Optimal. Leaf size=218 \[ -\frac {7 b^5 (3 b B-4 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{1024 c^{11/2}}+\frac {7 b^3 \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4} (3 b B-4 A c)}{1024 c^5}-\frac {7 b^2 \left (b x^2+c x^4\right )^{3/2} (3 b B-4 A c)}{384 c^4}+\frac {7 b x^2 \left (b x^2+c x^4\right )^{3/2} (3 b B-4 A c)}{320 c^3}-\frac {x^4 \left (b x^2+c x^4\right )^{3/2} (3 b B-4 A c)}{40 c^2}+\frac {B x^6 \left (b x^2+c x^4\right )^{3/2}}{12 c} \]
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Rubi [A] time = 0.38, antiderivative size = 218, 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 {7 b^2 \left (b x^2+c x^4\right )^{3/2} (3 b B-4 A c)}{384 c^4}+\frac {7 b^3 \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4} (3 b B-4 A c)}{1024 c^5}-\frac {7 b^5 (3 b B-4 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{1024 c^{11/2}}-\frac {x^4 \left (b x^2+c x^4\right )^{3/2} (3 b B-4 A c)}{40 c^2}+\frac {7 b x^2 \left (b x^2+c x^4\right )^{3/2} (3 b B-4 A c)}{320 c^3}+\frac {B x^6 \left (b x^2+c x^4\right )^{3/2}}{12 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^7 \left (A+B x^2\right ) \sqrt {b x^2+c x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^3 (A+B x) \sqrt {b x+c x^2} \, dx,x,x^2\right )\\ &=\frac {B x^6 \left (b x^2+c x^4\right )^{3/2}}{12 c}+\frac {\left (3 (-b B+A c)+\frac {3}{2} (-b B+2 A c)\right ) \operatorname {Subst}\left (\int x^3 \sqrt {b x+c x^2} \, dx,x,x^2\right )}{12 c}\\ &=-\frac {(3 b B-4 A c) x^4 \left (b x^2+c x^4\right )^{3/2}}{40 c^2}+\frac {B x^6 \left (b x^2+c x^4\right )^{3/2}}{12 c}+\frac {(7 b (3 b B-4 A c)) \operatorname {Subst}\left (\int x^2 \sqrt {b x+c x^2} \, dx,x,x^2\right )}{80 c^2}\\ &=\frac {7 b (3 b B-4 A c) x^2 \left (b x^2+c x^4\right )^{3/2}}{320 c^3}-\frac {(3 b B-4 A c) x^4 \left (b x^2+c x^4\right )^{3/2}}{40 c^2}+\frac {B x^6 \left (b x^2+c x^4\right )^{3/2}}{12 c}-\frac {\left (7 b^2 (3 b B-4 A c)\right ) \operatorname {Subst}\left (\int x \sqrt {b x+c x^2} \, dx,x,x^2\right )}{128 c^3}\\ &=-\frac {7 b^2 (3 b B-4 A c) \left (b x^2+c x^4\right )^{3/2}}{384 c^4}+\frac {7 b (3 b B-4 A c) x^2 \left (b x^2+c x^4\right )^{3/2}}{320 c^3}-\frac {(3 b B-4 A c) x^4 \left (b x^2+c x^4\right )^{3/2}}{40 c^2}+\frac {B x^6 \left (b x^2+c x^4\right )^{3/2}}{12 c}+\frac {\left (7 b^3 (3 b B-4 A c)\right ) \operatorname {Subst}\left (\int \sqrt {b x+c x^2} \, dx,x,x^2\right )}{256 c^4}\\ &=\frac {7 b^3 (3 b B-4 A c) \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{1024 c^5}-\frac {7 b^2 (3 b B-4 A c) \left (b x^2+c x^4\right )^{3/2}}{384 c^4}+\frac {7 b (3 b B-4 A c) x^2 \left (b x^2+c x^4\right )^{3/2}}{320 c^3}-\frac {(3 b B-4 A c) x^4 \left (b x^2+c x^4\right )^{3/2}}{40 c^2}+\frac {B x^6 \left (b x^2+c x^4\right )^{3/2}}{12 c}-\frac {\left (7 b^5 (3 b B-4 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+c x^2}} \, dx,x,x^2\right )}{2048 c^5}\\ &=\frac {7 b^3 (3 b B-4 A c) \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{1024 c^5}-\frac {7 b^2 (3 b B-4 A c) \left (b x^2+c x^4\right )^{3/2}}{384 c^4}+\frac {7 b (3 b B-4 A c) x^2 \left (b x^2+c x^4\right )^{3/2}}{320 c^3}-\frac {(3 b B-4 A c) x^4 \left (b x^2+c x^4\right )^{3/2}}{40 c^2}+\frac {B x^6 \left (b x^2+c x^4\right )^{3/2}}{12 c}-\frac {\left (7 b^5 (3 b B-4 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^5}\\ &=\frac {7 b^3 (3 b B-4 A c) \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{1024 c^5}-\frac {7 b^2 (3 b B-4 A c) \left (b x^2+c x^4\right )^{3/2}}{384 c^4}+\frac {7 b (3 b B-4 A c) x^2 \left (b x^2+c x^4\right )^{3/2}}{320 c^3}-\frac {(3 b B-4 A c) x^4 \left (b x^2+c x^4\right )^{3/2}}{40 c^2}+\frac {B x^6 \left (b x^2+c x^4\right )^{3/2}}{12 c}-\frac {7 b^5 (3 b B-4 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{1024 c^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 193, normalized size = 0.89 \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^4 c \left (2 A+B x^2\right )+56 b^3 c^2 x^2 \left (5 A+3 B x^2\right )-16 b^2 c^3 x^4 \left (14 A+9 B x^2\right )+64 b c^4 x^6 \left (3 A+2 B x^2\right )+256 c^5 x^8 \left (6 A+5 B x^2\right )+315 b^5 B\right )-105 b^{9/2} (3 b B-4 A c) \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )\right )}{15360 c^{11/2} x \sqrt {\frac {c x^2}{b}+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.74, size = 187, normalized size = 0.86 \begin {gather*} \frac {7 \left (3 b^6 B-4 A b^5 c\right ) \log \left (-2 \sqrt {c} \sqrt {b x^2+c x^4}+b+2 c x^2\right )}{2048 c^{11/2}}+\frac {\sqrt {b x^2+c x^4} \left (-420 A b^4 c+280 A b^3 c^2 x^2-224 A b^2 c^3 x^4+192 A b c^4 x^6+1536 A c^5 x^8+315 b^5 B-210 b^4 B c x^2+168 b^3 B c^2 x^4-144 b^2 B c^3 x^6+128 b B c^4 x^8+1280 B c^5 x^{10}\right )}{15360 c^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 368, normalized size = 1.69 \begin {gather*} \left [-\frac {105 \, {\left (3 \, B b^{6} - 4 \, 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 (B b c^{5} + 12 \, A c^{6}\right )} x^{8} + 315 \, B b^{5} c - 420 \, A b^{4} c^{2} - 48 \, {\left (3 \, B b^{2} c^{4} - 4 \, A b c^{5}\right )} x^{6} + 56 \, {\left (3 \, B b^{3} c^{3} - 4 \, A b^{2} c^{4}\right )} x^{4} - 70 \, {\left (3 \, B b^{4} c^{2} - 4 \, A b^{3} c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{30720 \, c^{6}}, \frac {105 \, {\left (3 \, B b^{6} - 4 \, 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 (B b c^{5} + 12 \, A c^{6}\right )} x^{8} + 315 \, B b^{5} c - 420 \, A b^{4} c^{2} - 48 \, {\left (3 \, B b^{2} c^{4} - 4 \, A b c^{5}\right )} x^{6} + 56 \, {\left (3 \, B b^{3} c^{3} - 4 \, A b^{2} c^{4}\right )} x^{4} - 70 \, {\left (3 \, B b^{4} c^{2} - 4 \, A b^{3} c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{15360 \, c^{6}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 245, normalized size = 1.12 \begin {gather*} \frac {1}{15360} \, {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, B x^{2} \mathrm {sgn}\relax (x) + \frac {B b c^{9} \mathrm {sgn}\relax (x) + 12 \, A c^{10} \mathrm {sgn}\relax (x)}{c^{10}}\right )} x^{2} - \frac {3 \, {\left (3 \, B b^{2} c^{8} \mathrm {sgn}\relax (x) - 4 \, A b c^{9} \mathrm {sgn}\relax (x)\right )}}{c^{10}}\right )} x^{2} + \frac {7 \, {\left (3 \, B b^{3} c^{7} \mathrm {sgn}\relax (x) - 4 \, A b^{2} c^{8} \mathrm {sgn}\relax (x)\right )}}{c^{10}}\right )} x^{2} - \frac {35 \, {\left (3 \, B b^{4} c^{6} \mathrm {sgn}\relax (x) - 4 \, A b^{3} c^{7} \mathrm {sgn}\relax (x)\right )}}{c^{10}}\right )} x^{2} + \frac {105 \, {\left (3 \, B b^{5} c^{5} \mathrm {sgn}\relax (x) - 4 \, A b^{4} c^{6} \mathrm {sgn}\relax (x)\right )}}{c^{10}}\right )} \sqrt {c x^{2} + b} x + \frac {7 \, {\left (3 \, B b^{6} \mathrm {sgn}\relax (x) - 4 \, 