Optimal. Leaf size=130 \[ \frac {128 c^3 \sqrt {b x^2+c x^4}}{35 b^5 x^2}-\frac {64 c^2 \sqrt {b x^2+c x^4}}{35 b^4 x^4}+\frac {48 c \sqrt {b x^2+c x^4}}{35 b^3 x^6}-\frac {8 \sqrt {b x^2+c x^4}}{7 b^2 x^8}+\frac {1}{b x^6 \sqrt {b x^2+c x^4}} \]
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Rubi [A] time = 0.23, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2015, 2016, 2014} \begin {gather*} \frac {128 c^3 \sqrt {b x^2+c x^4}}{35 b^5 x^2}-\frac {64 c^2 \sqrt {b x^2+c x^4}}{35 b^4 x^4}+\frac {48 c \sqrt {b x^2+c x^4}}{35 b^3 x^6}-\frac {8 \sqrt {b x^2+c x^4}}{7 b^2 x^8}+\frac {1}{b x^6 \sqrt {b x^2+c x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2014
Rule 2015
Rule 2016
Rubi steps
\begin {align*} \int \frac {1}{x^5 \left (b x^2+c x^4\right )^{3/2}} \, dx &=\frac {1}{b x^6 \sqrt {b x^2+c x^4}}+\frac {8 \int \frac {1}{x^7 \sqrt {b x^2+c x^4}} \, dx}{b}\\ &=\frac {1}{b x^6 \sqrt {b x^2+c x^4}}-\frac {8 \sqrt {b x^2+c x^4}}{7 b^2 x^8}-\frac {(48 c) \int \frac {1}{x^5 \sqrt {b x^2+c x^4}} \, dx}{7 b^2}\\ &=\frac {1}{b x^6 \sqrt {b x^2+c x^4}}-\frac {8 \sqrt {b x^2+c x^4}}{7 b^2 x^8}+\frac {48 c \sqrt {b x^2+c x^4}}{35 b^3 x^6}+\frac {\left (192 c^2\right ) \int \frac {1}{x^3 \sqrt {b x^2+c x^4}} \, dx}{35 b^3}\\ &=\frac {1}{b x^6 \sqrt {b x^2+c x^4}}-\frac {8 \sqrt {b x^2+c x^4}}{7 b^2 x^8}+\frac {48 c \sqrt {b x^2+c x^4}}{35 b^3 x^6}-\frac {64 c^2 \sqrt {b x^2+c x^4}}{35 b^4 x^4}-\frac {\left (128 c^3\right ) \int \frac {1}{x \sqrt {b x^2+c x^4}} \, dx}{35 b^4}\\ &=\frac {1}{b x^6 \sqrt {b x^2+c x^4}}-\frac {8 \sqrt {b x^2+c x^4}}{7 b^2 x^8}+\frac {48 c \sqrt {b x^2+c x^4}}{35 b^3 x^6}-\frac {64 c^2 \sqrt {b x^2+c x^4}}{35 b^4 x^4}+\frac {128 c^3 \sqrt {b x^2+c x^4}}{35 b^5 x^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 68, normalized size = 0.52 \begin {gather*} \frac {-5 b^4+8 b^3 c x^2-16 b^2 c^2 x^4+64 b c^3 x^6+128 c^4 x^8}{35 b^5 x^6 \sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.30, size = 77, normalized size = 0.59 \begin {gather*} \frac {\sqrt {b x^2+c x^4} \left (-5 b^4+8 b^3 c x^2-16 b^2 c^2 x^4+64 b c^3 x^6+128 c^4 x^8\right )}{35 b^5 x^8 \left (b+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 2.05, size = 76, normalized size = 0.58 \begin {gather*} \frac {{\left (128 \, c^{4} x^{8} + 64 \, b c^{3} x^{6} - 16 \, b^{2} c^{2} x^{4} + 8 \, b^{3} c x^{2} - 5 \, b^{4}\right )} \sqrt {c x^{4} + b x^{2}}}{35 \, {\left (b^{5} c x^{10} + b^{6} x^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} x^{5}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 72, normalized size = 0.55 \begin {gather*} -\frac {\left (c \,x^{2}+b \right ) \left (-128 c^{4} x^{8}-64 c^{3} x^{6} b +16 c^{2} x^{4} b^{2}-8 c \,x^{2} b^{3}+5 b^{4}\right )}{35 \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} b^{5} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.47, size = 113, normalized size = 0.87 \begin {gather*} \frac {128 \, c^{4} x^{2}}{35 \, \sqrt {c x^{4} + b x^{2}} b^{5}} + \frac {64 \, c^{3}}{35 \, \sqrt {c x^{4} + b x^{2}} b^{4}} - \frac {16 \, c^{2}}{35 \, \sqrt {c x^{4} + b x^{2}} b^{3} x^{2}} + \frac {8 \, c}{35 \, \sqrt {c x^{4} + b x^{2}} b^{2} x^{4}} - \frac {1}{7 \, \sqrt {c x^{4} + b x^{2}} b x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.41, size = 114, normalized size = 0.88 \begin {gather*} \frac {13\,c\,\sqrt {c\,x^4+b\,x^2}}{35\,b^3\,x^6}-\frac {\sqrt {c\,x^4+b\,x^2}}{7\,b^2\,x^8}-\frac {29\,c^2\,\sqrt {c\,x^4+b\,x^2}}{35\,b^4\,x^4}+\frac {\sqrt {c\,x^4+b\,x^2}\,\left (\frac {93\,c^3}{35\,b^4}+\frac {128\,c^4\,x^2}{35\,b^5}\right )}{x^2\,\left (c\,x^2+b\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 \frac {1}{x^{5} \left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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