\(\int (a+b x^2) (c+d x^2)^2 (e+f x^2)^2 \, dx\) [10]

Optimal result

Integrand size = 26, antiderivative size = 158 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx=a c^2 e^2 x+\frac {1}{3} c e (b c e+2 a (d e+c f)) x^3+\frac {1}{5} \left (2 b c e (d e+c f)+a \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x^5+\frac {1}{7} \left (2 a d f (d e+c f)+b \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x^7+\frac {1}{9} d f (a d f+2 b (d e+c f)) x^9+\frac {1}{11} b d^2 f^2 x^{11} \]

[Out]

a*c^2*e^2*x+1/3*c*e*(b*c*e+2*a*(c*f+d*e))*x^3+1/5*(2*b*c*e*(c*f+d*e)+a*(c^2*f^2+4*c*d*e*f+d^2*e^2))*x^5+1/7*(2
*a*d*f*(c*f+d*e)+b*(c^2*f^2+4*c*d*e*f+d^2*e^2))*x^7+1/9*d*f*(a*d*f+2*b*(c*f+d*e))*x^9+1/11*b*d^2*f^2*x^11

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {535} \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx=\frac {1}{7} x^7 \left (2 a d f (c f+d e)+b \left (c^2 f^2+4 c d e f+d^2 e^2\right )\right )+\frac {1}{5} x^5 \left (a \left (c^2 f^2+4 c d e f+d^2 e^2\right )+2 b c e (c f+d e)\right )+\frac {1}{9} d f x^9 (a d f+2 b (c f+d e))+\frac {1}{3} c e x^3 (2 a (c f+d e)+b c e)+a c^2 e^2 x+\frac {1}{11} b d^2 f^2 x^{11} \]

[In]

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

[Out]

a*c^2*e^2*x + (c*e*(b*c*e + 2*a*(d*e + c*f))*x^3)/3 + ((2*b*c*e*(d*e + c*f) + a*(d^2*e^2 + 4*c*d*e*f + c^2*f^2
))*x^5)/5 + ((2*a*d*f*(d*e + c*f) + b*(d^2*e^2 + 4*c*d*e*f + c^2*f^2))*x^7)/7 + (d*f*(a*d*f + 2*b*(d*e + c*f))
*x^9)/9 + (b*d^2*f^2*x^11)/11

Rule 535

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^(r_.), x_Symbol] :>
Int[ExpandIntegrand[(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && I
GtQ[p, 0] && IGtQ[q, 0] && IGtQ[r, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (a c^2 e^2+c e (b c e+2 a (d e+c f)) x^2+\left (2 b c e (d e+c f)+a \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x^4+\left (2 a d f (d e+c f)+b \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x^6+d f (a d f+2 b (d e+c f)) x^8+b d^2 f^2 x^{10}\right ) \, dx \\ & = a c^2 e^2 x+\frac {1}{3} c e (b c e+2 a (d e+c f)) x^3+\frac {1}{5} \left (2 b c e (d e+c f)+a \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x^5+\frac {1}{7} \left (2 a d f (d e+c f)+b \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x^7+\frac {1}{9} d f (a d f+2 b (d e+c f)) x^9+\frac {1}{11} b d^2 f^2 x^{11} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx=a c^2 e^2 x+\frac {1}{3} c e (b c e+2 a (d e+c f)) x^3+\frac {1}{5} \left (2 b c e (d e+c f)+a \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x^5+\frac {1}{7} \left (2 a d f (d e+c f)+b \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x^7+\frac {1}{9} d f (a d f+2 b (d e+c f)) x^9+\frac {1}{11} b d^2 f^2 x^{11} \]

[In]

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

[Out]

a*c^2*e^2*x + (c*e*(b*c*e + 2*a*(d*e + c*f))*x^3)/3 + ((2*b*c*e*(d*e + c*f) + a*(d^2*e^2 + 4*c*d*e*f + c^2*f^2
))*x^5)/5 + ((2*a*d*f*(d*e + c*f) + b*(d^2*e^2 + 4*c*d*e*f + c^2*f^2))*x^7)/7 + (d*f*(a*d*f + 2*b*(d*e + c*f))
*x^9)/9 + (b*d^2*f^2*x^11)/11

Maple [A] (verified)

