Optimal. Leaf size=128 \[ a^2 A d x+\frac {1}{3} a (2 A c d+a C d+a B e) x^3+\frac {1}{4} a^2 C e x^4+\frac {1}{5} c (A c d+2 a (C d+B e)) x^5+\frac {1}{3} a c C e x^6+\frac {1}{7} c^2 (C d+B e) x^7+\frac {1}{8} c^2 C e x^8+\frac {(B d+A e) \left (a+c x^2\right )^3}{6 c} \]
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Rubi [A]
time = 0.10, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1596, 1824}
\begin {gather*} a^2 A d x+\frac {1}{4} a^2 C e x^4+\frac {1}{5} c x^5 (2 a (B e+C d)+A c d)+\frac {1}{3} a x^3 (a B e+a C d+2 A c d)+\frac {\left (a+c x^2\right )^3 (A e+B d)}{6 c}+\frac {1}{3} a c C e x^6+\frac {1}{7} c^2 x^7 (B e+C d)+\frac {1}{8} c^2 C e x^8 \end {gather*}
Antiderivative was successfully verified.
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Rule 1596
Rule 1824
Rubi steps
\begin {align*} \int (d+e x) \left (a+c x^2\right )^2 \left (A+B x+C x^2\right ) \, dx &=\frac {(B d+A e) \left (a+c x^2\right )^3}{6 c}+\int \left (a+c x^2\right )^2 \left (-(B d+A e) x+(d+e x) \left (A+B x+C x^2\right )\right ) \, dx\\ &=\frac {(B d+A e) \left (a+c x^2\right )^3}{6 c}+\int \left (a^2 A d+a (2 A c d+a C d+a B e) x^2+a^2 C e x^3+c (A c d+2 a (C d+B e)) x^4+2 a c C e x^5+c^2 (C d+B e) x^6+c^2 C e x^7\right ) \, dx\\ &=a^2 A d x+\frac {1}{3} a (2 A c d+a C d+a B e) x^3+\frac {1}{4} a^2 C e x^4+\frac {1}{5} c (A c d+2 a (C d+B e)) x^5+\frac {1}{3} a c C e x^6+\frac {1}{7} c^2 (C d+B e) x^7+\frac {1}{8} c^2 C e x^8+\frac {(B d+A e) \left (a+c x^2\right )^3}{6 c}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 144, normalized size = 1.12 \begin {gather*} a^2 A d x+\frac {1}{2} a^2 (B d+A e) x^2+\frac {1}{3} a (2 A c d+a C d+a B e) x^3+\frac {1}{4} a (2 B c d+2 A c e+a C e) x^4+\frac {1}{5} c (A c d+2 a C d+2 a B e) x^5+\frac {1}{6} c (B c d+A c e+2 a C e) x^6+\frac {1}{7} c^2 (C d+B e) x^7+\frac {1}{8} c^2 C e x^8 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 151, normalized size = 1.18
method | result | size |
default | \(\frac {c^{2} C e \,x^{8}}{8}+\frac {\left (c^{2} e B +c^{2} d C \right ) x^{7}}{7}+\frac {\left (c^{2} e A +c^{2} d B +2 a c e C \right ) x^{6}}{6}+\frac {\left (A \,c^{2} d +2 B a c e +2 a c d C \right ) x^{5}}{5}+\frac {\left (2 a c e A +2 a c d B +e \,a^{2} C \right ) x^{4}}{4}+\frac {\left (2 a c d A +e \,a^{2} B +d \,a^{2} C \right ) x^{3}}{3}+\frac {\left (e \,a^{2} A +d \,a^{2} B \right ) x^{2}}{2}+a^{2} A d x\) | \(151\) |
norman | \(\frac {c^{2} C e \,x^{8}}{8}+\left (\frac {1}{7} c^{2} e B +\frac {1}{7} c^{2} d C \right ) x^{7}+\left (\frac {1}{6} c^{2} e A +\frac {1}{6} c^{2} d B +\frac {1}{3} a c e C \right ) x^{6}+\left (\frac {1}{5} A \,c^{2} d +\frac {2}{5} B a c e +\frac {2}{5} a c d C \right ) x^{5}+\left (\frac {1}{2} a c e A +\frac {1}{2} a c d B +\frac {1}{4} e \,a^{2} C \right ) x^{4}+\left (\frac {2}{3} a c d A +\frac {1}{3} e \,a^{2} B +\frac {1}{3} d \,a^{2} C \right ) x^{3}+\left (\frac {1}{2} e \,a^{2} A +\frac {1}{2} d \,a^{2} B \right ) x^{2}+a^{2} A d x\) | \(155\) |
gosper | \(\frac {1}{8} c^{2} C e \,x^{8}+\frac {1}{7} x^{7} c^{2} e B +\frac {1}{7} x^{7} c^{2} d C +\frac {1}{6} x^{6} c^{2} e A +\frac {1}{6} x^{6} c^{2} d B +\frac {1}{3} a c C e \,x^{6}+\frac {1}{5} x^{5} A \,c^{2} d +\frac {2}{5} x^{5} B a c e +\frac {2}{5} x^{5} a c d C +\frac {1}{2} x^{4} a c e A +\frac {1}{2} x^{4} a c d B +\frac {1}{4} a^{2} C e \,x^{4}+\frac {2}{3} x^{3} a c d A +\frac {1}{3} x^{3} e \,a^{2} B +\frac {1}{3} x^{3} d \,a^{2} C +\frac {1}{2} x^{2} e \,a^{2} A +\frac {1}{2} x^{2} d \,a^{2} B +a^{2} A d x\) | \(173\) |
