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

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {(c+d) (a+b x)^2}{2 b e} \end {gather*}

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*}

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Mathematica [A]
time = 0.00, size = 19, normalized size = 0.95 \begin {gather*} \frac {(c+d) \left (a x+\frac {b x^2}{2}\right )}{e} \end {gather*}

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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Mathics [A]
time = 1.76, size = 16, normalized size = 0.80 \begin {gather*} \frac {x \left (2 a+b x\right ) \left (c+d\right )}{2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x (2 a + b x) (c + d) / (2 e)

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Maple [A]
time = 0.01, size = 18, normalized size = 0.90

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.25, size = 17, normalized size = 0.85 \begin {gather*} \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} {\left (c + d\right )} e^{\left (-1\right )} \end {gather*}

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^(-1)

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Fricas [A]
time = 0.30, size = 26, normalized size = 1.30 \begin {gather*} \frac {1}{2} \, {\left ({\left (b c + b d\right )} x^{2} + 2 \, {\left (a c + a d\right )} x\right )} e^{\left (-1\right )} \end {gather*}

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^(-1)

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Sympy [A]
time = 0.03, size = 22, normalized size = 1.10 \begin {gather*} \frac {x^{2} \left (b c + b d\right )}{2 e} + \frac {x \left (a c + a d\right )}{e} \end {gather*}

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 = 0.00, size = 17, normalized size = 0.85 \begin {gather*} \left (c+d\right ) \mathrm {e}^{-1} \left (\frac {1}{2} b x^{2}+a x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*x^2 + 2*a*x)*(c + d)*e^(-1)

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Mupad [B]
time = 0.07, size = 16, normalized size = 0.80 \begin {gather*} \frac {x\,\left (c+d\right )\,\left (2\,a+b\,x\right )}{2\,e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c + d)*(a + b*x))/e,x)

[Out]

(x*(c + d)*(2*a + b*x))/(2*e)

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