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 = 0.18225, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 1833
Rule 1252
Rule 764
Rule 261
Rule 365
Rule 364
Rule 1336
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*}
Mathematica [A] time = 0.114865, 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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Maple [F] time = 0.222, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( f{x}^{3}+e{x}^{2}+dx+c \right ) \left ( b{x}^{4}+a \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (f x^{3} + e x^{2} + d x + c\right )}{\left (b x^{4} + a\right )}^{p} x^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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