Time = 3.43 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.07

method	result	size
default	\(\frac {b \,d^{2} f^{2} x^{11}}{11}+\frac {\left (\left (a \,d^{2}+2 b c d \right ) f^{2}+2 b \,d^{2} e f \right ) x^{9}}{9}+\frac {\left (\left (2 a c d +b \,c^{2}\right ) f^{2}+2 \left (a \,d^{2}+2 b c d \right ) e f +b \,d^{2} e^{2}\right ) x^{7}}{7}+\frac {\left (c^{2} a \,f^{2}+2 \left (2 a c d +b \,c^{2}\right ) e f +\left (a \,d^{2}+2 b c d \right ) e^{2}\right ) x^{5}}{5}+\frac {\left (2 c^{2} a e f +\left (2 a c d +b \,c^{2}\right ) e^{2}\right ) x^{3}}{3}+a \,c^{2} e^{2} x\)	\(169\)
norman	\(\frac {b \,d^{2} f^{2} x^{11}}{11}+\left (\frac {1}{9} a \,d^{2} f^{2}+\frac {2}{9} b c d \,f^{2}+\frac {2}{9} b \,d^{2} e f \right ) x^{9}+\left (\frac {2}{7} a c d \,f^{2}+\frac {2}{7} a \,d^{2} e f +\frac {1}{7} b \,c^{2} f^{2}+\frac {4}{7} b c d e f +\frac {1}{7} b \,d^{2} e^{2}\right ) x^{7}+\left (\frac {1}{5} c^{2} a \,f^{2}+\frac {4}{5} a c d e f +\frac {1}{5} a \,d^{2} e^{2}+\frac {2}{5} b \,c^{2} e f +\frac {2}{5} b c d \,e^{2}\right ) x^{5}+\left (\frac {2}{3} c^{2} a e f +\frac {2}{3} a c d \,e^{2}+\frac {1}{3} b \,c^{2} e^{2}\right ) x^{3}+a \,c^{2} e^{2} x\)	\(175\)
gosper	\(\frac {1}{11} b \,d^{2} f^{2} x^{11}+\frac {1}{9} x^{9} a \,d^{2} f^{2}+\frac {2}{9} x^{9} b c d \,f^{2}+\frac {2}{9} x^{9} b \,d^{2} e f +\frac {2}{7} x^{7} a c d \,f^{2}+\frac {2}{7} x^{7} a \,d^{2} e f +\frac {1}{7} x^{7} b \,c^{2} f^{2}+\frac {4}{7} x^{7} b c d e f +\frac {1}{7} x^{7} b \,d^{2} e^{2}+\frac {1}{5} x^{5} c^{2} a \,f^{2}+\frac {4}{5} x^{5} a c d e f +\frac {1}{5} x^{5} a \,d^{2} e^{2}+\frac {2}{5} x^{5} b \,c^{2} e f +\frac {2}{5} x^{5} b c d \,e^{2}+\frac {2}{3} x^{3} c^{2} a e f +\frac {2}{3} x^{3} a c d \,e^{2}+\frac {1}{3} x^{3} b \,c^{2} e^{2}+a \,c^{2} e^{2} x\)	\(203\)
risch	\(\frac {1}{11} b \,d^{2} f^{2} x^{11}+\frac {1}{9} x^{9} a \,d^{2} f^{2}+\frac {2}{9} x^{9} b c d \,f^{2}+\frac {2}{9} x^{9} b \,d^{2} e f +\frac {2}{7} x^{7} a c d \,f^{2}+\frac {2}{7} x^{7} a \,d^{2} e f +\frac {1}{7} x^{7} b \,c^{2} f^{2}+\frac {4}{7} x^{7} b c d e f +\frac {1}{7} x^{7} b \,d^{2} e^{2}+\frac {1}{5} x^{5} c^{2} a \,f^{2}+\frac {4}{5} x^{5} a c d e f +\frac {1}{5} x^{5} a \,d^{2} e^{2}+\frac {2}{5} x^{5} b \,c^{2} e f +\frac {2}{5} x^{5} b c d \,e^{2}+\frac {2}{3} x^{3} c^{2} a e f +\frac {2}{3} x^{3} a c d \,e^{2}+\frac {1}{3} x^{3} b \,c^{2} e^{2}+a \,c^{2} e^{2} x\)	\(203\)
parallelrisch	\(\frac {1}{11} b \,d^{2} f^{2} x^{11}+\frac {1}{9} x^{9} a \,d^{2} f^{2}+\frac {2}{9} x^{9} b c d \,f^{2}+\frac {2}{9} x^{9} b \,d^{2} e f +\frac {2}{7} x^{7} a c d \,f^{2}+\frac {2}{7} x^{7} a \,d^{2} e f +\frac {1}{7} x^{7} b \,c^{2} f^{2}+\frac {4}{7} x^{7} b c d e f +\frac {1}{7} x^{7} b \,d^{2} e^{2}+\frac {1}{5} x^{5} c^{2} a \,f^{2}+\frac {4}{5} x^{5} a c d e f +\frac {1}{5} x^{5} a \,d^{2} e^{2}+\frac {2}{5} x^{5} b \,c^{2} e f +\frac {2}{5} x^{5} b c d \,e^{2}+\frac {2}{3} x^{3} c^{2} a e f +\frac {2}{3} x^{3} a c d \,e^{2}+\frac {1}{3} x^{3} b \,c^{2} e^{2}+a \,c^{2} e^{2} x\)	\(203\)

