Optimal. Leaf size=30 \[ \frac {\left (a+b (c+d x)^4\right )^{p+1}}{4 b d (p+1)} \]
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Rubi [A] time = 0.03, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {372, 261} \[ \frac {\left (a+b (c+d x)^4\right )^{p+1}}{4 b d (p+1)} \]
Antiderivative was successfully verified.
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Rule 261
Rule 372
Rubi steps
\begin {align*} \int (c+d x)^3 \left (a+b (c+d x)^4\right )^p \, dx &=\frac {\operatorname {Subst}\left (\int x^3 \left (a+b x^4\right )^p \, dx,x,c+d x\right )}{d}\\ &=\frac {\left (a+b (c+d x)^4\right )^{1+p}}{4 b d (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 30, normalized size = 1.00 \[ \frac {\left (a+b (c+d x)^4\right )^{p+1}}{4 b d (p+1)} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.07, size = 104, normalized size = 3.47 \[ \frac {{\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a\right )} {\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a\right )}^{p}}{4 \, {\left (b d p + b d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 62, normalized size = 2.07 \[ \frac {{\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a\right )}^{p + 1}}{4 \, b d {\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 63, normalized size = 2.10 \[ \frac {\left (b \,d^{4} x^{4}+4 b c \,d^{3} x^{3}+6 b \,c^{2} d^{2} x^{2}+4 b \,c^{3} d x +b \,c^{4}+a \right )^{p +1}}{4 \left (p +1\right ) b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 28, normalized size = 0.93 \[ \frac {{\left ({\left (d x + c\right )}^{4} b + a\right )}^{p + 1}}{4 \, b d {\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.41, size = 88, normalized size = 2.93 \[ {\left (a+b\,{\left (c+d\,x\right )}^4\right )}^p\,\left (\frac {d^3\,x^4}{4\,\left (p+1\right )}+\frac {c^3\,x}{p+1}+\frac {b\,c^4+a}{4\,b\,d\,\left (p+1\right )}+\frac {3\,c^2\,d\,x^2}{2\,\left (p+1\right )}+\frac {c\,d^2\,x^3}{p+1}\right ) \]
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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