3.5.54 \(\int (d+e x) (a+c x^2) \, dx\) [454]

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

[Out]

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

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Rubi [A]
time = 0.00, 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 = {655} \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 655

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*}

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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

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Maple [A]
time = 0.11, size = 27, normalized size = 0.87

method	result	size
gosper	\(\frac {1}{4} c e \,x^{4}+\frac {1}{3} c d \,x^{3}+\frac {1}{2} a e \,x^{2}+a d x\)	\(27\)
default	\(\frac {1}{4} c e \,x^{4}+\frac {1}{3} c d \,x^{3}+\frac {1}{2} a e \,x^{2}+a d x\)	\(27\)
norman	\(\frac {1}{4} c e \,x^{4}+\frac {1}{3} c d \,x^{3}+\frac {1}{2} a e \,x^{2}+a d x\)	\(27\)
risch	\(\frac {1}{4} c e \,x^{4}+\frac {1}{3} c d \,x^{3}+\frac {1}{2} a e \,x^{2}+a d x\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.42, 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="maxima")

[Out]

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

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Fricas [A]
time = 2.38, size = 28, normalized size = 0.90 \begin {gather*} \frac {1}{3} \, c d x^{3} + a d x + \frac {1}{4} \, {\left (c x^{4} + 2 \, a x^{2}\right )} e \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]

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

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Sympy [A]
time = 0.01, 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

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Giac [A]
time = 1.19, 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

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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

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