risch | \(\frac {1}{8} c^{2} C e \,x^{8}+\frac {1}{7} x^{7} c^{2} e B +\frac {1}{7} x^{7} c^{2} d C +\frac {1}{6} x^{6} c^{2} e A +\frac {1}{6} x^{6} c^{2} d B +\frac {1}{3} a c C e \,x^{6}+\frac {1}{5} x^{5} A \,c^{2} d +\frac {2}{5} x^{5} B a c e +\frac {2}{5} x^{5} a c d C +\frac {1}{2} x^{4} a c e A +\frac {1}{2} x^{4} a c d B +\frac {1}{4} a^{2} C e \,x^{4}+\frac {2}{3} x^{3} a c d A +\frac {1}{3} x^{3} e \,a^{2} B +\frac {1}{3} x^{3} d \,a^{2} C +\frac {1}{2} x^{2} e \,a^{2} A +\frac {1}{2} x^{2} d \,a^{2} B +a^{2} A d x\) | \(173\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 161, normalized size = 1.26 \begin {gather*} \frac {1}{8} \, C c^{2} x^{8} e + \frac {1}{7} \, {\left (C c^{2} d + B c^{2} e\right )} x^{7} + \frac {1}{6} \, {\left (B c^{2} d + 2 \, C a c e + A c^{2} e\right )} x^{6} + \frac {1}{5} \, {\left (2 \, B a c e + {\left (2 \, C a c + A c^{2}\right )} d\right )} x^{5} + A a^{2} d x + \frac {1}{4} \, {\left (2 \, B a c d + C a^{2} e + 2 \, A a c e\right )} x^{4} + \frac {1}{3} \, {\left (B a^{2} e + {\left (C a^{2} + 2 \, A a c\right )} d\right )} x^{3} + \frac {1}{2} \, {\left (B a^{2} d + A a^{2} e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 162, normalized size = 1.27 \begin {gather*} \frac {1}{7} \, C c^{2} d x^{7} + \frac {1}{6} \, B c^{2} d x^{6} + \frac {1}{2} \, B a c d x^{4} + \frac {1}{5} \, {\left (2 \, C a c + A c^{2}\right )} d x^{5} + \frac {1}{2} \, B a^{2} d x^{2} + A a^{2} d x + \frac {1}{3} \, {\left (C a^{2} + 2 \, A a c\right )} d x^{3} + \frac {1}{840} \, {\left (105 \, C c^{2} x^{8} + 120 \, B c^{2} x^{7} + 336 \, B a c x^{5} + 140 \, {\left (2 \, C a c + A c^{2}\right )} x^{6} + 280 \, B a^{2} x^{3} + 420 \, A a^{2} x^{2} + 210 \, {\left (C a^{2} + 2 \, A a c\right )} x^{4}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 180, normalized size = 1.41 \begin {gather*} A a^{2} d x + \frac {C c^{2} e x^{8}}{8} + x^{7} \left (\frac {B c^{2} e}{7} + \frac {C c^{2} d}{7}\right ) + x^{6} \left (\frac {A c^{2} e}{6} + \frac {B c^{2} d}{6} + \frac {C a c e}{3}\right ) + x^{5} \left (\frac {A c^{2} d}{5} + \frac {2 B a c e}{5} + \frac {2 C a c d}{5}\right ) + x^{4} \left (\frac {A a c e}{2} + \frac {B a c d}{2} + \frac {C a^{2} e}{4}\right ) + x^{3} \cdot \left (\frac {2 A a c d}{3} + \frac {B a^{2} e}{3} + \frac {C a^{2} d}{3}\right ) + x^{2} \left (\frac {A a^{2} e}{2} + \frac {B a^{2} d}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.67, size = 181, normalized size = 1.41 \begin {gather*} \frac {1}{8} \, C c^{2} x^{8} e + \frac {1}{7} \, C c^{2} d x^{7} + \frac {1}{7} \, B c^{2} x^{7} e + \frac {1}{6} \, B c^{2} d x^{6} + \frac {1}{3} \, C a c x^{6} e + \frac {1}{6} \, A c^{2} x^{6} e + \frac {2}{5} \, C a c d x^{5} + \frac {1}{5} \, A c^{2} d x^{5} + \frac {2}{5} \, B a c x^{5} e + \frac {1}{2} \, B a c d x^{4} + \frac {1}{4} \, C a^{2} x^{4} e + \frac {1}{2} \, A a c x^{4} e + \frac {1}{3} \, C a^{2} d x^{3} + \frac {2}{3} \, A a c d x^{3} + \frac {1}{3} \, B a^{2} x^{3} e + \frac {1}{2} \, B a^{2} d x^{2} + \frac {1}{2} \, A a^{2} x^{2} e + A a^{2} d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.69, size = 140, normalized size = 1.09 \begin {gather*} x^3\,\left (\frac {B\,a^2\,e}{3}+\frac {C\,a^2\,d}{3}+\frac {2\,A\,a\,c\,d}{3}\right )+x^6\,\left (\frac {A\,c^2\,e}{6}+\frac {B\,c^2\,d}{6}+\frac {C\,a\,c\,e}{3}\right )+\frac {c\,x^5\,\left (A\,c\,d+2\,B\,a\,e+2\,C\,a\,d\right )}{5}+\frac {a\,x^4\,\left (2\,A\,c\,e+2\,B\,c\,d+C\,a\,e\right )}{4}+\frac {a^2\,x^2\,\left (A\,e+B\,d\right )}{2}+\frac {c^2\,x^7\,\left (B\,e+C\,d\right )}{7}+A\,a^2\,d\,x+\frac {C\,c^2\,e\,x^8}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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