Optimal. Leaf size=257 \[ -\frac {b 2^{p-2} \left (6 a c-b^2 (p+3)\right ) \left (a+b x^2+c x^4\right )^{p+1} \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}}\right )^{-p-1} \, _2F_1\left (-p,p+1;p+2;\frac {2 c x^2+b+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{c^3 (p+1) (2 p+3) \sqrt {b^2-4 a c}}+\frac {\left (-2 a c (2 p+3)+b^2 (p+2) (p+3)-2 b c (p+1) (p+3) x^2\right ) \left (a+b x^2+c x^4\right )^{p+1}}{8 c^3 (p+1) (p+2) (2 p+3)}+\frac {x^4 \left (a+b x^2+c x^4\right )^{p+1}}{4 c (p+2)} \]
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Rubi [A] time = 0.37, antiderivative size = 257, 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, 779, 624} \[ -\frac {b 2^{p-2} \left (6 a c-b^2 (p+3)\right ) \left (a+b x^2+c x^4\right )^{p+1} \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}}\right )^{-p-1} \, _2F_1\left (-p,p+1;p+2;\frac {2 c x^2+b+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{c^3 (p+1) (2 p+3) \sqrt {b^2-4 a c}}+\frac {\left (-2 a c (2 p+3)+b^2 (p+2) (p+3)-2 b c (p+1) (p+3) x^2\right ) \left (a+b x^2+c x^4\right )^{p+1}}{8 c^3 (p+1) (p+2) (2 p+3)}+\frac {x^4 \left (a+b x^2+c x^4\right )^{p+1}}{4 c (p+2)} \]
Antiderivative was successfully verified.
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Rule 624
Rule 742
Rule 779
Rule 1114
Rubi steps
\begin {align*} \int x^7 \left (a+b x^2+c x^4\right )^p \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^3 \left (a+b x+c x^2\right )^p \, dx,x,x^2\right )\\ &=\frac {x^4 \left (a+b x^2+c x^4\right )^{1+p}}{4 c (2+p)}+\frac {\operatorname {Subst}\left (\int x (-2 a-b (3+p) x) \left (a+b x+c x^2\right )^p \, dx,x,x^2\right )}{4 c (2+p)}\\ &=\frac {x^4 \left (a+b x^2+c x^4\right )^{1+p}}{4 c (2+p)}+\frac {\left (b^2 (2+p) (3+p)-2 a c (3+2 p)-2 b c (1+p) (3+p) x^2\right ) \left (a+b x^2+c x^4\right )^{1+p}}{8 c^3 (1+p) (2+p) (3+2 p)}+\frac {\left (b \left (6 a c-b^2 (3+p)\right )\right ) \operatorname {Subst}\left (\int \left (a+b x+c x^2\right )^p \, dx,x,x^2\right )}{8 c^3 (3+2 p)}\\ &=\frac {x^4 \left (a+b x^2+c x^4\right )^{1+p}}{4 c (2+p)}+\frac {\left (b^2 (2+p) (3+p)-2 a c (3+2 p)-2 b c (1+p) (3+p) x^2\right ) \left (a+b x^2+c x^4\right )^{1+p}}{8 c^3 (1+p) (2+p) (3+2 p)}-\frac {2^{-2+p} b \left (6 a c-b^2 (3+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^3 \sqrt {b^2-4 a c} (1+p) (3+2 p)}\\ \end {align*}
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Mathematica [C] time = 0.23, size = 162, normalized size = 0.63 \[ \frac {1}{8} x^8 \left (\frac {-\sqrt {b^2-4 a c}+b+2 c x^2}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}+b}\right )^{-p} \left (a+b x^2+c x^4\right )^p F_1\left (4;-p,-p;5;-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{\sqrt {b^2-4 a c}-b}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c x^{4} + b x^{2} + a\right )}^{p} x^{7}, x\right ) \]
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 \[ \int {\left (c x^{4} + b x^{2} + a\right )}^{p} x^{7}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int x^{7} \left (c \,x^{4}+b \,x^{2}+a \right )^{p}\, dx \]
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 \[ \int {\left (c x^{4} + b x^{2} + a\right )}^{p} x^{7}\,{d x} \]
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 \[ \int x^7\,{\left (c\,x^4+b\,x^2+a\right )}^p \,d x \]
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 \[ \int x^{7} \left (a + b x^{2} + c x^{4}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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