Optimal. Leaf size=134 \[ \frac {\left (a+b x^2\right ) (f x)^{m+1} (b d-a e) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )}{a b f (m+1) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {e \left (a+b x^2\right ) (f x)^{m+1}}{b f (m+1) \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.09, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {1250, 459, 364} \[ \frac {\left (a+b x^2\right ) (f x)^{m+1} (b d-a e) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )}{a b f (m+1) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {e \left (a+b x^2\right ) (f x)^{m+1}}{b f (m+1) \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 459
Rule 1250
Rubi steps
\begin {align*} \int \frac {(f x)^m \left (d+e x^2\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx &=\frac {\left (a b+b^2 x^2\right ) \int \frac {(f x)^m \left (d+e x^2\right )}{a b+b^2 x^2} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {e (f x)^{1+m} \left (a+b x^2\right )}{b f (1+m) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (\left (-b^2 d (1+m)+a b e (1+m)\right ) \left (a b+b^2 x^2\right )\right ) \int \frac {(f x)^m}{a b+b^2 x^2} \, dx}{b^2 (1+m) \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {e (f x)^{1+m} \left (a+b x^2\right )}{b f (1+m) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {(b d-a e) (f x)^{1+m} \left (a+b x^2\right ) \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{a b f (1+m) \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 78, normalized size = 0.58 \[ -\frac {x \left (a+b x^2\right ) (f x)^m \left ((a e-b d) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )-a e\right )}{a b (m+1) \sqrt {\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x^{2} + d\right )} \left (f x\right )^{m}}{\sqrt {b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}}, x\right ) \]
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 \[ \int \frac {{\left (e x^{2} + d\right )} \left (f x\right )^{m}}{\sqrt {b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \,x^{2}+d \right ) \left (f x \right )^{m}}{\sqrt {b^{2} x^{4}+2 a b \,x^{2}+a^{2}}}\, dx \]
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 \[ \int \frac {{\left (e x^{2} + d\right )} \left (f x\right )^{m}}{\sqrt {b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (f\,x\right )}^m\,\left (e\,x^2+d\right )}{\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}} \,d x \]
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 \[ \int \frac {\left (f x\right )^{m} \left (d + e x^{2}\right )}{\sqrt {\left (a + b x^{2}\right )^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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