Optimal. Leaf size=61 \[ -\frac {(2 b B-7 A c) \left (b x^2+c x^4\right )^{5/2}}{35 c^2 x^5}+\frac {B \left (b x^2+c x^4\right )^{5/2}}{7 c x^3} \]
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Rubi [A]
time = 0.11, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2064, 2039}
\begin {gather*} \frac {B \left (b x^2+c x^4\right )^{5/2}}{7 c x^3}-\frac {\left (b x^2+c x^4\right )^{5/2} (2 b B-7 A c)}{35 c^2 x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 2039
Rule 2064
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{x^2} \, dx &=\frac {B \left (b x^2+c x^4\right )^{5/2}}{7 c x^3}-\frac {(2 b B-7 A c) \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^2} \, dx}{7 c}\\ &=-\frac {(2 b B-7 A c) \left (b x^2+c x^4\right )^{5/2}}{35 c^2 x^5}+\frac {B \left (b x^2+c x^4\right )^{5/2}}{7 c x^3}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 41, normalized size = 0.67 \begin {gather*} \frac {\left (x^2 \left (b+c x^2\right )\right )^{5/2} \left (-2 b B+7 A c+5 B c x^2\right )}{35 c^2 x^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 45, normalized size = 0.74
method | result | size |
gosper | \(\frac {\left (c \,x^{2}+b \right ) \left (5 B c \,x^{2}+7 A c -2 B b \right ) \left (x^{4} c +b \,x^{2}\right )^{\frac {3}{2}}}{35 c^{2} x^{3}}\) | \(45\) |
default | \(\frac {\left (c \,x^{2}+b \right ) \left (5 B c \,x^{2}+7 A c -2 B b \right ) \left (x^{4} c +b \,x^{2}\right )^{\frac {3}{2}}}{35 c^{2} x^{3}}\) | \(45\) |
trager | \(\frac {\left (5 B \,c^{3} x^{6}+7 A \,c^{3} x^{4}+8 B b \,c^{2} x^{4}+14 A b \,c^{2} x^{2}+B \,b^{2} c \,x^{2}+7 A \,b^{2} c -2 B \,b^{3}\right ) \sqrt {x^{4} c +b \,x^{2}}}{35 c^{2} x}\) | \(83\) |
risch | \(\frac {\sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \left (5 B \,c^{3} x^{6}+7 A \,c^{3} x^{4}+8 B b \,c^{2} x^{4}+14 A b \,c^{2} x^{2}+B \,b^{2} c \,x^{2}+7 A \,b^{2} c -2 B \,b^{3}\right )}{35 x \,c^{2}}\) | \(83\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 80, normalized size = 1.31 \begin {gather*} \frac {{\left (c^{2} x^{4} + 2 \, b c x^{2} + b^{2}\right )} \sqrt {c x^{2} + b} A}{5 \, c} + \frac {{\left (5 \, c^{3} x^{6} + 8 \, b c^{2} x^{4} + b^{2} c x^{2} - 2 \, b^{3}\right )} \sqrt {c x^{2} + b} B}{35 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.58, size = 80, normalized size = 1.31 \begin {gather*} \frac {{\left (5 \, B c^{3} x^{6} + {\left (8 \, B b c^{2} + 7 \, A c^{3}\right )} x^{4} - 2 \, B b^{3} + 7 \, A b^{2} c + {\left (B b^{2} c + 14 \, A b c^{2}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{35 \, c^{2} x} \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 {\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}} \left (A + B x^{2}\right )}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.34, size = 72, normalized size = 1.18 \begin {gather*} \frac {{\left (2 \, B b^{\frac {7}{2}} - 7 \, A b^{\frac {5}{2}} c\right )} \mathrm {sgn}\left (x\right )}{35 \, c^{2}} + \frac {5 \, {\left (c x^{2} + b\right )}^{\frac {7}{2}} B \mathrm {sgn}\left (x\right ) - 7 \, {\left (c x^{2} + b\right )}^{\frac {5}{2}} B b \mathrm {sgn}\left (x\right ) + 7 \, {\left (c x^{2} + b\right )}^{\frac {5}{2}} A c \mathrm {sgn}\left (x\right )}{35 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.23, size = 83, normalized size = 1.36 \begin {gather*} \frac {\sqrt {c\,x^4+b\,x^2}\,\left (\frac {x^4\,\left (7\,A\,c^3+8\,B\,b\,c^2\right )}{35\,c^2}-\frac {2\,B\,b^3-7\,A\,b^2\,c}{35\,c^2}+\frac {B\,c\,x^6}{7}+\frac {b\,x^2\,\left (14\,A\,c+B\,b\right )}{35\,c}\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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