\(\int (d+e x^2) (a+b x^2+c x^4) \, dx\) [247]

Optimal result

Integrand size = 20, antiderivative size = 42 \[ \int \left (d+e x^2\right ) \left (a+b x^2+c x^4\right ) \, dx=a d x+\frac {1}{3} (b d+a e) x^3+\frac {1}{5} (c d+b e) x^5+\frac {1}{7} c e x^7 \]

[Out]

a*d*x+1/3*(a*e+b*d)*x^3+1/5*(b*e+c*d)*x^5+1/7*c*e*x^7

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1167} \[ \int \left (d+e x^2\right ) \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{3} x^3 (a e+b d)+a d x+\frac {1}{5} x^5 (b e+c d)+\frac {1}{7} c e x^7 \]

[In]

Int[(d + e*x^2)*(a + b*x^2 + c*x^4),x]

[Out]

a*d*x + ((b*d + a*e)*x^3)/3 + ((c*d + b*e)*x^5)/5 + (c*e*x^7)/7

Rule 1167

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rubi steps \begin{align*} \text {integral}& = \int \left (a d+(b d+a e) x^2+(c d+b e) x^4+c e x^6\right ) \, dx \\ & = a d x+\frac {1}{3} (b d+a e) x^3+\frac {1}{5} (c d+b e) x^5+\frac {1}{7} c e x^7 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int \left (d+e x^2\right ) \left (a+b x^2+c x^4\right ) \, dx=a d x+\frac {1}{3} (b d+a e) x^3+\frac {1}{5} (c d+b e) x^5+\frac {1}{7} c e x^7 \]

[In]

Integrate[(d + e*x^2)*(a + b*x^2 + c*x^4),x]

[Out]

a*d*x + ((b*d + a*e)*x^3)/3 + ((c*d + b*e)*x^5)/5 + (c*e*x^7)/7

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.88

method	result	size
default	\(a d x +\frac {\left (a e +b d \right ) x^{3}}{3}+\frac {\left (b e +c d \right ) x^{5}}{5}+\frac {c e \,x^{7}}{7}\)	\(37\)
norman	\(\frac {c e \,x^{7}}{7}+\left (\frac {b e}{5}+\frac {c d}{5}\right ) x^{5}+\left (\frac {a e}{3}+\frac {b d}{3}\right ) x^{3}+a d x\)	\(39\)
gosper	\(\frac {1}{7} c e \,x^{7}+\frac {1}{5} x^{5} b e +\frac {1}{5} c d \,x^{5}+\frac {1}{3} a e \,x^{3}+\frac {1}{3} x^{3} b d +a d x\)	\(41\)
risch	\(\frac {1}{7} c e \,x^{7}+\frac {1}{5} x^{5} b e +\frac {1}{5} c d \,x^{5}+\frac {1}{3} a e \,x^{3}+\frac {1}{3} x^{3} b d +a d x\)	\(41\)
parallelrisch	\(\frac {1}{7} c e \,x^{7}+\frac {1}{5} x^{5} b e +\frac {1}{5} c d \,x^{5}+\frac {1}{3} a e \,x^{3}+\frac {1}{3} x^{3} b d +a d x\)	\(41\)

[In]

int((e*x^2+d)*(c*x^4+b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

a*d*x+1/3*(a*e+b*d)*x^3+1/5*(b*e+c*d)*x^5+1/7*c*e*x^7

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.86 \[ \int \left (d+e x^2\right ) \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{7} \, c e x^{7} + \frac {1}{5} \, {\left (c d + b e\right )} x^{5} + \frac {1}{3} \, {\left (b d + a e\right )} x^{3} + a d x \]

[In]

integrate((e*x^2+d)*(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

1/7*c*e*x^7 + 1/5*(c*d + b*e)*x^5 + 1/3*(b*d + a*e)*x^3 + a*d*x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.93 \[ \int \left (d+e x^2\right ) \left (a+b x^2+c x^4\right ) \, dx=a d x + \frac {c e x^{7}}{7} + x^{5} \left (\frac {b e}{5} + \frac {c d}{5}\right ) + x^{3} \left (\frac {a e}{3} + \frac {b d}{3}\right ) \]

[In]

integrate((e*x**2+d)*(c*x**4+b*x**2+a),x)

[Out]

a*d*x + c*e*x**7/7 + x**5*(b*e/5 + c*d/5) + x**3*(a*e/3 + b*d/3)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.86 \[ \int \left (d+e x^2\right ) \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{7} \, c e x^{7} + \frac {1}{5} \, {\left (c d + b e\right )} x^{5} + \frac {1}{3} \, {\left (b d + a e\right )} x^{3} + a d x \]

[In]

integrate((e*x^2+d)*(c*x^4+b*x^2+a),x, algorithm="maxima")

[Out]

1/7*c*e*x^7 + 1/5*(c*d + b*e)*x^5 + 1/3*(b*d + a*e)*x^3 + a*d*x

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.95 \[ \int \left (d+e x^2\right ) \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{7} \, c e x^{7} + \frac {1}{5} \, c d x^{5} + \frac {1}{5} \, b e x^{5} + \frac {1}{3} \, b d x^{3} + \frac {1}{3} \, a e x^{3} + a d x \]

[In]

integrate((e*x^2+d)*(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

1/7*c*e*x^7 + 1/5*c*d*x^5 + 1/5*b*e*x^5 + 1/3*b*d*x^3 + 1/3*a*e*x^3 + a*d*x

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.90 \[ \int \left (d+e x^2\right ) \left (a+b x^2+c x^4\right ) \, dx=\frac {c\,e\,x^7}{7}+\left (\frac {b\,e}{5}+\frac {c\,d}{5}\right )\,x^5+\left (\frac {a\,e}{3}+\frac {b\,d}{3}\right )\,x^3+a\,d\,x \]

[In]

int((d + e*x^2)*(a + b*x^2 + c*x^4),x)

[Out]

x^3*((a*e)/3 + (b*d)/3) + x^5*((b*e)/5 + (c*d)/5) + a*d*x + (c*e*x^7)/7