Optimal. Leaf size=195 \[ -\frac {3 \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{16 a^{7/2}}+\frac {b \left (15 b^2-52 a c\right ) \sqrt {a+b x^2+c x^4}}{8 a^3 x^2 \left (b^2-4 a c\right )}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a+b x^2+c x^4}}{4 a^2 x^4 \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{a x^4 \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}} \]
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Rubi [A] time = 0.21, antiderivative size = 195, 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 = {1114, 740, 834, 806, 724, 206} \[ \frac {b \left (15 b^2-52 a c\right ) \sqrt {a+b x^2+c x^4}}{8 a^3 x^2 \left (b^2-4 a c\right )}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a+b x^2+c x^4}}{4 a^2 x^4 \left (b^2-4 a c\right )}-\frac {3 \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{16 a^{7/2}}+\frac {-2 a c+b^2+b c x^2}{a x^4 \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 724
Rule 740
Rule 806
Rule 834
Rule 1114
Rubi steps
\begin {align*} \int \frac {1}{x^5 \left (a+b x^2+c x^4\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^3 \left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=\frac {b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) x^4 \sqrt {a+b x^2+c x^4}}-\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} \left (-5 b^2+12 a c\right )-2 b c x}{x^3 \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{a \left (b^2-4 a c\right )}\\ &=\frac {b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) x^4 \sqrt {a+b x^2+c x^4}}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a+b x^2+c x^4}}{4 a^2 \left (b^2-4 a c\right ) x^4}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {1}{4} b \left (15 b^2-52 a c\right )-\frac {1}{2} c \left (5 b^2-12 a c\right ) x}{x^2 \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{2 a^2 \left (b^2-4 a c\right )}\\ &=\frac {b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) x^4 \sqrt {a+b x^2+c x^4}}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a+b x^2+c x^4}}{4 a^2 \left (b^2-4 a c\right ) x^4}+\frac {b \left (15 b^2-52 a c\right ) \sqrt {a+b x^2+c x^4}}{8 a^3 \left (b^2-4 a c\right ) x^2}+\frac {\left (3 \left (5 b^2-4 a c\right )\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{16 a^3}\\ &=\frac {b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) x^4 \sqrt {a+b x^2+c x^4}}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a+b x^2+c x^4}}{4 a^2 \left (b^2-4 a c\right ) x^4}+\frac {b \left (15 b^2-52 a c\right ) \sqrt {a+b x^2+c x^4}}{8 a^3 \left (b^2-4 a c\right ) x^2}-\frac {\left (3 \left (5 b^2-4 a c\right )\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 )}{8 a^3}\\ &=\frac {b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) x^4 \sqrt {a+b x^2+c x^4}}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a+b x^2+c x^4}}{4 a^2 \left (b^2-4 a c\right ) x^4}+\frac {b \left (15 b^2-52 a c\right ) \sqrt {a+b x^2+c x^4}}{8 a^3 \left (b^2-4 a c\right ) x^2}-\frac {3 \left (5 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 )}{16 a^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 179, normalized size = 0.92 \[ \frac {3 \left (16 a^2 c^2-24 a b^2 c+5 b^4\right ) \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )+\frac {2 \sqrt {a} \left (-8 a^3 c+2 a^2 \left (b^2+10 b c x^2-12 c^2 x^4\right )+a b x^2 \left (-5 b^2+62 b c x^2+52 c^2 x^4\right )-15 b^3 x^4 \left (b+c x^2\right )\right )}{x^4 \sqrt {a+b x^2+c x^4}}}{16 a^{7/2} \left (4 a c-b^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.40, size = 615, normalized size = 3.15 \[ \left [-\frac {3 \, {\left ({\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{8} + {\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{6} + {\left (5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} x^{4}\right )} \sqrt {a} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{4}}\right ) - 4 \, {\left ({\left (15 \, a b^{3} c - 52 \, a^{2} b c^{2}\right )} x^{6} - 2 \, a^{3} b^{2} + 8 \, a^{4} c + {\left (15 \, a b^{4} - 62 \, a^{2} b^{2} c + 24 \, a^{3} c^{2}\right )} x^{4} + 5 \, {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{32 \, {\left ({\left (a^{4} b^{2} c - 4 \, a^{5} c^{2}\right )} x^{8} + {\left (a^{4} b^{3} - 4 \, a^{5} b c\right )} x^{6} + {\left (a^{5} b^{2} - 4 \, a^{6} c\right )} x^{4}\right )}}, \frac {3 \, {\left ({\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{8} + {\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{6} + {\left (5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} x^{4}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{4} + a b x^{2} + a^{2}\right )}}\right ) + 2 \, {\left ({\left (15 \, a b^{3} c - 52 \, a^{2} b c^{2}\right )} x^{6} - 2 \, a^{3} b^{2} + 8 \, a^{4} c + {\left (15 \, a b^{4} - 62 \, a^{2} b^{2} c + 24 \, a^{3} c^{2}\right )} x^{4} + 5 \, {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{16 \, {\left ({\left (a^{4} b^{2} c - 4 \, a^{5} c^{2}\right )} x^{8} + {\left (a^{4} b^{3} - 4 \, a^{5} b c\right )} x^{6} + {\left (a^{5} b^{2} - 4 \, a^{6} c\right )} x^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 350, normalized size = 1.79 \[ \frac {\frac {{\left (a^{3} b^{3} c - 3 \, a^{4} b c^{2}\right )} x^{2}}{a^{6} b^{2} - 4 \, a^{7} c} + \frac {a^{3} b^{4} - 4 \, a^{4} b^{2} c + 2 \, a^{5} c^{2}}{a^{6} b^{2} - 4 \, a^{7} c}}{\sqrt {c x^{4} + b x^{2} + a}} + \frac {3 \, {\left (5 \, b^{2} - 4 \, a c\right )} \arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a} a^{3}} - \frac {7 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} b^{2} - 4 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{3} a c + 8 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} a b \sqrt {c} - 9 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a b^{2} - 4 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} a^{2} c - 16 \, a^{2} b \sqrt {c}}{8 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} - a\right )}^{2} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 314, normalized size = 1.61 \[ \frac {13 b \,c^{2} x^{2}}{2 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}\, a^{2}}-\frac {15 b^{3} c \,x^{2}}{8 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}\, a^{3}}+\frac {13 b^{2} c}{4 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}\, a^{2}}-\frac {15 b^{4}}{16 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}\, a^{3}}+\frac {3 c \ln \left (\frac {b \,x^{2}+2 a +2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {a}}{x^{2}}\right )}{4 a^{\frac {5}{2}}}-\frac {15 b^{2} \ln \left (\frac {b \,x^{2}+2 a +2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {a}}{x^{2}}\right )}{16 a^{\frac {7}{2}}}-\frac {3 c}{4 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a^{2}}+\frac {15 b^{2}}{16 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a^{3}}+\frac {5 b}{8 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a^{2} x^{2}}-\frac {1}{4 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a \,x^{4}} \]
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 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {1}{x^5\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{x^{5} \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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