Optimal. Leaf size=206 \[ \frac{c (d+e x)^8 \left (a B e^2-2 A c d e+5 B c d^2\right )}{4 e^6}+\frac{(d+e x)^6 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{6 e^6}-\frac{(d+e x)^5 \left (a e^2+c d^2\right )^2 (B d-A e)}{5 e^6}-\frac{2 c (d+e x)^7 \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{7 e^6}-\frac{c^2 (d+e x)^9 (5 B d-A e)}{9 e^6}+\frac{B c^2 (d+e x)^{10}}{10 e^6} \]
[Out]
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Rubi [A] time = 0.661174, 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 \[ \frac{c (d+e x)^8 \left (a B e^2-2 A c d e+5 B c d^2\right )}{4 e^6}+\frac{(d+e x)^6 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{6 e^6}-\frac{(d+e x)^5 \left (a e^2+c d^2\right )^2 (B d-A e)}{5 e^6}-\frac{2 c (d+e x)^7 \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{7 e^6}-\frac{c^2 (d+e x)^9 (5 B d-A e)}{9 e^6}+\frac{B c^2 (d+e x)^{10}}{10 e^6} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)*(d + e*x)^4*(a + c*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 80.6075, size = 204, normalized size = 0.99 \[ \frac{B c^{2} \left (d + e x\right )^{10}}{10 e^{6}} + \frac{c^{2} \left (d + e x\right )^{9} \left (A e - 5 B d\right )}{9 e^{6}} + \frac{c \left (d + e x\right )^{8} \left (- 2 A c d e + B a e^{2} + 5 B c d^{2}\right )}{4 e^{6}} + \frac{2 c \left (d + e x\right )^{7} \left (A a e^{3} + 3 A c d^{2} e - 3 B a d e^{2} - 5 B c d^{3}\right )}{7 e^{6}} + \frac{\left (d + e x\right )^{6} \left (a e^{2} + c d^{2}\right ) \left (- 4 A c d e + B a e^{2} + 5 B c d^{2}\right )}{6 e^{6}} + \frac{\left (d + e x\right )^{5} \left (A e - B d\right ) \left (a e^{2} + c d^{2}\right )^{2}}{5 e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**4*(c*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 0.147991, size = 314, normalized size = 1.52 \[ \frac{1}{6} x^6 \left (a^2 B e^4+8 a A c d e^3+12 a B c d^2 e^2+4 A c^2 d^3 e+B c^2 d^4\right )+\frac{1}{5} x^5 \left (a^2 A e^4+4 a^2 B d e^3+12 a A c d^2 e^2+8 a B c d^3 e+A c^2 d^4\right )+\frac{1}{2} a^2 d^3 x^2 (4 A e+B d)+a^2 A d^4 x+\frac{1}{4} c e^2 x^8 \left (a B e^2+2 A c d e+3 B c d^2\right )+\frac{2}{3} a d^2 x^3 \left (3 a A e^2+2 a B d e+A c d^2\right )+\frac{2}{7} c e x^7 \left (a A e^3+4 a B d e^2+3 A c d^2 e+2 B c d^3\right )+\frac{1}{2} a d x^4 \left (2 a A e^3+3 a B d e^2+4 A c d^2 e+B c d^3\right )+\frac{1}{9} c^2 e^3 x^9 (A e+4 B d)+\frac{1}{10} B c^2 e^4 x^{10} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)*(d + e*x)^4*(a + c*x^2)^2,x]
[Out]
