Optimal. Leaf size=171 \[ -\frac {b \left (7 b^2-12 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{256 c^4}+\frac {x^4 \left (a+b x^2+c x^4\right )^{3/2}}{10 c}+\frac {\left (35 b^2-32 a c-42 b c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{480 c^3}+\frac {b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{512 c^{9/2}} \]
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Rubi [A]
time = 0.11, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1128, 756, 793,
626, 635, 212} \begin {gather*} \frac {b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{512 c^{9/2}}-\frac {b \left (7 b^2-12 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{256 c^4}+\frac {\left (-32 a c+35 b^2-42 b c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{480 c^3}+\frac {x^4 \left (a+b x^2+c x^4\right )^{3/2}}{10 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 626
Rule 635
Rule 756
Rule 793
Rule 1128
Rubi steps
\begin {align*} \int x^7 \sqrt {a+b x^2+c x^4} \, dx &=\frac {1}{2} \text {Subst}\left (\int x^3 \sqrt {a+b x+c x^2} \, dx,x,x^2\right )\\ &=\frac {x^4 \left (a+b x^2+c x^4\right )^{3/2}}{10 c}+\frac {\text {Subst}\left (\int x \left (-2 a-\frac {7 b x}{2}\right ) \sqrt {a+b x+c x^2} \, dx,x,x^2\right )}{10 c}\\ &=\frac {x^4 \left (a+b x^2+c x^4\right )^{3/2}}{10 c}+\frac {\left (35 b^2-32 a c-42 b c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{480 c^3}-\frac {\left (b \left (7 b^2-12 a c\right )\right ) \text {Subst}\left (\int \sqrt {a+b x+c x^2} \, dx,x,x^2\right )}{64 c^3}\\ &=-\frac {b \left (7 b^2-12 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{256 c^4}+\frac {x^4 \left (a+b x^2+c x^4\right )^{3/2}}{10 c}+\frac {\left (35 b^2-32 a c-42 b c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{480 c^3}+\frac {\left (b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{512 c^4}\\ &=-\frac {b \left (7 b^2-12 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{256 c^4}+\frac {x^4 \left (a+b x^2+c x^4\right )^{3/2}}{10 c}+\frac {\left (35 b^2-32 a c-42 b c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{480 c^3}+\frac {\left (b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right )\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x^2}{\sqrt {a+b x^2+c x^4}}\right )}{256 c^4}\\ &=-\frac {b \left (7 b^2-12 a c\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{256 c^4}+\frac {x^4 \left (a+b x^2+c x^4\right )^{3/2}}{10 c}+\frac {\left (35 b^2-32 a c-42 b c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{480 c^3}+\frac {b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{512 c^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.30, size = 166, normalized size = 0.97 \begin {gather*} \frac {\sqrt {a+b x^2+c x^4} \left (-105 b^4+70 b^3 c x^2+4 b^2 c \left (115 a-14 c x^4\right )+8 b c^2 x^2 \left (-29 a+6 c x^4\right )+128 c^2 \left (-2 a^2+a c x^4+3 c^2 x^8\right )\right )}{3840 c^4}-\frac {\left (7 b^5-40 a b^3 c+48 a^2 b c^2\right ) \log \left (c^4 \left (b+2 c x^2-2 \sqrt {c} \sqrt {a+b x^2+c x^4}\right )\right )}{512 c^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 296, normalized size = 1.73
method | result | size |
risch | \(-\frac {\left (-384 c^{4} x^{8}-48 b \,c^{3} x^{6}-128 a \,c^{3} x^{4}+56 b^{2} c^{2} x^{4}+232 a b \,c^{2} x^{2}-70 b^{3} c \,x^{2}+256 a^{2} c^{2}-460 a \,b^{2} c +105 b^{4}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{3840 c^{4}}+\frac {3 b \,a^{2} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{32 c^{\frac {5}{2}}}-\frac {5 b^{3} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right ) a}{64 c^{\frac {7}{2}}}+\frac {7 b^{5} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{512 c^{\frac {9}{2}}}\) | \(211\) |
