Optimal. Leaf size=117 \[ \frac {\left (c d^2+a e^2\right )^2 (d+e x)^4}{4 e^5}-\frac {4 c d \left (c d^2+a e^2\right ) (d+e x)^5}{5 e^5}+\frac {c \left (3 c d^2+a e^2\right ) (d+e x)^6}{3 e^5}-\frac {4 c^2 d (d+e x)^7}{7 e^5}+\frac {c^2 (d+e x)^8}{8 e^5} \]
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Rubi [A]
time = 0.07, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {711}
\begin {gather*} \frac {c (d+e x)^6 \left (a e^2+3 c d^2\right )}{3 e^5}-\frac {4 c d (d+e x)^5 \left (a e^2+c d^2\right )}{5 e^5}+\frac {(d+e x)^4 \left (a e^2+c d^2\right )^2}{4 e^5}+\frac {c^2 (d+e x)^8}{8 e^5}-\frac {4 c^2 d (d+e x)^7}{7 e^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 711
Rubi steps
\begin {align*} \int (d+e x)^3 \left (a+c x^2\right )^2 \, dx &=\int \left (\frac {\left (c d^2+a e^2\right )^2 (d+e x)^3}{e^4}-\frac {4 c d \left (c d^2+a e^2\right ) (d+e x)^4}{e^4}+\frac {2 c \left (3 c d^2+a e^2\right ) (d+e x)^5}{e^4}-\frac {4 c^2 d (d+e x)^6}{e^4}+\frac {c^2 (d+e x)^7}{e^4}\right ) \, dx\\ &=\frac {\left (c d^2+a e^2\right )^2 (d+e x)^4}{4 e^5}-\frac {4 c d \left (c d^2+a e^2\right ) (d+e x)^5}{5 e^5}+\frac {c \left (3 c d^2+a e^2\right ) (d+e x)^6}{3 e^5}-\frac {4 c^2 d (d+e x)^7}{7 e^5}+\frac {c^2 (d+e x)^8}{8 e^5}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 117, normalized size = 1.00 \begin {gather*} \frac {1}{4} a^2 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+\frac {1}{30} a c x^3 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )+\frac {1}{280} c^2 x^5 \left (56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.43, size = 131, normalized size = 1.12
method | result | size |
norman | \(\frac {c^{2} e^{3} x^{8}}{8}+\frac {3 d \,e^{2} c^{2} x^{7}}{7}+\left (\frac {1}{3} e^{3} a c +\frac {1}{2} d^{2} e \,c^{2}\right ) x^{6}+\left (\frac {6}{5} d \,e^{2} a c +\frac {1}{5} c^{2} d^{3}\right ) x^{5}+\left (\frac {1}{4} a^{2} e^{3}+\frac {3}{2} d^{2} e a c \right ) x^{4}+\left (a^{2} d \,e^{2}+\frac {2}{3} d^{3} a c \right ) x^{3}+\frac {3 d^{2} e \,a^{2} x^{2}}{2}+a^{2} d^{3} x\) | \(128\) |
default | \(\frac {c^{2} e^{3} x^{8}}{8}+\frac {3 d \,e^{2} c^{2} x^{7}}{7}+\frac {\left (2 e^{3} a c +3 d^{2} e \,c^{2}\right ) x^{6}}{6}+\frac {\left (6 d \,e^{2} a c +c^{2} d^{3}\right ) x^{5}}{5}+\frac {\left (a^{2} e^{3}+6 d^{2} e a c \right ) x^{4}}{4}+\frac {\left (3 a^{2} d \,e^{2}+2 d^{3} a c \right ) x^{3}}{3}+\frac {3 d^{2} e \,a^{2} x^{2}}{2}+a^{2} d^{3} x\) | \(131\) |
gosper | \(\frac {1}{8} c^{2} e^{3} x^{8}+\frac {3}{7} d \,e^{2} c^{2} x^{7}+\frac {1}{3} x^{6} e^{3} a c +\frac {1}{2} x^{6} d^{2} e \,c^{2}+\frac {6}{5} x^{5} d \,e^{2} a c +\frac {1}{5} x^{5} c^{2} d^{3}+\frac {1}{4} x^{4} a^{2} e^{3}+\frac {3}{2} x^{4} d^{2} e a c +x^{3} a^{2} d \,e^{2}+\frac {2}{3} x^{3} d^{3} a c +\frac {3}{2} d^{2} e \,a^{2} x^{2}+a^{2} d^{3} x\) | \(132\) |
