Optimal. Leaf size=117 \[ \frac {\left (c d^2+a e^2\right )^2 (d+e x)^5}{5 e^5}-\frac {2 c d \left (c d^2+a e^2\right ) (d+e x)^6}{3 e^5}+\frac {2 c \left (3 c d^2+a e^2\right ) (d+e x)^7}{7 e^5}-\frac {c^2 d (d+e x)^8}{2 e^5}+\frac {c^2 (d+e x)^9}{9 e^5} \]
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Rubi [A]
time = 0.09, 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 {2 c (d+e x)^7 \left (a e^2+3 c d^2\right )}{7 e^5}-\frac {2 c d (d+e x)^6 \left (a e^2+c d^2\right )}{3 e^5}+\frac {(d+e x)^5 \left (a e^2+c d^2\right )^2}{5 e^5}+\frac {c^2 (d+e x)^9}{9 e^5}-\frac {c^2 d (d+e x)^8}{2 e^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 711
Rubi steps
\begin {align*} \int (d+e x)^4 \left (a+c x^2\right )^2 \, dx &=\int \left (\frac {\left (c d^2+a e^2\right )^2 (d+e x)^4}{e^4}-\frac {4 c d \left (c d^2+a e^2\right ) (d+e x)^5}{e^4}+\frac {2 c \left (3 c d^2+a e^2\right ) (d+e x)^6}{e^4}-\frac {4 c^2 d (d+e x)^7}{e^4}+\frac {c^2 (d+e x)^8}{e^4}\right ) \, dx\\ &=\frac {\left (c d^2+a e^2\right )^2 (d+e x)^5}{5 e^5}-\frac {2 c d \left (c d^2+a e^2\right ) (d+e x)^6}{3 e^5}+\frac {2 c \left (3 c d^2+a e^2\right ) (d+e x)^7}{7 e^5}-\frac {c^2 d (d+e x)^8}{2 e^5}+\frac {c^2 (d+e x)^9}{9 e^5}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 167, normalized size = 1.43 \begin {gather*} a^2 d^4 x+2 a^2 d^3 e x^2+\frac {2}{3} a d^2 \left (c d^2+3 a e^2\right ) x^3+a d e \left (2 c d^2+a e^2\right ) x^4+\frac {1}{5} \left (c^2 d^4+12 a c d^2 e^2+a^2 e^4\right ) x^5+\frac {2}{3} c d e \left (c d^2+2 a e^2\right ) x^6+\frac {2}{7} c e^2 \left (3 c d^2+a e^2\right ) x^7+\frac {1}{2} c^2 d e^3 x^8+\frac {1}{9} c^2 e^4 x^9 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 169, normalized size = 1.44
method | result | size |
norman | \(\frac {e^{4} c^{2} x^{9}}{9}+\frac {d \,e^{3} c^{2} x^{8}}{2}+\left (\frac {2}{7} a c \,e^{4}+\frac {6}{7} d^{2} e^{2} c^{2}\right ) x^{7}+\left (\frac {4}{3} d \,e^{3} a c +\frac {2}{3} d^{3} e \,c^{2}\right ) x^{6}+\left (\frac {1}{5} a^{2} e^{4}+\frac {12}{5} a c \,d^{2} e^{2}+\frac {1}{5} c^{2} d^{4}\right ) x^{5}+\left (d \,e^{3} a^{2}+2 d^{3} e a c \right ) x^{4}+\left (2 d^{2} e^{2} a^{2}+\frac {2}{3} a c \,d^{4}\right ) x^{3}+2 d^{3} e \,a^{2} x^{2}+a^{2} d^{4} x\) | \(165\) |
default | \(\frac {e^{4} c^{2} x^{9}}{9}+\frac {d \,e^{3} c^{2} x^{8}}{2}+\frac {\left (2 a c \,e^{4}+6 d^{2} e^{2} c^{2}\right ) x^{7}}{7}+\frac {\left (8 d \,e^{3} a c +4 d^{3} e \,c^{2}\right ) x^{6}}{6}+\frac {\left (a^{2} e^{4}+12 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) x^{5}}{5}+\frac {\left (4 d \,e^{3} a^{2}+8 d^{3} e a c \right ) x^{4}}{4}+\frac {\left (6 d^{2} e^{2} a^{2}+2 a c \,d^{4}\right ) x^{3}}{3}+2 d^{3} e \,a^{2} x^{2}+a^{2} d^{4} x\) | \(169\) |
gosper | \(\frac {1}{9} e^{4} c^{2} x^{9}+\frac {1}{2} d \,e^{3} c^{2} x^{8}+\frac {2}{7} x^{7} a c \,e^{4}+\frac {6}{7} x^{7} d^{2} e^{2} c^{2}+\frac {4}{3} x^{6} d \,e^{3} a c +\frac {2}{3} x^{6} d^{3} e \,c^{2}+\frac {1}{5} x^{5} a^{2} e^{4}+\frac {12}{5} x^{5} a c \,d^{2} e^{2}+\frac {1}{5} x^{5} c^{2} d^{4}+a^{2} d \,e^{3} x^{4}+2 a c \,d^{3} e \,x^{4}+2 x^{3} d^{2} e^{2} a^{2}+\frac {2}{3} x^{3} a c \,d^{4}+2 d^{3} e \,a^{2} x^{2}+a^{2} d^{4} x\) | \(173\) |
