Optimal. Leaf size=223 \[ -\frac {b^6 (9 b B-14 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{2048 c^{11/2}}+\frac {b^4 \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4} (9 b B-14 A c)}{2048 c^5}-\frac {b^2 \left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{3/2} (9 b B-14 A c)}{768 c^4}+\frac {b \left (b x^2+c x^4\right )^{5/2} (9 b B-14 A c)}{240 c^3}-\frac {x^2 \left (b x^2+c x^4\right )^{5/2} (9 b B-14 A c)}{168 c^2}+\frac {B x^4 \left (b x^2+c x^4\right )^{5/2}}{14 c} \]
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Rubi [A] time = 0.40, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {2034, 794, 670, 640, 612, 620, 206} \begin {gather*} \frac {b^4 \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4} (9 b B-14 A c)}{2048 c^5}-\frac {b^2 \left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{3/2} (9 b B-14 A c)}{768 c^4}-\frac {b^6 (9 b B-14 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{2048 c^{11/2}}+\frac {b \left (b x^2+c x^4\right )^{5/2} (9 b B-14 A c)}{240 c^3}-\frac {x^2 \left (b x^2+c x^4\right )^{5/2} (9 b B-14 A c)}{168 c^2}+\frac {B x^4 \left (b x^2+c x^4\right )^{5/2}}{14 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 620
Rule 640
Rule 670
Rule 794
Rule 2034
Rubi steps
\begin {align*} \int x^5 \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^2 (A+B x) \left (b x+c x^2\right )^{3/2} \, dx,x,x^2\right )\\ &=\frac {B x^4 \left (b x^2+c x^4\right )^{5/2}}{14 c}+\frac {\left (2 (-b B+A c)+\frac {5}{2} (-b B+2 A c)\right ) \operatorname {Subst}\left (\int x^2 \left (b x+c x^2\right )^{3/2} \, dx,x,x^2\right )}{14 c}\\ &=-\frac {(9 b B-14 A c) x^2 \left (b x^2+c x^4\right )^{5/2}}{168 c^2}+\frac {B x^4 \left (b x^2+c x^4\right )^{5/2}}{14 c}+\frac {(b (9 b B-14 A c)) \operatorname {Subst}\left (\int x \left (b x+c x^2\right )^{3/2} \, dx,x,x^2\right )}{48 c^2}\\ &=\frac {b (9 b B-14 A c) \left (b x^2+c x^4\right )^{5/2}}{240 c^3}-\frac {(9 b B-14 A c) x^2 \left (b x^2+c x^4\right )^{5/2}}{168 c^2}+\frac {B x^4 \left (b x^2+c x^4\right )^{5/2}}{14 c}-\frac {\left (b^2 (9 b B-14 A c)\right ) \operatorname {Subst}\left (\int \left (b x+c x^2\right )^{3/2} \, dx,x,x^2\right )}{96 c^3}\\ &=-\frac {b^2 (9 b B-14 A c) \left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{768 c^4}+\frac {b (9 b B-14 A c) \left (b x^2+c x^4\right )^{5/2}}{240 c^3}-\frac {(9 b B-14 A c) x^2 \left (b x^2+c x^4\right )^{5/2}}{168 c^2}+\frac {B x^4 \left (b x^2+c x^4\right )^{5/2}}{14 c}+\frac {\left (b^4 (9 b B-14 A c)\right ) \operatorname {Subst}\left (\int \sqrt {b x+c x^2} \, dx,x,x^2\right )}{512 c^4}\\ &=\frac {b^4 (9 b B-14 A c) \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{2048 c^5}-\frac {b^2 (9 b B-14 A c) \left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{768 c^4}+\frac {b (9 b B-14 A c) \left (b x^2+c x^4\right )^{5/2}}{240 c^3}-\frac {(9 b B-14 A c) x^2 \left (b x^2+c x^4\right )^{5/2}}{168 c^2}+\frac {B x^4 \left (b x^2+c x^4\right )^{5/2}}{14 c}-\frac {\left (b^6 (9 b B-14 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+c x^2}} \, dx,x,x^2\right )}{4096 c^5}\\ &=\frac {b^4 (9 b B-14 A c) \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{2048 c^5}-\frac {b^2 (9 b B-14 A c) \left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{768 c^4}+\frac {b (9 b B-14 A c) \left (b x^2+c x^4\right )^{5/2}}{240 c^3}-\frac {(9 b B-14 A c) x^2 \left (b x^2+c x^4\right )^{5/2}}{168 c^2}+\frac {B x^4 \left (b x^2+c x^4\right )^{5/2}}{14 c}-\frac {\left (b^6 (9 b B-14 