\(\int (a+b x^2)^2 (c+d x^2)^2 \, dx\) [154]

Optimal result

Integrand size = 19, antiderivative size = 82 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=a^2 c^2 x+\frac {2}{3} a c (b c+a d) x^3+\frac {1}{5} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^5+\frac {2}{7} b d (b c+a d) x^7+\frac {1}{9} b^2 d^2 x^9 \]

[Out]

a^2*c^2*x+2/3*a*c*(a*d+b*c)*x^3+1/5*(a^2*d^2+4*a*b*c*d+b^2*c^2)*x^5+2/7*b*d*(a*d+b*c)*x^7+1/9*b^2*d^2*x^9

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {380} \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {1}{5} x^5 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 x+\frac {2}{7} b d x^7 (a d+b c)+\frac {2}{3} a c x^3 (a d+b c)+\frac {1}{9} b^2 d^2 x^9 \]

[In]

Int[(a + b*x^2)^2*(c + d*x^2)^2,x]

[Out]

a^2*c^2*x + (2*a*c*(b*c + a*d)*x^3)/3 + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^5)/5 + (2*b*d*(b*c + a*d)*x^7)/7 +
(b^2*d^2*x^9)/9

Rule 380

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

Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 c^2+2 a c (b c+a d) x^2+\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^4+2 b d (b c+a d) x^6+b^2 d^2 x^8\right ) \, dx \\ & = a^2 c^2 x+\frac {2}{3} a c (b c+a d) x^3+\frac {1}{5} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^5+\frac {2}{7} b d (b c+a d) x^7+\frac {1}{9} b^2 d^2 x^9 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=a^2 c^2 x+\frac {2}{3} a c (b c+a d) x^3+\frac {1}{5} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^5+\frac {2}{7} b d (b c+a d) x^7+\frac {1}{9} b^2 d^2 x^9 \]

[In]

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

[Out]

a^2*c^2*x + (2*a*c*(b*c + a*d)*x^3)/3 + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^5)/5 + (2*b*d*(b*c + a*d)*x^7)/7 +
(b^2*d^2*x^9)/9

Maple [A] (verified)

Time = 2.58 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.05

method	result	size
norman	\(\frac {b^{2} d^{2} x^{9}}{9}+\left (\frac {2}{7} a b \,d^{2}+\frac {2}{7} b^{2} c d \right ) x^{7}+\left (\frac {1}{5} a^{2} d^{2}+\frac {4}{5} a b c d +\frac {1}{5} b^{2} c^{2}\right ) x^{5}+\left (\frac {2}{3} a^{2} c d +\frac {2}{3} b \,c^{2} a \right ) x^{3}+a^{2} c^{2} x\)	\(86\)
default	\(\frac {b^{2} d^{2} x^{9}}{9}+\frac {\left (2 a b \,d^{2}+2 b^{2} c d \right ) x^{7}}{7}+\frac {\left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) x^{5}}{5}+\frac {\left (2 a^{2} c d +2 b \,c^{2} a \right ) x^{3}}{3}+a^{2} c^{2} x\)	\(87\)
gosper	\(\frac {1}{9} b^{2} d^{2} x^{9}+\frac {2}{7} x^{7} a b \,d^{2}+\frac {2}{7} x^{7} b^{2} c d +\frac {1}{5} x^{5} a^{2} d^{2}+\frac {4}{5} x^{5} a b c d +\frac {1}{5} x^{5} b^{2} c^{2}+\frac {2}{3} x^{3} a^{2} c d +\frac {2}{3} x^{3} b \,c^{2} a +a^{2} c^{2} x\)	\(92\)
risch	\(\frac {1}{9} b^{2} d^{2} x^{9}+\frac {2}{7} x^{7} a b \,d^{2}+\frac {2}{7} x^{7} b^{2} c d +\frac {1}{5} x^{5} a^{2} d^{2}+\frac {4}{5} x^{5} a b c d +\frac {1}{5} x^{5} b^{2} c^{2}+\frac {2}{3} x^{3} a^{2} c d +\frac {2}{3} x^{3} b \,c^{2} a +a^{2} c^{2} x\)	\(92\)
parallelrisch	\(\frac {1}{9} b^{2} d^{2} x^{9}+\frac {2}{7} x^{7} a b \,d^{2}+\frac {2}{7} x^{7} b^{2} c d +\frac {1}{5} x^{5} a^{2} d^{2}+\frac {4}{5} x^{5} a b c d +\frac {1}{5} x^{5} b^{2} c^{2}+\frac {2}{3} x^{3} a^{2} c d +\frac {2}{3} x^{3} b \,c^{2} a +a^{2} c^{2} x\)	\(92\)