risch | \(\frac {1}{9} e^{4} c^{2} x^{9}+\frac {1}{2} d \,e^{3} c^{2} x^{8}+\frac {2}{7} x^{7} a c \,e^{4}+\frac {6}{7} x^{7} d^{2} e^{2} c^{2}+\frac {4}{3} x^{6} d \,e^{3} a c +\frac {2}{3} x^{6} d^{3} e \,c^{2}+\frac {1}{5} x^{5} a^{2} e^{4}+\frac {12}{5} x^{5} a c \,d^{2} e^{2}+\frac {1}{5} x^{5} c^{2} d^{4}+a^{2} d \,e^{3} x^{4}+2 a c \,d^{3} e \,x^{4}+2 x^{3} d^{2} e^{2} a^{2}+\frac {2}{3} x^{3} a c \,d^{4}+2 d^{3} e \,a^{2} x^{2}+a^{2} d^{4} x\) | \(173\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 157, normalized size = 1.34 \begin {gather*} \frac {1}{9} \, c^{2} x^{9} e^{4} + \frac {1}{2} \, c^{2} d x^{8} e^{3} + \frac {2}{7} \, {\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{7} + 2 \, a^{2} d^{3} x^{2} e + a^{2} d^{4} x + \frac {2}{3} \, {\left (c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (c^{2} d^{4} + 12 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{5} + {\left (2 \, a c d^{3} e + a^{2} d e^{3}\right )} x^{4} + \frac {2}{3} \, {\left (a c d^{4} + 3 \, a^{2} d^{2} e^{2}\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.51, size = 162, normalized size = 1.38 \begin {gather*} \frac {1}{5} \, c^{2} d^{4} x^{5} + \frac {2}{3} \, a c d^{4} x^{3} + a^{2} d^{4} x + \frac {1}{315} \, {\left (35 \, c^{2} x^{9} + 90 \, a c x^{7} + 63 \, a^{2} x^{5}\right )} e^{4} + \frac {1}{6} \, {\left (3 \, c^{2} d x^{8} + 8 \, a c d x^{6} + 6 \, a^{2} d x^{4}\right )} e^{3} + \frac {2}{35} \, {\left (15 \, c^{2} d^{2} x^{7} + 42 \, a c d^{2} x^{5} + 35 \, a^{2} d^{2} x^{3}\right )} e^{2} + \frac {2}{3} \, {\left (c^{2} d^{3} x^{6} + 3 \, a c d^{3} x^{4} + 3 \, a^{2} d^{3} 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 = 182, normalized size = 1.56 \begin {gather*} a^{2} d^{4} x + 2 a^{2} d^{3} e x^{2} + \frac {c^{2} d e^{3} x^{8}}{2} + \frac {c^{2} e^{4} x^{9}}{9} + x^{7} \cdot \left (\frac {2 a c e^{4}}{7} + \frac {6 c^{2} d^{2} e^{2}}{7}\right ) + x^{6} \cdot \left (\frac {4 a c d e^{3}}{3} + \frac {2 c^{2} d^{3} e}{3}\right ) + x^{5} \left (\frac {a^{2} e^{4}}{5} + \frac {12 a c d^{2} e^{2}}{5} + \frac {c^{2} d^{4}}{5}\right ) + x^{4} \left (a^{2} d e^{3} + 2 a c d^{3} e\right ) + x^{3} \cdot \left (2 a^{2} d^{2} e^{2} + \frac {2 a c d^{4}}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.20, size = 166, normalized size = 1.42 \begin {gather*} \frac {1}{9} \, c^{2} x^{9} e^{4} + \frac {1}{2} \, c^{2} d x^{8} e^{3} + \frac {6}{7} \, c^{2} d^{2} x^{7} e^{2} + \frac {2}{3} \, c^{2} d^{3} x^{6} e + \frac {1}{5} \, c^{2} d^{4} x^{5} + \frac {2}{7} \, a c x^{7} e^{4} + \frac {4}{3} \, a c d x^{6} e^{3} + \frac {12}{5} \, a c d^{2} x^{5} e^{2} + 2 \, a c d^{3} x^{4} e + \frac {2}{3} \, a c d^{4} x^{3} + \frac {1}{5} \, a^{2} x^{5} e^{4} + a^{2} d x^{4} e^{3} + 2 \, a^{2} d^{2} x^{3} e^{2} + 2 \, a^{2} d^{3} x^{2} e + a^{2} d^{4} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 160, normalized size = 1.37 \begin {gather*} x^5\,\left (\frac {a^2\,e^4}{5}+\frac {12\,a\,c\,d^2\,e^2}{5}+\frac {c^2\,d^4}{5}\right )+x^3\,\left (2\,a^2\,d^2\,e^2+\frac {2\,c\,a\,d^4}{3}\right )+x^7\,\left (\frac {6\,c^2\,d^2\,e^2}{7}+\frac {2\,a\,c\,e^4}{7}\right )+a^2\,d^4\,x+\frac {c^2\,e^4\,x^9}{9}+2\,a^2\,d^3\,e\,x^2+\frac {c^2\,d\,e^3\,x^8}{2}+a\,d\,e\,x^4\,\left (2\,c\,d^2+a\,e^2\right )+\frac {2\,c\,d\,e\,x^6\,\left (c\,d^2+2\,a\,e^2\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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