3.1288 \(\int (A+B x) (d+e x) \left (a+c x^2\right ) \, dx\)

Optimal. Leaf size=62 \[ \frac{1}{3} x^3 (a B e+A c d)+\frac{1}{2} a x^2 (A e+B d)+a A d x+\frac{1}{4} c x^4 (A e+B d)+\frac{1}{5} B c e x^5 \]

[Out]

a*A*d*x + (a*(B*d + A*e)*x^2)/2 + ((A*c*d + a*B*e)*x^3)/3 + (c*(B*d + A*e)*x^4)/
4 + (B*c*e*x^5)/5

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Rubi [A]  time = 0.129126, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{1}{3} x^3 (a B e+A c d)+\frac{1}{2} a x^2 (A e+B d)+a A d x+\frac{1}{4} c x^4 (A e+B d)+\frac{1}{5} B c e x^5 \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)*(d + e*x)*(a + c*x^2),x]

[Out]

a*A*d*x + (a*(B*d + A*e)*x^2)/2 + ((A*c*d + a*B*e)*x^3)/3 + (c*(B*d + A*e)*x^4)/
4 + (B*c*e*x^5)/5

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B c e x^{5}}{5} + a d \int A\, dx + a \left (A e + B d\right ) \int x\, dx + \frac{c x^{4} \left (A e + B d\right )}{4} + x^{3} \left (\frac{A c d}{3} + \frac{B a e}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)*(c*x**2+a),x)

[Out]

B*c*e*x**5/5 + a*d*Integral(A, x) + a*(A*e + B*d)*Integral(x, x) + c*x**4*(A*e +
B*d)/4 + x**3*(A*c*d/3 + B*a*e/3)

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Mathematica [A]  time = 0.035058, size = 62, normalized size = 1. \[ \frac{1}{3} x^3 (a B e+A c d)+\frac{1}{2} a x^2 (A e+B d)+a A d x+\frac{1}{4} c x^4 (A e+B d)+\frac{1}{5} B c e x^5 \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)*(d + e*x)*(a + c*x^2),x]

[Out]

a*A*d*x + (a*(B*d + A*e)*x^2)/2 + ((A*c*d + a*B*e)*x^3)/3 + (c*(B*d + A*e)*x^4)/
4 + (B*c*e*x^5)/5

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Maple [A]  time = 0.001, size = 55, normalized size = 0.9 \[ aAdx+{\frac{a \left ( Ae+Bd \right ){x}^{2}}{2}}+{\frac{ \left ( Acd+aBe \right ){x}^{3}}{3}}+{\frac{c \left ( Ae+Bd \right ){x}^{4}}{4}}+{\frac{Bce{x}^{5}}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)*(c*x^2+a),x)

[Out]

a*A*d*x+1/2*a*(A*e+B*d)*x^2+1/3*(A*c*d+B*a*e)*x^3+1/4*c*(A*e+B*d)*x^4+1/5*B*c*e*
x^5

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Maxima [A]  time = 0.693955, size = 76, normalized size = 1.23 \[ \frac{1}{5} \, B c e x^{5} + \frac{1}{4} \,{\left (B c d + A c e\right )} x^{4} + A a d x + \frac{1}{3} \,{\left (A c d + B a e\right )} x^{3} + \frac{1}{2} \,{\left (B a d + A a e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)*(B*x + A)*(e*x + d),x, algorithm="maxima")

[Out]

1/5*B*c*e*x^5 + 1/4*(B*c*d + A*c*e)*x^4 + A*a*d*x + 1/3*(A*c*d + B*a*e)*x^3 + 1/
2*(B*a*d + A*a*e)*x^2

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Fricas [A]  time = 0.243798, size = 1, normalized size = 0.02 \[ \frac{1}{5} x^{5} e c B + \frac{1}{4} x^{4} d c B + \frac{1}{4} x^{4} e c A + \frac{1}{3} x^{3} e a B + \frac{1}{3} x^{3} d c A + \frac{1}{2} x^{2} d a B + \frac{1}{2} x^{2} e a A + x d a A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)*(B*x + A)*(e*x + d),x, algorithm="fricas")

[Out]

1/5*x^5*e*c*B + 1/4*x^4*d*c*B + 1/4*x^4*e*c*A + 1/3*x^3*e*a*B + 1/3*x^3*d*c*A +
1/2*x^2*d*a*B + 1/2*x^2*e*a*A + x*d*a*A

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Sympy [A]  time = 0.112023, size = 66, normalized size = 1.06 \[ A a d x + \frac{B c e x^{5}}{5} + x^{4} \left (\frac{A c e}{4} + \frac{B c d}{4}\right ) + x^{3} \left (\frac{A c d}{3} + \frac{B a e}{3}\right ) + x^{2} \left (\frac{A a e}{2} + \frac{B a d}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)*(c*x**2+a),x)

[Out]

A*a*d*x + B*c*e*x**5/5 + x**4*(A*c*e/4 + B*c*d/4) + x**3*(A*c*d/3 + B*a*e/3) + x
**2*(A*a*e/2 + B*a*d/2)

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GIAC/XCAS [A]  time = 0.286711, size = 89, normalized size = 1.44 \[ \frac{1}{5} \, B c x^{5} e + \frac{1}{4} \, B c d x^{4} + \frac{1}{4} \, A c x^{4} e + \frac{1}{3} \, A c d x^{3} + \frac{1}{3} \, B a x^{3} e + \frac{1}{2} \, B a d x^{2} + \frac{1}{2} \, A a x^{2} e + A a d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)*(B*x + A)*(e*x + d),x, algorithm="giac")

[Out]

1/5*B*c*x^5*e + 1/4*B*c*d*x^4 + 1/4*A*c*x^4*e + 1/3*A*c*d*x^3 + 1/3*B*a*x^3*e +
1/2*B*a*d*x^2 + 1/2*A*a*x^2*e + A*a*d*x