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 = 0.0036228, 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, Rules used = {9} \[ \frac{(c+d) (a+b x)^2}{2 b e} \]

Antiderivative was successfully verified.

[In]

Int[((c + d)*(a + b*x))/e,x]

[Out]

((c + d)*(a + b*x)^2)/(2*b*e)

Rule 9

Int[(a_)*((b_) + (c_.)*(x_)), x_Symbol] :> Simp[(a*(b + c*x)^2)/(2*c), x] /; FreeQ[{a, b, c}, x]

Rubi steps

\begin{align*} \int \frac{(c+d) (a+b x)}{e} \, dx &=\frac{(c+d) (a+b x)^2}{2 b e}\\ \end{align*}

Mathematica [A]  time = 0.0007703, 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 = 0.001, size = 18, normalized size = 0.9 \begin{align*}{\frac{c+d}{e} \left ( ax+{\frac{b{x}^{2}}{2}} \right ) } \end{align*}

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 = 1.04736, size = 24, normalized size = 1.2 \begin{align*} \frac{{\left (b x^{2} + 2 \, a x\right )}{\left (c + d\right )}}{2 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*x^2 + 2*a*x)*(c + d)/e

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Fricas [A]  time = 1.53868, size = 58, normalized size = 2.9 \begin{align*} \frac{{\left (b c + b d\right )} x^{2} + 2 \,{\left (a c + a d\right )} x}{2 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d)*(b*x+a)/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 = 0.068977, size = 22, normalized size = 1.1 \begin{align*} \frac{x^{2} \left (b c + b d\right )}{2 e} + \frac{x \left (a c + a d\right )}{e} \end{align*}

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 [A]  time = 1.17999, size = 23, normalized size = 1.15 \begin{align*} \frac{1}{2} \,{\left (b x^{2} + 2 \, a x\right )}{\left (c + d\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*x^2 + 2*a*x)*(c + d)*e^(-1)