Optimal. Leaf size=334 \[ -\frac{c (d+e x)^9 \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{9 e^8}+\frac{3 c^2 (d+e x)^{11} \left (a B e^2-2 A c d e+7 B c d^2\right )}{11 e^8}-\frac{c^2 (d+e x)^{10} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{10 e^8}+\frac{(d+e x)^7 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{7 e^8}-\frac{(d+e x)^6 \left (a e^2+c d^2\right )^3 (B d-A e)}{6 e^8}-\frac{3 c (d+e x)^8 \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{8 e^8}-\frac{c^3 (d+e x)^{12} (7 B d-A e)}{12 e^8}+\frac{B c^3 (d+e x)^{13}}{13 e^8} \]
[Out]
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Rubi [A] time = 1.37349, antiderivative size = 334, 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 \[ \frac{c (d+e x)^9 \left (3 a^2 B e^4-12 a A c d e^3+30 a B c d^2 e^2-20 A c^2 d^3 e+35 B c^2 d^4\right )}{9 e^8}+\frac{3 c^2 (d+e x)^{11} \left (a B e^2-2 A c d e+7 B c d^2\right )}{11 e^8}-\frac{c^2 (d+e x)^{10} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{10 e^8}+\frac{(d+e x)^7 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{7 e^8}-\frac{(d+e x)^6 \left (a e^2+c d^2\right )^3 (B d-A e)}{6 e^8}-\frac{3 c (d+e x)^8 \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{8 e^8}-\frac{c^3 (d+e x)^{12} (7 B d-A e)}{12 e^8}+\frac{B c^3 (d+e x)^{13}}{13 e^8} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)*(d + e*x)^5*(a + c*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 153.93, size = 343, normalized size = 1.03 \[ \frac{B c^{3} \left (d + e x\right )^{13}}{13 e^{8}} + \frac{c^{3} \left (d + e x\right )^{12} \left (A e - 7 B d\right )}{12 e^{8}} + \frac{3 c^{2} \left (d + e x\right )^{11} \left (- 2 A c d e + B a e^{2} + 7 B c d^{2}\right )}{11 e^{8}} + \frac{c^{2} \left (d + e x\right )^{10} \left (3 A a e^{3} + 15 A c d^{2} e - 15 B a d e^{2} - 35 B c d^{3}\right )}{10 e^{8}} + \frac{c \left (d + e x\right )^{9} \left (- 12 A a c d e^{3} - 20 A c^{2} d^{3} e + 3 B a^{2} e^{4} + 30 B a c d^{2} e^{2} + 35 B c^{2} d^{4}\right )}{9 e^{8}} + \frac{3 c \left (d + e x\right )^{8} \left (a e^{2} + c d^{2}\right ) \left (A a e^{3} + 5 A c d^{2} e - 3 B a d e^{2} - 7 B c d^{3}\right )}{8 e^{8}} + \frac{\left (d + e x\right )^{7} \left (a e^{2} + c d^{2}\right )^{2} \left (- 6 A c d e + B a e^{2} + 7 B c d^{2}\right )}{7 e^{8}} + \frac{\left (d + e x\right )^{6} \left (A e - B d\right ) \left (a e^{2} + c d^{2}\right )^{3}}{6 e^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**5*(c*x**2+a)**3,x)
[Out]
