Optimal. Leaf size=214 \[ \frac {5 b^{3/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (9 b B-7 A c) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{42 c^{13/4} \sqrt {b x^2+c x^4}}-\frac {5 \sqrt {b x^2+c x^4} (9 b B-7 A c)}{21 c^3 \sqrt {x}}+\frac {x^{3/2} \sqrt {b x^2+c x^4} (9 b B-7 A c)}{7 b c^2}-\frac {x^{11/2} (b B-A c)}{b c \sqrt {b x^2+c x^4}} \]
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Rubi [A] time = 0.33, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2037, 2024, 2032, 329, 220} \[ \frac {5 b^{3/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (9 b B-7 A c) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{42 c^{13/4} \sqrt {b x^2+c x^4}}+\frac {x^{3/2} \sqrt {b x^2+c x^4} (9 b B-7 A c)}{7 b c^2}-\frac {5 \sqrt {b x^2+c x^4} (9 b B-7 A c)}{21 c^3 \sqrt {x}}-\frac {x^{11/2} (b B-A c)}{b c \sqrt {b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 329
Rule 2024
Rule 2032
Rule 2037
Rubi steps
\begin {align*} \int \frac {x^{13/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx &=-\frac {(b B-A c) x^{11/2}}{b c \sqrt {b x^2+c x^4}}+\frac {\left (\frac {9 b B}{2}-\frac {7 A c}{2}\right ) \int \frac {x^{9/2}}{\sqrt {b x^2+c x^4}} \, dx}{b c}\\ &=-\frac {(b B-A c) x^{11/2}}{b c \sqrt {b x^2+c x^4}}+\frac {(9 b B-7 A c) x^{3/2} \sqrt {b x^2+c x^4}}{7 b c^2}-\frac {(5 (9 b B-7 A c)) \int \frac {x^{5/2}}{\sqrt {b x^2+c x^4}} \, dx}{14 c^2}\\ &=-\frac {(b B-A c) x^{11/2}}{b c \sqrt {b x^2+c x^4}}-\frac {5 (9 b B-7 A c) \sqrt {b x^2+c x^4}}{21 c^3 \sqrt {x}}+\frac {(9 b B-7 A c) x^{3/2} \sqrt {b x^2+c x^4}}{7 b c^2}+\frac {(5 b (9 b B-7 A c)) \int \frac {\sqrt {x}}{\sqrt {b x^2+c x^4}} \, dx}{42 c^3}\\ &=-\frac {(b B-A c) x^{11/2}}{b c \sqrt {b x^2+c x^4}}-\frac {5 (9 b B-7 A c) \sqrt {b x^2+c x^4}}{21 c^3 \sqrt {x}}+\frac {(9 b B-7 A c) x^{3/2} \sqrt {b x^2+c x^4}}{7 b c^2}+\frac {\left (5 b (9 b B-7 A c) x \sqrt {b+c x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x^2}} \, dx}{42 c^3 \sqrt {b x^2+c x^4}}\\ &=-\frac {(b B-A c) x^{11/2}}{b c \sqrt {b x^2+c x^4}}-\frac {5 (9 b B-7 A c) \sqrt {b x^2+c x^4}}{21 c^3 \sqrt {x}}+\frac {(9 b B-7 A c) x^{3/2} \sqrt {b x^2+c x^4}}{7 b c^2}+\frac {\left (5 b (9 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 )}{21 c^3 \sqrt {b x^2+c x^4}}\\ &=-\frac {(b B-A c) x^{11/2}}{b c \sqrt {b x^2+c x^4}}-\frac {5 (9 b B-7 A c) \sqrt {b x^2+c x^4}}{21 c^3 \sqrt {x}}+\frac {(9 b B-7 A c) x^{3/2} \sqrt {b x^2+c x^4}}{7 b c^2}+\frac {5 b^{3/4} (9 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 )}{42 c^{13/4} \sqrt {b x^2+c x^4}}\\ \end {align*}
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Mathematica [C] time = 0.17, size = 110, normalized size = 0.51 \[ \frac {x^{3/2} \left (5 b \sqrt {\frac {c x^2}{b}+1} (9 b B-7 A c) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {c x^2}{b}\right )+b c \left (35 A-18 B x^2\right )+2 c^2 x^2 \left (7 A+3 B x^2\right )-45 b^2 B\right )}{21 c^3 \sqrt {x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x^{4} + A x^{2}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}}{c^{2} x^{4} + 2 \, b c x^{2} + b^{2}}, x\right ) \]
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 \[ \int \frac {{\left (B x^{2} + A\right )} x^{\frac {13}{2}}}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 255, normalized size = 1.19 \[ -\frac {\left (c \,x^{2}+b \right ) \left (-12 B \,c^{3} x^{5}-28 A \,c^{3} x^{3}+36 B b \,c^{2} x^{3}-70 A b \,c^{2} x +90 B \,b^{2} c x +35 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \sqrt {-b c}\, A b c \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-45 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \sqrt {-b c}\, B \,b^{2} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )\right ) x^{\frac {5}{2}}}{42 \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} c^{4}} \]
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 \[ \int \frac {{\left (B x^{2} + A\right )} x^{\frac {13}{2}}}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^{13/2}\,\left (B\,x^2+A\right )}{{\left (c\,x^4+b\,x^2\right )}^{3/2}} \,d x \]
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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