[In]

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

[Out]

1/11*b*d^2*f^2*x^11+1/9*((a*d^2+2*b*c*d)*f^2+2*b*d^2*e*f)*x^9+1/7*((2*a*c*d+b*c^2)*f^2+2*(a*d^2+2*b*c*d)*e*f+b
*d^2*e^2)*x^7+1/5*(c^2*a*f^2+2*(2*a*c*d+b*c^2)*e*f+(a*d^2+2*b*c*d)*e^2)*x^5+1/3*(2*c^2*a*e*f+(2*a*c*d+b*c^2)*e
^2)*x^3+a*c^2*e^2*x

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.06 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx=\frac {1}{11} \, b d^{2} f^{2} x^{11} + \frac {1}{9} \, {\left (2 \, b d^{2} e f + {\left (2 \, b c d + a d^{2}\right )} f^{2}\right )} x^{9} + \frac {1}{7} \, {\left (b d^{2} e^{2} + 2 \, {\left (2 \, b c d + a d^{2}\right )} e f + {\left (b c^{2} + 2 \, a c d\right )} f^{2}\right )} x^{7} + a c^{2} e^{2} x + \frac {1}{5} \, {\left (a c^{2} f^{2} + {\left (2 \, b c d + a d^{2}\right )} e^{2} + 2 \, {\left (b c^{2} + 2 \, a c d\right )} e f\right )} x^{5} + \frac {1}{3} \, {\left (2 \, a c^{2} e f + {\left (b c^{2} + 2 \, a c d\right )} e^{2}\right )} x^{3} \]

[In]

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

[Out]

1/11*b*d^2*f^2*x^11 + 1/9*(2*b*d^2*e*f + (2*b*c*d + a*d^2)*f^2)*x^9 + 1/7*(b*d^2*e^2 + 2*(2*b*c*d + a*d^2)*e*f
+ (b*c^2 + 2*a*c*d)*f^2)*x^7 + a*c^2*e^2*x + 1/5*(a*c^2*f^2 + (2*b*c*d + a*d^2)*e^2 + 2*(b*c^2 + 2*a*c*d)*e*f
)*x^5 + 1/3*(2*a*c^2*e*f + (b*c^2 + 2*a*c*d)*e^2)*x^3

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.37 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx=a c^{2} e^{2} x + \frac {b d^{2} f^{2} x^{11}}{11} + x^{9} \left (\frac {a d^{2} f^{2}}{9} + \frac {2 b c d f^{2}}{9} + \frac {2 b d^{2} e f}{9}\right ) + x^{7} \cdot \left (\frac {2 a c d f^{2}}{7} + \frac {2 a d^{2} e f}{7} + \frac {b c^{2} f^{2}}{7} + \frac {4 b c d e f}{7} + \frac {b d^{2} e^{2}}{7}\right ) + x^{5} \left (\frac {a c^{2} f^{2}}{5} + \frac {4 a c d e f}{5} + \frac {a d^{2} e^{2}}{5} + \frac {2 b c^{2} e f}{5} + \frac {2 b c d e^{2}}{5}\right ) + x^{3} \cdot \left (\frac {2 a c^{2} e f}{3} + \frac {2 a c d e^{2}}{3} + \frac {b c^{2} e^{2}}{3}\right ) \]

[In]

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

[Out]

a*c**2*e**2*x + b*d**2*f**2*x**11/11 + x**9*(a*d**2*f**2/9 + 2*b*c*d*f**2/9 + 2*b*d**2*e*f/9) + x**7*(2*a*c*d*
f**2/7 + 2*a*d**2*e*f/7 + b*c**2*f**2/7 + 4*b*c*d*e*f/7 + b*d**2*e**2/7) + x**5*(a*c**2*f**2/5 + 4*a*c*d*e*f/5
+ a*d**2*e**2/5 + 2*b*c**2*e*f/5 + 2*b*c*d*e**2/5) + x**3*(2*a*c**2*e*f/3 + 2*a*c*d*e**2/3 + b*c**2*e**2/3)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.06 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx=\frac {1}{11} \, b d^{2} f^{2} x^{11} + \frac {1}{9} \, {\left (2 \, b d^{2} e f + {\left (2 \, b c d + a d^{2}\right )} f^{2}\right )} x^{9} + \frac {1}{7} \, {\left (b d^{2} e^{2} + 2 \, {\left (2 \, b c d + a d^{2}\right )} e f + {\left (b c^{2} + 2 \, a c d\right )} f^{2}\right )} x^{7} + a c^{2} e^{2} x + \frac {1}{5} \, {\left (a c^{2} f^{2} + {\left (2 \, b c d + a d^{2}\right )} e^{2} + 2 \, {\left (b c^{2} + 2 \, a c d\right )} e f\right )} x^{5} + \frac {1}{3} \, {\left (2 \, a c^{2} e f + {\left (b c^{2} + 2 \, a c d\right )} e^{2}\right )} x^{3} \]

