Optimal. Leaf size=31 \[ \frac {a (c+d x)^4}{4 d}+\frac {b (c+d x)^6}{6 d} \]
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Rubi [A] time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {372, 14} \[ \frac {a (c+d x)^4}{4 d}+\frac {b (c+d x)^6}{6 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 372
Rubi steps
\begin {align*} \int (c+d x)^3 \left (a+b (c+d x)^2\right ) \, dx &=\frac {\operatorname {Subst}\left (\int x^3 \left (a+b x^2\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a x^3+b x^5\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {a (c+d x)^4}{4 d}+\frac {b (c+d x)^6}{6 d}\\ \end {align*}
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Mathematica [B] time = 0.03, size = 77, normalized size = 2.48 \[ \frac {1}{12} x (2 c+d x) \left (3 a \left (2 c^2+2 c d x+d^2 x^2\right )+2 b \left (3 c^4+6 c^3 d x+7 c^2 d^2 x^2+4 c d^3 x^3+d^4 x^4\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.66, size = 93, normalized size = 3.00 \[ \frac {1}{6} x^{6} d^{5} b + x^{5} d^{4} c b + \frac {5}{2} x^{4} d^{3} c^{2} b + \frac {10}{3} x^{3} d^{2} c^{3} b + \frac {5}{2} x^{2} d c^{4} b + \frac {1}{4} x^{4} d^{3} a + x c^{5} b + x^{3} d^{2} c a + \frac {3}{2} x^{2} d c^{2} a + x c^{3} a \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 86, normalized size = 2.77 \[ \frac {1}{2} \, {\left (d x^{2} + 2 \, c x\right )} b c^{4} + \frac {1}{2} \, {\left (d x^{2} + 2 \, c x\right )}^{2} b c^{2} d + \frac {1}{6} \, {\left (d x^{2} + 2 \, c x\right )}^{3} b d^{2} + \frac {1}{2} \, {\left (d x^{2} + 2 \, c x\right )} a c^{2} + \frac {1}{4} \, {\left (d x^{2} + 2 \, c x\right )}^{2} a d \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 112, normalized size = 3.61 \[ \frac {b \,d^{5} x^{6}}{6}+b c \,d^{4} x^{5}+\left (b \,c^{2}+a \right ) c^{3} x +\frac {\left (9 b \,c^{2} d^{3}+\left (b \,c^{2}+a \right ) d^{3}\right ) x^{4}}{4}+\frac {\left (7 b \,c^{3} d^{2}+3 \left (b \,c^{2}+a \right ) c \,d^{2}\right ) x^{3}}{3}+\frac {\left (2 b \,c^{4} d +3 \left (b \,c^{2}+a \right ) c^{2} d \right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.53, size = 86, normalized size = 2.77 \[ \frac {1}{6} \, b d^{5} x^{6} + b c d^{4} x^{5} + \frac {1}{4} \, {\left (10 \, b c^{2} + a\right )} d^{3} x^{4} + \frac {1}{3} \, {\left (10 \, b c^{3} + 3 \, a c\right )} d^{2} x^{3} + \frac {1}{2} \, {\left (5 \, b c^{4} + 3 \, a c^{2}\right )} d x^{2} + {\left (b c^{5} + a c^{3}\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 86, normalized size = 2.77 \[ x\,\left (b\,c^5+a\,c^3\right )+\frac {d^3\,x^4\,\left (10\,b\,c^2+a\right )}{4}+\frac {b\,d^5\,x^6}{6}+\frac {c^2\,d\,x^2\,\left (5\,b\,c^2+3\,a\right )}{2}+\frac {c\,d^2\,x^3\,\left (10\,b\,c^2+3\,a\right )}{3}+b\,c\,d^4\,x^5 \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.11, size = 99, normalized size = 3.19 \[ b c d^{4} x^{5} + \frac {b d^{5} x^{6}}{6} + x^{4} \left (\frac {a d^{3}}{4} + \frac {5 b c^{2} d^{3}}{2}\right ) + x^{3} \left (a c d^{2} + \frac {10 b c^{3} d^{2}}{3}\right ) + x^{2} \left (\frac {3 a c^{2} d}{2} + \frac {5 b c^{4} d}{2}\right ) + x \left (a c^{3} + b c^{5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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