default | \(\frac {x^{4} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{10 c}-\frac {7 b \,x^{2} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{80 c^{2}}+\frac {7 b^{2} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{96 c^{3}}-\frac {7 b^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{128 c^{3}}-\frac {7 b^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{256 c^{4}}-\frac {5 b^{3} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right ) a}{64 c^{\frac {7}{2}}}+\frac {7 b^{5} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{512 c^{\frac {9}{2}}}+\frac {3 b a \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{32 c^{2}}+\frac {3 b^{2} a \sqrt {c \,x^{4}+b \,x^{2}+a}}{64 c^{3}}+\frac {3 b \,a^{2} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{32 c^{\frac {5}{2}}}-\frac {a \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{15 c^{2}}\) | \(296\) |
elliptic | \(\frac {x^{4} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{10 c}-\frac {7 b \,x^{2} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{80 c^{2}}+\frac {7 b^{2} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{96 c^{3}}-\frac {7 b^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{128 c^{3}}-\frac {7 b^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{256 c^{4}}-\frac {5 b^{3} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right ) a}{64 c^{\frac {7}{2}}}+\frac {7 b^{5} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{512 c^{\frac {9}{2}}}+\frac {3 b a \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{32 c^{2}}+\frac {3 b^{2} a \sqrt {c \,x^{4}+b \,x^{2}+a}}{64 c^{3}}+\frac {3 b \,a^{2} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{32 c^{\frac {5}{2}}}-\frac {a \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{15 c^{2}}\) | \(296\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 367, normalized size = 2.15 \begin {gather*} \left [\frac {15 \, {\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (384 \, c^{5} x^{8} + 48 \, b c^{4} x^{6} - 105 \, b^{4} c + 460 \, a b^{2} c^{2} - 256 \, a^{2} c^{3} - 8 \, {\left (7 \, b^{2} c^{3} - 16 \, a c^{4}\right )} x^{4} + 2 \, {\left (35 \, b^{3} c^{2} - 116 \, a b c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{15360 \, c^{5}}, -\frac {15 \, {\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) - 2 \, {\left (384 \, c^{5} x^{8} + 48 \, b c^{4} x^{6} - 105 \, b^{4} c + 460 \, a b^{2} c^{2} - 256 \, a^{2} c^{3} - 8 \, {\left (7 \, b^{2} c^{3} - 16 \, a c^{4}\right )} x^{4} + 2 \, {\left (35 \, b^{3} c^{2} - 116 \, a b c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{7680 \, c^{5}}\right ] \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 {a + b x^{2} + c x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.91, size = 172, normalized size = 1.01 \begin {gather*} \frac {1}{3840} \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, x^{2} + \frac {b}{c}\right )} x^{2} - \frac {7 \, b^{2} c^{2} - 16 \, a c^{3}}{c^{4}}\right )} x^{2} + \frac {35 \, b^{3} c - 116 \, a b c^{2}}{c^{4}}\right )} x^{2} - \frac {105 \, b^{4} - 460 \, a b^{2} c + 256 \, a^{2} c^{2}}{c^{4}}\right )} - \frac {{\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} - b \right |}\right )}{512 \, c^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.31, size = 315, normalized size = 1.84 \begin {gather*} \frac {x^4\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{10\,c}+\frac {7\,b\,\left (\frac {a\,\left (\left (\frac {b}{4\,c}+\frac {x^2}{2}\right )\,\sqrt {c\,x^4+b\,x^2+a}+\frac {\ln \left (\sqrt {c\,x^4+b\,x^2+a}+\frac {c\,x^2+\frac {b}{2}}{\sqrt {c}}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{4\,c}-\frac {x^2\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{4\,c}+\frac {5\,b\,\left (\frac {\left (8\,c\,\left (c\,x^4+a\right )-3\,b^2+2\,b\,c\,x^2\right )\,\sqrt {c\,x^4+b\,x^2+a}}{24\,c^2}+\frac {\ln \left (2\,\sqrt {c\,x^4+b\,x^2+a}+\frac {2\,c\,x^2+b}{\sqrt {c}}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}\right )}{8\,c}\right )}{20\,c}-\frac {a\,\left (\frac {\left (8\,c\,\left (c\,x^4+a\right )-3\,b^2+2\,b\,c\,x^2\right )\,\sqrt {c\,x^4+b\,x^2+a}}{24\,c^2}+\frac {\ln \left (2\,\sqrt {c\,x^4+b\,x^2+a}+\frac {2\,c\,x^2+b}{\sqrt {c}}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}\right )}{5\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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