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

Optimal. Leaf size=57 \[ \frac {(d+e x)^3 \left (a e^2+c d^2\right )}{3 e^3}+\frac {c (d+e x)^5}{5 e^3}-\frac {c d (d+e x)^4}{2 e^3} \]

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Rubi [A]  time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {697} \begin {gather*} \frac {(d+e x)^3 \left (a e^2+c d^2\right )}{3 e^3}+\frac {c (d+e x)^5}{5 e^3}-\frac {c d (d+e x)^4}{2 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2*(a + c*x^2),x]

[Out]

((c*d^2 + a*e^2)*(d + e*x)^3)/(3*e^3) - (c*d*(d + e*x)^4)/(2*e^3) + (c*(d + e*x)^5)/(5*e^3)

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int (d+e x)^2 \left (a+c x^2\right ) \, dx &=\int \left (\frac {\left (c d^2+a e^2\right ) (d+e x)^2}{e^2}-\frac {2 c d (d+e x)^3}{e^2}+\frac {c (d+e x)^4}{e^2}\right ) \, dx\\ &=\frac {\left (c d^2+a e^2\right ) (d+e x)^3}{3 e^3}-\frac {c d (d+e x)^4}{2 e^3}+\frac {c (d+e x)^5}{5 e^3}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 53, normalized size = 0.93 \begin {gather*} \frac {1}{3} x^3 \left (a e^2+c d^2\right )+a d^2 x+a d e x^2+\frac {1}{2} c d e x^4+\frac {1}{5} c e^2 x^5 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^2*(a + c*x^2),x]

[Out]

a*d^2*x + a*d*e*x^2 + ((c*d^2 + a*e^2)*x^3)/3 + (c*d*e*x^4)/2 + (c*e^2*x^5)/5

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.36, size = 49, normalized size = 0.86 \begin {gather*} \frac {1}{5} x^{5} e^{2} c + \frac {1}{2} x^{4} e d c + \frac {1}{3} x^{3} d^{2} c + \frac {1}{3} x^{3} e^{2} a + x^{2} e d a + x d^{2} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^5*e^2*c + 1/2*x^4*e*d*c + 1/3*x^3*d^2*c + 1/3*x^3*e^2*a + x^2*e*d*a + x*d^2*a

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giac [A]  time = 0.15, size = 49, normalized size = 0.86 \begin {gather*} \frac {1}{5} \, c x^{5} e^{2} + \frac {1}{2} \, c d x^{4} e + \frac {1}{3} \, c d^{2} x^{3} + \frac {1}{3} \, a x^{3} e^{2} + a d x^{2} e + a d^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c*x^5*e^2 + 1/2*c*d*x^4*e + 1/3*c*d^2*x^3 + 1/3*a*x^3*e^2 + a*d*x^2*e + a*d^2*x

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maple [A]  time = 0.04, size = 48, normalized size = 0.84 \begin {gather*} \frac {c \,e^{2} x^{5}}{5}+\frac {c d e \,x^{4}}{2}+a d e \,x^{2}+a \,d^{2} x +\frac {\left (a \,e^{2}+c \,d^{2}\right ) x^{3}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c*e^2*x^5+1/2*c*d*e*x^4+1/3*(a*e^2+c*d^2)*x^3+a*d*e*x^2+a*d^2*x

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maxima [A]  time = 1.34, size = 47, normalized size = 0.82 \begin {gather*} \frac {1}{5} \, c e^{2} x^{5} + \frac {1}{2} \, c d e x^{4} + a d e x^{2} + a d^{2} x + \frac {1}{3} \, {\left (c d^{2} + a e^{2}\right )} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c*e^2*x^5 + 1/2*c*d*e*x^4 + a*d*e*x^2 + a*d^2*x + 1/3*(c*d^2 + a*e^2)*x^3

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mupad [B]  time = 0.02, size = 48, normalized size = 0.84 \begin {gather*} x^3\,\left (\frac {c\,d^2}{3}+\frac {a\,e^2}{3}\right )+\frac {c\,e^2\,x^5}{5}+a\,d^2\,x+a\,d\,e\,x^2+\frac {c\,d\,e\,x^4}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)*(d + e*x)^2,x)

[Out]

x^3*((a*e^2)/3 + (c*d^2)/3) + (c*e^2*x^5)/5 + a*d^2*x + a*d*e*x^2 + (c*d*e*x^4)/2

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sympy [A]  time = 0.07, size = 51, normalized size = 0.89 \begin {gather*} a d^{2} x + a d e x^{2} + \frac {c d e x^{4}}{2} + \frac {c e^{2} x^{5}}{5} + x^{3} \left (\frac {a e^{2}}{3} + \frac {c d^{2}}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*d**2*x + a*d*e*x**2 + c*d*e*x**4/2 + c*e**2*x**5/5 + x**3*(a*e**2/3 + c*d**2/3)

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