Optimal. Leaf size=317 \[ \frac {d (f x)^{m+1} \sqrt {a+b x^2+c x^4} F_1\left (\frac {m+1}{2};-\frac {1}{2},-\frac {1}{2};\frac {m+3}{2};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{f (m+1) \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}}+\frac {e (f x)^{m+3} \sqrt {a+b x^2+c x^4} F_1\left (\frac {m+3}{2};-\frac {1}{2},-\frac {1}{2};\frac {m+5}{2};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{f^3 (m+3) \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}} \]
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Rubi [A] time = 0.36, antiderivative size = 317, 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} \[ \frac {d (f x)^{m+1} \sqrt {a+b x^2+c x^4} F_1\left (\frac {m+1}{2};-\frac {1}{2},-\frac {1}{2};\frac {m+3}{2};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{f (m+1) \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}}+\frac {e (f x)^{m+3} \sqrt {a+b x^2+c x^4} F_1\left (\frac {m+3}{2};-\frac {1}{2},-\frac {1}{2};\frac {m+5}{2};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{f^3 (m+3) \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}} \]
Antiderivative was successfully verified.
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Rule 510
Rule 1141
Rule 1335
Rubi steps
\begin {align*} \int (f x)^m \left (d+e x^2\right ) \sqrt {a+b x^2+c x^4} \, dx &=\int \left (d (f x)^m \sqrt {a+b x^2+c x^4}+\frac {e (f x)^{2+m} \sqrt {a+b x^2+c x^4}}{f^2}\right ) \, dx\\ &=d \int (f x)^m \sqrt {a+b x^2+c x^4} \, dx+\frac {e \int (f x)^{2+m} \sqrt {a+b x^2+c x^4} \, dx}{f^2}\\ &=\frac {\left (d \sqrt {a+b x^2+c x^4}\right ) \int (f x)^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}}} \, dx}{\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}}}}+\frac {\left (e \sqrt {a+b x^2+c x^4}\right ) \int (f x)^{2+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}}} \, dx}{f^2 \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}}}}\\ &=\frac {d (f x)^{1+m} \sqrt {a+b x^2+c x^4} F_1\left (\frac {1+m}{2};-\frac {1}{2},-\frac {1}{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 )}{f (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}}}}+\frac {e (f x)^{3+m} \sqrt {a+b x^2+c x^4} F_1\left (\frac {3+m}{2};-\frac {1}{2},-\frac {1}{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 )}{f^3 (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}}}}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 267, normalized size = 0.84 \[ \frac {x (f x)^m \sqrt {a+b x^2+c x^4} \left (d (m+3) F_1\left (\frac {m+1}{2};-\frac {1}{2},-\frac {1}{2};\frac {m+3}{2};-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{\sqrt {b^2-4 a c}-b}\right )+e (m+1) x^2 F_1\left (\frac {m+3}{2};-\frac {1}{2},-\frac {1}{2};\frac {m+5}{2};-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{\sqrt {b^2-4 a c}-b}\right )\right )}{(m+1) (m+3) \sqrt {\frac {-\sqrt {b^2-4 a c}+b+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}+b}}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {c x^{4} + b x^{2} + a} {\left (e x^{2} + d\right )} \left (f x\right )^{m}, x\right ) \]
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 \[ \int \sqrt {c x^{4} + b x^{2} + a} {\left (e x^{2} + d\right )} \left (f x\right )^{m}\,{d x} \]
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 \[ \int \left (e \,x^{2}+d \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (f x \right )^{m}\, dx \]
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 \[ \int \sqrt {c x^{4} + b x^{2} + a} {\left (e x^{2} + d\right )} \left (f x\right )^{m}\,{d x} \]
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 \[ \int {\left (f\,x\right )}^m\,\left (e\,x^2+d\right )\,\sqrt {c\,x^4+b\,x^2+a} \,d x \]
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 \[ \int \left (f x\right )^{m} \left (d + e x^{2}\right ) \sqrt {a + b x^{2} + c x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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