3.1.17 \(\int (a+b x^2) (c+d x^2)^3 (e+f x^2)^2 \, dx\)

Optimal. Leaf size=226 \[ \frac {1}{9} d x^9 \left (a d f (3 c f+2 d e)+b \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )\right )+\frac {1}{7} x^7 \left (a d \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )+b c \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )\right )+\frac {1}{5} c x^5 \left (a \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )+b c e (2 c f+3 d e)\right )+\frac {1}{3} c^2 e x^3 (2 a c f+3 a d e+b c e)+\frac {1}{11} d^2 f x^{11} (a d f+3 b c f+2 b d e)+a c^3 e^2 x+\frac {1}{13} b d^3 f^2 x^{13} \]

________________________________________________________________________________________

Rubi [A]  time = 0.22, antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {521} \begin {gather*} \frac {1}{9} d x^9 \left (a d f (3 c f+2 d e)+b \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )\right )+\frac {1}{7} x^7 \left (a d \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )+b c \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )\right )+\frac {1}{5} c x^5 \left (a \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )+b c e (2 c f+3 d e)\right )+\frac {1}{3} c^2 e x^3 (2 a c f+3 a d e+b c e)+\frac {1}{11} d^2 f x^{11} (a d f+3 b c f+2 b d e)+a c^3 e^2 x+\frac {1}{13} b d^3 f^2 x^{13} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)*(c + d*x^2)^3*(e + f*x^2)^2,x]

[Out]

a*c^3*e^2*x + (c^2*e*(b*c*e + 3*a*d*e + 2*a*c*f)*x^3)/3 + (c*(b*c*e*(3*d*e + 2*c*f) + a*(3*d^2*e^2 + 6*c*d*e*f
+ c^2*f^2))*x^5)/5 + ((b*c*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2) + a*d*(d^2*e^2 + 6*c*d*e*f + 3*c^2*f^2))*x^7)/7
+ (d*(a*d*f*(2*d*e + 3*c*f) + b*(d^2*e^2 + 6*c*d*e*f + 3*c^2*f^2))*x^9)/9 + (d^2*f*(2*b*d*e + 3*b*c*f + a*d*f)
*x^11)/11 + (b*d^3*f^2*x^13)/13

Rule 521

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*} \int \left (a+b x^2\right ) \left (c+d x^2\right )^3 \left (e+f x^2\right )^2 \, dx &=\int \left (a c^3 e^2+c^2 e (b c e+3 a d e+2 a c f) x^2+c \left (b c e (3 d e+2 c f)+a \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )\right ) x^4+\left (b c \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )+a d \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x^6+d \left (a d f (2 d e+3 c f)+b \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x^8+d^2 f (2 b d e+3 b c f+a d f) x^{10}+b d^3 f^2 x^{12}\right ) \, dx\\ &=a c^3 e^2 x+\frac {1}{3} c^2 e (b c e+3 a d e+2 a c f) x^3+\frac {1}{5} c \left (b c e (3 d e+2 c f)+a \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )\right ) x^5+\frac {1}{7} \left (b c \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )+a d \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x^7+\frac {1}{9} d \left (a d f (2 d e+3 c f)+b \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x^9+\frac {1}{11} d^2 f (2 b d e+3 b c f+a d f) x^{11}+\frac {1}{13} b d^3 f^2 x^{13}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.09, size = 226, normalized size = 1.00 \begin {gather*} \frac {1}{9} d x^9 \left (a d f (3 c f+2 d e)+b \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )\right )+\frac {1}{7} x^7 \left (a d \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )+b c \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )\right )+\frac {1}{5} c x^5 \left (a \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )+b c e (2 c f+3 d e)\right )+\frac {1}{3} c^2 e x^3 (2 a c f+3 a d e+b c e)+\frac {1}{11} d^2 f x^{11} (a d f+3 b c f+2 b d e)+a c^3 e^2 x+\frac {1}{13} b d^3 f^2 x^{13} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)*(c + d*x^2)^3*(e + f*x^2)^2,x]

[Out]

a*c^3*e^2*x + (c^2*e*(b*c*e + 3*a*d*e + 2*a*c*f)*x^3)/3 + (c*(b*c*e*(3*d*e + 2*c*f) + a*(3*d^2*e^2 + 6*c*d*e*f
+ c^2*f^2))*x^5)/5 + ((b*c*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2) + a*d*(d^2*e^2 + 6*c*d*e*f + 3*c^2*f^2))*x^7)/7
+ (d*(a*d*f*(2*d*e + 3*c*f) + b*(d^2*e^2 + 6*c*d*e*f + 3*c^2*f^2))*x^9)/9 + (d^2*f*(2*b*d*e + 3*b*c*f + a*d*f)
*x^11)/11 + (b*d^3*f^2*x^13)/13