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Mathematica [A] time = 0.275174, size = 542, normalized size = 1.62 \[ \frac{1}{2} a^3 d^4 x^2 (5 A e+B d)+a^3 A d^5 x+\frac{1}{9} c e x^9 \left (B \left (3 a^2 e^4+30 a c d^2 e^2+5 c^2 d^4\right )+5 A c d e \left (3 a e^2+2 c d^2\right )\right )+\frac{1}{7} x^7 \left (A c d \left (15 a^2 e^4+30 a c d^2 e^2+c^2 d^4\right )+a B e \left (a^2 e^4+30 a c d^2 e^2+15 c^2 d^4\right )\right )+\frac{1}{5} a d x^5 \left (A \left (5 a^2 e^4+30 a c d^2 e^2+3 c^2 d^4\right )+5 a B d e \left (2 a e^2+3 c d^2\right )\right )+\frac{1}{8} c x^8 \left (A e \left (3 a^2 e^4+30 a c d^2 e^2+5 c^2 d^4\right )+B \left (15 a^2 d e^4+30 a c d^3 e^2+c^2 d^5\right )\right )+\frac{1}{6} a x^6 \left (A e \left (a^2 e^4+30 a c d^2 e^2+15 c^2 d^4\right )+B \left (5 a^2 d e^4+30 a c d^3 e^2+3 c^2 d^5\right )\right )+\frac{1}{3} a^2 d^3 x^3 \left (10 a A e^2+5 a B d e+3 A c d^2\right )+\frac{1}{4} a^2 d^2 x^4 \left (10 a A e^3+10 a B d e^2+15 A c d^2 e+3 B c d^3\right )+\frac{1}{11} c^2 e^3 x^{11} \left (3 a B e^2+5 A c d e+10 B c d^2\right )+\frac{1}{10} c^2 e^2 x^{10} \left (3 a A e^3+15 a B d e^2+10 A c d^2 e+10 B c d^3\right )+\frac{1}{12} c^3 e^4 x^{12} (A e+5 B d)+\frac{1}{13} B c^3 e^5 x^{13} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)*(d + e*x)^5*(a + c*x^2)^3,x]
[Out]
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Maple [A] time = 0.003, size = 557, normalized size = 1.7 \[{\frac{B{e}^{5}{c}^{3}{x}^{13}}{13}}+{\frac{ \left ( A{e}^{5}+5\,Bd{e}^{4} \right ){c}^{3}{x}^{12}}{12}}+{\frac{ \left ( \left ( 5\,Ad{e}^{4}+10\,B{d}^{2}{e}^{3} \right ){c}^{3}+3\,B{e}^{5}a{c}^{2} \right ){x}^{11}}{11}}+{\frac{ \left ( \left ( 10\,A{d}^{2}{e}^{3}+10\,B{d}^{3}{e}^{2} \right ){c}^{3}+3\, \left ( A{e}^{5}+5\,Bd{e}^{4} \right ) a{c}^{2} \right ){x}^{10}}{10}}+{\frac{ \left ( \left ( 10\,A{d}^{3}{e}^{2}+5\,B{d}^{4}e \right ){c}^{3}+3\, \left ( 5\,Ad{e}^{4}+10\,B{d}^{2}{e}^{3} \right ) a{c}^{2}+3\,B{e}^{5}{a}^{2}c \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 5\,A{d}^{4}e+B{d}^{5} \right ){c}^{3}+3\, \left ( 10\,A{d}^{2}{e}^{3}+10\,B{d}^{3}{e}^{2} \right ) a{c}^{2}+3\, \left ( A{e}^{5}+5\,Bd{e}^{4} \right ){a}^{2}c \right ){x}^{8}}{8}}+{\frac{ \left ( A{d}^{5}{c}^{3}+3\, \left ( 10\,A{d}^{3}{e}^{2}+5\,B{d}^{4}e \right ) a{c}^{2}+3\, \left ( 5\,Ad{e}^{4}+10\,B{d}^{2}{e}^{3} \right ){a}^{2}c+B{e}^{5}{a}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 3\, \left ( 5\,A{d}^{4}e+B{d}^{5} \right ) a{c}^{2}+3\, \left ( 10\,A{d}^{2}{e}^{3}+10\,B{d}^{3}{e}^{2} \right ){a}^{2}c+ \left ( A{e}^{5}+5\,Bd{e}^{4} \right ){a}^{3} \right ){x}^{6}}{6}}+{\frac{ \left ( 3\,A{d}^{5}a{c}^{2}+3\, \left ( 10\,A{d}^{3}{e}^{2}+5\,B{d}^{4}e \right ){a}^{2}c+ \left ( 5\,Ad{e}^{4}+10\,B{d}^{2}{e}^{3} \right ){a}^{3} \right ){x}^{5}}{5}}+{\frac{ \left ( 3\, \left ( 