Optimal. Leaf size=274 \[ c x \left (1+\frac {2 b x^2}{c-\sqrt {-4 a b+c^2}}\right )^{-p} \left (1+\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}}\right )^{-p} \left (a+c x^2+b x^4\right )^p F_1\left (\frac {1}{2};-p,-p;\frac {3}{2};-\frac {2 b x^2}{c-\sqrt {-4 a b+c^2}},-\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}}\right )+\frac {1}{3} e x^3 \left (1+\frac {2 b x^2}{c-\sqrt {-4 a b+c^2}}\right )^{-p} \left (1+\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}}\right )^{-p} \left (a+c x^2+b x^4\right )^p F_1\left (\frac {3}{2};-p,-p;\frac {5}{2};-\frac {2 b x^2}{c-\sqrt {-4 a b+c^2}},-\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}}\right ) \]
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Rubi [A]
time = 0.13, 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 = {1217, 1119,
440, 1155, 524} \begin {gather*} \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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 440
Rule 524
Rule 1119
Rule 1155
Rule 1217
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 = 0.40, size = 232, normalized size = 0.85 \begin {gather*} \frac {1}{3} x \left (\frac {c-\sqrt {-4 a b+c^2}+2 b x^2}{c-\sqrt {-4 a b+c^2}}\right )^{-p} \left (\frac {c+\sqrt {-4 a b+c^2}+2 b x^2}{c+\sqrt {-4 a b+c^2}}\right )^{-p} \left (a+c x^2+b x^4\right )^p \left (3 c 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 )+e x^2 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 )\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.01, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 \left (c + e x^{2}\right ) \left (a + b x^{4} + c x^{2}\right )^{p}\, dx \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*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (e\,x^2+c\right )\,{\left (b\,x^4+c\,x^2+a\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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