Optimal. Leaf size=206 \[ \frac{2 c (d+e x)^7 \left (a B e^2-2 A c d e+5 B c d^2\right )}{7 e^6}+\frac{(d+e x)^5 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{5 e^6}-\frac{(d+e x)^4 \left (a e^2+c d^2\right )^2 (B d-A e)}{4 e^6}-\frac{c (d+e x)^6 \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{3 e^6}-\frac{c^2 (d+e x)^8 (5 B d-A e)}{8 e^6}+\frac{B c^2 (d+e x)^9}{9 e^6} \]
[Out]
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Rubi [A] time = 0.520853, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{2 c (d+e x)^7 \left (a B e^2-2 A c d e+5 B c d^2\right )}{7 e^6}+\frac{(d+e x)^5 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{5 e^6}-\frac{(d+e x)^4 \left (a e^2+c d^2\right )^2 (B d-A e)}{4 e^6}-\frac{c (d+e x)^6 \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{3 e^6}-\frac{c^2 (d+e x)^8 (5 B d-A e)}{8 e^6}+\frac{B c^2 (d+e x)^9}{9 e^6} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)*(d + e*x)^3*(a + c*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 75.3665, size = 204, normalized size = 0.99 \[ \frac{B c^{2} \left (d + e x\right )^{9}}{9 e^{6}} + \frac{c^{2} \left (d + e x\right )^{8} \left (A e - 5 B d\right )}{8 e^{6}} + \frac{2 c \left (d + e x\right )^{7} \left (- 2 A c d e + B a e^{2} + 5 B c d^{2}\right )}{7 e^{6}} + \frac{c \left (d + e x\right )^{6} \left (A a e^{3} + 3 A c d^{2} e - 3 B a d e^{2} - 5 B c d^{3}\right )}{3 e^{6}} + \frac{\left (d + e x\right )^{5} \left (a e^{2} + c d^{2}\right ) \left (- 4 A c d e + B a e^{2} + 5 B c d^{2}\right )}{5 e^{6}} + \frac{\left (d + e x\right )^{4} \left (A e - B d\right ) \left (a e^{2} + c d^{2}\right )^{2}}{4 e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**3*(c*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 0.118284, size = 244, normalized size = 1.18 \[ \frac{1}{5} x^5 \left (a^2 B e^3+6 a A c d e^2+6 a B c d^2 e+A c^2 d^3\right )+\frac{1}{2} a^2 d^2 x^2 (3 A e+B d)+a^2 A d^3 x+\frac{1}{7} c e x^7 \left (2 a B e^2+3 A c d e+3 B c d^2\right )+\frac{1}{3} a d x^3 \left (3 a A e^2+3 a B d e+2 A c d^2\right )+\frac{1}{6} c x^6 \left (2 a A e^3+6 a B d e^2+3 A c d^2 e+B c d^3\right )+\frac{1}{4} a x^4 \left (a A e^3+3 a B d e^2+6 A c d^2 e+2 B c d^3\right )+\frac{1}{8} c^2 e^2 x^8 (A e+3 B d)+\frac{1}{9} B c^2 e^3 x^9 \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)*(d + e*x)^3*(a + c*x^2)^2,x]
[Out]
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Maple [A] time = 0.001, size = 252, normalized size = 1.2 \[{\frac{B{c}^{2}{e}^{3}{x}^{9}}{9}}+{\frac{ \left ( A{e}^{3}+3\,Bd{e}^{2} \right ){c}^{2}{x}^{8}}{8}}+{\frac{ \left ( \left ( 3\,Ad{e}^{2}+3\,B{d}^{2}e \right ){c}^{2}+2\,B{e}^{3}ac \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 3\,A{d}^{2}e+B{d}^{3} \right ){c}^{2}+2\, \left ( A{e}^{3}+3\,Bd{e}^{2} \right ) ac \right ){x}^{6}}{6}}+{\frac{ \left ( A{d}^{3}{c}^{2}+2\, \left ( 3\,Ad{e}^{2}+3\,B{d}^{2}e \right ) ac+{a}^{2}B{e}^{3} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\, \left ( 3\,A{d}^{2}e+B{d}^{3} \right ) ac+ \left ( A{e}^{3}+3\,Bd{e}^{2} \right ){a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( 2\,A{d}^{3}ac+ \left ( 3\,Ad{e}^{2}+3\,B{d}^{2}e \right ){a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,A{d}^{2}e+B{d}^{3} \right ){a}^{2}{x}^{2}}{2}}+A{d}^{3}{a}^{2}x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^3*(c*x^2+a)^2,x)
[Out]
