Optimal. Leaf size=143 \[ \frac {c x \left (a+b x^4\right )^{p+1} \, _2F_1\left (1,p+\frac {5}{4};\frac {5}{4};-\frac {b x^4}{a}\right )}{a}+\frac {d x^2 \left (a+b x^4\right )^{p+1} \, _2F_1\left (1,p+\frac {3}{2};\frac {3}{2};-\frac {b x^4}{a}\right )}{2 a}+\frac {e x^3 \left (a+b x^4\right )^{p+1} \, _2F_1\left (1,p+\frac {7}{4};\frac {7}{4};-\frac {b x^4}{a}\right )}{3 a}+\frac {f \left (a+b x^4\right )^{p+1}}{4 b (p+1)} \]
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Rubi [A] time = 0.13, antiderivative size = 170, normalized size of antiderivative = 1.19, number of steps used = 12, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {1885, 1204, 246, 245, 365, 364, 1248, 641} \[ c x \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac {1}{4},-p;\frac {5}{4};-\frac {b x^4}{a}\right )+\frac {1}{2} d x^2 \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^4}{a}\right )+\frac {1}{3} e x^3 \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac {3}{4},-p;\frac {7}{4};-\frac {b x^4}{a}\right )+\frac {f \left (a+b x^4\right )^{p+1}}{4 b (p+1)} \]
Antiderivative was successfully verified.
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Rule 245
Rule 246
Rule 364
Rule 365
Rule 641
Rule 1204
Rule 1248
Rule 1885
Rubi steps
\begin {align*} \int \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^p \, dx &=\int \left (\left (c+e x^2\right ) \left (a+b x^4\right )^p+x \left (d+f x^2\right ) \left (a+b x^4\right )^p\right ) \, dx\\ &=\int \left (c+e x^2\right ) \left (a+b x^4\right )^p \, dx+\int x \left (d+f x^2\right ) \left (a+b x^4\right )^p \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int (d+f x) \left (a+b x^2\right )^p \, dx,x,x^2\right )+\int \left (c \left (a+b x^4\right )^p+e x^2 \left (a+b x^4\right )^p\right ) \, dx\\ &=\frac {f \left (a+b x^4\right )^{1+p}}{4 b (1+p)}+c \int \left (a+b x^4\right )^p \, dx+\frac {1}{2} d \operatorname {Subst}\left (\int \left (a+b x^2\right )^p \, dx,x,x^2\right )+e \int x^2 \left (a+b x^4\right )^p \, dx\\ &=\frac {f \left (a+b x^4\right )^{1+p}}{4 b (1+p)}+\left (c \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p}\right ) \int \left (1+\frac {b x^4}{a}\right )^p \, dx+\frac {1}{2} \left (d \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p}\right ) \operatorname {Subst}\left (\int \left (1+\frac {b x^2}{a}\right )^p \, dx,x,x^2\right )+\left (e \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p}\right ) \int x^2 \left (1+\frac {b x^4}{a}\right )^p \, dx\\ &=\frac {f \left (a+b x^4\right )^{1+p}}{4 b (1+p)}+c x \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \, _2F_1\left (\frac {1}{4},-p;\frac {5}{4};-\frac {b x^4}{a}\right )+\frac {1}{2} d x^2 \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^4}{a}\right )+\frac {1}{3} e x^3 \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \, _2F_1\left (\frac {3}{4},-p;\frac {7}{4};-\frac {b x^4}{a}\right )\\ \end {align*}
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Mathematica [A] time = 0.12, size = 147, normalized size = 1.03 \[ \frac {1}{12} \left (a+b x^4\right )^p \left (12 c x \left (\frac {b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac {1}{4},-p;\frac {5}{4};-\frac {b x^4}{a}\right )+6 d x^2 \left (\frac {b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^4}{a}\right )+4 e x^3 \left (\frac {b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac {3}{4},-p;\frac {7}{4};-\frac {b x^4}{a}\right )+\frac {3 f \left (a+b x^4\right )}{b (p+1)}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{4} + a\right )}^{p}, x\right ) \]
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 \[ \int {\left (f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{4} + a\right )}^{p}\,{d x} \]
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 \[ \int \left (f \,x^{3}+e \,x^{2}+d x +c \right ) \left (b \,x^{4}+a \right )^{p}\, dx \]
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 \[ \int {\left (f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{4} + a\right )}^{p}\,{d x} \]
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 \[ \int {\left (b\,x^4+a\right )}^p\,\left (f\,x^3+e\,x^2+d\,x+c\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 49.57, size = 141, normalized size = 0.99 \[ \frac {a^{p} c x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, - p \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {a^{p} d x^{2} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - p \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{2} + \frac {a^{p} e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, - p \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} + f \left (\begin {cases} \frac {a^{p} x^{4}}{4} & \text {for}\: b = 0 \\\frac {\begin {cases} \frac {\left (a + b x^{4}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (a + b x^{4} \right )} & \text {otherwise} \end {cases}}{4 b} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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