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 {11}{2}}} - \frac {7 \, {\left (3 \, B b^{6} \log \left ({\left | b \right |}\right ) - 4 \, A b^{5} c \log \left ({\left | b \right |}\right )\right )} \mathrm {sgn}\relax (x)}{2048 \, 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 = 290, normalized size = 1.33 \begin {gather*} \frac {\sqrt {c \,x^{4}+b \,x^{2}}\, \left (1280 \left (c \,x^{2}+b \right )^{\frac {3}{2}} B \,c^{\frac {9}{2}} x^{9}+1536 \left (c \,x^{2}+b \right )^{\frac {3}{2}} A \,c^{\frac {9}{2}} x^{7}-1152 \left (c \,x^{2}+b \right )^{\frac {3}{2}} B b \,c^{\frac {7}{2}} x^{7}-1344 \left (c \,x^{2}+b \right )^{\frac {3}{2}} A b \,c^{\frac {7}{2}} x^{5}+1008 \left (c \,x^{2}+b \right )^{\frac {3}{2}} B \,b^{2} c^{\frac {5}{2}} x^{5}+1120 \left (c \,x^{2}+b \right )^{\frac {3}{2}} A \,b^{2} c^{\frac {5}{2}} x^{3}-840 \left (c \,x^{2}+b \right )^{\frac {3}{2}} B \,b^{3} c^{\frac {3}{2}} x^{3}+420 A \,b^{5} c \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right )-315 B \,b^{6} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right )+420 \sqrt {c \,x^{2}+b}\, A \,b^{4} c^{\frac {3}{2}} x -315 \sqrt {c \,x^{2}+b}\, B \,b^{5} \sqrt {c}\, x -840 \left (c \,x^{2}+b \right )^{\frac {3}{2}} A \,b^{3} c^{\frac {3}{2}} x +630 \left (c \,x^{2}+b \right )^{\frac {3}{2}} B \,b^{4} \sqrt {c}\, x \right )}{15360 \sqrt {c \,x^{2}+b}\, c^{\frac {11}{2}} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.62, size = 321, normalized size = 1.47 \begin {gather*} \frac {1}{7680} \, {\left (\frac {768 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} x^{4}}{c} - \frac {420 \, \sqrt {c x^{4} + b x^{2}} b^{3} x^{2}}{c^{3}} - \frac {672 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} b x^{2}}{c^{2}} + \frac {105 \, b^{5} \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^{4}}{c^{4}} + \frac {560 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} b^{2}}{c^{3}}\right )} A + \frac {1}{30720} \, {\left (\frac {2560 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} x^{6}}{c} - \frac {2304 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} b x^{4}}{c^{2}} + \frac {1260 \, \sqrt {c x^{4} + b x^{2}} b^{4} x^{2}}{c^{4}} + \frac {2016 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} b^{2} x^{2}}{c^{3}} - \frac {315 \, b^{6} \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^{5}}{c^{5}} - \frac {1680 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} b^{3}}{c^{4}}\right )} B \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.47, size = 289, normalized size = 1.33 \begin {gather*} \frac {A\,x^4\,{\left (c\,x^4+b\,x^2\right )}^{3/2}}{10\,c}+\frac {B\,x^6\,{\left (c\,x^4+b\,x^2\right )}^{3/2}}{12\,c}-\frac {3\,B\,b\,\left (\frac {7\,b\,\left (\frac {5\,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 )}{8\,c}-\frac {x^2\,{\left (c\,x^4+b\,x^2\right )}^{3/2}}{4\,c}\right )}{10\,c}+\frac {x^4\,{\left (c\,x^4+b\,x^2\right )}^{3/2}}{5\,c}\right )}{8\,c}+\frac {7\,A\,b\,\left (\frac {5\,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 )}{8\,c}-\frac {x^2\,{\left (c\,x^4+b\,x^2\right )}^{3/2}}{4\,c}\right )}{20\,c} \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^{7} \sqrt {x^{2} \left (b + c x^{2}\right )} \left (A + B x^{2}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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