Optimal. Leaf size=326 \[ -\frac {2 b^{5/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (b B-3 A c) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 c^{7/4} \sqrt {b x^2+c x^4}}+\frac {4 b^{5/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (b B-3 A c) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 c^{7/4} \sqrt {b x^2+c x^4}}-\frac {4 b x^{3/2} \left (b+c x^2\right ) (b B-3 A c)}{15 c^{3/2} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {2 \sqrt {x} \sqrt {b x^2+c x^4} (b B-3 A c)}{15 c}+\frac {2 B \left (b x^2+c x^4\right )^{3/2}}{9 c x^{3/2}} \]
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Rubi [A] time = 0.37, antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2039, 2021, 2032, 329, 305, 220, 1196} \[ -\frac {2 b^{5/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (b B-3 A c) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 c^{7/4} \sqrt {b x^2+c x^4}}+\frac {4 b^{5/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (b B-3 A c) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 c^{7/4} \sqrt {b x^2+c x^4}}-\frac {4 b x^{3/2} \left (b+c x^2\right ) (b B-3 A c)}{15 c^{3/2} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {2 \sqrt {x} \sqrt {b x^2+c x^4} (b B-3 A c)}{15 c}+\frac {2 B \left (b x^2+c x^4\right )^{3/2}}{9 c x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 305
Rule 329
Rule 1196
Rule 2021
Rule 2032
Rule 2039
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \sqrt {b x^2+c x^4}}{\sqrt {x}} \, dx &=\frac {2 B \left (b x^2+c x^4\right )^{3/2}}{9 c x^{3/2}}-\frac {\left (2 \left (\frac {3 b B}{2}-\frac {9 A c}{2}\right )\right ) \int \frac {\sqrt {b x^2+c x^4}}{\sqrt {x}} \, dx}{9 c}\\ &=-\frac {2 (b B-3 A c) \sqrt {x} \sqrt {b x^2+c x^4}}{15 c}+\frac {2 B \left (b x^2+c x^4\right )^{3/2}}{9 c x^{3/2}}-\frac {(2 b (b B-3 A c)) \int \frac {x^{3/2}}{\sqrt {b x^2+c x^4}} \, dx}{15 c}\\ &=-\frac {2 (b B-3 A c) \sqrt {x} \sqrt {b x^2+c x^4}}{15 c}+\frac {2 B \left (b x^2+c x^4\right )^{3/2}}{9 c x^{3/2}}-\frac {\left (2 b (b B-3 A c) x \sqrt {b+c x^2}\right ) \int \frac {\sqrt {x}}{\sqrt {b+c x^2}} \, dx}{15 c \sqrt {b x^2+c x^4}}\\ &=-\frac {2 (b B-3 A c) \sqrt {x} \sqrt {b x^2+c x^4}}{15 c}+\frac {2 B \left (b x^2+c x^4\right )^{3/2}}{9 c x^{3/2}}-\frac {\left (4 b (b B-3 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 )}{15 c \sqrt {b x^2+c x^4}}\\ &=-\frac {2 (b B-3 A c) \sqrt {x} \sqrt {b x^2+c x^4}}{15 c}+\frac {2 B \left (b x^2+c x^4\right )^{3/2}}{9 c x^{3/2}}-\frac {\left (4 b^{3/2} (b B-3 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 )}{15 c^{3/2} \sqrt {b x^2+c x^4}}+\frac {\left (4 b^{3/2} (b B-3 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 )}{15 c^{3/2} \sqrt {b x^2+c x^4}}\\ &=-\frac {4 b (b B-3 A c) x^{3/2} \left (b+c x^2\right )}{15 c^{3/2} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {2 (b B-3 A c) \sqrt {x} \sqrt {b x^2+c x^4}}{15 c}+\frac {2 B \left (b x^2+c x^4\right )^{3/2}}{9 c x^{3/2}}+\frac {4 b^{5/4} (b B-3 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 )}{15 c^{7/4} \sqrt {b x^2+c x^4}}-\frac {2 b^{5/4} (b B-3 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 )}{15 c^{7/4} \sqrt {b x^2+c x^4}}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 94, normalized size = 0.29 \[ \frac {2 \sqrt {x} \sqrt {x^2 \left (b+c x^2\right )} \left ((3 A c-b B) \, _2F_1\left (-\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {c x^2}{b}\right )+B \sqrt {\frac {c x^2}{b}+1} \left (b+c x^2\right )\right )}{9 c \sqrt {\frac {c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{4} + b x^{2}} {\left (B x^{2} + A\right )}}{\sqrt {x}}, x\right ) \]
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 \[ \int \frac {\sqrt {c x^{4} + b x^{2}} {\left (B x^{2} + A\right )}}{\sqrt {x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 422, normalized size = 1.29 \[ \frac {2 \sqrt {c \,x^{4}+b \,x^{2}}\, \left (5 B \,c^{3} x^{6}+9 A \,c^{3} x^{4}+7 B b \,c^{2} x^{4}+9 A b \,c^{2} x^{2}+2 B \,b^{2} c \,x^{2}+18 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, A \,b^{2} c \EllipticE \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-9 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, A \,b^{2} c \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-6 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, B \,b^{3} \EllipticE \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )+3 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, B \,b^{3} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )\right )}{45 \left (c \,x^{2}+b \right ) c^{2} x^{\frac {3}{2}}} \]
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 \[ \int \frac {\sqrt {c x^{4} + b x^{2}} {\left (B x^{2} + A\right )}}{\sqrt {x}}\,{d x} \]
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 \[ \int \frac {\left (B\,x^2+A\right )\,\sqrt {c\,x^4+b\,x^2}}{\sqrt {x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x^{2} \left (b + c x^{2}\right )} \left (A + B x^{2}\right )}{\sqrt {x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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