Optimal. Leaf size=143 \[ \frac {f \left (a+b x^4\right )^{1+p}}{4 b (1+p)}+\frac {c x \left (a+b x^4\right )^{1+p} \, _2F_1\left (1,\frac {5}{4}+p;\frac {5}{4};-\frac {b x^4}{a}\right )}{a}+\frac {d x^2 \left (a+b x^4\right )^{1+p} \, _2F_1\left (1,\frac {3}{2}+p;\frac {3}{2};-\frac {b x^4}{a}\right )}{2 a}+\frac {e x^3 \left (a+b x^4\right )^{1+p} \, _2F_1\left (1,\frac {7}{4}+p;\frac {7}{4};-\frac {b x^4}{a}\right )}{3 a} \]
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Rubi [A]
time = 0.10, 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 = {1899, 1218,
252, 251, 372, 371, 1262, 655} \begin {gather*} 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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 251
Rule 252
Rule 371
Rule 372
Rule 655
Rule 1218
Rule 1262
Rule 1899
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} \text {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 \text {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 ) \text {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 = 0.66, size = 147, normalized size = 1.03 \begin {gather*} \frac {1}{12} \left (a+b x^4\right )^p \left (\frac {3 f \left (a+b x^4\right )}{b (1+p)}+12 c x \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 )+6 d x^2 \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 )+4 e x^3 \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 )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.37, size = 27, normalized size = 0.19 \begin {gather*} {\rm integral}\left ({\left (f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{4} + a\right )}^{p}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 20.50, size = 141, normalized size = 0.99 \begin {gather*} \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 ) \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (b\,x^4+a\right )}^p\,\left (f\,x^3+e\,x^2+d\,x+c\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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