Optimal. Leaf size=138 \[ -\frac {8 c^2 \left (b+2 c x^2\right ) (7 b B-8 A c)}{35 b^5 \sqrt {b x^2+c x^4}}+\frac {2 c (7 b B-8 A c)}{35 b^3 x^2 \sqrt {b x^2+c x^4}}-\frac {7 b B-8 A c}{35 b^2 x^4 \sqrt {b x^2+c x^4}}-\frac {A}{7 b x^6 \sqrt {b x^2+c x^4}} \]
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Rubi [A] time = 0.27, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2034, 792, 658, 613} \begin {gather*} -\frac {8 c^2 \left (b+2 c x^2\right ) (7 b B-8 A c)}{35 b^5 \sqrt {b x^2+c x^4}}+\frac {2 c (7 b B-8 A c)}{35 b^3 x^2 \sqrt {b x^2+c x^4}}-\frac {7 b B-8 A c}{35 b^2 x^4 \sqrt {b x^2+c x^4}}-\frac {A}{7 b x^6 \sqrt {b x^2+c x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 613
Rule 658
Rule 792
Rule 2034
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x^5 \left (b x^2+c x^4\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {A+B x}{x^3 \left (b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=-\frac {A}{7 b x^6 \sqrt {b x^2+c x^4}}+\frac {\left (\frac {1}{2} (b B-2 A c)-3 (-b B+A c)\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )}{7 b}\\ &=-\frac {A}{7 b x^6 \sqrt {b x^2+c x^4}}-\frac {7 b B-8 A c}{35 b^2 x^4 \sqrt {b x^2+c x^4}}-\frac {(3 c (7 b B-8 A c)) \operatorname {Subst}\left (\int \frac {1}{x \left (b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )}{35 b^2}\\ &=-\frac {A}{7 b x^6 \sqrt {b x^2+c x^4}}-\frac {7 b B-8 A c}{35 b^2 x^4 \sqrt {b x^2+c x^4}}+\frac {2 c (7 b B-8 A c)}{35 b^3 x^2 \sqrt {b x^2+c x^4}}+\frac {\left (4 c^2 (7 b B-8 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )}{35 b^3}\\ &=-\frac {A}{7 b x^6 \sqrt {b x^2+c x^4}}-\frac {7 b B-8 A c}{35 b^2 x^4 \sqrt {b x^2+c x^4}}+\frac {2 c (7 b B-8 A c)}{35 b^3 x^2 \sqrt {b x^2+c x^4}}-\frac {8 c^2 (7 b B-8 A c) \left (b+2 c x^2\right )}{35 b^5 \sqrt {b x^2+c x^4}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 75, normalized size = 0.54 \begin {gather*} \frac {x^2 \left (b^3-2 b^2 c x^2+8 b c^2 x^4+16 c^3 x^6\right ) (8 A c-7 b B)-5 A b^4}{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 = 0.51, size = 123, normalized size = 0.89 \begin {gather*} \frac {\sqrt {b x^2+c x^4} \left (-5 A b^4+8 A b^3 c x^2-16 A b^2 c^2 x^4+64 A b c^3 x^6+128 A c^4 x^8-7 b^4 B x^2+14 b^3 B c x^4-56 b^2 B c^2 x^6-112 b B c^3 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 = 0.43, size = 121, normalized size = 0.88 \begin {gather*} -\frac {{\left (16 \, {\left (7 \, B b c^{3} - 8 \, A c^{4}\right )} x^{8} + 8 \, {\left (7 \, B b^{2} c^{2} - 8 \, A b c^{3}\right )} x^{6} + 5 \, A b^{4} - 2 \, {\left (7 \, B b^{3} c - 8 \, A b^{2} c^{2}\right )} x^{4} + {\left (7 \, B b^{4} - 8 \, A b^{3} c\right )} x^{2}\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 {B x^{2} + A}{{\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 = 0.05, size = 118, normalized size = 0.86 \begin {gather*} -\frac {\left (c \,x^{2}+b \right ) \left (-128 A \,c^{4} x^{8}+112 B b \,c^{3} x^{8}-64 A b \,c^{3} x^{6}+56 B \,b^{2} c^{2} x^{6}+16 A \,b^{2} c^{2} x^{4}-14 B \,b^{3} c \,x^{4}-8 A \,b^{3} c \,x^{2}+7 B \,b^{4} x^{2}+5 A \,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 = 1.48, size = 208, normalized size = 1.51 \begin {gather*} -\frac {1}{5} \, B {\left (\frac {16 \, c^{3} x^{2}}{\sqrt {c x^{4} + b x^{2}} b^{4}} + \frac {8 \, c^{2}}{\sqrt {c x^{4} + b x^{2}} b^{3}} - \frac {2 \, c}{\sqrt {c x^{4} + b x^{2}} b^{2} x^{2}} + \frac {1}{\sqrt {c x^{4} + b x^{2}} b x^{4}}\right )} + \frac {1}{35} \, A {\left (\frac {128 \, c^{4} x^{2}}{\sqrt {c x^{4} + b x^{2}} b^{5}} + \frac {64 \, c^{3}}{\sqrt {c x^{4} + b x^{2}} b^{4}} - \frac {16 \, c^{2}}{\sqrt {c x^{4} + b x^{2}} b^{3} x^{2}} + \frac {8 \, c}{\sqrt {c x^{4} + b x^{2}} b^{2} x^{4}} - \frac {5}{\sqrt {c x^{4} + b x^{2}} b x^{6}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.56, size = 173, normalized size = 1.25 \begin {gather*} -\frac {\left (7\,B\,b^2-13\,A\,b\,c\right )\,\sqrt {c\,x^4+b\,x^2}}{35\,b^4\,x^6}-\frac {\left (x^2\,\left (\frac {58\,A\,c^4-42\,B\,b\,c^3}{35\,b^5}-\frac {2\,c^3\,\left (93\,A\,c-77\,B\,b\right )}{35\,b^5}\right )-\frac {c^2\,\left (93\,A\,c-77\,B\,b\right )}{35\,b^4}\right )\,\sqrt {c\,x^4+b\,x^2}}{x^2\,\left (c\,x^2+b\right )}-\frac {A\,\sqrt {c\,x^4+b\,x^2}}{7\,b^2\,x^8}-\frac {c\,\left (29\,A\,c-21\,B\,b\right )\,\sqrt {c\,x^4+b\,x^2}}{35\,b^4\,x^4} \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 {A + B x^{2}}{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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