\(\int \frac {a c+(b c+a d) x+b d x^2}{a+b x} \, dx\) [1763]

Optimal result

Integrand size = 27, antiderivative size = 12 \[ \int \frac {a c+(b c+a d) x+b d x^2}{a+b x} \, dx=c x+\frac {d x^2}{2} \]

[Out]

c*x+1/2*d*x^2

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {24} \[ \int \frac {a c+(b c+a d) x+b d x^2}{a+b x} \, dx=c x+\frac {d x^2}{2} \]

[In]

Int[(a*c + (b*c + a*d)*x + b*d*x^2)/(a + b*x),x]

[Out]

c*x + (d*x^2)/2

Rule 24

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((A_.) + (B_.)*(v_) + (C_.)*(v_)^2), x_Symbol] :> Dist[1/b^2, Int[u*(a + b*
v)^(m + 1)*Simp[b*B - a*C + b*C*v, x], x], x] /; FreeQ[{a, b, A, B, C}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0] &&
LeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (b^2 c+b^2 d x\right ) \, dx}{b^2} \\ & = c x+\frac {d x^2}{2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {a c+(b c+a d) x+b d x^2}{a+b x} \, dx=c x+\frac {d x^2}{2} \]

[In]

Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)/(a + b*x),x]

[Out]

c*x + (d*x^2)/2

Maple [A] (verified)

Time = 2.03 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92

method	result	size
gosper	\(\frac {x \left (d x +2 c \right )}{2}\)	\(11\)
default	\(c x +\frac {1}{2} d \,x^{2}\)	\(11\)
norman	\(c x +\frac {1}{2} d \,x^{2}\)	\(11\)
risch	\(c x +\frac {1}{2} d \,x^{2}\)	\(11\)
parallelrisch	\(c x +\frac {1}{2} d \,x^{2}\)	\(11\)
parts	\(c x +\frac {1}{2} d \,x^{2}\)	\(11\)

[In]

int((b*d*x^2+(a*d+b*c)*x+a*c)/(b*x+a),x,method=_RETURNVERBOSE)

[Out]

1/2*x*(d*x+2*c)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {a c+(b c+a d) x+b d x^2}{a+b x} \, dx=\frac {1}{2} \, d x^{2} + c x \]

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x^2)/(b*x+a),x, algorithm="fricas")

[Out]

1/2*d*x^2 + c*x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {a c+(b c+a d) x+b d x^2}{a+b x} \, dx=c x + \frac {d x^{2}}{2} \]

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x**2)/(b*x+a),x)

[Out]

c*x + d*x**2/2

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {a c+(b c+a d) x+b d x^2}{a+b x} \, dx=\frac {1}{2} \, d x^{2} + c x \]

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x^2)/(b*x+a),x, algorithm="maxima")

[Out]

1/2*d*x^2 + c*x

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {a c+(b c+a d) x+b d x^2}{a+b x} \, dx=\frac {1}{2} \, d x^{2} + c x \]

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x^2)/(b*x+a),x, algorithm="giac")

[Out]

1/2*d*x^2 + c*x

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {a c+(b c+a d) x+b d x^2}{a+b x} \, dx=\frac {d\,x^2}{2}+c\,x \]

[In]

int((a*c + x*(a*d + b*c) + b*d*x^2)/(a + b*x),x)

[Out]

c*x + (d*x^2)/2