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Maple [A] time = 0.001, size = 327, normalized size = 1.6 \[{\frac{B{e}^{4}{c}^{2}{x}^{10}}{10}}+{\frac{ \left ( A{e}^{4}+4\,Bd{e}^{3} \right ){c}^{2}{x}^{9}}{9}}+{\frac{ \left ( \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ){c}^{2}+2\,B{e}^{4}ac \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ){c}^{2}+2\, \left ( A{e}^{4}+4\,Bd{e}^{3} \right ) ac \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 4\,A{d}^{3}e+B{d}^{4} \right ){c}^{2}+2\, \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ) ac+B{e}^{4}{a}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( A{d}^{4}{c}^{2}+2\, \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ) ac+ \left ( A{e}^{4}+4\,Bd{e}^{3} \right ){a}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\, \left ( 4\,A{d}^{3}e+B{d}^{4} \right ) ac+ \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ){a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( 2\,A{d}^{4}ac+ \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ){a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,A{d}^{3}e+B{d}^{4} \right ){a}^{2}{x}^{2}}{2}}+A{d}^{4}{a}^{2}x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^4*(c*x^2+a)^2,x)
[Out]
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Maxima [A] time = 0.711724, size = 448, normalized size = 2.17 \[ \frac{1}{10} \, B c^{2} e^{4} x^{10} + \frac{1}{9} \,{\left (4 \, B c^{2} d e^{3} + A c^{2} e^{4}\right )} x^{9} + \frac{1}{4} \,{\left (3 \, B c^{2} d^{2} e^{2} + 2 \, A c^{2} d e^{3} + B a c e^{4}\right )} x^{8} + A a^{2} d^{4} x + \frac{2}{7} \,{\left (2 \, B c^{2} d^{3} e + 3 \, A c^{2} d^{2} e^{2} + 4 \, B a c d e^{3} + A a c e^{4}\right )} x^{7} + \frac{1}{6} \,{\left (B c^{2} d^{4} + 4 \, A c^{2} d^{3} e + 12 \, B a c d^{2} e^{2} + 8 \, A a c d e^{3} + B a^{2} e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (A c^{2} d^{4} + 8 \, B a c d^{3} e + 12 \, A a c d^{2} e^{2} + 4 \, B a^{2} d e^{3} + A a^{2} e^{4}\right )} x^{5} + \frac{1}{2} \,{\left (B a c d^{4} + 4 \, A a c d^{3} e + 3 \, B a^{2} d^{2} e^{2} + 2 \, A a^{2} d e^{3}\right )} x^{4} + \frac{2}{3} \,{\left (A a c d^{4} + 2 \, B a^{2} d^{3} e + 3 \, A a^{2} d^{2} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B a^{2} d^{4} + 4 \, A a^{2} d^{3} e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(B*x + A)*(e*x + d)^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.270524, size = 1, normalized size = 0. \[ \frac{1}{10} x^{10} e^{4} c^{2} B + \frac{4}{9} x^{9} e^{3} d c^{2} B + \frac{1}{9} x^{9} e^{4} c^{2} A + \frac{3}{4} x^{8} e^{2} d^{2} c^{2} B + \frac{1}{4} x^{8} e^{4} c a B + \frac{1}{2} x^{8} e^{3} d c^{2} A + \frac{4}{7} x^{7} e d^{3} c^{2} B + \frac{8}{7} x^{7} e^{3} d c a B + \frac{6}{7} x^{7} e^{2} d^{2} c^{2} A + \frac{2}{7} x^{7} e^{4} c a A + \frac{1}{6} x^{6} d^{4} c^{2} B + 2 x^{6} e^{2} d^{2} c a B + \frac{1}{6} x^{6} e^{4} a^{2} B + \frac{2}{3} x^{6} e d^{3} c^{2} A + \frac{4}{3} x^{6} e^{3} d c a A + \frac{8}{5} x^{5} e d^{3} c a B + \frac{4}{5} x^{5} e^{3} d a^{2} B + \frac{1}{5} x^{5} d^{4} c^{2} A + \frac{12}{5} x^{5} e^{2} d^{2} c