Optimal. Leaf size=134 \[ \frac {\left (-8 a c+3 b^2-2 b c x^2\right ) \sqrt {a+b x^2+c x^4}}{2 c^2 \left (b^2-4 a c\right )}+\frac {x^4 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {3 b \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{4 c^{5/2}} \]
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Rubi [A] time = 0.11, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1114, 738, 779, 621, 206} \begin {gather*} \frac {\left (-8 a c+3 b^2-2 b c x^2\right ) \sqrt {a+b x^2+c x^4}}{2 c^2 \left (b^2-4 a c\right )}+\frac {x^4 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {3 b \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{4 c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 738
Rule 779
Rule 1114
Rubi steps
\begin {align*} \int \frac {x^7}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=\frac {x^4 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\operatorname {Subst}\left (\int \frac {x (4 a+2 b x)}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{b^2-4 a c}\\ &=\frac {x^4 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}+\frac {\left (3 b^2-8 a c-2 b c x^2\right ) \sqrt {a+b x^2+c x^4}}{2 c^2 \left (b^2-4 a c\right )}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{4 c^2}\\ &=\frac {x^4 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}+\frac {\left (3 b^2-8 a c-2 b c x^2\right ) \sqrt {a+b x^2+c x^4}}{2 c^2 \left (b^2-4 a c\right )}-\frac {(3 b) \operatorname {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 )}{2 c^2}\\ &=\frac {x^4 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}+\frac {\left (3 b^2-8 a c-2 b c x^2\right ) \sqrt {a+b x^2+c x^4}}{2 c^2 \left (b^2-4 a c\right )}-\frac {3 b \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{4 c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 137, normalized size = 1.02 \begin {gather*} \frac {\frac {2 \sqrt {c} \left (8 a^2 c+a \left (-3 b^2+10 b c x^2+4 c^2 x^4\right )-b^2 x^2 \left (3 b+c x^2\right )\right )}{\sqrt {a+b x^2+c x^4}}+3 b \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 )}{4 c^{5/2} \left (4 a c-b^2\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.58, size = 131, normalized size = 0.98 \begin {gather*} \frac {8 a^2 c-3 a b^2+10 a b c x^2+4 a c^2 x^4-3 b^3 x^2-b^2 c x^4}{2 c^2 \left (4 a c-b^2\right ) \sqrt {a+b x^2+c x^4}}+\frac {3 b \log \left (-2 c^{5/2} \sqrt {a+b x^2+c x^4}+b c^2+2 c^3 x^2\right )}{4 c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.58, size = 459, normalized size = 3.43 \begin {gather*} \left [\frac {3 \, {\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{4} + a b^{3} - 4 \, a^{2} b c + {\left (b^{4} - 4 \, a b^{2} c\right )} x^{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 ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + 3 \, a b^{2} c - 8 \, a^{2} c^{2} + {\left (3 \, b^{3} c - 10 \, a b c^{2}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{8 \, {\left (a b^{2} c^{3} - 4 \, a^{2} c^{4} + {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{2}\right )}}, \frac {3 \, {\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{4} + a b^{3} - 4 \, a^{2} b c + {\left (b^{4} - 4 \, a b^{2} c\right )} x^{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 ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + 3 \, a b^{2} c - 8 \, a^{2} c^{2} + {\left (3 \, b^{3} c - 10 \, a b c^{2}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{4 \, {\left (a b^{2} c^{3} - 4 \, a^{2} c^{4} + {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 154, normalized size = 1.15 \begin {gather*} \frac {{\left (\frac {{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2}}{b^{2} c^{2} - 4 \, a c^{3}} + \frac {3 \, b^{3} - 10 \, a b c}{b^{2} c^{2} - 4 \, a c^{3}}\right )} x^{2} + \frac {3 \, a b^{2} - 8 \, a^{2} c}{b^{2} c^{2} - 4 \, a c^{3}}}{2 \, \sqrt {c x^{4} + b x^{2} + a}} + \frac {3 \, b \log \left ({\left | -2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} - b \right |}\right )}{4 \, c^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 264, normalized size = 1.97 \begin {gather*} \frac {2 a b \,x^{2}}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}\, c}-\frac {3 b^{3} x^{2}}{4 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}\, c^{2}}+\frac {x^{4}}{2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, c}+\frac {a \,b^{2}}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}\, c^{2}}-\frac {3 b^{4}}{8 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}\, c^{3}}+\frac {3 b \,x^{2}}{4 \sqrt {c \,x^{4}+b \,x^{2}+a}\, c^{2}}-\frac {3 b \ln \left (\frac {c \,x^{2}+\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{4 c^{\frac {5}{2}}}+\frac {a}{\sqrt {c \,x^{4}+b \,x^{2}+a}\, c^{2}}-\frac {3 b^{2}}{8 \sqrt {c \,x^{4}+b \,x^{2}+a}\, c^{3}} \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 {x^7}{{\left (c\,x^4+b\,x^2+a\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 \frac {x^{7}}{\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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