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 )}{2048 c^5}\\ &=\frac {b^4 (9 b B-14 A c) \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{2048 c^5}-\frac {b^2 (9 b B-14 A c) \left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{768 c^4}+\frac {b (9 b B-14 A c) \left (b x^2+c x^4\right )^{5/2}}{240 c^3}-\frac {(9 b B-14 A c) x^2 \left (b x^2+c x^4\right )^{5/2}}{168 c^2}+\frac {B x^4 \left (b x^2+c x^4\right )^{5/2}}{14 c}-\frac {b^6 (9 b B-14 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{2048 c^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 215, normalized size = 0.96 \begin {gather*} \frac {\sqrt {x^2 \left (b+c x^2\right )} \left (\sqrt {c} x \sqrt {\frac {c x^2}{b}+1} \left (-210 b^5 c \left (7 A+3 B x^2\right )+28 b^4 c^2 x^2 \left (35 A+18 B x^2\right )-16 b^3 c^3 x^4 \left (49 A+27 B x^2\right )+96 b^2 c^4 x^6 \left (7 A+4 B x^2\right )+256 b c^5 x^8 \left (91 A+75 B x^2\right )+2560 c^6 x^{10} \left (7 A+6 B x^2\right )+945 b^6 B\right )-105 b^{11/2} (9 b B-14 A c) \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )\right )}{215040 c^{11/2} x \sqrt {\frac {c x^2}{b}+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.82, size = 211, normalized size = 0.95 \begin {gather*} \frac {\left (9 b^7 B-14 A b^6 c\right ) \log \left (-2 \sqrt {c} \sqrt {b x^2+c x^4}+b+2 c x^2\right )}{4096 c^{11/2}}+\frac {\sqrt {b x^2+c x^4} \left (-1470 A b^5 c+980 A b^4 c^2 x^2-784 A b^3 c^3 x^4+672 A b^2 c^4 x^6+23296 A b c^5 x^8+17920 A c^6 x^{10}+945 b^6 B-630 b^5 B c x^2+504 b^4 B c^2 x^4-432 b^3 B c^3 x^6+384 b^2 B c^4 x^8+19200 b B c^5 x^{10}+15360 B c^6 x^{12}\right )}{215040 c^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 418, normalized size = 1.87 \begin {gather*} \left [-\frac {105 \, {\left (9 \, B b^{7} - 14 \, A b^{6} c\right )} \sqrt {c} \log \left (-2 \, c x^{2} - b - 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right ) - 2 \, {\left (15360 \, B c^{7} x^{12} + 1280 \, {\left (15 \, B b c^{6} + 14 \, A c^{7}\right )} x^{10} + 128 \, {\left (3 \, B b^{2} c^{5} + 182 \, A b c^{6}\right )} x^{8} + 945 \, B b^{6} c - 1470 \, A b^{5} c^{2} - 48 \, {\left (9 \, B b^{3} c^{4} - 14 \, A b^{2} c^{5}\right )} x^{6} + 56 \, {\left (9 \, B b^{4} c^{3} - 14 \, A b^{3} c^{4}\right )} x^{4} - 70 \, {\left (9 \, B b^{5} c^{2} - 14 \, A b^{4} c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{430080 \, c^{6}}, \frac {105 \, {\left (9 \, B b^{7} - 14 \, A b^{6} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-c}}{c x^{2} + b}\right ) + {\left (15360 \, B c^{7} x^{12} + 1280 \, {\left (15 \, B b c^{6} + 14 \, A c^{7}\right )} x^{10} + 128 \, {\left (3 \, B b^{2} c^{5} + 182 \, A b c^{6}\right )} x^{8} + 945 \, B b^{6} c - 1470 \, A b^{5} c^{2} - 48 \, {\left (9 \, B b^{3} c^{4} - 14 \, A b^{2} c^{5}\right )} x^{6} + 56 \, {\left (9 \, B b^{4} c^{3} - 14 \, A b^{3} c^{4}\right )} x^{4} - 70 \, {\left (9 \, B b^{5} c^{2} - 14 \, A b^{4} c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{215040 \, c^{6}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 280, normalized size = 1.26 \begin {gather*} \frac {1}{215040} \, {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (12 \, B c x^{2} \mathrm {sgn}\relax (x) + \frac {15 \, B b c^{12} \mathrm {sgn}\relax (x) + 14 \, A c^{13} \mathrm {sgn}\relax (x)}{c^{12}}\right )} x^{2} + \frac {3 \, B b^{2} c^{11} \mathrm {sgn}\relax (x) + 182 \, A b c^{12} \mathrm {sgn}\relax (x)}{c^{12}}\right )} x^{2} - \frac {3 \, {\left (9 \, B b^{3} c^{10} \mathrm {sgn}\relax (x) - 14 \, A b^{2} c^{11} \mathrm {sgn}\relax (x)\right )}}{c^{12}}\right )} x^{2} + \frac {7 \, {\left (9 \, B b^{4} c^{9} \mathrm {sgn}\relax (x) - 14 \, A b^{3} c^{10} \mathrm {sgn}\relax (x)\right )}}{c^{12}}\right )} x^{2} - \frac {35 \, {\left (9 \, B b^{5} c^{8} \mathrm {sgn}\relax (x) - 14 \, A b^{4} c^{9} \mathrm {sgn}\relax (x)\right )}}{c^{12}}\right )} x^{2} + \frac {105 \, {\left (9 \, B b^{6} c^{7} \mathrm {sgn}\relax (x) - 14 \, A b^{5} c^{8} \mathrm {sgn}\relax (x)\right )}}{c^{12}}\right )} \sqrt {c x^{2} + b} x + \frac {{\left (9 \, B b^{7} \mathrm {sgn}\relax (x) - 14 \, A b^{6} c \mathrm {sgn}\relax (x)\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + b} \right |}\right )}{2048 \, c^{\frac {11}{2}}} - \frac {{\left (9 \, B b^{7} \log \left ({\left | b \right |}\right ) - 14 \, A b^{6} c \log \left ({\left | b \right |}\right )\right )} \mathrm {sgn}\relax (x)}{4096 \, c^{\frac {11}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 328, normalized size = 1.47 \begin {gather*} \frac {\left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \left (15360 \left (c \,x^{2}+b \right )^{\frac {5}{2}} B \,c^{\frac {9}{2}} x^{9}+17920 \left (c \,x^{2}+b \right )^{\frac {5}{2}} A \,c^{\frac {9}{2}} x^{7}-11520 \left (c \,x^{2}+b \right )^{\frac {5}{2}} B b \,c^{\frac {7}{2}} x^{7}-12544 \left (c \,x^{2}+b \right )^{\frac {5}{2}} A b \,c^{\frac {7}{2}} x^{5}+8064 \left (c \,x^{2}+b \right )^{\frac {5}{2}} B \,b^{2} c^{\frac {5}{2}} x^{5}+1470 A \,b^{6} c \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right )-945 B \,b^{7} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right )+1470 \sqrt {c \,x^{2}+b}\, A \,b^{5} c^{\frac {3}{2}} x +7840 \left (c \,x^{2}+b \right )^{\frac {5}{2}} A \,b^{2} c^{\frac {5}{2}} x^{3}-945 \sqrt {c \,x^{2}+b}\, B \,b^{6} \sqrt {c}\, x -5040 \left (c \,x^{2}+b \right )^{\frac {5}{2}} B \,b^{3} c^{\frac {3}{2}} x^{3}+980 \left (c \,x^{2}+b \right )^{\frac {3}{2}} A \,b^{4} c^{\frac {3}{2}} x -630 \left (c \,x^{2}+b \right )^{\frac {3}{2}} B \,b^{5} \sqrt {c}\, x -3920 \left (c \,x^{2}+b \right )^{\frac {5}{2}} A \,b^{3} c^{\frac {3}{2}} x +2520 \left (c \,x^{2}+b \right )^{\frac {5}{2}} B \,b^{4} \sqrt {c}\, x \right )}{215040 \left (c \,x^{2}+b \right )^{\frac {3}{2}} c^{\frac {11}{2}} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.58, size = 363, normalized size = 1.63 \begin {gather*} -\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 )} A + \frac {1}{143360} \, {\left (\frac {10240 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {5}{2}} x^{4}}{c} + \frac {1260 \, \sqrt {c x^{4} + b x^{2}} b^{5} x^{2}}{c^{4}} - \frac {3360 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} b^{3} x^{2}}{c^{3}} - \frac {7680 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {5}{2}} b x^{2}}{c^{2}} - \frac {315 \, b^{7} \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right )}{c^{\frac {11}{2}}} + \frac {630 \, \sqrt {c x^{4} + b x^{2}} b^{6}}{c^{5}} - \frac {1680 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} b^{4}}{c^{4}} + \frac {5376 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {5}{2}} b^{2}}{c^{3}}\right )} B \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^5\,\left (B\,x^2+A\right )\,{\left (c\,x^4+b\,x^2\right )}^{3/2} \,d 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 x^{5} \left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}} \left (A + B x^{2}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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