[In]

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

[Out]

1/9*b^2*d^2*x^9+(2/7*a*b*d^2+2/7*b^2*c*d)*x^7+(1/5*a^2*d^2+4/5*a*b*c*d+1/5*b^2*c^2)*x^5+(2/3*a^2*c*d+2/3*b*c^2
*a)*x^3+a^2*c^2*x

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {1}{9} \, b^{2} d^{2} x^{9} + \frac {2}{7} \, {\left (b^{2} c d + a b d^{2}\right )} x^{7} + \frac {1}{5} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{5} + a^{2} c^{2} x + \frac {2}{3} \, {\left (a b c^{2} + a^{2} c d\right )} x^{3} \]

[In]

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

[Out]

1/9*b^2*d^2*x^9 + 2/7*(b^2*c*d + a*b*d^2)*x^7 + 1/5*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^5 + a^2*c^2*x + 2/3*(a*b
*c^2 + a^2*c*d)*x^3

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.18 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=a^{2} c^{2} x + \frac {b^{2} d^{2} x^{9}}{9} + x^{7} \cdot \left (\frac {2 a b d^{2}}{7} + \frac {2 b^{2} c d}{7}\right ) + x^{5} \left (\frac {a^{2} d^{2}}{5} + \frac {4 a b c d}{5} + \frac {b^{2} c^{2}}{5}\right ) + x^{3} \cdot \left (\frac {2 a^{2} c d}{3} + \frac {2 a b c^{2}}{3}\right ) \]

[In]

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

[Out]

a**2*c**2*x + b**2*d**2*x**9/9 + x**7*(2*a*b*d**2/7 + 2*b**2*c*d/7) + x**5*(a**2*d**2/5 + 4*a*b*c*d/5 + b**2*c
**2/5) + x**3*(2*a**2*c*d/3 + 2*a*b*c**2/3)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {1}{9} \, b^{2} d^{2} x^{9} + \frac {2}{7} \, {\left (b^{2} c d + a b d^{2}\right )} x^{7} + \frac {1}{5} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{5} + a^{2} c^{2} x + \frac {2}{3} \, {\left (a b c^{2} + a^{2} c d\right )} x^{3} \]

[In]

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

[Out]

1/9*b^2*d^2*x^9 + 2/7*(b^2*c*d + a*b*d^2)*x^7 + 1/5*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^5 + a^2*c^2*x + 2/3*(a*b
*c^2 + a^2*c*d)*x^3

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.11 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {1}{9} \, b^{2} d^{2} x^{9} + \frac {2}{7} \, b^{2} c d x^{7} + \frac {2}{7} \, a b d^{2} x^{7} + \frac {1}{5} \, b^{2} c^{2} x^{5} + \frac {4}{5} \, a b c d x^{5} + \frac {1}{5} \, a^{2} d^{2} x^{5} + \frac {2}{3} \, a b c^{2} x^{3} + \frac {2}{3} \, a^{2} c d x^{3} + a^{2} c^{2} x \]

[In]

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

[Out]

1/9*b^2*d^2*x^9 + 2/7*b^2*c*d*x^7 + 2/7*a*b*d^2*x^7 + 1/5*b^2*c^2*x^5 + 4/5*a*b*c*d*x^5 + 1/5*a^2*d^2*x^5 + 2/
3*a*b*c^2*x^3 + 2/3*a^2*c*d*x^3 + a^2*c^2*x

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.91 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=x^5\,\left (\frac {a^2\,d^2}{5}+\frac {4\,a\,b\,c\,d}{5}+\frac {b^2\,c^2}{5}\right )+a^2\,c^2\,x+\frac {b^2\,d^2\,x^9}{9}+\frac {2\,a\,c\,x^3\,\left (a\,d+b\,c\right )}{3}+\frac {2\,b\,d\,x^7\,\left (a\,d+b\,c\right )}{7} \]

[In]

int((a + b*x^2)^2*(c + d*x^2)^2,x)

[Out]

x^5*((a^2*d^2)/5 + (b^2*c^2)/5 + (4*a*b*c*d)/5) + a^2*c^2*x + (b^2*d^2*x^9)/9 + (2*a*c*x^3*(a*d + b*c))/3 + (2
*b*d*x^7*(a*d + b*c))/7