Optimal. Leaf size=108 \[ \frac{(d+e x)^5 \left (a B e^2-2 A c d e+3 B c d^2\right )}{5 e^4}-\frac{(d+e x)^4 \left (a e^2+c d^2\right ) (B d-A e)}{4 e^4}-\frac{c (d+e x)^6 (3 B d-A e)}{6 e^4}+\frac{B c (d+e x)^7}{7 e^4} \]
[Out]
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Rubi [A] time = 0.249301, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{(d+e x)^5 \left (a B e^2-2 A c d e+3 B c d^2\right )}{5 e^4}-\frac{(d+e x)^4 \left (a e^2+c d^2\right ) (B d-A e)}{4 e^4}-\frac{c (d+e x)^6 (3 B d-A e)}{6 e^4}+\frac{B c (d+e x)^7}{7 e^4} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)*(d + e*x)^3*(a + c*x^2),x]
[Out]
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Rubi in Sympy [A] time = 33.0648, size = 100, normalized size = 0.93 \[ \frac{B c \left (d + e x\right )^{7}}{7 e^{4}} + \frac{c \left (d + e x\right )^{6} \left (A e - 3 B d\right )}{6 e^{4}} + \frac{\left (d + e x\right )^{5} \left (- 2 A c d e + B a e^{2} + 3 B c d^{2}\right )}{5 e^{4}} + \frac{\left (d + e x\right )^{4} \left (A e - B d\right ) \left (a e^{2} + c d^{2}\right )}{4 e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**3*(c*x**2+a),x)
[Out]
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Mathematica [A] time = 0.0652644, size = 150, normalized size = 1.39 \[ \frac{1}{5} e x^5 \left (a B e^2+3 A c d e+3 B c d^2\right )+\frac{1}{3} d x^3 \left (3 a A e^2+3 a B d e+A c d^2\right )+\frac{1}{4} x^4 \left (a A e^3+3 a B d e^2+3 A c d^2 e+B c d^3\right )+\frac{1}{2} a d^2 x^2 (3 A e+B d)+a A d^3 x+\frac{1}{6} c e^2 x^6 (A e+3 B d)+\frac{1}{7} B c e^3 x^7 \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)*(d + e*x)^3*(a + c*x^2),x]
[Out]
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Maple [A] time = 0.001, size = 151, normalized size = 1.4 \[{\frac{B{e}^{3}c{x}^{7}}{7}}+{\frac{ \left ( A{e}^{3}+3\,Bd{e}^{2} \right ) c{x}^{6}}{6}}+{\frac{ \left ( \left ( 3\,Ad{e}^{2}+3\,B{d}^{2}e \right ) c+B{e}^{3}a \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 3\,A{d}^{2}e+B{d}^{3} \right ) c+ \left ( A{e}^{3}+3\,Bd{e}^{2} \right ) a \right ){x}^{4}}{4}}+{\frac{ \left ( A{d}^{3}c+ \left ( 3\,Ad{e}^{2}+3\,B{d}^{2}e \right ) a \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,A{d}^{2}e+B{d}^{3} \right ) a{x}^{2}}{2}}+A{d}^{3}ax \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^3*(c*x^2+a),x)
[Out]
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Maxima [A] time = 0.690269, size = 200, normalized size = 1.85 \[ \frac{1}{7} \, B c e^{3} x^{7} + \frac{1}{6} \,{\left (3 \, B c d e^{2} + A c e^{3}\right )} x^{6} + A a d^{3} x + \frac{1}{5} \,{\left (3 \, B c d^{2} e + 3 \, A c d e^{2} + B a e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (B c d^{3} + 3 \, A c d^{2} e + 3 \, B a d e^{2} + A a e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (A c d^{3} + 3 \, B a d^{2} e + 3 \, A a d e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B a d^{3} + 3 \, A a d^{2} e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)*(B*x + A)*(e*x + d)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.277646, size = 1, normalized size = 0.01 \[ \frac{1}{7} x^{7} e^{3} c B + \frac{1}{2} x^{6} e^{2} d c B + \frac{1}{6} x^{6} e^{3} c A + \frac{3}{5} x^{5} e d^{2} c B + \frac{1}{5} x^{5} e^{3} a B + \frac{3}{5} x^{5} e^{2} d c A + \frac{1}{4} x^{4} d^{3} c B + \frac{3}{4} x^{4} e^{2} d a B + \frac{3}{4} x^{4} e d^{2} c A + \frac{1}{4} x^{4} e^{3} a A + x^{3} e d^{2} a B + \frac{1}{3} x^{3} d^{3} c A + x^{3} e^{2} d a A + \frac{1}{2} x^{2} d^{3} a B + \frac{3}{2} x^{2} e d^{2} a A + x d^{3} a A \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)*(B*x + A)*(e*x + d)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.17488, size = 173, normalized size = 1.6 \[ A a d^{3} x + \frac{B c e^{3} x^{7}}{7} + x^{6} \left (\frac{A c e^{3}}{6} + \frac{B c d e^{2}}{2}\right ) + x^{5} \left (\frac{3 A c d e^{2}}{5} + \frac{B a e^{3}}{5} + \frac{3 B c d^{2} e}{5}\right ) + x^{4} \left (\frac{A a e^{3}}{4} + \frac{3 A c d^{2} e}{4} + \frac{3 B a d e^{2}}{4} + \frac{B c d^{3}}{4}\right ) + x^{3} \left (A a d e^{2} + \frac{A c d^{3}}{3} + B a d^{2} e\right ) + x^{2} \left (\frac{3 A a d^{2} e}{2} + \frac{B a d^{3}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)**3*(c*x**2+a),x)
[Out]
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GIAC/XCAS [A] time = 0.280496, size = 216, normalized size = 2. \[ \frac{1}{7} \, B c x^{7} e^{3} + \frac{1}{2} \, B c d x^{6} e^{2} + \frac{3}{5} \, B c d^{2} x^{5} e + \frac{1}{4} \, B c d^{3} x^{4} + \frac{1}{6} \, A c x^{6} e^{3} + \frac{3}{5} \, A c d x^{5} e^{2} + \frac{3}{4} \, A c d^{2} x^{4} e + \frac{1}{3} \, A c d^{3} x^{3} + \frac{1}{5} \, B a x^{5} e^{3} + \frac{3}{4} \, B a d x^{4} e^{2} + B a d^{2} x^{3} e + \frac{1}{2} \, B a d^{3} x^{2} + \frac{1}{4} \, A a x^{4} e^{3} + A a d x^{3} e^{2} + \frac{3}{2} \, A a d^{2} x^{2} e + A a d^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)*(B*x + A)*(e*x + d)^3,x, algorithm="giac")
[Out]