[In]

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

[Out]

1/11*b*d^2*f^2*x^11 + 1/9*(2*b*d^2*e*f + (2*b*c*d + a*d^2)*f^2)*x^9 + 1/7*(b*d^2*e^2 + 2*(2*b*c*d + a*d^2)*e*f
+ (b*c^2 + 2*a*c*d)*f^2)*x^7 + a*c^2*e^2*x + 1/5*(a*c^2*f^2 + (2*b*c*d + a*d^2)*e^2 + 2*(b*c^2 + 2*a*c*d)*e*f
)*x^5 + 1/3*(2*a*c^2*e*f + (b*c^2 + 2*a*c*d)*e^2)*x^3

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.28 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx=\frac {1}{11} \, b d^{2} f^{2} x^{11} + \frac {2}{9} \, b d^{2} e f x^{9} + \frac {2}{9} \, b c d f^{2} x^{9} + \frac {1}{9} \, a d^{2} f^{2} x^{9} + \frac {1}{7} \, b d^{2} e^{2} x^{7} + \frac {4}{7} \, b c d e f x^{7} + \frac {2}{7} \, a d^{2} e f x^{7} + \frac {1}{7} \, b c^{2} f^{2} x^{7} + \frac {2}{7} \, a c d f^{2} x^{7} + \frac {2}{5} \, b c d e^{2} x^{5} + \frac {1}{5} \, a d^{2} e^{2} x^{5} + \frac {2}{5} \, b c^{2} e f x^{5} + \frac {4}{5} \, a c d e f x^{5} + \frac {1}{5} \, a c^{2} f^{2} x^{5} + \frac {1}{3} \, b c^{2} e^{2} x^{3} + \frac {2}{3} \, a c d e^{2} x^{3} + \frac {2}{3} \, a c^{2} e f x^{3} + a c^{2} e^{2} x \]

[In]

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

[Out]

1/11*b*d^2*f^2*x^11 + 2/9*b*d^2*e*f*x^9 + 2/9*b*c*d*f^2*x^9 + 1/9*a*d^2*f^2*x^9 + 1/7*b*d^2*e^2*x^7 + 4/7*b*c*
d*e*f*x^7 + 2/7*a*d^2*e*f*x^7 + 1/7*b*c^2*f^2*x^7 + 2/7*a*c*d*f^2*x^7 + 2/5*b*c*d*e^2*x^5 + 1/5*a*d^2*e^2*x^5
+ 2/5*b*c^2*e*f*x^5 + 4/5*a*c*d*e*f*x^5 + 1/5*a*c^2*f^2*x^5 + 1/3*b*c^2*e^2*x^3 + 2/3*a*c*d*e^2*x^3 + 2/3*a*c^
2*e*f*x^3 + a*c^2*e^2*x

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx=x^5\,\left (\frac {2\,b\,c^2\,e\,f}{5}+\frac {a\,c^2\,f^2}{5}+\frac {2\,b\,c\,d\,e^2}{5}+\frac {4\,a\,c\,d\,e\,f}{5}+\frac {a\,d^2\,e^2}{5}\right )+x^7\,\left (\frac {b\,c^2\,f^2}{7}+\frac {4\,b\,c\,d\,e\,f}{7}+\frac {2\,a\,c\,d\,f^2}{7}+\frac {b\,d^2\,e^2}{7}+\frac {2\,a\,d^2\,e\,f}{7}\right )+\frac {b\,d^2\,f^2\,x^{11}}{11}+a\,c^2\,e^2\,x+\frac {c\,e\,x^3\,\left (2\,a\,c\,f+2\,a\,d\,e+b\,c\,e\right )}{3}+\frac {d\,f\,x^9\,\left (a\,d\,f+2\,b\,c\,f+2\,b\,d\,e\right )}{9} \]

[In]

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

[Out]

x^5*((a*c^2*f^2)/5 + (a*d^2*e^2)/5 + (2*b*c*d*e^2)/5 + (2*b*c^2*e*f)/5 + (4*a*c*d*e*f)/5) + x^7*((b*c^2*f^2)/7
+ (b*d^2*e^2)/7 + (2*a*c*d*f^2)/7 + (2*a*d^2*e*f)/7 + (4*b*c*d*e*f)/7) + (b*d^2*f^2*x^11)/11 + a*c^2*e^2*x +
(c*e*x^3*(2*a*c*f + 2*a*d*e + b*c*e))/3 + (d*f*x^9*(a*d*f + 2*b*c*f + 2*b*d*e))/9