Optimal. Leaf size=274 \[ \frac {1}{3} e x^3 \left (\frac {2 b x^2}{c-\sqrt {c^2-4 a b}}+1\right )^{-p} \left (a+b x^4+c x^2\right )^p \left (\frac {2 b x^2}{\sqrt {c^2-4 a b}+c}+1\right )^{-p} F_1\left (\frac {3}{2};-p,-p;\frac {5}{2};-\frac {2 b x^2}{c-\sqrt {c^2-4 a b}},-\frac {2 b x^2}{c+\sqrt {c^2-4 a b}}\right )+c x \left (\frac {2 b x^2}{c-\sqrt {c^2-4 a b}}+1\right )^{-p} \left (a+b x^4+c x^2\right )^p \left (\frac {2 b x^2}{\sqrt {c^2-4 a b}+c}+1\right )^{-p} F_1\left (\frac {1}{2};-p,-p;\frac {3}{2};-\frac {2 b x^2}{c-\sqrt {c^2-4 a b}},-\frac {2 b x^2}{c+\sqrt {c^2-4 a b}}\right ) \]
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Rubi [A] time = 0.22, antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1203, 1105, 429, 1141, 510} \[ \frac {1}{3} e x^3 \left (\frac {2 b x^2}{c-\sqrt {c^2-4 a b}}+1\right )^{-p} \left (a+b x^4+c x^2\right )^p \left (\frac {2 b x^2}{\sqrt {c^2-4 a b}+c}+1\right )^{-p} F_1\left (\frac {3}{2};-p,-p;\frac {5}{2};-\frac {2 b x^2}{c-\sqrt {c^2-4 a b}},-\frac {2 b x^2}{c+\sqrt {c^2-4 a b}}\right )+c x \left (\frac {2 b x^2}{c-\sqrt {c^2-4 a b}}+1\right )^{-p} \left (a+b x^4+c x^2\right )^p \left (\frac {2 b x^2}{\sqrt {c^2-4 a b}+c}+1\right )^{-p} F_1\left (\frac {1}{2};-p,-p;\frac {3}{2};-\frac {2 b x^2}{c-\sqrt {c^2-4 a b}},-\frac {2 b x^2}{c+\sqrt {c^2-4 a b}}\right ) \]
Antiderivative was successfully verified.
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Rule 429
Rule 510
Rule 1105
Rule 1141
Rule 1203
Rubi steps
\begin {align*} \int \left (c+e x^2\right ) \left (a+c x^2+b x^4\right )^p \, dx &=\int \left (c \left (a+c x^2+b x^4\right )^p+e x^2 \left (a+c x^2+b x^4\right )^p\right ) \, dx\\ &=c \int \left (a+c x^2+b x^4\right )^p \, dx+e \int x^2 \left (a+c x^2+b x^4\right )^p \, dx\\ &=\left (c \left (1+\frac {2 b x^2}{c-\sqrt {-4 a b+c^2}}\right )^{-p} \left (1+\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}}\right )^{-p} \left (a+c x^2+b x^4\right )^p\right ) \int \left (1+\frac {2 b x^2}{c-\sqrt {-4 a b+c^2}}\right )^p \left (1+\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}}\right )^p \, dx+\left (e \left (1+\frac {2 b x^2}{c-\sqrt {-4 a b+c^2}}\right )^{-p} \left (1+\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}}\right )^{-p} \left (a+c x^2+b x^4\right )^p\right ) \int x^2 \left (1+\frac {2 b x^2}{c-\sqrt {-4 a b+c^2}}\right )^p \left (1+\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}}\right )^p \, dx\\ &=c x \left (1+\frac {2 b x^2}{c-\sqrt {-4 a b+c^2}}\right )^{-p} \left (1+\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}}\right )^{-p} \left (a+c x^2+b x^4\right )^p F_1\left (\frac {1}{2};-p,-p;\frac {3}{2};-\frac {2 b x^2}{c-\sqrt {-4 a b+c^2}},-\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}}\right )+\frac {1}{3} e x^3 \left (1+\frac {2 b x^2}{c-\sqrt {-4 a b+c^2}}\right )^{-p} \left (1+\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}}\right )^{-p} \left (a+c x^2+b x^4\right )^p F_1\left (\frac {3}{2};-p,-p;\frac {5}{2};-\frac {2 b x^2}{c-\sqrt {-4 a b+c^2}},-\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.25, size = 232, normalized size = 0.85 \[ \frac {1}{3} x \left (\frac {-\sqrt {c^2-4 a b}+2 b x^2+c}{c-\sqrt {c^2-4 a b}}\right )^{-p} \left (\frac {\sqrt {c^2-4 a b}+2 b x^2+c}{\sqrt {c^2-4 a b}+c}\right )^{-p} \left (a+b x^4+c x^2\right )^p \left (e x^2 F_1\left (\frac {3}{2};-p,-p;\frac {5}{2};-\frac {2 b x^2}{c+\sqrt {c^2-4 a b}},\frac {2 b x^2}{\sqrt {c^2-4 a b}-c}\right )+3 c F_1\left (\frac {1}{2};-p,-p;\frac {3}{2};-\frac {2 b x^2}{c+\sqrt {c^2-4 a b}},\frac {2 b x^2}{\sqrt {c^2-4 a b}-c}\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (e x^{2} + c\right )} {\left (b x^{4} + c x^{2} + a\right )}^{p}, x\right ) \]
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 \[ \int {\left (e x^{2} + c\right )} {\left (b x^{4} + c x^{2} + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int \left (e \,x^{2}+c \right ) \left (b \,x^{4}+c \,x^{2}+a \right )^{p}\, dx \]
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 \[ \int {\left (e x^{2} + c\right )} {\left (b x^{4} + c x^{2} + a\right )}^{p}\,{d x} \]
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 \[ \int \left (e\,x^2+c\right )\,{\left (b\,x^4+c\,x^2+a\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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