Optimal. Leaf size=114 \[ -\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{16 c^{7/2}}+\frac {5 b^2 \sqrt {b x^2+c x^4}}{16 c^3}-\frac {5 b x^2 \sqrt {b x^2+c x^4}}{24 c^2}+\frac {x^4 \sqrt {b x^2+c x^4}}{6 c} \]
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Rubi [A] time = 0.13, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2018, 670, 640, 620, 206} \begin {gather*} \frac {5 b^2 \sqrt {b x^2+c x^4}}{16 c^3}-\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{16 c^{7/2}}-\frac {5 b x^2 \sqrt {b x^2+c x^4}}{24 c^2}+\frac {x^4 \sqrt {b x^2+c x^4}}{6 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 640
Rule 670
Rule 2018
Rubi steps
\begin {align*} \int \frac {x^7}{\sqrt {b x^2+c x^4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {b x+c x^2}} \, dx,x,x^2\right )\\ &=\frac {x^4 \sqrt {b x^2+c x^4}}{6 c}-\frac {(5 b) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {b x+c x^2}} \, dx,x,x^2\right )}{12 c}\\ &=-\frac {5 b x^2 \sqrt {b x^2+c x^4}}{24 c^2}+\frac {x^4 \sqrt {b x^2+c x^4}}{6 c}+\frac {\left (5 b^2\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {b x+c x^2}} \, dx,x,x^2\right )}{16 c^2}\\ &=\frac {5 b^2 \sqrt {b x^2+c x^4}}{16 c^3}-\frac {5 b x^2 \sqrt {b x^2+c x^4}}{24 c^2}+\frac {x^4 \sqrt {b x^2+c x^4}}{6 c}-\frac {\left (5 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+c x^2}} \, dx,x,x^2\right )}{32 c^3}\\ &=\frac {5 b^2 \sqrt {b x^2+c x^4}}{16 c^3}-\frac {5 b x^2 \sqrt {b x^2+c x^4}}{24 c^2}+\frac {x^4 \sqrt {b x^2+c x^4}}{6 c}-\frac {\left (5 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x^2}{\sqrt {b x^2+c x^4}}\right )}{16 c^3}\\ &=\frac {5 b^2 \sqrt {b x^2+c x^4}}{16 c^3}-\frac {5 b x^2 \sqrt {b x^2+c x^4}}{24 c^2}+\frac {x^4 \sqrt {b x^2+c x^4}}{6 c}-\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{16 c^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 100, normalized size = 0.88 \begin {gather*} \frac {x \left (\sqrt {c} x \left (15 b^3+5 b^2 c x^2-2 b c^2 x^4+8 c^3 x^6\right )-15 b^3 \sqrt {b+c x^2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b+c x^2}}\right )\right )}{48 c^{7/2} \sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.24, size = 93, normalized size = 0.82 \begin {gather*} \frac {5 b^3 \log \left (-2 c^{7/2} \sqrt {b x^2+c x^4}+b c^3+2 c^4 x^2\right )}{32 c^{7/2}}+\frac {\sqrt {b x^2+c x^4} \left (15 b^2-10 b c x^2+8 c^2 x^4\right )}{48 c^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 166, normalized size = 1.46 \begin {gather*} \left [\frac {15 \, b^{3} \sqrt {c} \log \left (-2 \, c x^{2} - b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right ) + 2 \, {\left (8 \, c^{3} x^{4} - 10 \, b c^{2} x^{2} + 15 \, b^{2} c\right )} \sqrt {c x^{4} + b x^{2}}}{96 \, c^{4}}, \frac {15 \, b^{3} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-c}}{c x^{2} + b}\right ) + {\left (8 \, c^{3} x^{4} - 10 \, b c^{2} x^{2} + 15 \, b^{2} c\right )} \sqrt {c x^{4} + b x^{2}}}{48 \, c^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 87, normalized size = 0.76 \begin {gather*} \frac {1}{48} \, \sqrt {c x^{4} + b x^{2}} {\left (2 \, x^{2} {\left (\frac {4 \, x^{2}}{c} - \frac {5 \, b}{c^{2}}\right )} + \frac {15 \, b^{2}}{c^{3}}\right )} + \frac {5 \, b^{3} \log \left ({\left | -2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2}}\right )} \sqrt {c} - b \right |}\right )}{32 \, c^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 105, normalized size = 0.92 \begin {gather*} \frac {\sqrt {c \,x^{2}+b}\, \left (8 \sqrt {c \,x^{2}+b}\, c^{\frac {7}{2}} x^{5}-10 \sqrt {c \,x^{2}+b}\, b \,c^{\frac {5}{2}} x^{3}-15 b^{3} c \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right )+15 \sqrt {c \,x^{2}+b}\, b^{2} c^{\frac {3}{2}} x \right ) x}{48 \sqrt {c \,x^{4}+b \,x^{2}}\, c^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.47, size = 100, normalized size = 0.88 \begin {gather*} \frac {\sqrt {c x^{4} + b x^{2}} x^{4}}{6 \, c} - \frac {5 \, \sqrt {c x^{4} + b x^{2}} b x^{2}}{24 \, c^{2}} - \frac {5 \, b^{3} \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right )}{32 \, c^{\frac {7}{2}}} + \frac {5 \, \sqrt {c x^{4} + b x^{2}} b^{2}}{16 \, c^{3}} \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}{\sqrt {c\,x^4+b\,x^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}}{\sqrt {x^{2} \left (b + c x^{2}\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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