\(\int \frac {(c+d) (a+b x)}{e} \, dx\) [156]

Optimal result

Integrand size = 12, antiderivative size = 20 \[ \int \frac {(c+d) (a+b x)}{e} \, dx=\frac {(c+d) (a+b x)^2}{2 b e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {9} \[ \int \frac {(c+d) (a+b x)}{e} \, dx=\frac {(c+d) (a+b x)^2}{2 b e} \]

[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*} \text {integral}& = \frac {(c+d) (a+b x)^2}{2 b e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {(c+d) (a+b x)}{e} \, dx=\frac {(c+d) \left (a x+\frac {b x^2}{2}\right )}{e} \]

[In]

Integrate[((c + d)*(a + b*x))/e,x]

[Out]

((c + d)*(a*x + (b*x^2)/2))/e

Maple [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85

method	result	size
gosper	\(\frac {x \left (b x +2 a \right ) \left (c +d \right )}{2 e}\)	\(17\)
default	\(\frac {\left (a x +\frac {1}{2} b \,x^{2}\right ) \left (c +d \right )}{e}\)	\(18\)
parallelrisch	\(\frac {\left (a x +\frac {1}{2} b \,x^{2}\right ) \left (c +d \right )}{e}\)	\(18\)
norman	\(\frac {a \left (c +d \right ) x}{e}+\frac {\left (c +d \right ) b \,x^{2}}{2 e}\)	\(23\)
risch	\(\frac {a x c}{e}+\frac {a x d}{e}+\frac {b \,x^{2} c}{2 e}+\frac {b \,x^{2} d}{2 e}\)	\(36\)

[In]

int((c+d)*(b*x+a)/e,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {(c+d) (a+b x)}{e} \, dx=\frac {{\left (b c + b d\right )} x^{2} + 2 \, {\left (a c + a d\right )} x}{2 \, e} \]

[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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(c+d) (a+b x)}{e} \, dx=\frac {x^{2} \left (b c + b d\right )}{2 e} + \frac {x \left (a c + a d\right )}{e} \]

[In]

integrate((c+d)*(b*x+a)/e,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {(c+d) (a+b x)}{e} \, dx=\frac {{\left (b x^{2} + 2 \, a x\right )} {\left (c + d\right )}}{2 \, e} \]

[In]

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

[Out]

1/2*(b*x^2 + 2*a*x)*(c + d)/e

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {(c+d) (a+b x)}{e} \, dx=\frac {{\left (b x^{2} + 2 \, a x\right )} {\left (c + d\right )}}{2 \, e} \]

[In]

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

[Out]

1/2*(b*x^2 + 2*a*x)*(c + d)/e

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {(c+d) (a+b x)}{e} \, dx=\frac {x\,\left (c+d\right )\,\left (2\,a+b\,x\right )}{2\,e} \]

[In]

int(((c + d)*(a + b*x))/e,x)

[Out]

(x*(c + d)*(2*a + b*x))/(2*e)