Optimal. Leaf size=167 \[ \frac {b^5 (7 b B-12 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{1024 c^{9/2}}-\frac {b^3 \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4} (7 b B-12 A c)}{1024 c^4}+\frac {b \left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{3/2} (7 b B-12 A c)}{384 c^3}-\frac {\left (b x^2+c x^4\right )^{5/2} \left (-12 A c+7 b B-10 B c x^2\right )}{120 c^2} \]
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Rubi [A] time = 0.25, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ -\frac {b^3 \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4} (7 b B-12 A c)}{1024 c^4}+\frac {b^5 (7 b B-12 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{1024 c^{9/2}}+\frac {b \left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{3/2} (7 b B-12 A c)}{384 c^3}-\frac {\left (b x^2+c x^4\right )^{5/2} \left (-12 A c+7 b B-10 B c x^2\right )}{120 c^2} \]
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 ) \left (b x^2+c x^4\right )^{3/2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x (A+B x) \left (b x+c x^2\right )^{3/2} \, dx,x,x^2\right )\\ &=-\frac {\left (7 b B-12 A c-10 B c x^2\right ) \left (b x^2+c x^4\right )^{5/2}}{120 c^2}+\frac {(b (7 b B-12 A c)) \operatorname {Subst}\left (\int \left (b x+c x^2\right )^{3/2} \, dx,x,x^2\right )}{48 c^2}\\ &=\frac {b (7 b B-12 A c) \left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{384 c^3}-\frac {\left (7 b B-12 A c-10 B c x^2\right ) \left (b x^2+c x^4\right )^{5/2}}{120 c^2}-\frac {\left (b^3 (7 b B-12 A c)\right ) \operatorname {Subst}\left (\int \sqrt {b x+c x^2} \, dx,x,x^2\right )}{256 c^3}\\ &=-\frac {b^3 (7 b B-12 A c) \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{1024 c^4}+\frac {b (7 b B-12 A c) \left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{384 c^3}-\frac {\left (7 b B-12 A c-10 B c x^2\right ) \left (b x^2+c x^4\right )^{5/2}}{120 c^2}+\frac {\left (b^5 (7 b B-12 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+c x^2}} \, dx,x,x^2\right )}{2048 c^4}\\ &=-\frac {b^3 (7 b B-12 A c) \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{1024 c^4}+\frac {b (7 b B-12 A c) \left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{384 c^3}-\frac {\left (7 b B-12 A c-10 B c x^2\right ) \left (b x^2+c x^4\right )^{5/2}}{120 c^2}+\frac {\left (b^5 (7 b B-12 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 )}{1024 c^4}\\ &=-\frac {b^3 (7 b B-12 A c) \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{1024 c^4}+\frac {b (7 b B-12 A c) \left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{384 c^3}-\frac {\left (7 b B-12 A c-10 B c x^2\right ) \left (b x^2+c x^4\right )^{5/2}}{120 c^2}+\frac {b^5 (7 b B-12 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{1024 c^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 193, normalized size = 1.16 \[ \frac {\sqrt {x^2 \left (b+c x^2\right )} \left (15 b^{9/2} (7 b B-12 A c) \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )+\sqrt {c} x \sqrt {\frac {c x^2}{b}+1} \left (10 b^4 c \left (18 A+7 B x^2\right )-8 b^3 c^2 x^2 \left (15 A+7 B x^2\right )+48 b^2 c^3 x^4 \left (2 A+B x^2\right )+64 b c^4 x^6 \left (33 A+26 B x^2\right )+256 c^5 x^8 \left (6 A+5 B x^2\right )-105 b^5 B\right )\right )}{15360 c^{9/2} x \sqrt {\frac {c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.29, size = 369, normalized size = 2.21 \[ \left [-\frac {15 \, {\left (7 \, B b^{6} - 12 \, A b^{5} c\right )} \sqrt {c} \log \left (-2 \, c x^{2} - b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right ) - 2 \, {\left (1280 \, B c^{6} x^{10} + 128 \, {\left (13 \, B b c^{5} + 12 \, A c^{6}\right )} x^{8} - 105 \, B b^{5} c + 180 \, A b^{4} c^{2} + 48 \, {\left (B b^{2} c^{4} + 44 \, A b c^{5}\right )} x^{6} - 8 \, {\left (7 \, B b^{3} c^{3} - 12 \, A b^{2} c^{4}\right )} x^{4} + 10 \, {\left (7 \, B b^{4} c^{2} - 12 \, A b^{3} c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{30720 \, c^{5}}, -\frac {15 \, {\left (7 \, B b^{6} - 12 \, A b^{5} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-c}}{c x^{2} + b}\right ) - {\left (1280 \, B c^{6} x^{10} + 128 \, {\left (13 \, B b c^{5} + 12 \, A c^{6}\right )} x^{8} - 105 \, B b^{5} c + 180 \, A b^{4} c^{2} + 48 \, {\left (B b^{2} c^{4} + 44 \, A b c^{5}\right )} x^{6} - 8 \, {\left (7 \, B b^{3} c^{3} - 12 \, A b^{2} c^{4}\right )} x^{4} + 10 \, {\left (7 \, B b^{4} c^{2} - 12 \, A b^{3} c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{15360 \, c^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 246, normalized size = 1.47 \[ \frac {1}{15360} \, {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, B c x^{2} \mathrm {sgn}\relax (x) + \frac {13 \, B b c^{10} \mathrm {sgn}\relax (x) + 12 \, A c^{11} \mathrm {sgn}\relax (x)}{c^{10}}\right )} x^{2} + \frac {3 \, {\left (B b^{2} c^{9} \mathrm {sgn}\relax (x) + 44 \, A b c^{10} \mathrm {sgn}\relax (x)\right )}}{c^{10}}\right )} x^{2} - \frac {7 \, B b^{3} c^{8} \mathrm {sgn}\relax (x) - 12 \, A b^{2} c^{9} \mathrm {sgn}\relax (x)}{c^{10}}\right )} x^{2} + \frac {5 \, {\left (7 \, B b^{4} c^{7} \mathrm {sgn}\relax (x) - 12 \, A b^{3} c^{8} \mathrm {sgn}\relax (x)\right )}}{c^{10}}\right )} x^{2} - \frac {15 \, {\left (7 \, B b^{5} c^{6} \mathrm {sgn}\relax (x) - 12 \, A b^{4} c^{7} \mathrm {sgn}\relax (x)\right )}}{c^{10}}\right )} \sqrt {c x^{2} + b} x - \frac {{\left (7 \, B b^{6} \mathrm {sgn}\relax (x) - 12 \, A b^{5} c \mathrm {sgn}\relax (x)\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + b} \right |}\right )}{1024 \, c^{\frac {9}{2}}} + \frac {{\left (7 \, B b^{6} \log \left ({\left | b \right |}\right ) - 12 \, A b^{5} c \log \left ({\left | b \right |}\right )\right )} \mathrm {sgn}\relax (x)}{2048 \, c^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 286, normalized size = 1.71 \[ \frac {\left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \left (1280 \left (c \,x^{2}+b \right )^{\frac {5}{2}} B \,c^{\frac {7}{2}} x^{7}+1536 \left (c \,x^{2}+b \right )^{\frac {5}{2}} A \,c^{\frac {7}{2}} x^{5}-896 \left (c \,x^{2}+b \right )^{\frac {5}{2}} B b \,c^{\frac {5}{2}} x^{5}-180 A \,b^{5} c \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right )+105 B \,b^{6} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right )-180 \sqrt {c \,x^{2}+b}\, A \,b^{4} c^{\frac {3}{2}} x -960 \left (c \,x^{2}+b \right )^{\frac {5}{2}} A b \,c^{\frac {5}{2}} x^{3}+105 \sqrt {c \,x^{2}+b}\, B \,b^{5} \sqrt {c}\, x +560 \left (c \,x^{2}+b \right )^{\frac {5}{2}} B \,b^{2} c^{\frac {3}{2}} x^{3}-120 \left (c \,x^{2}+b \right )^{\frac {3}{2}} A \,b^{3} c^{\frac {3}{2}} x +70 \left (c \,x^{2}+b \right )^{\frac {3}{2}} B \,b^{4} \sqrt {c}\, x +480 \left (c \,x^{2}+b \right )^{\frac {5}{2}} A \,b^{2} c^{\frac {3}{2}} x -280 \left (c \,x^{2}+b \right )^{\frac {5}{2}} B \,b^{3} \sqrt {c}\, x \right )}{15360 \left (c \,x^{2}+b \right )^{\frac {3}{2}} c^{\frac {9}{2}} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.56, size = 315, normalized size = 1.89 \[ \frac {1}{2560} \, {\left (\frac {60 \, \sqrt {c x^{4} + b x^{2}} b^{3} x^{2}}{c^{2}} - \frac {160 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} b x^{2}}{c} - \frac {15 \, b^{5} \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right )}{c^{\frac {7}{2}}} + \frac {30 \, \sqrt {c x^{4} + b x^{2}} b^{4}}{c^{3}} - \frac {80 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} b^{2}}{c^{2}} + \frac {256 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {5}{2}}}{c}\right )} A - \frac {1}{30720} \, {\left (\frac {420 \, \sqrt {c x^{4} + b x^{2}} b^{4} x^{2}}{c^{3}} - \frac {1120 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} b^{2} x^{2}}{c^{2}} - \frac {2560 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {5}{2}} x^{2}}{c} - \frac {105 \, b^{6} \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right )}{c^{\frac {9}{2}}} + \frac {210 \, \sqrt {c x^{4} + b x^{2}} b^{5}}{c^{4}} - \frac {560 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} b^{3}}{c^{3}} + \frac {1792 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {5}{2}} b}{c^{2}}\right )} B \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^3\,\left (B\,x^2+A\right )\,{\left (c\,x^4+b\,x^2\right )}^{3/2} \,d x \]
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 \[ \int x^{3} \left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}} \left (A + B x^{2}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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