a A + \frac{1}{5} x^{5} e^{4} a^{2} A + \frac{1}{2} x^{4} d^{4} c a B + \frac{3}{2} x^{4} e^{2} d^{2} a^{2} B + 2 x^{4} e d^{3} c a A + x^{4} e^{3} d a^{2} A + \frac{4}{3} x^{3} e d^{3} a^{2} B + \frac{2}{3} x^{3} d^{4} c a A + 2 x^{3} e^{2} d^{2} a^{2} A + \frac{1}{2} x^{2} d^{4} a^{2} B + 2 x^{2} e d^{3} a^{2} A + x d^{4} a^{2} A \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(B*x + A)*(e*x + d)^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.289103, size = 398, normalized size = 1.93 \[ A a^{2} d^{4} x + \frac{B c^{2} e^{4} x^{10}}{10} + x^{9} \left (\frac{A c^{2} e^{4}}{9} + \frac{4 B c^{2} d e^{3}}{9}\right ) + x^{8} \left (\frac{A c^{2} d e^{3}}{2} + \frac{B a c e^{4}}{4} + \frac{3 B c^{2} d^{2} e^{2}}{4}\right ) + x^{7} \left (\frac{2 A a c e^{4}}{7} + \frac{6 A c^{2} d^{2} e^{2}}{7} + \frac{8 B a c d e^{3}}{7} + \frac{4 B c^{2} d^{3} e}{7}\right ) + x^{6} \left (\frac{4 A a c d e^{3}}{3} + \frac{2 A c^{2} d^{3} e}{3} + \frac{B a^{2} e^{4}}{6} + 2 B a c d^{2} e^{2} + \frac{B c^{2} d^{4}}{6}\right ) + x^{5} \left (\frac{A a^{2} e^{4}}{5} + \frac{12 A a c d^{2} e^{2}}{5} + \frac{A c^{2} d^{4}}{5} + \frac{4 B a^{2} d e^{3}}{5} + \frac{8 B a c d^{3} e}{5}\right ) + x^{4} \left (A a^{2} d e^{3} + 2 A a c d^{3} e + \frac{3 B a^{2} d^{2} e^{2}}{2} + \frac{B a c d^{4}}{2}\right ) + x^{3} \left (2 A a^{2} d^{2} e^{2} + \frac{2 A a c d^{4}}{3} + \frac{4 B a^{2} d^{3} e}{3}\right ) + x^{2} \left (2 A a^{2} d^{3} e + \frac{B a^{2} d^{4}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)**4*(c*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.280522, size = 493, normalized size = 2.39 \[ \frac{1}{10} \, B c^{2} x^{10} e^{4} + \frac{4}{9} \, B c^{2} d x^{9} e^{3} + \frac{3}{4} \, B c^{2} d^{2} x^{8} e^{2} + \frac{4}{7} \, B c^{2} d^{3} x^{7} e + \frac{1}{6} \, B c^{2} d^{4} x^{6} + \frac{1}{9} \, A c^{2} x^{9} e^{4} + \frac{1}{2} \, A c^{2} d x^{8} e^{3} + \frac{6}{7} \, A c^{2} d^{2} x^{7} e^{2} + \frac{2}{3} \, A c^{2} d^{3} x^{6} e + \frac{1}{5} \, A c^{2} d^{4} x^{5} + \frac{1}{4} \, B a c x^{8} e^{4} + \frac{8}{7} \, B a c d x^{7} e^{3} + 2 \, B a c d^{2} x^{6} e^{2} + \frac{8}{5} \, B a c d^{3} x^{5} e + \frac{1}{2} \, B a c d^{4} x^{4} + \frac{2}{7} \, A a c x^{7} e^{4} + \frac{4}{3} \, A a c d x^{6} e^{3} + \frac{12}{5} \, A a c d^{2} x^{5} e^{2} + 2 \, A a c d^{3} x^{4} e + \frac{2}{3} \, A a c d^{4} x^{3} + \frac{1}{6} \, B a^{2} x^{6} e^{4} + \frac{4}{5} \, B a^{2} d x^{5} e^{3} + \frac{3}{2} \, B a^{2} d^{2} x^{4} e^{2} + \frac{4}{3} \, B a^{2} d^{3} x^{3} e + \frac{1}{2} \, B a^{2} d^{4} x^{2} + \frac{1}{5} \, A a^{2} x^{5} e^{4} + A a^{2} d x^{4} e^{3} + 2 \, A a^{2} d^{2} x^{3} e^{2} + 2 \, A a^{2} d^{3} x^{2} e + A a^{2} d^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(B*x + A)*(e*x + d)^4,x, algorithm="giac")
[Out]