Optimal. Leaf size=125 \[ -\frac {b^3 (5 b B-8 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{128 c^{7/2}}+\frac {b \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4} (5 b B-8 A c)}{128 c^3}-\frac {\left (b x^2+c x^4\right )^{3/2} \left (-8 A c+5 b B-6 B c x^2\right )}{48 c^2} \]
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Rubi [A] time = 0.20, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2034, 779, 612, 620, 206} \[ -\frac {b^3 (5 b B-8 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{128 c^{7/2}}+\frac {b \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4} (5 b B-8 A c)}{128 c^3}-\frac {\left (b x^2+c x^4\right )^{3/2} \left (-8 A c+5 b B-6 B c x^2\right )}{48 c^2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 620
Rule 779
Rule 2034
Rubi steps
\begin {align*} \int x^3 \left (A+B x^2\right ) \sqrt {b x^2+c x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x (A+B x) \sqrt {b x+c x^2} \, dx,x,x^2\right )\\ &=-\frac {\left (5 b B-8 A c-6 B c x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{48 c^2}+\frac {(b (5 b B-8 A c)) \operatorname {Subst}\left (\int \sqrt {b x+c x^2} \, dx,x,x^2\right )}{32 c^2}\\ &=\frac {b (5 b B-8 A c) \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{128 c^3}-\frac {\left (5 b B-8 A c-6 B c x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{48 c^2}-\frac {\left (b^3 (5 b B-8 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+c x^2}} \, dx,x,x^2\right )}{256 c^3}\\ &=\frac {b (5 b B-8 A c) \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{128 c^3}-\frac {\left (5 b B-8 A c-6 B c x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{48 c^2}-\frac {\left (b^3 (5 b B-8 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x^2}{\sqrt {b x^2+c x^4}}\right )}{128 c^3}\\ &=\frac {b (5 b B-8 A c) \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{128 c^3}-\frac {\left (5 b B-8 A c-6 B c x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{48 c^2}-\frac {b^3 (5 b B-8 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{128 c^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 151, normalized size = 1.21 \[ \frac {\sqrt {x^2 \left (b+c x^2\right )} \left (\sqrt {c} x \sqrt {\frac {c x^2}{b}+1} \left (-2 b^2 c \left (12 A+5 B x^2\right )+8 b c^2 x^2 \left (2 A+B x^2\right )+16 c^3 x^4 \left (4 A+3 B x^2\right )+15 b^3 B\right )-3 b^{5/2} (5 b B-8 A c) \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )\right )}{384 c^{7/2} x \sqrt {\frac {c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 272, normalized size = 2.18 \[ \left [-\frac {3 \, {\left (5 \, B b^{4} - 8 \, A b^{3} c\right )} \sqrt {c} \log \left (-2 \, c x^{2} - b - 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right ) - 2 \, {\left (48 \, B c^{4} x^{6} + 15 \, B b^{3} c - 24 \, A b^{2} c^{2} + 8 \, {\left (B b c^{3} + 8 \, A c^{4}\right )} x^{4} - 2 \, {\left (5 \, B b^{2} c^{2} - 8 \, A b c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{768 \, c^{4}}, \frac {3 \, {\left (5 \, B b^{4} - 8 \, A b^{3} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-c}}{c x^{2} + b}\right ) + {\left (48 \, B c^{4} x^{6} + 15 \, B b^{3} c - 24 \, A b^{2} c^{2} + 8 \, {\left (B b c^{3} + 8 \, A c^{4}\right )} x^{4} - 2 \, {\left (5 \, B b^{2} c^{2} - 8 \, A b c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{384 \, c^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 177, normalized size = 1.42 \[ \frac {1}{384} \, {\left (2 \, {\left (4 \, {\left (6 \, B x^{2} \mathrm {sgn}\relax (x) + \frac {B b c^{5} \mathrm {sgn}\relax (x) + 8 \, A c^{6} \mathrm {sgn}\relax (x)}{c^{6}}\right )} x^{2} - \frac {5 \, B b^{2} c^{4} \mathrm {sgn}\relax (x) - 8 \, A b c^{5} \mathrm {sgn}\relax (x)}{c^{6}}\right )} x^{2} + \frac {3 \, {\left (5 \, B b^{3} c^{3} \mathrm {sgn}\relax (x) - 8 \, A b^{2} c^{4} \mathrm {sgn}\relax (x)\right )}}{c^{6}}\right )} \sqrt {c x^{2} + b} x + \frac {{\left (5 \, B b^{4} \mathrm {sgn}\relax (x) - 8 \, A b^{3} c \mathrm {sgn}\relax (x)\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + b} \right |}\right )}{128 \, c^{\frac {7}{2}}} - \frac {{\left (5 \, B b^{4} \log \left ({\left | b \right |}\right ) - 8 \, A b^{3} c \log \left ({\left | b \right |}\right )\right )} \mathrm {sgn}\relax (x)}{256 \, c^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 206, normalized size = 1.65 \[ \frac {\sqrt {c \,x^{4}+b \,x^{2}}\, \left (48 \left (c \,x^{2}+b \right )^{\frac {3}{2}} B \,c^{\frac {5}{2}} x^{5}+64 \left (c \,x^{2}+b \right )^{\frac {3}{2}} A \,c^{\frac {5}{2}} x^{3}-40 \left (c \,x^{2}+b \right )^{\frac {3}{2}} B b \,c^{\frac {3}{2}} x^{3}+24 A \,b^{3} c \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right )-15 B \,b^{4} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right )+24 \sqrt {c \,x^{2}+b}\, A \,b^{2} c^{\frac {3}{2}} x -15 \sqrt {c \,x^{2}+b}\, B \,b^{3} \sqrt {c}\, x -48 \left (c \,x^{2}+b \right )^{\frac {3}{2}} A b \,c^{\frac {3}{2}} x +30 \left (c \,x^{2}+b \right )^{\frac {3}{2}} B \,b^{2} \sqrt {c}\, x \right )}{384 \sqrt {c \,x^{2}+b}\, c^{\frac {7}{2}} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.49, size = 225, normalized size = 1.80 \[ -\frac {1}{96} \, {\left (\frac {12 \, \sqrt {c x^{4} + b x^{2}} b x^{2}}{c} - \frac {3 \, b^{3} \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right )}{c^{\frac {5}{2}}} + \frac {6 \, \sqrt {c x^{4} + b x^{2}} b^{2}}{c^{2}} - \frac {16 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}{c}\right )} A + \frac {1}{768} \, {\left (\frac {60 \, \sqrt {c x^{4} + b x^{2}} b^{2} x^{2}}{c^{2}} + \frac {96 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} x^{2}}{c} - \frac {15 \, b^{4} \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right )}{c^{\frac {7}{2}}} + \frac {30 \, \sqrt {c x^{4} + b x^{2}} b^{3}}{c^{3}} - \frac {80 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} b}{c^{2}}\right )} B \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.74, size = 177, normalized size = 1.42 \[ \frac {B\,x^2\,{\left (c\,x^4+b\,x^2\right )}^{3/2}}{8\,c}-\frac {5\,B\,b\,\left (\frac {b^3\,\ln \left (b+2\,c\,x^2+2\,\sqrt {c}\,\relax |x|\,\sqrt {c\,x^2+b}\right )}{16\,c^{5/2}}+\frac {\sqrt {c\,x^4+b\,x^2}\,\left (-3\,b^2+2\,b\,c\,x^2+8\,c^2\,x^4\right )}{24\,c^2}\right )}{16\,c}+\frac {A\,b^3\,\ln \left (b+2\,c\,x^2+2\,\sqrt {c}\,\relax |x|\,\sqrt {c\,x^2+b}\right )}{32\,c^{5/2}}+\frac {A\,\sqrt {c\,x^4+b\,x^2}\,\left (-3\,b^2+2\,b\,c\,x^2+8\,c^2\,x^4\right )}{48\,c^2} \]
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 \[ \int x^{3} \sqrt {x^{2} \left (b + c x^{2}\right )} \left (A + B x^{2}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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