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Maxima [A] time = 0.709194, size = 351, normalized size = 1.7 \[ \frac{1}{9} \, B c^{2} e^{3} x^{9} + \frac{1}{8} \,{\left (3 \, B c^{2} d e^{2} + A c^{2} e^{3}\right )} x^{8} + \frac{1}{7} \,{\left (3 \, B c^{2} d^{2} e + 3 \, A c^{2} d e^{2} + 2 \, B a c e^{3}\right )} x^{7} + A a^{2} d^{3} x + \frac{1}{6} \,{\left (B c^{2} d^{3} + 3 \, A c^{2} d^{2} e + 6 \, B a c d e^{2} + 2 \, A a c e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (A c^{2} d^{3} + 6 \, B a c d^{2} e + 6 \, A a c d e^{2} + B a^{2} e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (2 \, B a c d^{3} + 6 \, A a c d^{2} e + 3 \, B a^{2} d e^{2} + A a^{2} e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (2 \, A a c d^{3} + 3 \, B a^{2} d^{2} e + 3 \, A a^{2} d e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B a^{2} d^{3} + 3 \, A a^{2} d^{2} e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(B*x + A)*(e*x + d)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.25121, size = 1, normalized size = 0. \[ \frac{1}{9} x^{9} e^{3} c^{2} B + \frac{3}{8} x^{8} e^{2} d c^{2} B + \frac{1}{8} x^{8} e^{3} c^{2} A + \frac{3}{7} x^{7} e d^{2} c^{2} B + \frac{2}{7} x^{7} e^{3} c a B + \frac{3}{7} x^{7} e^{2} d c^{2} A + \frac{1}{6} x^{6} d^{3} c^{2} B + x^{6} e^{2} d c a B + \frac{1}{2} x^{6} e d^{2} c^{2} A + \frac{1}{3} x^{6} e^{3} c a A + \frac{6}{5} x^{5} e d^{2} c a B + \frac{1}{5} x^{5} e^{3} a^{2} B + \frac{1}{5} x^{5} d^{3} c^{2} A + \frac{6}{5} x^{5} e^{2} d c a A + \frac{1}{2} x^{4} d^{3} c a B + \frac{3}{4} x^{4} e^{2} d a^{2} B + \frac{3}{2} x^{4} e d^{2} c a A + \frac{1}{4} x^{4} e^{3} a^{2} A + x^{3} e d^{2} a^{2} B + \frac{2}{3} x^{3} d^{3} c a A + x^{3} e^{2} d a^{2} A + \frac{1}{2} x^{2} d^{3} a^{2} B + \frac{3}{2} x^{2} e d^{2} a^{2} A + x d^{3} a^{2} A \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(B*x + A)*(e*x + d)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.250721, size = 303, normalized size = 1.47 \[ A a^{2} d^{3} x + \frac{B c^{2} e^{3} x^{9}}{9} + x^{8} \left (\frac{A c^{2} e^{3}}{8} + \frac{3 B c^{2} d e^{2}}{8}\right ) + x^{7} \left (\frac{3 A c^{2} d e^{2}}{7} + \frac{2 B a c e^{3}}{7} + \frac{3 B c^{2} d^{2} e}{7}\right ) + x^{6} \left (\frac{A a c e^{3}}{3} + \frac{A c^{2} d^{2} e}{2} + B a c d e^{2} + \frac{B c^{2} d^{3}}{6}\right ) + x^{5} \left (\frac{6 A a c d e^{2}}{5} + \frac{A c^{2} d^{3}}{5} + \frac{B a^{2} e^{3}}{5} + \frac{6 B a c d^{2} e}{5}\right ) + x^{4} \left (\frac{A a^{2} e^{3}}{4} + \frac{3 A a c d^{2} e}{2} + \frac{3 B a^{2} d e^{2}}{4} + \frac{B a c d^{3}}{2}\right ) + x^{3} \left (A a^{2} d e^{2} + \frac{2 A a c d^{3}}{3} + B a^{2} d^{2} e\right ) + x^{2} \left (\frac{3 A a^{2} d^{2} e}{2} + \frac{B a^{2} d^{3}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)**3*(c*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.285498, size = 379, normalized size = 1.84 \[ \frac{1}{9} \, B c^{2} x^{9} e^{3} + \frac{3}{8} \, B c^{2} d x^{8} e^{2} + \frac{3}{7} \, B c^{2} d^{2} x^{7} e + \frac{1}{6} \, B c^{2} d^{3} x^{6} + \frac{1}{8} \, A c^{2} x^{8} e^{3} + \frac{3}{7} \, A c^{2} d x^{7} e^{2} + \frac{1}{2} \, A c^{2} d^{2} x^{6} e + \frac{1}{5} \, A c^{2} d^{3} x^{5} + \frac{2}{7} \, B a c x^{7} e^{3} + B a c d x^{6} e^{2} + \frac{6}{5} \, B a c d^{2} x^{5} e + \frac{1}{2} \, B a c d^{3} x^{4} + \frac{1}{3} \, A a c x^{6} e^{3} + \frac{6}{5} \, A a c d x^{5} e^{2} + \frac{3}{2} \, A a c d^{2} x^{4} e + \frac{2}{3} \, A a c d^{3} x^{3} + \frac{1}{5} \, B a^{2} x^{5} e^{3} + \frac{3}{4} \, B a^{2} d x^{4} e^{2} + B a^{2} d^{2} x^{3} e + \frac{1}{2} \, B a^{2} d^{3} x^{2} + \frac{1}{4} \, A a^{2} x^{4} e^{3} + A a^{2} d x^{3} e^{2} + \frac{3}{2} \, A a^{2} d^{2} x^{2} e + A a^{2} d^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(B*x + A)*(e*x + d)^3,x, algorithm="giac")
[Out]