Optimal. Leaf size=104 \[ -\frac {2 \sqrt {b x^2+c x^4} (4 b B-3 A c)}{3 c^3 x}+\frac {x \sqrt {b x^2+c x^4} (4 b B-3 A c)}{3 b c^2}-\frac {x^5 (b B-A c)}{b c \sqrt {b x^2+c x^4}} \]
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Rubi [A] time = 0.20, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2037, 2016, 1588} \begin {gather*} \frac {x \sqrt {b x^2+c x^4} (4 b B-3 A c)}{3 b c^2}-\frac {2 \sqrt {b x^2+c x^4} (4 b B-3 A c)}{3 c^3 x}-\frac {x^5 (b B-A c)}{b c \sqrt {b x^2+c x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 1588
Rule 2016
Rule 2037
Rubi steps
\begin {align*} \int \frac {x^6 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx &=-\frac {(b B-A c) x^5}{b c \sqrt {b x^2+c x^4}}+\frac {(4 b B-3 A c) \int \frac {x^4}{\sqrt {b x^2+c x^4}} \, dx}{b c}\\ &=-\frac {(b B-A c) x^5}{b c \sqrt {b x^2+c x^4}}+\frac {(4 b B-3 A c) x \sqrt {b x^2+c x^4}}{3 b c^2}-\frac {(2 (4 b B-3 A c)) \int \frac {x^2}{\sqrt {b x^2+c x^4}} \, dx}{3 c^2}\\ &=-\frac {(b B-A c) x^5}{b c \sqrt {b x^2+c x^4}}-\frac {2 (4 b B-3 A c) \sqrt {b x^2+c x^4}}{3 c^3 x}+\frac {(4 b B-3 A c) x \sqrt {b x^2+c x^4}}{3 b c^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 60, normalized size = 0.58 \begin {gather*} \frac {x \left (b \left (6 A c-4 B c x^2\right )+c^2 x^2 \left (3 A+B x^2\right )-8 b^2 B\right )}{3 c^3 \sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.85, size = 71, normalized size = 0.68 \begin {gather*} \frac {\sqrt {b x^2+c x^4} \left (6 A b c+3 A c^2 x^2-8 b^2 B-4 b B c x^2+B c^2 x^4\right )}{3 c^3 x \left (b+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 68, normalized size = 0.65 \begin {gather*} \frac {{\left (B c^{2} x^{4} - 8 \, B b^{2} + 6 \, A b c - {\left (4 \, B b c - 3 \, A c^{2}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{3 \, {\left (c^{4} x^{3} + b c^{3} x\right )}} \end {gather*}
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 \begin {gather*} \int \frac {{\left (B x^{2} + A\right )} x^{6}}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 66, normalized size = 0.63 \begin {gather*} \frac {\left (c \,x^{2}+b \right ) \left (B \,c^{2} x^{4}+3 A \,c^{2} x^{2}-4 B b c \,x^{2}+6 A b c -8 B \,b^{2}\right ) x^{3}}{3 \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.56, size = 59, normalized size = 0.57 \begin {gather*} \frac {{\left (c x^{2} + 2 \, b\right )} A}{\sqrt {c x^{2} + b} c^{2}} + \frac {{\left (c^{2} x^{4} - 4 \, b c x^{2} - 8 \, b^{2}\right )} B}{3 \, \sqrt {c x^{2} + b} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 67, normalized size = 0.64 \begin {gather*} \frac {\sqrt {c\,x^4+b\,x^2}\,\left (-8\,B\,b^2-4\,B\,b\,c\,x^2+6\,A\,b\,c+B\,c^2\,x^4+3\,A\,c^2\,x^2\right )}{3\,c^3\,x\,\left (c\,x^2+b\right )} \end {gather*}
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 \begin {gather*} \int \frac {x^{6} \left (A + B x^{2}\right )}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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