Optimal. Leaf size=89 \[ \frac {1}{2} a^5 d x^2+\frac {5}{6} a^4 c d x^6+a^3 c^2 d x^{10}+\frac {5}{7} a^2 c^3 d x^{14}+\frac {5}{18} a c^4 d x^{18}+\frac {e \left (a+c x^4\right )^6}{24 c}+\frac {1}{22} c^5 d x^{22} \]
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Rubi [A] time = 0.08, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1248, 641, 194} \[ \frac {5}{7} a^2 c^3 d x^{14}+a^3 c^2 d x^{10}+\frac {5}{6} a^4 c d x^6+\frac {1}{2} a^5 d x^2+\frac {5}{18} a c^4 d x^{18}+\frac {e \left (a+c x^4\right )^6}{24 c}+\frac {1}{22} c^5 d x^{22} \]
Antiderivative was successfully verified.
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Rule 194
Rule 641
Rule 1248
Rubi steps
\begin {align*} \int x \left (d+e x^2\right ) \left (a+c x^4\right )^5 \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int (d+e x) \left (a+c x^2\right )^5 \, dx,x,x^2\right )\\ &=\frac {e \left (a+c x^4\right )^6}{24 c}+\frac {1}{2} d \operatorname {Subst}\left (\int \left (a+c x^2\right )^5 \, dx,x,x^2\right )\\ &=\frac {e \left (a+c x^4\right )^6}{24 c}+\frac {1}{2} d \operatorname {Subst}\left (\int \left (a^5+5 a^4 c x^2+10 a^3 c^2 x^4+10 a^2 c^3 x^6+5 a c^4 x^8+c^5 x^{10}\right ) \, dx,x,x^2\right )\\ &=\frac {1}{2} a^5 d x^2+\frac {5}{6} a^4 c d x^6+a^3 c^2 d x^{10}+\frac {5}{7} a^2 c^3 d x^{14}+\frac {5}{18} a c^4 d x^{18}+\frac {1}{22} c^5 d x^{22}+\frac {e \left (a+c x^4\right )^6}{24 c}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 146, normalized size = 1.64 \[ \frac {1}{2} a^5 d x^2+\frac {1}{4} a^5 e x^4+\frac {5}{6} a^4 c d x^6+\frac {5}{8} a^4 c e x^8+a^3 c^2 d x^{10}+\frac {5}{6} a^3 c^2 e x^{12}+\frac {5}{7} a^2 c^3 d x^{14}+\frac {5}{8} a^2 c^3 e x^{16}+\frac {5}{18} a c^4 d x^{18}+\frac {1}{4} a c^4 e x^{20}+\frac {1}{22} c^5 d x^{22}+\frac {1}{24} c^5 e x^{24} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 124, normalized size = 1.39 \[ \frac {1}{24} x^{24} e c^{5} + \frac {1}{22} x^{22} d c^{5} + \frac {1}{4} x^{20} e c^{4} a + \frac {5}{18} x^{18} d c^{4} a + \frac {5}{8} x^{16} e c^{3} a^{2} + \frac {5}{7} x^{14} d c^{3} a^{2} + \frac {5}{6} x^{12} e c^{2} a^{3} + x^{10} d c^{2} a^{3} + \frac {5}{8} x^{8} e c a^{4} + \frac {5}{6} x^{6} d c a^{4} + \frac {1}{4} x^{4} e a^{5} + \frac {1}{2} x^{2} d a^{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 130, normalized size = 1.46 \[ \frac {1}{24} \, c^{5} x^{24} e + \frac {1}{22} \, c^{5} d x^{22} + \frac {1}{4} \, a c^{4} x^{20} e + \frac {5}{18} \, a c^{4} d x^{18} + \frac {5}{8} \, a^{2} c^{3} x^{16} e + \frac {5}{7} \, a^{2} c^{3} d x^{14} + \frac {5}{6} \, a^{3} c^{2} x^{12} e + a^{3} c^{2} d x^{10} + \frac {5}{8} \, a^{4} c x^{8} e + \frac {5}{6} \, a^{4} c d x^{6} + \frac {1}{4} \, a^{5} x^{4} e + \frac {1}{2} \, a^{5} d x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 125, normalized size = 1.40 \[ \frac {1}{24} c^{5} e \,x^{24}+\frac {1}{22} c^{5} d \,x^{22}+\frac {1}{4} a \,c^{4} e \,x^{20}+\frac {5}{18} a \,c^{4} d \,x^{18}+\frac {5}{8} a^{2} c^{3} e \,x^{16}+\frac {5}{7} a^{2} c^{3} d \,x^{14}+\frac {5}{6} a^{3} c^{2} e \,x^{12}+a^{3} c^{2} d \,x^{10}+\frac {5}{8} a^{4} c e \,x^{8}+\frac {5}{6} a^{4} c d \,x^{6}+\frac {1}{4} a^{5} e \,x^{4}+\frac {1}{2} a^{5} d \,x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 124, normalized size = 1.39 \[ \frac {1}{24} \, c^{5} e x^{24} + \frac {1}{22} \, c^{5} d x^{22} + \frac {1}{4} \, a c^{4} e x^{20} + \frac {5}{18} \, a c^{4} d x^{18} + \frac {5}{8} \, a^{2} c^{3} e x^{16} + \frac {5}{7} \, a^{2} c^{3} d x^{14} + \frac {5}{6} \, a^{3} c^{2} e x^{12} + a^{3} c^{2} d x^{10} + \frac {5}{8} \, a^{4} c e x^{8} + \frac {5}{6} \, a^{4} c d x^{6} + \frac {1}{4} \, a^{5} e x^{4} + \frac {1}{2} \, a^{5} d x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 124, normalized size = 1.39 \[ \frac {e\,a^5\,x^4}{4}+\frac {d\,a^5\,x^2}{2}+\frac {5\,e\,a^4\,c\,x^8}{8}+\frac {5\,d\,a^4\,c\,x^6}{6}+\frac {5\,e\,a^3\,c^2\,x^{12}}{6}+d\,a^3\,c^2\,x^{10}+\frac {5\,e\,a^2\,c^3\,x^{16}}{8}+\frac {5\,d\,a^2\,c^3\,x^{14}}{7}+\frac {e\,a\,c^4\,x^{20}}{4}+\frac {5\,d\,a\,c^4\,x^{18}}{18}+\frac {e\,c^5\,x^{24}}{24}+\frac {d\,c^5\,x^{22}}{22} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 150, normalized size = 1.69 \[ \frac {a^{5} d x^{2}}{2} + \frac {a^{5} e x^{4}}{4} + \frac {5 a^{4} c d x^{6}}{6} + \frac {5 a^{4} c e x^{8}}{8} + a^{3} c^{2} d x^{10} + \frac {5 a^{3} c^{2} e x^{12}}{6} + \frac {5 a^{2} c^{3} d x^{14}}{7} + \frac {5 a^{2} c^{3} e x^{16}}{8} + \frac {5 a c^{4} d x^{18}}{18} + \frac {a c^{4} e x^{20}}{4} + \frac {c^{5} d x^{22}}{22} + \frac {c^{5} e x^{24}}{24} \]
Verification of antiderivative is not currently implemented for this CAS.
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