5\,A{d}^{4}e+B{d}^{5} \right ){a}^{2}c+ \left ( 10\,A{d}^{2}{e}^{3}+10\,B{d}^{3}{e}^{2} \right ){a}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,A{d}^{5}{a}^{2}c+ \left ( 10\,A{d}^{3}{e}^{2}+5\,B{d}^{4}e \right ){a}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ( 5\,A{d}^{4}e+B{d}^{5} \right ){a}^{3}{x}^{2}}{2}}+A{d}^{5}{a}^{3}x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^5*(c*x^2+a)^3,x)
[Out]
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Maxima [A] time = 0.726069, size = 788, normalized size = 2.36 \[ \frac{1}{13} \, B c^{3} e^{5} x^{13} + \frac{1}{12} \,{\left (5 \, B c^{3} d e^{4} + A c^{3} e^{5}\right )} x^{12} + \frac{1}{11} \,{\left (10 \, B c^{3} d^{2} e^{3} + 5 \, A c^{3} d e^{4} + 3 \, B a c^{2} e^{5}\right )} x^{11} + \frac{1}{10} \,{\left (10 \, B c^{3} d^{3} e^{2} + 10 \, A c^{3} d^{2} e^{3} + 15 \, B a c^{2} d e^{4} + 3 \, A a c^{2} e^{5}\right )} x^{10} + A a^{3} d^{5} x + \frac{1}{9} \,{\left (5 \, B c^{3} d^{4} e + 10 \, A c^{3} d^{3} e^{2} + 30 \, B a c^{2} d^{2} e^{3} + 15 \, A a c^{2} d e^{4} + 3 \, B a^{2} c e^{5}\right )} x^{9} + \frac{1}{8} \,{\left (B c^{3} d^{5} + 5 \, A c^{3} d^{4} e + 30 \, B a c^{2} d^{3} e^{2} + 30 \, A a c^{2} d^{2} e^{3} + 15 \, B a^{2} c d e^{4} + 3 \, A a^{2} c e^{5}\right )} x^{8} + \frac{1}{7} \,{\left (A c^{3} d^{5} + 15 \, B a c^{2} d^{4} e + 30 \, A a c^{2} d^{3} e^{2} + 30 \, B a^{2} c d^{2} e^{3} + 15 \, A a^{2} c d e^{4} + B a^{3} e^{5}\right )} x^{7} + \frac{1}{6} \,{\left (3 \, B a c^{2} d^{5} + 15 \, A a c^{2} d^{4} e + 30 \, B a^{2} c d^{3} e^{2} + 30 \, A a^{2} c d^{2} e^{3} + 5 \, B a^{3} d e^{4} + A a^{3} e^{5}\right )} x^{6} + \frac{1}{5} \,{\left (3 \, A a c^{2} d^{5} + 15 \, B a^{2} c d^{4} e + 30 \, A a^{2} c d^{3} e^{2} + 10 \, B a^{3} d^{2} e^{3} + 5 \, A a^{3} d e^{4}\right )} x^{5} + \frac{1}{4} \,{\left (3 \, B a^{2} c d^{5} + 15 \, A a^{2} c d^{4} e + 10 \, B a^{3} d^{3} e^{2} + 10 \, A a^{3} d^{2} e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (3 \, A a^{2} c d^{5} + 5 \, B a^{3} d^{4} e + 10 \, A a^{3} d^{3} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B a^{3} d^{5} + 5 \, A a^{3} d^{4} e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3*(B*x + A)*(e*x + d)^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.261602, size = 1, normalized size = 0. \[ \frac{1}{13} x^{13} e^{5} c^{3} B + \frac{5}{12} x^{12} e^{4} d c^{3} B + \frac{1}{12} x^{12} e^{5} c^{3} A + \frac{10}{11} x^{11} e^{3} d^{2} c^{3} B + \frac{3}{11} x^{11} e^{5} c^{2} a B + \frac{5}{11} x^{11} e^{4} d c^{3} A + x^{10} e^{2} d^{3} c^{3} B + \frac{3}{2} x^{10} e^{4} d c^{2} a B + x^{10} e^{3} d^{2} c^{3} A + \frac{3}{10} x^{10} e^{5} c^{2} a A + \frac{5}{9} x^{9} e d^{4} c^{3} B + \frac{10}{3} x^{9} e^{3} d^{2} c^{2} a B + \frac{1}{3} x^{9} e^{5} c a^{2} B + \frac{10}{9} x^{9} e^{2} d^{3} c^{3} A + \frac{5}{3} x^{9} e^{4} d c^{2} a A + \frac{1}{8} x^{8} d^{5} c^{3} B + \frac{15}{4} x^{8} e^{2} d^{3} c^{2} a B + \frac{15}{8} x^{8} e^{4} d c a^{2} B + \frac{5}{8} x^{8} e d^{4} c^{3} A + \frac{15}{4} x^{8} e^{3} d^{2} c^{2} a A + \frac{3}{8} x^{8} e^{5} c a^{2} A + \frac{15}{7} x^{7} e d^{4} c^{2} a B + \frac{30}{7} x^{7} e^{3} d^{2} c a^{2} B + \frac{1}{7} x^{7} e^{5} a^{3} B + \frac{1}{7} x^{7} d^{5} c^{3} A + \frac{30}{7} x^{7} e^{2} d^{3} c^{2} a A + \frac{15}{7} x^{7} e^{4} d c a^{2} A + \frac{1}{2} x^{6} d^{5} c^{2} a B + 5 x^{6} e^{2} d^{3} c a^{2} B + \frac{5}{6} x^{6} e^{4} d a^{3} B + \frac{5}{2} x^{6} e d^{4} c^{2} a A + 5 x^{6} e^{3} d^{2} c a^{2} A + \frac{1}{6} x^{6} e^{5} a^{3} A + 3 x^{5} e d^{4} c a^{2} B + 2 x^{5} e^{3} d^{2} a^{3} B + \frac{3}{5} x^{5} d^{5} c^{2} a A + 6 x^{5} e^{2} d^{3} c a^{2} A + x^{5} e^{4} d a^{3} A + \frac{3}{4} x^{4} d^{5} c a^{2} B + \frac{5}{2} x^{4} e^{2} d^{3} a^{3} B + \frac{15}{4} x^{4} e d^{4} c a^{2} A + \frac{5}{2} x^{4} e^{3} d^{2} a^{3} A + \frac{5}{3} x^{3} e d^{4} a^{3} B + x^{3} d^{5} c a^{2} A + \frac{10}{3} x^{3} e^{2} d^{3} a^{3} A + \frac{1}{2} x^{2} d^{5} a^{3} B + \frac{5}{2} x^{2} e d^{4} a^{3} A + x d^{5} a^{3} A \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3*(B*x + A)*(e*x + d)^5,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.382277, size = 694, normalized size = 2.08 \[ A a^{3} d^{5} x + \frac{B c^{3} e^{5} x^{13}}{13} + x^{12} \left (\frac{A c^{3} e^{5}}{12} + \frac{5 B c^{3} d e^{4}}{12}\right ) + x^{11} \left (\frac{5 A c^{3} d e^{4}}{11} + \frac{3 B a c^{2} e^{5}}{11} + \frac{10 B c^{3} d^{2} e^{3}}{11}\right ) + x^{10} \left (\frac{3 A a c^{2} e^{5}}{10} + A c^{3} d^{2} e^{3} + \frac{3 B a c^{2} d e^{4}}{2} + B c^{3} d^{3} e^{2}\right ) + x^{9} \left (\frac{5 A a c^{2} d e^{4}}{3} + \frac{10 A c^{3} d^{3} e^{2}}{9} + \frac{B a^{2} c e^{5}}{3} + \frac{10 B a c^{2} d^{2} e^{3}}{3} + \frac{5 B c^{3} d^{4} e}{9}\right ) + x^{8} \left (\frac{3 A a^{2} c e^{5}}{8} + \frac{15 A a c^{2} d^{2} e^{3}}{4} + \frac{5 A c^{3} d^{4} e}{8} + \frac{15 B a^{2} c d e^{4}}{8} + \frac{15 B a c^{2} d^{3} e^{2}}{4} + \frac{B c^{3} d^{5}}{8}\right ) + x^{7} \left (\frac{15 A a^{2} c d e^{4}}{7} + \frac{30 A a c^{2} d^{3} e^{2}}{7} + \frac{A c^{3} d^{5}}{7} + \frac{B a^{3} e^{5}}{7} + \frac{30 B a^{2} c d^{2} e^{3}}{7} + \frac{15 B a c^{2} d^{4} e}{7}\right ) + x^{6} \left (\frac{A a^{3} e^{5}}{6} + 5 A a^{2} c d^{2} e^{3} + \frac{5 A a c^{2} d^{4} e}{2} + \frac{5 B