Optimal. Leaf size=243 \[ \frac {b^{11/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (13 b B-15 A c) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{77 c^{17/4} \sqrt {b x^2+c x^4}}-\frac {2 b^2 \sqrt {b x^2+c x^4} (13 b B-15 A c)}{77 c^4 \sqrt {x}}+\frac {6 b x^{3/2} \sqrt {b x^2+c x^4} (13 b B-15 A c)}{385 c^3}-\frac {2 x^{7/2} \sqrt {b x^2+c x^4} (13 b B-15 A c)}{165 c^2}+\frac {2 B x^{11/2} \sqrt {b x^2+c x^4}}{15 c} \]
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Rubi [A] time = 0.37, antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2039, 2024, 2032, 329, 220} \[ -\frac {2 b^2 \sqrt {b x^2+c x^4} (13 b B-15 A c)}{77 c^4 \sqrt {x}}+\frac {b^{11/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (13 b B-15 A c) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{77 c^{17/4} \sqrt {b x^2+c x^4}}-\frac {2 x^{7/2} \sqrt {b x^2+c x^4} (13 b B-15 A c)}{165 c^2}+\frac {6 b x^{3/2} \sqrt {b x^2+c x^4} (13 b B-15 A c)}{385 c^3}+\frac {2 B x^{11/2} \sqrt {b x^2+c x^4}}{15 c} \]
Antiderivative was successfully verified.
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Rule 220
Rule 329
Rule 2024
Rule 2032
Rule 2039
Rubi steps
\begin {align*} \int \frac {x^{13/2} \left (A+B x^2\right )}{\sqrt {b x^2+c x^4}} \, dx &=\frac {2 B x^{11/2} \sqrt {b x^2+c x^4}}{15 c}-\frac {\left (2 \left (\frac {13 b B}{2}-\frac {15 A c}{2}\right )\right ) \int \frac {x^{13/2}}{\sqrt {b x^2+c x^4}} \, dx}{15 c}\\ &=-\frac {2 (13 b B-15 A c) x^{7/2} \sqrt {b x^2+c x^4}}{165 c^2}+\frac {2 B x^{11/2} \sqrt {b x^2+c x^4}}{15 c}+\frac {(3 b (13 b B-15 A c)) \int \frac {x^{9/2}}{\sqrt {b x^2+c x^4}} \, dx}{55 c^2}\\ &=\frac {6 b (13 b B-15 A c) x^{3/2} \sqrt {b x^2+c x^4}}{385 c^3}-\frac {2 (13 b B-15 A c) x^{7/2} \sqrt {b x^2+c x^4}}{165 c^2}+\frac {2 B x^{11/2} \sqrt {b x^2+c x^4}}{15 c}-\frac {\left (3 b^2 (13 b B-15 A c)\right ) \int \frac {x^{5/2}}{\sqrt {b x^2+c x^4}} \, dx}{77 c^3}\\ &=-\frac {2 b^2 (13 b B-15 A c) \sqrt {b x^2+c x^4}}{77 c^4 \sqrt {x}}+\frac {6 b (13 b B-15 A c) x^{3/2} \sqrt {b x^2+c x^4}}{385 c^3}-\frac {2 (13 b B-15 A c) x^{7/2} \sqrt {b x^2+c x^4}}{165 c^2}+\frac {2 B x^{11/2} \sqrt {b x^2+c x^4}}{15 c}+\frac {\left (b^3 (13 b B-15 A c)\right ) \int \frac {\sqrt {x}}{\sqrt {b x^2+c x^4}} \, dx}{77 c^4}\\ &=-\frac {2 b^2 (13 b B-15 A c) \sqrt {b x^2+c x^4}}{77 c^4 \sqrt {x}}+\frac {6 b (13 b B-15 A c) x^{3/2} \sqrt {b x^2+c x^4}}{385 c^3}-\frac {2 (13 b B-15 A c) x^{7/2} \sqrt {b x^2+c x^4}}{165 c^2}+\frac {2 B x^{11/2} \sqrt {b x^2+c x^4}}{15 c}+\frac {\left (b^3 (13 b B-15 A c) x \sqrt {b+c x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x^2}} \, dx}{77 c^4 \sqrt {b x^2+c x^4}}\\ &=-\frac {2 b^2 (13 b B-15 A c) \sqrt {b x^2+c x^4}}{77 c^4 \sqrt {x}}+\frac {6 b (13 b B-15 A c) x^{3/2} \sqrt {b x^2+c x^4}}{385 c^3}-\frac {2 (13 b B-15 A c) x^{7/2} \sqrt {b x^2+c x^4}}{165 c^2}+\frac {2 B x^{11/2} \sqrt {b x^2+c x^4}}{15 c}+\frac {\left (2 b^3 (13 b B-15 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 )}{77 c^4 \sqrt {b x^2+c x^4}}\\ &=-\frac {2 b^2 (13 b B-15 A c) \sqrt {b x^2+c x^4}}{77 c^4 \sqrt {x}}+\frac {6 b (13 b B-15 A c) x^{3/2} \sqrt {b x^2+c x^4}}{385 c^3}-\frac {2 (13 b B-15 A c) x^{7/2} \sqrt {b x^2+c x^4}}{165 c^2}+\frac {2 B x^{11/2} \sqrt {b x^2+c x^4}}{15 c}+\frac {b^{11/4} (13 b B-15 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 )}{77 c^{17/4} \sqrt {b x^2+c x^4}}\\ \end {align*}
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Mathematica [C] time = 0.18, size = 143, normalized size = 0.59 \[ \frac {2 x^{3/2} \left (15 b^3 \sqrt {\frac {c x^2}{b}+1} (13 b B-15 A c) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {c x^2}{b}\right )-\left (b+c x^2\right ) \left (-9 b^2 c \left (25 A+13 B x^2\right )+b c^2 x^2 \left (135 A+91 B x^2\right )-7 c^3 x^4 \left (15 A+11 B x^2\right )+195 b^3 B\right )\right )}{1155 c^4 \sqrt {x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.03, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x^{6} + A x^{4}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}}{c x^{2} + b}, x\right ) \]
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 \[ \int \frac {{\left (B x^{2} + A\right )} x^{\frac {13}{2}}}{\sqrt {c x^{4} + b x^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 298, normalized size = 1.23 \[ -\frac {\left (-154 B \,c^{5} x^{9}-210 A \,c^{5} x^{7}+28 B b \,c^{4} x^{7}+60 A b \,c^{4} x^{5}-52 B \,b^{2} c^{3} x^{5}-180 A \,b^{2} c^{3} x^{3}+156 B \,b^{3} c^{2} x^{3}-450 A \,b^{3} c^{2} x +390 B \,b^{4} c x +225 \sqrt {-b c}\, \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, A \,b^{3} c \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-195 \sqrt {-b c}\, \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, B \,b^{4} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )\right ) \sqrt {x}}{1155 \sqrt {c \,x^{4}+b \,x^{2}}\, c^{5}} \]
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 \[ \int \frac {{\left (B x^{2} + A\right )} x^{\frac {13}{2}}}{\sqrt {c x^{4} + b x^{2}}}\,{d x} \]
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 \[ \int \frac {x^{13/2}\,\left (B\,x^2+A\right )}{\sqrt {c\,x^4+b\,x^2}} \,d x \]
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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