Optimal. Leaf size=51 \[ \frac {a^2 (c+d x)^4}{4 d}+\frac {a b (c+d x)^6}{3 d}+\frac {b^2 (c+d x)^8}{8 d} \]
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Rubi [A] time = 0.07, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {372, 266, 43} \[ \frac {a^2 (c+d x)^4}{4 d}+\frac {a b (c+d x)^6}{3 d}+\frac {b^2 (c+d x)^8}{8 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 372
Rubi steps
\begin {align*} \int (c+d x)^3 \left (a+b (c+d x)^2\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int x^3 \left (a+b x^2\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int x (a+b x)^2 \, dx,x,(c+d x)^2\right )}{2 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^2 x+2 a b x^2+b^2 x^3\right ) \, dx,x,(c+d x)^2\right )}{2 d}\\ &=\frac {a^2 (c+d x)^4}{4 d}+\frac {a b (c+d x)^6}{3 d}+\frac {b^2 (c+d x)^8}{8 d}\\ \end {align*}
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Mathematica [B] time = 0.03, size = 172, normalized size = 3.37 \[ \frac {1}{4} d^3 x^4 \left (a^2+20 a b c^2+35 b^2 c^4\right )+\frac {1}{3} c d^2 x^3 \left (3 a^2+20 a b c^2+21 b^2 c^4\right )+\frac {1}{2} c^2 d x^2 \left (3 a^2+10 a b c^2+7 b^2 c^4\right )+\frac {1}{6} b d^5 x^6 \left (2 a+21 b c^2\right )+b c d^4 x^5 \left (2 a+7 b c^2\right )+c^3 x \left (a+b c^2\right )^2+b^2 c d^6 x^7+\frac {1}{8} b^2 d^7 x^8 \]
Antiderivative was successfully verified.
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fricas [B] time = 0.78, size = 207, normalized size = 4.06 \[ \frac {1}{8} x^{8} d^{7} b^{2} + x^{7} d^{6} c b^{2} + \frac {7}{2} x^{6} d^{5} c^{2} b^{2} + 7 x^{5} d^{4} c^{3} b^{2} + \frac {35}{4} x^{4} d^{3} c^{4} b^{2} + \frac {1}{3} x^{6} d^{5} b a + 7 x^{3} d^{2} c^{5} b^{2} + 2 x^{5} d^{4} c b a + \frac {7}{2} x^{2} d c^{6} b^{2} + 5 x^{4} d^{3} c^{2} b a + x c^{7} b^{2} + \frac {20}{3} x^{3} d^{2} c^{3} b a + 5 x^{2} d c^{4} b a + \frac {1}{4} x^{4} d^{3} a^{2} + 2 x c^{5} b a + x^{3} d^{2} c a^{2} + \frac {3}{2} x^{2} d c^{2} a^{2} + x c^{3} a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 173, normalized size = 3.39 \[ \frac {1}{2} \, {\left (d x^{2} + 2 \, c x\right )} b^{2} c^{6} + \frac {3}{4} \, {\left (d x^{2} + 2 \, c x\right )}^{2} b^{2} c^{4} d + \frac {1}{2} \, {\left (d x^{2} + 2 \, c x\right )}^{3} b^{2} c^{2} d^{2} + \frac {1}{8} \, {\left (d x^{2} + 2 \, c x\right )}^{4} b^{2} d^{3} + {\left (d x^{2} + 2 \, c x\right )} a b c^{4} + {\left (d x^{2} + 2 \, c x\right )}^{2} a b c^{2} d + \frac {1}{3} \, {\left (d x^{2} + 2 \, c x\right )}^{3} a b d^{2} + \frac {1}{2} \, {\left (d x^{2} + 2 \, c x\right )} a^{2} c^{2} + \frac {1}{4} \, {\left (d x^{2} + 2 \, c x\right )}^{2} a^{2} d \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 324, normalized size = 6.35 \[ \frac {b^{2} d^{7} x^{8}}{8}+b^{2} c \,d^{6} x^{7}+\frac {\left (15 b^{2} c^{2} d^{5}+\left (4 b^{2} c^{2} d^{2}+2 \left (b \,c^{2}+a \right ) b \,d^{2}\right ) d^{3}\right ) x^{6}}{6}+\frac {\left (13 b^{2} c^{3} d^{4}+4 \left (b \,c^{2}+a \right ) b c \,d^{4}+3 \left (4 b^{2} c^{2} d^{2}+2 \left (b \,c^{2}+a \right ) b \,d^{2}\right ) c \,d^{2}\right ) x^{5}}{5}+\left (b \,c^{2}+a \right )^{2} c^{3} x +\frac {\left (4 b^{2} c^{4} d^{3}+12 \left (b \,c^{2}+a \right ) b \,c^{2} d^{3}+3 \left (4 b^{2} c^{2} d^{2}+2 \left (b \,c^{2}+a \right ) b \,d^{2}\right ) c^{2} d +\left (b \,c^{2}+a \right )^{2} d^{3}\right ) x^{4}}{4}+\frac {\left (12 \left (b \,c^{2}+a \right ) b \,c^{3} d^{2}+\left (4 b^{2} c^{2} d^{2}+2 \left (b \,c^{2}+a \right ) b \,d^{2}\right ) c^{3}+3 \left (b \,c^{2}+a \right )^{2} c \,d^{2}\right ) x^{3}}{3}+\frac {\left (4 \left (b \,c^{2}+a \right ) b \,c^{4} d +3 \left (b \,c^{2}+a \right )^{2} c^{2} d \right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.51, size = 176, normalized size = 3.45 \[ \frac {1}{8} \, b^{2} d^{7} x^{8} + b^{2} c d^{6} x^{7} + \frac {1}{6} \, {\left (21 \, b^{2} c^{2} + 2 \, a b\right )} d^{5} x^{6} + {\left (7 \, b^{2} c^{3} + 2 \, a b c\right )} d^{4} x^{5} + \frac {1}{4} \, {\left (35 \, b^{2} c^{4} + 20 \, a b c^{2} + a^{2}\right )} d^{3} x^{4} + \frac {1}{3} \, {\left (21 \, b^{2} c^{5} + 20 \, a b c^{3} + 3 \, a^{2} c\right )} d^{2} x^{3} + \frac {1}{2} \, {\left (7 \, b^{2} c^{6} + 10 \, a b c^{4} + 3 \, a^{2} c^{2}\right )} d x^{2} + {\left (b^{2} c^{7} + 2 \, a b c^{5} + a^{2} c^{3}\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 162, normalized size = 3.18 \[ c^3\,x\,{\left (b\,c^2+a\right )}^2+\frac {b^2\,d^7\,x^8}{8}+\frac {d^3\,x^4\,\left (a^2+20\,a\,b\,c^2+35\,b^2\,c^4\right )}{4}+\frac {b\,d^5\,x^6\,\left (21\,b\,c^2+2\,a\right )}{6}+b^2\,c\,d^6\,x^7+\frac {c^2\,d\,x^2\,\left (3\,a^2+10\,a\,b\,c^2+7\,b^2\,c^4\right )}{2}+\frac {c\,d^2\,x^3\,\left (3\,a^2+20\,a\,b\,c^2+21\,b^2\,c^4\right )}{3}+b\,c\,d^4\,x^5\,\left (7\,b\,c^2+2\,a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.16, size = 209, normalized size = 4.10 \[ b^{2} c d^{6} x^{7} + \frac {b^{2} d^{7} x^{8}}{8} + x^{6} \left (\frac {a b d^{5}}{3} + \frac {7 b^{2} c^{2} d^{5}}{2}\right ) + x^{5} \left (2 a b c d^{4} + 7 b^{2} c^{3} d^{4}\right ) + x^{4} \left (\frac {a^{2} d^{3}}{4} + 5 a b c^{2} d^{3} + \frac {35 b^{2} c^{4} d^{3}}{4}\right ) + x^{3} \left (a^{2} c d^{2} + \frac {20 a b c^{3} d^{2}}{3} + 7 b^{2} c^{5} d^{2}\right ) + x^{2} \left (\frac {3 a^{2} c^{2} d}{2} + 5 a b c^{4} d + \frac {7 b^{2} c^{6} d}{2}\right ) + x \left (a^{2} c^{3} + 2 a b c^{5} + b^{2} c^{7}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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