Optimal. Leaf size=175 \[ \frac {c \left (a+b x^4\right )^{p+1}}{4 b (p+1)}+\frac {1}{5} d x^5 \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac {5}{4},-p;\frac {9}{4};-\frac {b x^4}{a}\right )+\frac {1}{6} e x^6 \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac {3}{2},-p;\frac {5}{2};-\frac {b x^4}{a}\right )+\frac {1}{7} f x^7 \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac {7}{4},-p;\frac {11}{4};-\frac {b x^4}{a}\right ) \]
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Rubi [A] time = 0.18, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1833, 1252, 764, 261, 365, 364, 1336} \[ \frac {c \left (a+b x^4\right )^{p+1}}{4 b (p+1)}+\frac {1}{5} d x^5 \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac {5}{4},-p;\frac {9}{4};-\frac {b x^4}{a}\right )+\frac {1}{6} e x^6 \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac {3}{2},-p;\frac {5}{2};-\frac {b x^4}{a}\right )+\frac {1}{7} f x^7 \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \, _2F_1\left (\frac {7}{4},-p;\frac {11}{4};-\frac {b x^4}{a}\right ) \]
Antiderivative was successfully verified.
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Rule 261
Rule 364
Rule 365
Rule 764
Rule 1252
Rule 1336
Rule 1833
Rubi steps
\begin {align*} \int x^3 \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^p \, dx &=\int \left (x^3 \left (c+e x^2\right ) \left (a+b x^4\right )^p+x^4 \left (d+f x^2\right ) \left (a+b x^4\right )^p\right ) \, dx\\ &=\int x^3 \left (c+e x^2\right ) \left (a+b x^4\right )^p \, dx+\int x^4 \left (d+f x^2\right ) \left (a+b x^4\right )^p \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int x (c+e x) \left (a+b x^2\right )^p \, dx,x,x^2\right )+\int \left (d x^4 \left (a+b x^4\right )^p+f x^6 \left (a+b x^4\right )^p\right ) \, dx\\ &=\frac {1}{2} c \operatorname {Subst}\left (\int x \left (a+b x^2\right )^p \, dx,x,x^2\right )+d \int x^4 \left (a+b x^4\right )^p \, dx+\frac {1}{2} e \operatorname {Subst}\left (\int x^2 \left (a+b x^2\right )^p \, dx,x,x^2\right )+f \int x^6 \left (a+b x^4\right )^p \, dx\\ &=\frac {c \left (a+b x^4\right )^{1+p}}{4 b (1+p)}+\left (d \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p}\right ) \int x^4 \left (1+\frac {b x^4}{a}\right )^p \, dx+\frac {1}{2} \left (e \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p}\right ) \operatorname {Subst}\left (\int x^2 \left (1+\frac {b x^2}{a}\right )^p \, dx,x,x^2\right )+\left (f \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p}\right ) \int x^6 \left (1+\frac {b x^4}{a}\right )^p \, dx\\ &=\frac {c \left (a+b x^4\right )^{1+p}}{4 b (1+p)}+\frac {1}{5} d x^5 \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \, _2F_1\left (\frac {5}{4},-p;\frac {9}{4};-\frac {b x^4}{a}\right )+\frac {1}{6} e x^6 \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \, _2F_1\left (\frac {3}{2},-p;\frac {5}{2};-\frac {b x^4}{a}\right )+\frac {1}{7} f x^7 \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \, _2F_1\left (\frac {7}{4},-p;\frac {11}{4};-\frac {b x^4}{a}\right )\\ \end {align*}
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Mathematica [A] time = 0.14, size = 145, normalized size = 0.83 \[ \frac {\left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \left (105 c \left (a+b x^4\right ) \left (\frac {b x^4}{a}+1\right )^p+84 b d (p+1) x^5 \, _2F_1\left (\frac {5}{4},-p;\frac {9}{4};-\frac {b x^4}{a}\right )+70 b e (p+1) x^6 \, _2F_1\left (\frac {3}{2},-p;\frac {5}{2};-\frac {b x^4}{a}\right )+60 b f (p+1) x^7 \, _2F_1\left (\frac {7}{4},-p;\frac {11}{4};-\frac {b x^4}{a}\right )\right )}{420 b (p+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (f x^{6} + e x^{5} + d x^{4} + c x^{3}\right )} {\left (b x^{4} + a\right )}^{p}, x\right ) \]
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 \[ \int {\left (f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{4} + a\right )}^{p} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.48, size = 0, normalized size = 0.00 \[ \int \left (f \,x^{3}+e \,x^{2}+d x +c \right ) x^{3} \left (b \,x^{4}+a \right )^{p}\, dx \]
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 \[ \frac {{\left (b x^{4} + a\right )}^{p + 1} c}{4 \, b {\left (p + 1\right )}} + \int {\left (f x^{6} + e x^{5} + d x^{4}\right )} {\left (b x^{4} + a\right )}^{p}\,{d x} \]
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 \[ \int x^3\,{\left (b\,x^4+a\right )}^p\,\left (f\,x^3+e\,x^2+d\,x+c\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 111.18, size = 143, normalized size = 0.82 \[ \frac {a^{p} d x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, - p \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} + \frac {a^{p} e x^{6} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, - p \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{6} + \frac {a^{p} f x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{4}, - p \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} + c \left (\begin {cases} \frac {a^{p} x^{4}}{4} & \text {for}\: b = 0 \\\frac {\begin {cases} \frac {\left (a + b x^{4}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (a + b x^{4} \right )} & \text {otherwise} \end {cases}}{4 b} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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