Optimal. Leaf size=162 \[ \frac {3 b \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{512 a^{7/2}}-\frac {3 b \left (b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{256 a^3 x^4}+\frac {b \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 a^2 x^8}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 a x^{10}} \]
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Rubi [A] time = 0.14, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1114, 730, 720, 724, 206} \begin {gather*} -\frac {3 b \left (b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{256 a^3 x^4}+\frac {3 b \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{512 a^{7/2}}+\frac {b \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 a^2 x^8}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 a x^{10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 720
Rule 724
Rule 730
Rule 1114
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^{11}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^6} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 a x^{10}}-\frac {b \operatorname {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^5} \, dx,x,x^2\right )}{4 a}\\ &=\frac {b \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 a^2 x^8}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 a x^{10}}+\frac {\left (3 b \left (b^2-4 a c\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x+c x^2}}{x^3} \, dx,x,x^2\right )}{64 a^2}\\ &=-\frac {3 b \left (b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{256 a^3 x^4}+\frac {b \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 a^2 x^8}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 a x^{10}}-\frac {\left (3 b \left (b^2-4 a c\right )^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{512 a^3}\\ &=-\frac {3 b \left (b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{256 a^3 x^4}+\frac {b \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 a^2 x^8}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 a x^{10}}+\frac {\left (3 b \left (b^2-4 a c\right )^2\right ) \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x^2}{\sqrt {a+b x^2+c x^4}}\right )}{256 a^3}\\ &=-\frac {3 b \left (b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}}{256 a^3 x^4}+\frac {b \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 a^2 x^8}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 a x^{10}}+\frac {3 b \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{512 a^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 167, normalized size = 1.03 \begin {gather*} \frac {b \left (16 a^{3/2} \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}-3 x^4 \left (b^2-4 a c\right ) \left (2 \sqrt {a} \left (2 a+b x^2\right ) \sqrt {a+b x^2+c x^4}-x^4 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )\right )\right )}{512 a^{7/2} x^8}-\frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 a x^{10}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.36, size = 174, normalized size = 1.07 \begin {gather*} \frac {\sqrt {a+b x^2+c x^4} \left (-128 a^4-176 a^3 b x^2-256 a^3 c x^4-8 a^2 b^2 x^4-56 a^2 b c x^6-128 a^2 c^2 x^8+10 a b^3 x^6+100 a b^2 c x^8-15 b^4 x^8\right )}{1280 a^3 x^{10}}-\frac {3 \left (16 a^2 b c^2-8 a b^3 c+b^5\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x^2-\sqrt {a+b x^2+c x^4}}{\sqrt {a}}\right )}{256 a^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.56, size = 383, normalized size = 2.36 \begin {gather*} \left [\frac {15 \, {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \sqrt {a} x^{10} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{4}}\right ) - 4 \, {\left ({\left (15 \, a b^{4} - 100 \, a^{2} b^{2} c + 128 \, a^{3} c^{2}\right )} x^{8} + 176 \, a^{4} b x^{2} - 2 \, {\left (5 \, a^{2} b^{3} - 28 \, a^{3} b c\right )} x^{6} + 128 \, a^{5} + 8 \, {\left (a^{3} b^{2} + 32 \, a^{4} c\right )} x^{4}\right )} \sqrt {c x^{4} + b x^{2} + a}}{5120 \, a^{4} x^{10}}, -\frac {15 \, {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \sqrt {-a} x^{10} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{4} + a b x^{2} + a^{2}\right )}}\right ) + 2 \, {\left ({\left (15 \, a b^{4} - 100 \, a^{2} b^{2} c + 128 \, a^{3} c^{2}\right )} x^{8} + 176 \, a^{4} b x^{2} - 2 \, {\left (5 \, a^{2} b^{3} - 28 \, a^{3} b c\right )} x^{6} + 128 \, a^{5} + 8 \, {\left (a^{3} b^{2} + 32 \, a^{4} c\right )} x^{4}\right )} \sqrt {c x^{4} + b x^{2} + a}}{2560 \, a^{4} x^{10}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.53, size = 832, normalized size = 5.14 \begin {gather*} -\frac {3 \, {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}}{\sqrt {-a}}\right )}{256 \, \sqrt {-a} a^{3}} + \frac {15 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{9} b^{5} - 120 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{9} a b^{3} c + 240 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{9} a^{2} b c^{2} + 1280 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{8} a^{3} c^{\frac {5}{2}} - 70 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{7} a b^{5} + 560 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{7} a^{2} b^{3} c + 2720 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{7} a^{3} b c^{2} + 5120 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{6} a^{3} b^{2} c^{\frac {3}{2}} + 128 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} a^{2} b^{5} + 2560 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} a^{3} b^{3} c + 3840 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{5} a^{4} b c^{2} + 1280 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{4} a^{3} b^{4} \sqrt {c} + 2560 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{4} a^{4} b^{2} c^{\frac {3}{2}} + 2560 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{4} a^{5} c^{\frac {5}{2}} + 70 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a^{3} b^{5} + 2000 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a^{4} b^{3} c + 2400 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a^{5} b c^{2} + 2560 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} a^{5} b^{2} c^{\frac {3}{2}} - 15 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{4} b^{5} + 120 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{5} b^{3} c + 1040 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{6} b c^{2} + 256 \, a^{7} c^{\frac {5}{2}}}{1280 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} - a\right )}^{5} a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 337, normalized size = 2.08 \begin {gather*} \frac {3 b \,c^{2} \ln \left (\frac {b \,x^{2}+2 a +2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {a}}{x^{2}}\right )}{32 a^{\frac {3}{2}}}-\frac {3 b^{3} c \ln \left (\frac {b \,x^{2}+2 a +2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {a}}{x^{2}}\right )}{64 a^{\frac {5}{2}}}+\frac {3 b^{5} \ln \left (\frac {b \,x^{2}+2 a +2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {a}}{x^{2}}\right )}{512 a^{\frac {7}{2}}}-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, c^{2}}{10 a \,x^{2}}+\frac {5 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{2} c}{64 a^{2} x^{2}}-\frac {3 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{4}}{256 a^{3} x^{2}}-\frac {7 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b c}{160 a \,x^{4}}+\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{3}}{128 a^{2} x^{4}}-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{2}}{160 a \,x^{6}}-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, c}{5 x^{6}}-\frac {11 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b}{80 x^{8}}-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, a}{10 x^{10}} \end {gather*}
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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{x^{11}} \,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 \frac {\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}{x^{11}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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