Optimal. Leaf size=269 \[ \frac {c (g x)^{m+1} \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+1}{4},-p;\frac {m+5}{4};-\frac {b x^4}{a}\right )}{g (m+1)}+\frac {d (g x)^{m+2} \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+2}{4},-p;\frac {m+6}{4};-\frac {b x^4}{a}\right )}{g^2 (m+2)}+\frac {e (g x)^{m+3} \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+3}{4},-p;\frac {m+7}{4};-\frac {b x^4}{a}\right )}{g^3 (m+3)}+\frac {f (g x)^{m+4} \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+4}{4},-p;\frac {m+8}{4};-\frac {b x^4}{a}\right )}{g^4 (m+4)} \]
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Rubi [A] time = 0.26, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1833, 1336, 365, 364} \[ \frac {c (g x)^{m+1} \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+1}{4},-p;\frac {m+5}{4};-\frac {b x^4}{a}\right )}{g (m+1)}+\frac {d (g x)^{m+2} \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+2}{4},-p;\frac {m+6}{4};-\frac {b x^4}{a}\right )}{g^2 (m+2)}+\frac {e (g x)^{m+3} \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+3}{4},-p;\frac {m+7}{4};-\frac {b x^4}{a}\right )}{g^3 (m+3)}+\frac {f (g x)^{m+4} \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+4}{4},-p;\frac {m+8}{4};-\frac {b x^4}{a}\right )}{g^4 (m+4)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 1336
Rule 1833
Rubi steps
\begin {align*} \int (g x)^m \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^p \, dx &=\int \left ((g x)^m \left (c+e x^2\right ) \left (a+b x^4\right )^p+\frac {(g x)^{1+m} \left (d+f x^2\right ) \left (a+b x^4\right )^p}{g}\right ) \, dx\\ &=\frac {\int (g x)^{1+m} \left (d+f x^2\right ) \left (a+b x^4\right )^p \, dx}{g}+\int (g x)^m \left (c+e x^2\right ) \left (a+b x^4\right )^p \, dx\\ &=\frac {\int \left (d (g x)^{1+m} \left (a+b x^4\right )^p+\frac {f (g x)^{3+m} \left (a+b x^4\right )^p}{g^2}\right ) \, dx}{g}+\int \left (c (g x)^m \left (a+b x^4\right )^p+\frac {e (g x)^{2+m} \left (a+b x^4\right )^p}{g^2}\right ) \, dx\\ &=c \int (g x)^m \left (a+b x^4\right )^p \, dx+\frac {f \int (g x)^{3+m} \left (a+b x^4\right )^p \, dx}{g^3}+\frac {e \int (g x)^{2+m} \left (a+b x^4\right )^p \, dx}{g^2}+\frac {d \int (g x)^{1+m} \left (a+b x^4\right )^p \, dx}{g}\\ &=\left (c \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p}\right ) \int (g x)^m \left (1+\frac {b x^4}{a}\right )^p \, dx+\frac {\left (f \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p}\right ) \int (g x)^{3+m} \left (1+\frac {b x^4}{a}\right )^p \, dx}{g^3}+\frac {\left (e \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p}\right ) \int (g x)^{2+m} \left (1+\frac {b x^4}{a}\right )^p \, dx}{g^2}+\frac {\left (d \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p}\right ) \int (g x)^{1+m} \left (1+\frac {b x^4}{a}\right )^p \, dx}{g}\\ &=\frac {c (g x)^{1+m} \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \, _2F_1\left (\frac {1+m}{4},-p;\frac {5+m}{4};-\frac {b x^4}{a}\right )}{g (1+m)}+\frac {d (g x)^{2+m} \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \, _2F_1\left (\frac {2+m}{4},-p;\frac {6+m}{4};-\frac {b x^4}{a}\right )}{g^2 (2+m)}+\frac {e (g x)^{3+m} \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \, _2F_1\left (\frac {3+m}{4},-p;\frac {7+m}{4};-\frac {b x^4}{a}\right )}{g^3 (3+m)}+\frac {f (g x)^{4+m} \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \, _2F_1\left (\frac {4+m}{4},-p;\frac {8+m}{4};-\frac {b x^4}{a}\right )}{g^4 (4+m)}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 174, normalized size = 0.65 \[ x (g x)^m \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \left (\frac {c \, _2F_1\left (\frac {m+1}{4},-p;\frac {m+5}{4};-\frac {b x^4}{a}\right )}{m+1}+x \left (\frac {d \, _2F_1\left (\frac {m+2}{4},-p;\frac {m+6}{4};-\frac {b x^4}{a}\right )}{m+2}+x \left (\frac {e \, _2F_1\left (\frac {m+3}{4},-p;\frac {m+7}{4};-\frac {b x^4}{a}\right )}{m+3}+\frac {f x \, _2F_1\left (\frac {m+4}{4},-p;\frac {m+8}{4};-\frac {b x^4}{a}\right )}{m+4}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{4} + a\right )}^{p} \left (g x\right )^{m}, x\right ) \]
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 \[ \int {\left (f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{4} + a\right )}^{p} \left (g x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.47, size = 0, normalized size = 0.00 \[ \int \left (f \,x^{3}+e \,x^{2}+d x +c \right ) \left (g x \right )^{m} \left (b \,x^{4}+a \right )^{p}\, dx \]
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 \[ \int {\left (f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{4} + a\right )}^{p} \left (g x\right )^{m}\,{d x} \]
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 \[ \int {\left (g\,x\right )}^m\,{\left (b\,x^4+a\right )}^p\,\left (f\,x^3+e\,x^2+d\,x+c\right ) \,d x \]
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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