3.4.74 \(\int (d+e x) (a+c x^2) \, dx\)

Optimal. Leaf size=31 \[ \frac {e \left (a+c x^2\right )^2}{4 c}+a d x+\frac {1}{3} c d x^3 \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {641} \begin {gather*} \frac {e \left (a+c x^2\right )^2}{4 c}+a d x+\frac {1}{3} c d x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)*(a + c*x^2),x]

[Out]

a*d*x + (c*d*x^3)/3 + (e*(a + c*x^2)^2)/(4*c)

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rubi steps

\begin {align*} \int (d+e x) \left (a+c x^2\right ) \, dx &=\frac {e \left (a+c x^2\right )^2}{4 c}+d \int \left (a+c x^2\right ) \, dx\\ &=a d x+\frac {1}{3} c d x^3+\frac {e \left (a+c x^2\right )^2}{4 c}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 32, normalized size = 1.03 \begin {gather*} a d x+\frac {1}{2} a e x^2+\frac {1}{3} c d x^3+\frac {1}{4} c e x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)*(a + c*x^2),x]

[Out]

a*d*x + (a*e*x^2)/2 + (c*d*x^3)/3 + (c*e*x^4)/4

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x) \left (a+c x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)*(a + c*x^2),x]

[Out]

IntegrateAlgebraic[(d + e*x)*(a + c*x^2), x]

________________________________________________________________________________________

fricas [A]  time = 0.37, size = 26, normalized size = 0.84 \begin {gather*} \frac {1}{4} x^{4} e c + \frac {1}{3} x^{3} d c + \frac {1}{2} x^{2} e a + x d a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4*e*c + 1/3*x^3*d*c + 1/2*x^2*e*a + x*d*a

________________________________________________________________________________________

giac [A]  time = 0.15, size = 28, normalized size = 0.90 \begin {gather*} \frac {1}{4} \, c x^{4} e + \frac {1}{3} \, c d x^{3} + \frac {1}{2} \, a x^{2} e + a d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*c*x^4*e + 1/3*c*d*x^3 + 1/2*a*x^2*e + a*d*x

________________________________________________________________________________________

maple [A]  time = 0.04, size = 27, normalized size = 0.87 \begin {gather*} \frac {1}{4} c e \,x^{4}+\frac {1}{3} c d \,x^{3}+\frac {1}{2} a e \,x^{2}+a d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*c*e*x^4+1/3*c*d*x^3+1/2*a*e*x^2+a*d*x

________________________________________________________________________________________

maxima [A]  time = 1.33, size = 26, normalized size = 0.84 \begin {gather*} \frac {1}{4} \, c e x^{4} + \frac {1}{3} \, c d x^{3} + \frac {1}{2} \, a e x^{2} + a d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*c*e*x^4 + 1/3*c*d*x^3 + 1/2*a*e*x^2 + a*d*x

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 26, normalized size = 0.84 \begin {gather*} \frac {c\,e\,x^4}{4}+\frac {c\,d\,x^3}{3}+\frac {a\,e\,x^2}{2}+a\,d\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)*(d + e*x),x)

[Out]

a*d*x + (a*e*x^2)/2 + (c*d*x^3)/3 + (c*e*x^4)/4

________________________________________________________________________________________

sympy [A]  time = 0.06, size = 29, normalized size = 0.94 \begin {gather*} a d x + \frac {a e x^{2}}{2} + \frac {c d x^{3}}{3} + \frac {c e x^{4}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(c*x**2+a),x)

[Out]

a*d*x + a*e*x**2/2 + c*d*x**3/3 + c*e*x**4/4

________________________________________________________________________________________