Optimal. Leaf size=104 \[ -\frac {(b B-A c) x^5}{b c \sqrt {b x^2+c x^4}}-\frac {2 (4 b B-3 A c) \sqrt {b x^2+c x^4}}{3 c^3 x}+\frac {(4 b B-3 A c) x \sqrt {b x^2+c x^4}}{3 b c^2} \]
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Rubi [A]
time = 0.13, 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 = {2062, 2041,
1602} \begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 1602
Rule 2041
Rule 2062
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 (-8 b^2 B+c^2 x^2 \left (3 A+B x^2\right )+b \left (6 A c-4 B c x^2\right )\right )}{3 c^3 \sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 66, normalized size = 0.63
method | result | size |
gosper | \(\frac {\left (c \,x^{2}+b \right ) \left (B \,c^{2} x^{4}+3 A \,c^{2} x^{2}-4 b B \,x^{2} c +6 A b c -8 b^{2} B \right ) x^{3}}{3 c^{3} \left (x^{4} c +b \,x^{2}\right )^{\frac {3}{2}}}\) | \(66\) |
default | \(\frac {\left (c \,x^{2}+b \right ) \left (B \,c^{2} x^{4}+3 A \,c^{2} x^{2}-4 b B \,x^{2} c +6 A b c -8 b^{2} B \right ) x^{3}}{3 c^{3} \left (x^{4} c +b \,x^{2}\right )^{\frac {3}{2}}}\) | \(66\) |
trager | \(\frac {\left (B \,c^{2} x^{4}+3 A \,c^{2} x^{2}-4 b B \,x^{2} c +6 A b c -8 b^{2} B \right ) \sqrt {x^{4} c +b \,x^{2}}}{3 \left (c \,x^{2}+b \right ) c^{3} x}\) | \(68\) |
risch | \(\frac {\left (B c \,x^{2}+3 A c -5 B b \right ) \left (c \,x^{2}+b \right ) x}{3 c^{3} \sqrt {x^{2} \left (c \,x^{2}+b \right )}}+\frac {b \left (A c -B b \right ) x}{c^{3} \sqrt {x^{2} \left (c \,x^{2}+b \right )}}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, 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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Fricas [A]
time = 1.48, 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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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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Giac [A]
time = 1.39, size = 107, normalized size = 1.03 \begin {gather*} \frac {2 \, {\left (4 \, B b^{2} - 3 \, A b c\right )} \mathrm {sgn}\left (x\right )}{3 \, \sqrt {b} c^{3}} - \frac {B b^{2} - A b c}{\sqrt {c x^{2} + b} c^{3} \mathrm {sgn}\left (x\right )} + \frac {{\left (c x^{2} + b\right )}^{\frac {3}{2}} B c^{6} - 6 \, \sqrt {c x^{2} + b} B b c^{6} + 3 \, \sqrt {c x^{2} + b} A c^{7}}{3 \, c^{9} \mathrm {sgn}\left (x\right )} \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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