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a+b x^2\right ) \left (c+d x^2\right )^3 \left (e+f x^2\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2)*(c + d*x^2)^3*(e + f*x^2)^2,x]

[Out]

IntegrateAlgebraic[(a + b*x^2)*(c + d*x^2)^3*(e + f*x^2)^2, x]

________________________________________________________________________________________

fricas [A]  time = 1.22, size = 289, normalized size = 1.28 \begin {gather*} \frac {1}{13} x^{13} f^{2} d^{3} b + \frac {2}{11} x^{11} f e d^{3} b + \frac {3}{11} x^{11} f^{2} d^{2} c b + \frac {1}{11} x^{11} f^{2} d^{3} a + \frac {1}{9} x^{9} e^{2} d^{3} b + \frac {2}{3} x^{9} f e d^{2} c b + \frac {1}{3} x^{9} f^{2} d c^{2} b + \frac {2}{9} x^{9} f e d^{3} a + \frac {1}{3} x^{9} f^{2} d^{2} c a + \frac {3}{7} x^{7} e^{2} d^{2} c b + \frac {6}{7} x^{7} f e d c^{2} b + \frac {1}{7} x^{7} f^{2} c^{3} b + \frac {1}{7} x^{7} e^{2} d^{3} a + \frac {6}{7} x^{7} f e d^{2} c a + \frac {3}{7} x^{7} f^{2} d c^{2} a + \frac {3}{5} x^{5} e^{2} d c^{2} b + \frac {2}{5} x^{5} f e c^{3} b + \frac {3}{5} x^{5} e^{2} d^{2} c a + \frac {6}{5} x^{5} f e d c^{2} a + \frac {1}{5} x^{5} f^{2} c^{3} a + \frac {1}{3} x^{3} e^{2} c^{3} b + x^{3} e^{2} d c^{2} a + \frac {2}{3} x^{3} f e c^{3} a + x e^{2} c^{3} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*x^13*f^2*d^3*b + 2/11*x^11*f*e*d^3*b + 3/11*x^11*f^2*d^2*c*b + 1/11*x^11*f^2*d^3*a + 1/9*x^9*e^2*d^3*b +
2/3*x^9*f*e*d^2*c*b + 1/3*x^9*f^2*d*c^2*b + 2/9*x^9*f*e*d^3*a + 1/3*x^9*f^2*d^2*c*a + 3/7*x^7*e^2*d^2*c*b + 6/
7*x^7*f*e*d*c^2*b + 1/7*x^7*f^2*c^3*b + 1/7*x^7*e^2*d^3*a + 6/7*x^7*f*e*d^2*c*a + 3/7*x^7*f^2*d*c^2*a + 3/5*x^
5*e^2*d*c^2*b + 2/5*x^5*f*e*c^3*b + 3/5*x^5*e^2*d^2*c*a + 6/5*x^5*f*e*d*c^2*a + 1/5*x^5*f^2*c^3*a + 1/3*x^3*e^
2*c^3*b + x^3*e^2*d*c^2*a + 2/3*x^3*f*e*c^3*a + x*e^2*c^3*a