a^{3} d e^{4}}{6} + 5 B a^{2} c d^{3} e^{2} + \frac{B a c^{2} d^{5}}{2}\right ) + x^{5} \left (A a^{3} d e^{4} + 6 A a^{2} c d^{3} e^{2} + \frac{3 A a c^{2} d^{5}}{5} + 2 B a^{3} d^{2} e^{3} + 3 B a^{2} c d^{4} e\right ) + x^{4} \left (\frac{5 A a^{3} d^{2} e^{3}}{2} + \frac{15 A a^{2} c d^{4} e}{4} + \frac{5 B a^{3} d^{3} e^{2}}{2} + \frac{3 B a^{2} c d^{5}}{4}\right ) + x^{3} \left (\frac{10 A a^{3} d^{3} e^{2}}{3} + A a^{2} c d^{5} + \frac{5 B a^{3} d^{4} e}{3}\right ) + x^{2} \left (\frac{5 A a^{3} d^{4} e}{2} + \frac{B a^{3} d^{5}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)**5*(c*x**2+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.282803, size = 856, normalized size = 2.56 \[ \frac{1}{13} \, B c^{3} x^{13} e^{5} + \frac{5}{12} \, B c^{3} d x^{12} e^{4} + \frac{10}{11} \, B c^{3} d^{2} x^{11} e^{3} + B c^{3} d^{3} x^{10} e^{2} + \frac{5}{9} \, B c^{3} d^{4} x^{9} e + \frac{1}{8} \, B c^{3} d^{5} x^{8} + \frac{1}{12} \, A c^{3} x^{12} e^{5} + \frac{5}{11} \, A c^{3} d x^{11} e^{4} + A c^{3} d^{2} x^{10} e^{3} + \frac{10}{9} \, A c^{3} d^{3} x^{9} e^{2} + \frac{5}{8} \, A c^{3} d^{4} x^{8} e + \frac{1}{7} \, A c^{3} d^{5} x^{7} + \frac{3}{11} \, B a c^{2} x^{11} e^{5} + \frac{3}{2} \, B a c^{2} d x^{10} e^{4} + \frac{10}{3} \, B a c^{2} d^{2} x^{9} e^{3} + \frac{15}{4} \, B a c^{2} d^{3} x^{8} e^{2} + \frac{15}{7} \, B a c^{2} d^{4} x^{7} e + \frac{1}{2} \, B a c^{2} d^{5} x^{6} + \frac{3}{10} \, A a c^{2} x^{10} e^{5} + \frac{5}{3} \, A a c^{2} d x^{9} e^{4} + \frac{15}{4} \, A a c^{2} d^{2} x^{8} e^{3} + \frac{30}{7} \, A a c^{2} d^{3} x^{7} e^{2} + \frac{5}{2} \, A a c^{2} d^{4} x^{6} e + \frac{3}{5} \, A a c^{2} d^{5} x^{5} + \frac{1}{3} \, B a^{2} c x^{9} e^{5} + \frac{15}{8} \, B a^{2} c d x^{8} e^{4} + \frac{30}{7} \, B a^{2} c d^{2} x^{7} e^{3} + 5 \, B a^{2} c d^{3} x^{6} e^{2} + 3 \, B a^{2} c d^{4} x^{5} e + \frac{3}{4} \, B a^{2} c d^{5} x^{4} + \frac{3}{8} \, A a^{2} c x^{8} e^{5} + \frac{15}{7} \, A a^{2} c d x^{7} e^{4} + 5 \, A a^{2} c d^{2} x^{6} e^{3} + 6 \, A a^{2} c d^{3} x^{5} e^{2} + \frac{15}{4} \, A a^{2} c d^{4} x^{4} e + A a^{2} c d^{5} x^{3} + \frac{1}{7} \, B a^{3} x^{7} e^{5} + \frac{5}{6} \, B a^{3} d x^{6} e^{4} + 2 \, B a^{3} d^{2} x^{5} e^{3} + \frac{5}{2} \, B a^{3} d^{3} x^{4} e^{2} + \frac{5}{3} \, B a^{3} d^{4} x^{3} e + \frac{1}{2} \, B a^{3} d^{5} x^{2} + \frac{1}{6} \, A a^{3} x^{6} e^{5} + A a^{3} d x^{5} e^{4} + \frac{5}{2} \, A a^{3} d^{2} x^{4} e^{3} + \frac{10}{3} \, A a^{3} d^{3} x^{3} e^{2} + \frac{5}{2} \, A a^{3} d^{4} x^{2} e + A a^{3} d^{5} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3*(B*x + A)*(e*x + d)^5,x, algorithm="giac")
[Out]