Optimal. Leaf size=160 \[ \frac {b 2^{p-1} \left (a+b x^2+c x^4\right )^{p+1} \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}}\right )^{-p-1} \, _2F_1\left (-p,p+1;p+2;\frac {2 c x^2+b+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{c (p+1) \sqrt {b^2-4 a c}}+\frac {\left (a+b x^2+c x^4\right )^{p+1}}{4 c (p+1)} \]
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Rubi [A] time = 0.13, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1114, 640, 624} \[ \frac {b 2^{p-1} \left (a+b x^2+c x^4\right )^{p+1} \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}}\right )^{-p-1} \, _2F_1\left (-p,p+1;p+2;\frac {2 c x^2+b+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{c (p+1) \sqrt {b^2-4 a c}}+\frac {\left (a+b x^2+c x^4\right )^{p+1}}{4 c (p+1)} \]
Antiderivative was successfully verified.
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Rule 624
Rule 640
Rule 1114
Rubi steps
\begin {align*} \int x^3 \left (a+b x^2+c x^4\right )^p \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x \left (a+b x+c x^2\right )^p \, dx,x,x^2\right )\\ &=\frac {\left (a+b x^2+c x^4\right )^{1+p}}{4 c (1+p)}-\frac {b \operatorname {Subst}\left (\int \left (a+b x+c x^2\right )^p \, dx,x,x^2\right )}{4 c}\\ &=\frac {\left (a+b x^2+c x^4\right )^{1+p}}{4 c (1+p)}+\frac {2^{-1+p} b \left (-\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{\sqrt {b^2-4 a c}}\right )^{-1-p} \left (a+b x^2+c x^4\right )^{1+p} \, _2F_1\left (-p,1+p;2+p;\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{2 \sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c} (1+p)}\\ \end {align*}
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Mathematica [C] time = 0.20, size = 162, normalized size = 1.01 \[ \frac {1}{4} x^4 \left (\frac {-\sqrt {b^2-4 a c}+b+2 c x^2}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}+b}\right )^{-p} \left (a+b x^2+c x^4\right )^p F_1\left (2;-p,-p;3;-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{\sqrt {b^2-4 a c}-b}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c x^{4} + b x^{2} + a\right )}^{p} x^{3}, x\right ) \]
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 \[ \int {\left (c x^{4} + b x^{2} + a\right )}^{p} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int x^{3} \left (c \,x^{4}+b \,x^{2}+a \right )^{p}\, dx \]
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 \[ \int {\left (c x^{4} + b x^{2} + a\right )}^{p} x^{3}\,{d x} \]
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 \[ \int x^3\,{\left (c\,x^4+b\,x^2+a\right )}^p \,d x \]
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 \[ \int x^{3} \left (a + b x^{2} + c x^{4}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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