Optimal. Leaf size=96 \[ \frac {2 b \left (b x^2+c x^4\right )^{5/2} (4 b B-9 A c)}{315 c^3 x^5}-\frac {\left (b x^2+c x^4\right )^{5/2} (4 b B-9 A c)}{63 c^2 x^3}+\frac {B \left (b x^2+c x^4\right )^{5/2}}{9 c x} \]
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Rubi [A] time = 0.07, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1145, 2002, 2014} \[ -\frac {\left (b x^2+c x^4\right )^{5/2} (4 b B-9 A c)}{63 c^2 x^3}+\frac {2 b \left (b x^2+c x^4\right )^{5/2} (4 b B-9 A c)}{315 c^3 x^5}+\frac {B \left (b x^2+c x^4\right )^{5/2}}{9 c x} \]
Antiderivative was successfully verified.
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Rule 1145
Rule 2002
Rule 2014
Rubi steps
\begin {align*} \int \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx &=\frac {B \left (b x^2+c x^4\right )^{5/2}}{9 c x}-\frac {(4 b B-9 A c) \int \left (b x^2+c x^4\right )^{3/2} \, dx}{9 c}\\ &=-\frac {(4 b B-9 A c) \left (b x^2+c x^4\right )^{5/2}}{63 c^2 x^3}+\frac {B \left (b x^2+c x^4\right )^{5/2}}{9 c x}+\frac {(2 b (4 b B-9 A c)) \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^2} \, dx}{63 c^2}\\ &=\frac {2 b (4 b B-9 A c) \left (b x^2+c x^4\right )^{5/2}}{315 c^3 x^5}-\frac {(4 b B-9 A c) \left (b x^2+c x^4\right )^{5/2}}{63 c^2 x^3}+\frac {B \left (b x^2+c x^4\right )^{5/2}}{9 c x}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 71, normalized size = 0.74 \[ \frac {x \left (b+c x^2\right )^3 \left (-2 b c \left (9 A+10 B x^2\right )+5 c^2 x^2 \left (9 A+7 B x^2\right )+8 b^2 B\right )}{315 c^3 \sqrt {x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 106, normalized size = 1.10 \[ \frac {{\left (35 \, B c^{4} x^{8} + 5 \, {\left (10 \, B b c^{3} + 9 \, A c^{4}\right )} x^{6} + 8 \, B b^{4} - 18 \, A b^{3} c + 3 \, {\left (B b^{2} c^{2} + 24 \, A b c^{3}\right )} x^{4} - {\left (4 \, B b^{3} c - 9 \, A b^{2} c^{2}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{315 \, c^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 105, normalized size = 1.09 \[ -\frac {2 \, {\left (4 \, B b^{\frac {9}{2}} - 9 \, A b^{\frac {7}{2}} c\right )} \mathrm {sgn}\relax (x)}{315 \, c^{3}} + \frac {35 \, {\left (c x^{2} + b\right )}^{\frac {9}{2}} B \mathrm {sgn}\relax (x) - 90 \, {\left (c x^{2} + b\right )}^{\frac {7}{2}} B b \mathrm {sgn}\relax (x) + 63 \, {\left (c x^{2} + b\right )}^{\frac {5}{2}} B b^{2} \mathrm {sgn}\relax (x) + 45 \, {\left (c x^{2} + b\right )}^{\frac {7}{2}} A c \mathrm {sgn}\relax (x) - 63 \, {\left (c x^{2} + b\right )}^{\frac {5}{2}} A b c \mathrm {sgn}\relax (x)}{315 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 67, normalized size = 0.70 \[ -\frac {\left (c \,x^{2}+b \right ) \left (-35 B \,c^{2} x^{4}-45 A \,c^{2} x^{2}+20 B b c \,x^{2}+18 A b c -8 B \,b^{2}\right ) \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}{315 c^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.56, size = 105, normalized size = 1.09 \[ \frac {{\left (5 \, c^{3} x^{6} + 8 \, b c^{2} x^{4} + b^{2} c x^{2} - 2 \, b^{3}\right )} \sqrt {c x^{2} + b} A}{35 \, c^{2}} + \frac {{\left (35 \, c^{4} x^{8} + 50 \, b c^{3} x^{6} + 3 \, b^{2} c^{2} x^{4} - 4 \, b^{3} c x^{2} + 8 \, b^{4}\right )} \sqrt {c x^{2} + b} B}{315 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 103, normalized size = 1.07 \[ \frac {\sqrt {c\,x^4+b\,x^2}\,\left (\frac {8\,B\,b^4-18\,A\,b^3\,c}{315\,c^3}+\frac {x^6\,\left (45\,A\,c^4+50\,B\,b\,c^3\right )}{315\,c^3}+\frac {B\,c\,x^8}{9}+\frac {b^2\,x^2\,\left (9\,A\,c-4\,B\,b\right )}{315\,c^2}+\frac {b\,x^4\,\left (24\,A\,c+B\,b\right )}{105\,c}\right )}{x} \]
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 \[ \int \left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}} \left (A + B x^{2}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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