3.156 \(\int \frac{(c+d) (a+b x)}{e} \, dx\)

Optimal. Leaf size=20 \[ \frac{(c+d) (a+b x)^2}{2 b e} \]

[Out]

((c + d)*(a + b*x)^2)/(2*b*e)

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Rubi [A]  time = 0.0133103, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{(c+d) (a+b x)^2}{2 b e} \]

Antiderivative was successfully verified.

[In]  Int[((c + d)*(a + b*x))/e,x]

[Out]

((c + d)*(a + b*x)^2)/(2*b*e)

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Rubi in Sympy [A]  time = 2.32039, size = 14, normalized size = 0.7 \[ \frac{\left (a + b x\right )^{2} \left (c + d\right )}{2 b e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c+d)*(b*x+a)/e,x)

[Out]

(a + b*x)**2*(c + d)/(2*b*e)

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Mathematica [A]  time = 0.00134553, size = 19, normalized size = 0.95 \[ \frac{(c+d) \left (a x+\frac{b x^2}{2}\right )}{e} \]

Antiderivative was successfully verified.

[In]  Integrate[((c + d)*(a + b*x))/e,x]

[Out]

((c + d)*(a*x + (b*x^2)/2))/e

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Maple [A]  time = 0.002, size = 18, normalized size = 0.9 \[{\frac{c+d}{e} \left ( ax+{\frac{b{x}^{2}}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c+d)*(b*x+a)/e,x)

[Out]

(c+d)/e*(a*x+1/2*b*x^2)

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Maxima [A]  time = 1.34811, size = 24, normalized size = 1.2 \[ \frac{{\left (b x^{2} + 2 \, a x\right )}{\left (c + d\right )}}{2 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(c + d)/e,x, algorithm="maxima")

[Out]

1/2*(b*x^2 + 2*a*x)*(c + d)/e

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Fricas [A]  time = 0.196058, size = 36, normalized size = 1.8 \[ \frac{{\left (b c + b d\right )} x^{2} + 2 \,{\left (a c + a d\right )} x}{2 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(c + d)/e,x, algorithm="fricas")

[Out]

1/2*((b*c + b*d)*x^2 + 2*(a*c + a*d)*x)/e

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Sympy [A]  time = 0.080573, size = 22, normalized size = 1.1 \[ \frac{x^{2} \left (b c + b d\right )}{2 e} + \frac{x \left (a c + a d\right )}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c+d)*(b*x+a)/e,x)

[Out]

x**2*(b*c + b*d)/(2*e) + x*(a*c + a*d)/e

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GIAC/XCAS [A]  time = 0.223422, size = 23, normalized size = 1.15 \[ \frac{1}{2} \,{\left (b x^{2} + 2 \, a x\right )}{\left (c + d\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(c + d)/e,x, algorithm="giac")

[Out]

1/2*(b*x^2 + 2*a*x)*(c + d)*e^(-1)