Optimal. Leaf size=369 \[ -\frac {7 b^{9/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (11 b B-13 A c) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{195 c^{15/4} \sqrt {b x^2+c x^4}}+\frac {14 b^{9/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (11 b B-13 A c) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{195 c^{15/4} \sqrt {b x^2+c x^4}}-\frac {14 b^2 x^{3/2} \left (b+c x^2\right ) (11 b B-13 A c)}{195 c^{7/2} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}+\frac {14 b \sqrt {x} \sqrt {b x^2+c x^4} (11 b B-13 A c)}{585 c^3}-\frac {2 x^{5/2} \sqrt {b x^2+c x^4} (11 b B-13 A c)}{117 c^2}+\frac {2 B x^{9/2} \sqrt {b x^2+c x^4}}{13 c} \]
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Rubi [A] time = 0.44, antiderivative size = 369, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2039, 2024, 2032, 329, 305, 220, 1196} \[ -\frac {14 b^2 x^{3/2} \left (b+c x^2\right ) (11 b B-13 A c)}{195 c^{7/2} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {7 b^{9/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (11 b B-13 A c) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{195 c^{15/4} \sqrt {b x^2+c x^4}}+\frac {14 b^{9/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (11 b B-13 A c) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{195 c^{15/4} \sqrt {b x^2+c x^4}}-\frac {2 x^{5/2} \sqrt {b x^2+c x^4} (11 b B-13 A c)}{117 c^2}+\frac {14 b \sqrt {x} \sqrt {b x^2+c x^4} (11 b B-13 A c)}{585 c^3}+\frac {2 B x^{9/2} \sqrt {b x^2+c x^4}}{13 c} \]
Antiderivative was successfully verified.
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Rule 220
Rule 305
Rule 329
Rule 1196
Rule 2024
Rule 2032
Rule 2039
Rubi steps
\begin {align*} \int \frac {x^{11/2} \left (A+B x^2\right )}{\sqrt {b x^2+c x^4}} \, dx &=\frac {2 B x^{9/2} \sqrt {b x^2+c x^4}}{13 c}-\frac {\left (2 \left (\frac {11 b B}{2}-\frac {13 A c}{2}\right )\right ) \int \frac {x^{11/2}}{\sqrt {b x^2+c x^4}} \, dx}{13 c}\\ &=-\frac {2 (11 b B-13 A c) x^{5/2} \sqrt {b x^2+c x^4}}{117 c^2}+\frac {2 B x^{9/2} \sqrt {b x^2+c x^4}}{13 c}+\frac {(7 b (11 b B-13 A c)) \int \frac {x^{7/2}}{\sqrt {b x^2+c x^4}} \, dx}{117 c^2}\\ &=\frac {14 b (11 b B-13 A c) \sqrt {x} \sqrt {b x^2+c x^4}}{585 c^3}-\frac {2 (11 b B-13 A c) x^{5/2} \sqrt {b x^2+c x^4}}{117 c^2}+\frac {2 B x^{9/2} \sqrt {b x^2+c x^4}}{13 c}-\frac {\left (7 b^2 (11 b B-13 A c)\right ) \int \frac {x^{3/2}}{\sqrt {b x^2+c x^4}} \, dx}{195 c^3}\\ &=\frac {14 b (11 b B-13 A c) \sqrt {x} \sqrt {b x^2+c x^4}}{585 c^3}-\frac {2 (11 b B-13 A c) x^{5/2} \sqrt {b x^2+c x^4}}{117 c^2}+\frac {2 B x^{9/2} \sqrt {b x^2+c x^4}}{13 c}-\frac {\left (7 b^2 (11 b B-13 A c) x \sqrt {b+c x^2}\right ) \int \frac {\sqrt {x}}{\sqrt {b+c x^2}} \, dx}{195 c^3 \sqrt {b x^2+c x^4}}\\ &=\frac {14 b (11 b B-13 A c) \sqrt {x} \sqrt {b x^2+c x^4}}{585 c^3}-\frac {2 (11 b B-13 A c) x^{5/2} \sqrt {b x^2+c x^4}}{117 c^2}+\frac {2 B x^{9/2} \sqrt {b x^2+c x^4}}{13 c}-\frac {\left (14 b^2 (11 b B-13 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 )}{195 c^3 \sqrt {b x^2+c x^4}}\\ &=\frac {14 b (11 b B-13 A c) \sqrt {x} \sqrt {b x^2+c x^4}}{585 c^3}-\frac {2 (11 b B-13 A c) x^{5/2} \sqrt {b x^2+c x^4}}{117 c^2}+\frac {2 B x^{9/2} \sqrt {b x^2+c x^4}}{13 c}-\frac {\left (14 b^{5/2} (11 b B-13 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 )}{195 c^{7/2} \sqrt {b x^2+c x^4}}+\frac {\left (14 b^{5/2} (11 b B-13 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 )}{195 c^{7/2} \sqrt {b x^2+c x^4}}\\ &=-\frac {14 b^2 (11 b B-13 A c) x^{3/2} \left (b+c x^2\right )}{195 c^{7/2} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}+\frac {14 b (11 b B-13 A c) \sqrt {x} \sqrt {b x^2+c x^4}}{585 c^3}-\frac {2 (11 b B-13 A c) x^{5/2} \sqrt {b x^2+c x^4}}{117 c^2}+\frac {2 B x^{9/2} \sqrt {b x^2+c x^4}}{13 c}+\frac {14 b^{9/4} (11 b B-13 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 )}{195 c^{15/4} \sqrt {b x^2+c x^4}}-\frac {7 b^{9/4} (11 b B-13 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 )}{195 c^{15/4} \sqrt {b x^2+c x^4}}\\ \end {align*}
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Mathematica [C] time = 0.15, size = 122, normalized size = 0.33 \[ \frac {2 x^{5/2} \left (\left (b+c x^2\right ) \left (-b c \left (91 A+55 B x^2\right )+5 c^2 x^2 \left (13 A+9 B x^2\right )+77 b^2 B\right )+7 b^2 \sqrt {\frac {c x^2}{b}+1} (13 A c-11 b B) \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {c x^2}{b}\right )\right )}{585 c^3 \sqrt {x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.06, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x^{5} + A x^{3}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}}{c x^{2} + b}, x\right ) \]
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 \[ \int \frac {{\left (B x^{2} + A\right )} x^{\frac {11}{2}}}{\sqrt {c x^{4} + b x^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 437, normalized size = 1.18 \[ \frac {\left (90 B \,c^{4} x^{8}+130 A \,c^{4} x^{6}-20 B b \,c^{3} x^{6}-52 A b \,c^{3} x^{4}+44 B \,b^{2} c^{2} x^{4}-182 A \,b^{2} c^{2} x^{2}+154 B \,b^{3} c \,x^{2}+546 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, A \,b^{3} c \EllipticE \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-273 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, A \,b^{3} c \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-462 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, B \,b^{4} \EllipticE \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )+231 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, B \,b^{4} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )\right ) \sqrt {x}}{585 \sqrt {c \,x^{4}+b \,x^{2}}\, c^{4}} \]
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 \[ \int \frac {{\left (B x^{2} + A\right )} x^{\frac {11}{2}}}{\sqrt {c x^{4} + b x^{2}}}\,{d x} \]
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 \[ \int \frac {x^{11/2}\,\left (B\,x^2+A\right )}{\sqrt {c\,x^4+b\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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