3.452 \(\int (d+e x)^3 (a+c x^2) \, dx\)

Optimal. Leaf size=57 \[ \frac {(d+e x)^4 \left (a e^2+c d^2\right )}{4 e^3}+\frac {c (d+e x)^6}{6 e^3}-\frac {2 c d (d+e x)^5}{5 e^3} \]

[Out]

1/4*(a*e^2+c*d^2)*(e*x+d)^4/e^3-2/5*c*d*(e*x+d)^5/e^3+1/6*c*(e*x+d)^6/e^3

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Rubi [A]  time = 0.05, 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} \[ \frac {(d+e x)^4 \left (a e^2+c d^2\right )}{4 e^3}+\frac {c (d+e x)^6}{6 e^3}-\frac {2 c d (d+e x)^5}{5 e^3} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*(a + c*x^2),x]

[Out]

((c*d^2 + a*e^2)*(d + e*x)^4)/(4*e^3) - (2*c*d*(d + e*x)^5)/(5*e^3) + (c*(d + e*x)^6)/(6*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)^3 \left (a+c x^2\right ) \, dx &=\int \left (\frac {\left (c d^2+a e^2\right ) (d+e x)^3}{e^2}-\frac {2 c d (d+e x)^4}{e^2}+\frac {c (d+e x)^5}{e^2}\right ) \, dx\\ &=\frac {\left (c d^2+a e^2\right ) (d+e x)^4}{4 e^3}-\frac {2 c d (d+e x)^5}{5 e^3}+\frac {c (d+e x)^6}{6 e^3}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 74, normalized size = 1.30 \[ \frac {1}{4} a x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+\frac {1}{60} c x^3 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3*(a + c*x^2),x]

[Out]

(a*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3))/4 + (c*x^3*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3)
)/60

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fricas [A]  time = 0.99, size = 73, normalized size = 1.28 \[ \frac {1}{6} x^{6} e^{3} c + \frac {3}{5} x^{5} e^{2} d c + \frac {3}{4} x^{4} e d^{2} c + \frac {1}{4} x^{4} e^{3} a + \frac {1}{3} x^{3} d^{3} c + x^{3} e^{2} d a + \frac {3}{2} x^{2} e d^{2} a + x d^{3} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(c*x^2+a),x, algorithm="fricas")

[Out]

1/6*x^6*e^3*c + 3/5*x^5*e^2*d*c + 3/4*x^4*e*d^2*c + 1/4*x^4*e^3*a + 1/3*x^3*d^3*c + x^3*e^2*d*a + 3/2*x^2*e*d^
2*a + x*d^3*a

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giac [A]  time = 0.17, size = 71, normalized size = 1.25 \[ \frac {1}{6} \, c x^{6} e^{3} + \frac {3}{5} \, c d x^{5} e^{2} + \frac {3}{4} \, c d^{2} x^{4} e + \frac {1}{3} \, c d^{3} x^{3} + \frac {1}{4} \, a x^{4} e^{3} + a d x^{3} e^{2} + \frac {3}{2} \, a d^{2} x^{2} e + a d^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(c*x^2+a),x, algorithm="giac")

[Out]

1/6*c*x^6*e^3 + 3/5*c*d*x^5*e^2 + 3/4*c*d^2*x^4*e + 1/3*c*d^3*x^3 + 1/4*a*x^4*e^3 + a*d*x^3*e^2 + 3/2*a*d^2*x^
2*e + a*d^3*x

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maple [A]  time = 0.04, size = 73, normalized size = 1.28 \[ \frac {c \,e^{3} x^{6}}{6}+\frac {3 c d \,e^{2} x^{5}}{5}+\frac {3 a \,d^{2} e \,x^{2}}{2}+a \,d^{3} x +\frac {\left (e^{3} a +3 d^{2} e c \right ) x^{4}}{4}+\frac {\left (3 d \,e^{2} a +d^{3} c \right ) x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(c*x^2+a),x)

[Out]

1/6*e^3*c*x^6+3/5*d*e^2*c*x^5+1/4*(a*e^3+3*c*d^2*e)*x^4+1/3*(3*a*d*e^2+c*d^3)*x^3+3/2*d^2*e*a*x^2+d^3*a*x

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maxima [A]  time = 1.33, size = 72, normalized size = 1.26 \[ \frac {1}{6} \, c e^{3} x^{6} + \frac {3}{5} \, c d e^{2} x^{5} + \frac {3}{2} \, a d^{2} e x^{2} + a d^{3} x + \frac {1}{4} \, {\left (3 \, c d^{2} e + a e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (c d^{3} + 3 \, a d e^{2}\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(c*x^2+a),x, algorithm="maxima")

[Out]

1/6*c*e^3*x^6 + 3/5*c*d*e^2*x^5 + 3/2*a*d^2*e*x^2 + a*d^3*x + 1/4*(3*c*d^2*e + a*e^3)*x^4 + 1/3*(c*d^3 + 3*a*d
*e^2)*x^3

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mupad [B]  time = 0.03, size = 71, normalized size = 1.25 \[ x^3\,\left (\frac {c\,d^3}{3}+a\,d\,e^2\right )+x^4\,\left (\frac {3\,c\,d^2\,e}{4}+\frac {a\,e^3}{4}\right )+\frac {c\,e^3\,x^6}{6}+a\,d^3\,x+\frac {3\,a\,d^2\,e\,x^2}{2}+\frac {3\,c\,d\,e^2\,x^5}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)*(d + e*x)^3,x)

[Out]

x^3*((c*d^3)/3 + a*d*e^2) + x^4*((a*e^3)/4 + (3*c*d^2*e)/4) + (c*e^3*x^6)/6 + a*d^3*x + (3*a*d^2*e*x^2)/2 + (3
*c*d*e^2*x^5)/5

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sympy [A]  time = 0.08, size = 80, normalized size = 1.40 \[ a d^{3} x + \frac {3 a d^{2} e x^{2}}{2} + \frac {3 c d e^{2} x^{5}}{5} + \frac {c e^{3} x^{6}}{6} + x^{4} \left (\frac {a e^{3}}{4} + \frac {3 c d^{2} e}{4}\right ) + x^{3} \left (a d e^{2} + \frac {c d^{3}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(c*x**2+a),x)

[Out]

a*d**3*x + 3*a*d**2*e*x**2/2 + 3*c*d*e**2*x**5/5 + c*e**3*x**6/6 + x**4*(a*e**3/4 + 3*c*d**2*e/4) + x**3*(a*d*
e**2 + c*d**3/3)

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