\(\int \frac {a c+a d x+b c x^3+b d x^4}{a+b x^3} \, dx\) [56]

Optimal result

Integrand size = 30, antiderivative size = 12 \[ \int \frac {a c+a d x+b c x^3+b d x^4}{a+b x^3} \, dx=c x+\frac {d x^2}{2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {1600} \[ \int \frac {a c+a d x+b c x^3+b d x^4}{a+b x^3} \, dx=c x+\frac {d x^2}{2} \]

[In]

Int[(a*c + a*d*x + b*c*x^3 + b*d*x^4)/(a + b*x^3),x]

[Out]

c*x + (d*x^2)/2

Rule 1600

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rubi steps \begin{align*} \text {integral}& = \int (c+d x) \, dx \\ & = 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 \[ \int \frac {a c+a d x+b c x^3+b d x^4}{a+b x^3} \, dx=c x+\frac {d x^2}{2} \]

[In]

Integrate[(a*c + a*d*x + b*c*x^3 + b*d*x^4)/(a + b*x^3),x]

[Out]

c*x + (d*x^2)/2

Maple [A] (verified)

Time = 1.42 (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^4+b*c*x^3+a*d*x+a*c)/(b*x^3+a),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.65 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {a c+a d x+b c x^3+b d x^4}{a+b x^3} \, dx=\frac {1}{2} \, d x^{2} + c x \]

[In]

integrate((b*d*x^4+b*c*x^3+a*d*x+a*c)/(b*x^3+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 \[ \int \frac {a c+a d x+b c x^3+b d x^4}{a+b x^3} \, dx=c x + \frac {d x^{2}}{2} \]

[In]

integrate((b*d*x**4+b*c*x**3+a*d*x+a*c)/(b*x**3+a),x)

[Out]

c*x + d*x**2/2

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {a c+a d x+b c x^3+b d x^4}{a+b x^3} \, dx=\frac {1}{2} \, d x^{2} + c x \]

[In]

integrate((b*d*x^4+b*c*x^3+a*d*x+a*c)/(b*x^3+a),x, algorithm="maxima")

[Out]

1/2*d*x^2 + c*x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {a c+a d x+b c x^3+b d x^4}{a+b x^3} \, dx=\frac {1}{2} \, d x^{2} + c x \]

[In]

integrate((b*d*x^4+b*c*x^3+a*d*x+a*c)/(b*x^3+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 \[ \int \frac {a c+a d x+b c x^3+b d x^4}{a+b x^3} \, dx=\frac {d\,x^2}{2}+c\,x \]

[In]

int((a*c + a*d*x + b*c*x^3 + b*d*x^4)/(a + b*x^3),x)

[Out]

c*x + (d*x^2)/2