\(\int (d+e x)^2 (a+c x^2) \, dx\) [453]

Optimal result

Integrand size = 15, antiderivative size = 57 \[ \int (d+e x)^2 \left (a+c x^2\right ) \, dx=\frac {\left (c d^2+a e^2\right ) (d+e x)^3}{3 e^3}-\frac {c d (d+e x)^4}{2 e^3}+\frac {c (d+e x)^5}{5 e^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {711} \[ \int (d+e x)^2 \left (a+c x^2\right ) \, dx=\frac {(d+e x)^3 \left (a e^2+c d^2\right )}{3 e^3}+\frac {c (d+e x)^5}{5 e^3}-\frac {c d (d+e x)^4}{2 e^3} \]

[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 711

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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.93 \[ \int (d+e x)^2 \left (a+c x^2\right ) \, dx=a d^2 x+a d e x^2+\frac {1}{3} \left (c d^2+a e^2\right ) x^3+\frac {1}{2} c d e x^4+\frac {1}{5} c e^2 x^5 \]

[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

Maple [A] (verified)

Time = 2.24 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.84

method	result	size
default	\(\frac {c \,e^{2} x^{5}}{5}+\frac {x^{4} c d e}{2}+\frac {\left (e^{2} a +c \,d^{2}\right ) x^{3}}{3}+a d e \,x^{2}+x a \,d^{2}\)	\(48\)
norman	\(\frac {c \,e^{2} x^{5}}{5}+\frac {x^{4} c d e}{2}+\left (\frac {e^{2} a}{3}+\frac {c \,d^{2}}{3}\right ) x^{3}+a d e \,x^{2}+x a \,d^{2}\)	\(49\)
gosper	\(\frac {1}{5} c \,e^{2} x^{5}+\frac {1}{2} x^{4} c d e +\frac {1}{3} x^{3} e^{2} a +\frac {1}{3} c \,d^{2} x^{3}+a d e \,x^{2}+x a \,d^{2}\)	\(50\)
risch	\(\frac {1}{5} c \,e^{2} x^{5}+\frac {1}{2} x^{4} c d e +\frac {1}{3} x^{3} e^{2} a +\frac {1}{3} c \,d^{2} x^{3}+a d e \,x^{2}+x a \,d^{2}\)	\(50\)
parallelrisch	\(\frac {1}{5} c \,e^{2} x^{5}+\frac {1}{2} x^{4} c d e +\frac {1}{3} x^{3} e^{2} a +\frac {1}{3} c \,d^{2} x^{3}+a d e \,x^{2}+x a \,d^{2}\)	\(50\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.57 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.82 \[ \int (d+e x)^2 \left (a+c x^2\right ) \, dx=\frac {1}{5} \, c e^{2} x^{5} + \frac {1}{2} \, c d e x^{4} + a d e x^{2} + a d^{2} x + \frac {1}{3} \, {\left (c d^{2} + a e^{2}\right )} x^{3} \]

[In]

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

[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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.89 \[ \int (d+e x)^2 \left (a+c x^2\right ) \, dx=a d^{2} x + a d e x^{2} + \frac {c d e x^{4}}{2} + \frac {c e^{2} x^{5}}{5} + x^{3} \left (\frac {a e^{2}}{3} + \frac {c d^{2}}{3}\right ) \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.82 \[ \int (d+e x)^2 \left (a+c x^2\right ) \, dx=\frac {1}{5} \, c e^{2} x^{5} + \frac {1}{2} \, c d e x^{4} + a d e x^{2} + a d^{2} x + \frac {1}{3} \, {\left (c d^{2} + a e^{2}\right )} x^{3} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.86 \[ \int (d+e x)^2 \left (a+c x^2\right ) \, dx=\frac {1}{5} \, c e^{2} x^{5} + \frac {1}{2} \, c d e x^{4} + \frac {1}{3} \, c d^{2} x^{3} + \frac {1}{3} \, a e^{2} x^{3} + a d e x^{2} + a d^{2} x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.84 \[ \int (d+e x)^2 \left (a+c x^2\right ) \, dx=x^3\,\left (\frac {c\,d^2}{3}+\frac {a\,e^2}{3}\right )+\frac {c\,e^2\,x^5}{5}+a\,d^2\,x+a\,d\,e\,x^2+\frac {c\,d\,e\,x^4}{2} \]

[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