c x^4}} \, dx}{2 b c}\\ &=-\frac {(b B-A c) x^{3/2}}{b c \sqrt {b x^2+c x^4}}+\frac {\left ((b B+A c) x \sqrt {b+c x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x^2}} \, dx}{2 b c \sqrt {b x^2+c x^4}}\\ &=-\frac {(b B-A c) x^{3/2}}{b c \sqrt {b x^2+c x^4}}+\frac {\left ((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 )}{b c \sqrt {b x^2+c x^4}}\\ &=-\frac {(b B-A c) x^{3/2}}{b c \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 )}{2 b^{5/4} c^{5/4} \sqrt {b x^2+c x^4}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 76, normalized size = 0.55 \[ \frac {x^{3/2} \left (\sqrt {\frac {c x^2}{b}+1} (A c+b B) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {c x^2}{b}\right )+A c-b B\right )}{b c \sqrt {x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{4} + b x^{2}} {\left (B x^{2} + A\right )} \sqrt {x}}{c^{2} x^{6} + 2 \, b c x^{4} + b^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (B x^{2} + A\right )} x^{\frac {5}{2}}}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 222, normalized size = 1.62 \[ \frac {\left (c \,x^{2}+b \right ) \left (2 A \,c^{2} x -2 B b c x +\sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-b c}\, A c \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-b c}\, B b \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )\right ) x^{\frac {5}{2}}}{2 \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} b \,c^{2}} \]
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 \[ \int \frac {{\left (B x^{2} + A\right )} x^{\frac {5}{2}}}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^{5/2}\,\left (B\,x^2+A\right )}{{\left (c\,x^4+b\,x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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