risch | \(\frac {1}{8} c^{2} e^{3} x^{8}+\frac {3}{7} d \,e^{2} c^{2} x^{7}+\frac {1}{3} x^{6} e^{3} a c +\frac {1}{2} x^{6} d^{2} e \,c^{2}+\frac {6}{5} x^{5} d \,e^{2} a c +\frac {1}{5} x^{5} c^{2} d^{3}+\frac {1}{4} x^{4} a^{2} e^{3}+\frac {3}{2} x^{4} d^{2} e a c +x^{3} a^{2} d \,e^{2}+\frac {2}{3} x^{3} d^{3} a c +\frac {3}{2} d^{2} e \,a^{2} x^{2}+a^{2} d^{3} x\) | \(132\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 127, normalized size = 1.09 \begin {gather*} \frac {1}{8} \, c^{2} x^{8} e^{3} + \frac {3}{7} \, c^{2} d x^{7} e^{2} + \frac {1}{6} \, {\left (3 \, c^{2} d^{2} e + 2 \, a c e^{3}\right )} x^{6} + \frac {3}{2} \, a^{2} d^{2} x^{2} e + a^{2} d^{3} x + \frac {1}{5} \, {\left (c^{2} d^{3} + 6 \, a c d e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (6 \, a c d^{2} e + a^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (2 \, a c d^{3} + 3 \, a^{2} d e^{2}\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.66, size = 125, normalized size = 1.07 \begin {gather*} \frac {1}{5} \, c^{2} d^{3} x^{5} + \frac {2}{3} \, a c d^{3} x^{3} + a^{2} d^{3} x + \frac {1}{24} \, {\left (3 \, c^{2} x^{8} + 8 \, a c x^{6} + 6 \, a^{2} x^{4}\right )} e^{3} + \frac {1}{35} \, {\left (15 \, c^{2} d x^{7} + 42 \, a c d x^{5} + 35 \, a^{2} d x^{3}\right )} e^{2} + \frac {1}{2} \, {\left (c^{2} d^{2} x^{6} + 3 \, a c d^{2} x^{4} + 3 \, a^{2} d^{2} x^{2}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 141, normalized size = 1.21 \begin {gather*} a^{2} d^{3} x + \frac {3 a^{2} d^{2} e x^{2}}{2} + \frac {3 c^{2} d e^{2} x^{7}}{7} + \frac {c^{2} e^{3} x^{8}}{8} + x^{6} \left (\frac {a c e^{3}}{3} + \frac {c^{2} d^{2} e}{2}\right ) + x^{5} \cdot \left (\frac {6 a c d e^{2}}{5} + \frac {c^{2} d^{3}}{5}\right ) + x^{4} \left (\frac {a^{2} e^{3}}{4} + \frac {3 a c d^{2} e}{2}\right ) + x^{3} \left (a^{2} d e^{2} + \frac {2 a c d^{3}}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.61, size = 128, normalized size = 1.09 \begin {gather*} \frac {1}{8} \, c^{2} x^{8} e^{3} + \frac {3}{7} \, c^{2} d x^{7} e^{2} + \frac {1}{2} \, c^{2} d^{2} x^{6} e + \frac {1}{5} \, c^{2} d^{3} x^{5} + \frac {1}{3} \, a c x^{6} e^{3} + \frac {6}{5} \, a c d x^{5} e^{2} + \frac {3}{2} \, a c d^{2} x^{4} e + \frac {2}{3} \, a c d^{3} x^{3} + \frac {1}{4} \, a^{2} x^{4} e^{3} + a^{2} d x^{3} e^{2} + \frac {3}{2} \, a^{2} d^{2} x^{2} e + a^{2} d^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 127, normalized size = 1.09 \begin {gather*} x^3\,\left (a^2\,d\,e^2+\frac {2\,c\,a\,d^3}{3}\right )+x^4\,\left (\frac {a^2\,e^3}{4}+\frac {3\,c\,a\,d^2\,e}{2}\right )+x^5\,\left (\frac {c^2\,d^3}{5}+\frac {6\,a\,c\,d\,e^2}{5}\right )+x^6\,\left (\frac {c^2\,d^2\,e}{2}+\frac {a\,c\,e^3}{3}\right )+a^2\,d^3\,x+\frac {c^2\,e^3\,x^8}{8}+\frac {3\,a^2\,d^2\,e\,x^2}{2}+\frac {3\,c^2\,d\,e^2\,x^7}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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