Optimal. Leaf size=323 \[ \frac {d (f x)^{m+1} \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1} F_1\left (\frac {m+1}{2};\frac {3}{2},\frac {3}{2};\frac {m+3}{2};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{a f (m+1) \sqrt {a+b x^2+c x^4}}+\frac {e (f x)^{m+3} \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1} F_1\left (\frac {m+3}{2};\frac {3}{2},\frac {3}{2};\frac {m+5}{2};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{a f^3 (m+3) \sqrt {a+b x^2+c x^4}} \]
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Rubi [A] time = 0.39, antiderivative size = 323, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1335, 1141, 510} \[ \frac {d (f x)^{m+1} \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1} F_1\left (\frac {m+1}{2};\frac {3}{2},\frac {3}{2};\frac {m+3}{2};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{a f (m+1) \sqrt {a+b x^2+c x^4}}+\frac {e (f x)^{m+3} \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1} F_1\left (\frac {m+3}{2};\frac {3}{2},\frac {3}{2};\frac {m+5}{2};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{a f^3 (m+3) \sqrt {a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 510
Rule 1141
Rule 1335
Rubi steps
\begin {align*} \int \frac {(f x)^m \left (d+e x^2\right )}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx &=\int \left (\frac {d (f x)^m}{\left (a+b x^2+c x^4\right )^{3/2}}+\frac {e (f x)^{2+m}}{f^2 \left (a+b x^2+c x^4\right )^{3/2}}\right ) \, dx\\ &=d \int \frac {(f x)^m}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx+\frac {e \int \frac {(f x)^{2+m}}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx}{f^2}\\ &=\frac {\left (d \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}\right ) \int \frac {(f x)^m}{\left (1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}\right )^{3/2} \left (1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )^{3/2}} \, dx}{a \sqrt {a+b x^2+c x^4}}+\frac {\left (e \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}\right ) \int \frac {(f x)^{2+m}}{\left (1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}\right )^{3/2} \left (1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )^{3/2}} \, dx}{a f^2 \sqrt {a+b x^2+c x^4}}\\ &=\frac {d (f x)^{1+m} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}} F_1\left (\frac {1+m}{2};\frac {3}{2},\frac {3}{2};\frac {3+m}{2};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{a f (1+m) \sqrt {a+b x^2+c x^4}}+\frac {e (f x)^{3+m} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}} F_1\left (\frac {3+m}{2};\frac {3}{2},\frac {3}{2};\frac {5+m}{2};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{a f^3 (3+m) \sqrt {a+b x^2+c x^4}}\\ \end {align*}
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Mathematica [A] time = 0.41, size = 307, normalized size = 0.95 \[ \frac {x (f x)^m \left (\sqrt {b^2-4 a c}-b-2 c x^2\right ) \sqrt {\frac {-\sqrt {b^2-4 a c}+b+2 c x^2}{b-\sqrt {b^2-4 a c}}} \left (\frac {\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}+b}\right )^{3/2} \left (d (m+3) F_1\left (\frac {m+1}{2};\frac {3}{2},\frac {3}{2};\frac {m+3}{2};-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{\sqrt {b^2-4 a c}-b}\right )+e (m+1) x^2 F_1\left (\frac {m+3}{2};\frac {3}{2},\frac {3}{2};\frac {m+5}{2};-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{\sqrt {b^2-4 a c}-b}\right )\right )}{(m+1) (m+3) \left (\sqrt {b^2-4 a c}-b\right ) \left (a+b x^2+c x^4\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (e x^{2} + d\right )} \left (f x\right )^{m}}{c^{2} x^{8} + 2 \, b c x^{6} + {\left (b^{2} + 2 \, a c\right )} x^{4} + 2 \, a b x^{2} + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (e x^{2} + d\right )} \left (f x\right )^{m}}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \,x^{2}+d \right ) \left (f x \right )^{m}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x^{2} + d\right )} \left (f x\right )^{m}}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (f\,x\right )}^m\,\left (e\,x^2+d\right )}{{\left (c\,x^4+b\,x^2+a\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (f x\right )^{m} \left (d + e x^{2}\right )}{\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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