Optimal. Leaf size=223 \[ \frac {2^{p-1} \left (2 a c-b^2 (p+2)\right ) \left (a+b x^2+c x^4\right )^{p+1} \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}}\right )^{-p-1} \, _2F_1\left (-p,p+1;p+2;\frac {2 c x^2+b+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{c^2 (p+1) (2 p+3) \sqrt {b^2-4 a c}}-\frac {b (p+2) \left (a+b x^2+c x^4\right )^{p+1}}{4 c^2 (p+1) (2 p+3)}+\frac {x^2 \left (a+b x^2+c x^4\right )^{p+1}}{2 c (2 p+3)} \]
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Rubi [A] time = 0.21, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1114, 742, 640, 624} \[ \frac {2^{p-1} \left (2 a c-b^2 (p+2)\right ) \left (a+b x^2+c x^4\right )^{p+1} \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}}\right )^{-p-1} \, _2F_1\left (-p,p+1;p+2;\frac {2 c x^2+b+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{c^2 (p+1) (2 p+3) \sqrt {b^2-4 a c}}-\frac {b (p+2) \left (a+b x^2+c x^4\right )^{p+1}}{4 c^2 (p+1) (2 p+3)}+\frac {x^2 \left (a+b x^2+c x^4\right )^{p+1}}{2 c (2 p+3)} \]
Antiderivative was successfully verified.
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Rule 624
Rule 640
Rule 742
Rule 1114
Rubi steps
\begin {align*} \int x^5 \left (a+b x^2+c x^4\right )^p \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^2 \left (a+b x+c x^2\right )^p \, dx,x,x^2\right )\\ &=\frac {x^2 \left (a+b x^2+c x^4\right )^{1+p}}{2 c (3+2 p)}+\frac {\operatorname {Subst}\left (\int (-a-b (2+p) x) \left (a+b x+c x^2\right )^p \, dx,x,x^2\right )}{2 c (3+2 p)}\\ &=-\frac {b (2+p) \left (a+b x^2+c x^4\right )^{1+p}}{4 c^2 (1+p) (3+2 p)}+\frac {x^2 \left (a+b x^2+c x^4\right )^{1+p}}{2 c (3+2 p)}-\frac {\left (2 a c-b^2 (2+p)\right ) \operatorname {Subst}\left (\int \left (a+b x+c x^2\right )^p \, dx,x,x^2\right )}{4 c^2 (3+2 p)}\\ &=-\frac {b (2+p) \left (a+b x^2+c x^4\right )^{1+p}}{4 c^2 (1+p) (3+2 p)}+\frac {x^2 \left (a+b x^2+c x^4\right )^{1+p}}{2 c (3+2 p)}+\frac {2^{-1+p} \left (2 a c-b^2 (2+p)\right ) \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^2 \sqrt {b^2-4 a c} (1+p) (3+2 p)}\\ \end {align*}
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Mathematica [C] time = 0.22, size = 162, normalized size = 0.73 \[ \frac {1}{6} x^6 \left (\frac {-\sqrt {b^2-4 a c}+b+2 c x^2}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}+b}\right )^{-p} \left (a+b x^2+c x^4\right )^p F_1\left (3;-p,-p;4;-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{\sqrt {b^2-4 a c}-b}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c x^{4} + b x^{2} + a\right )}^{p} x^{5}, x\right ) \]
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 \[ \int {\left (c x^{4} + b x^{2} + a\right )}^{p} x^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int x^{5} \left (c \,x^{4}+b \,x^{2}+a \right )^{p}\, dx \]
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 \[ \int {\left (c x^{4} + b x^{2} + a\right )}^{p} x^{5}\,{d x} \]
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 \[ \int x^5\,{\left (c\,x^4+b\,x^2+a\right )}^p \,d x \]
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 \[ \int x^{5} \left (a + b x^{2} + c x^{4}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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