________________________________________________________________________________________

giac [A]  time = 0.35, size = 289, normalized size = 1.28 \begin {gather*} \frac {1}{13} \, b d^{3} f^{2} x^{13} + \frac {3}{11} \, b c d^{2} f^{2} x^{11} + \frac {1}{11} \, a d^{3} f^{2} x^{11} + \frac {2}{11} \, b d^{3} f x^{11} e + \frac {1}{3} \, b c^{2} d f^{2} x^{9} + \frac {1}{3} \, a c d^{2} f^{2} x^{9} + \frac {2}{3} \, b c d^{2} f x^{9} e + \frac {2}{9} \, a d^{3} f x^{9} e + \frac {1}{9} \, b d^{3} x^{9} e^{2} + \frac {1}{7} \, b c^{3} f^{2} x^{7} + \frac {3}{7} \, a c^{2} d f^{2} x^{7} + \frac {6}{7} \, b c^{2} d f x^{7} e + \frac {6}{7} \, a c d^{2} f x^{7} e + \frac {3}{7} \, b c d^{2} x^{7} e^{2} + \frac {1}{7} \, a d^{3} x^{7} e^{2} + \frac {1}{5} \, a c^{3} f^{2} x^{5} + \frac {2}{5} \, b c^{3} f x^{5} e + \frac {6}{5} \, a c^{2} d f x^{5} e + \frac {3}{5} \, b c^{2} d x^{5} e^{2} + \frac {3}{5} \, a c d^{2} x^{5} e^{2} + \frac {2}{3} \, a c^{3} f x^{3} e + \frac {1}{3} \, b c^{3} x^{3} e^{2} + a c^{2} d x^{3} e^{2} + a c^{3} x e^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*b*d^3*f^2*x^13 + 3/11*b*c*d^2*f^2*x^11 + 1/11*a*d^3*f^2*x^11 + 2/11*b*d^3*f*x^11*e + 1/3*b*c^2*d*f^2*x^9
+ 1/3*a*c*d^2*f^2*x^9 + 2/3*b*c*d^2*f*x^9*e + 2/9*a*d^3*f*x^9*e + 1/9*b*d^3*x^9*e^2 + 1/7*b*c^3*f^2*x^7 + 3/7*
a*c^2*d*f^2*x^7 + 6/7*b*c^2*d*f*x^7*e + 6/7*a*c*d^2*f*x^7*e + 3/7*b*c*d^2*x^7*e^2 + 1/7*a*d^3*x^7*e^2 + 1/5*a*
c^3*f^2*x^5 + 2/5*b*c^3*f*x^5*e + 6/5*a*c^2*d*f*x^5*e + 3/5*b*c^2*d*x^5*e^2 + 3/5*a*c*d^2*x^5*e^2 + 2/3*a*c^3*
f*x^3*e + 1/3*b*c^3*x^3*e^2 + a*c^2*d*x^3*e^2 + a*c^3*x*e^2

________________________________________________________________________________________

maple [A]  time = 0.00, size = 244, normalized size = 1.08 \begin {gather*} \frac {b \,d^{3} f^{2} x^{13}}{13}+\frac {\left (2 b \,d^{3} e f +\left (a \,d^{3}+3 b c \,d^{2}\right ) f^{2}\right ) x^{11}}{11}+\frac {\left (b \,d^{3} e^{2}+2 \left (a \,d^{3}+3 b c \,d^{2}\right ) e f +\left (3 a c \,d^{2}+3 b \,c^{2} d \right ) f^{2}\right ) x^{9}}{9}+a \,c^{3} e^{2} x +\frac {\left (\left (a \,d^{3}+3 b c \,d^{2}\right ) e^{2}+2 \left (3 a c \,d^{2}+3 b \,c^{2} d \right ) e f +\left (3 a \,c^{2} d +b \,c^{3}\right ) f^{2}\right ) x^{7}}{7}+\frac {\left (a \,c^{3} f^{2}+\left (3 a c \,d^{2}+3 b \,c^{2} d \right ) e^{2}+2 \left (3 a \,c^{2} d +b \,c^{3}\right ) e f \right ) x^{5}}{5}+\frac {\left (2 a \,c^{3} e f +\left (3 a \,c^{2} d +b \,c^{3}\right ) e^{2}\right ) x^{3}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*b*d^3*f^2*x^13+1/11*((a*d^3+3*b*c*d^2)*f^2+2*b*d^3*e*f)*x^11+1/9*((3*a*c*d^2+3*b*c^2*d)*f^2+2*(a*d^3+3*b*
c*d^2)*e*f+b*d^3*e^2)*x^9+1/7*((3*a*c^2*d+b*c^3)*f^2+2*(3*a*c*d^2+3*b*c^2*d)*e*f+(a*d^3+3*b*c*d^2)*e^2)*x^7+1/
5*(a*c^3*f^2+2*(3*a*c^2*d+b*c^3)*e*f+(3*a*c*d^2+3*b*c^2*d)*e^2)*x^5+1/3*(2*a*c^3*e*f+(3*a*c^2*d+b*c^3)*e^2)*x^
3+a*c^3*e^2*x

________________________________________________________________________________________

maxima [A]  time = 0.84, size = 239, normalized size = 1.06 \begin {gather*} \frac {1}{13} \, b d^{3} f^{2} x^{13} + \frac {1}{11} \, {\left (2 \, b d^{3} e f + {\left (3 \, b c d^{2} + a d^{3}\right )} f^{2}\right )} x^{11} + \frac {1}{9} \, {\left (b d^{3} e^{2} + 2 \, {\left (3 \, b c d^{2} + a d^{3}\right )} e f + 3 \, {\left (b c^{2} d + a c d^{2}\right )} f^{2}\right )} x^{9} + \frac {1}{7} \, {\left ({\left (3 \, b c d^{2} + a d^{3}\right )} e^{2} + 6 \, {\left (b c^{2} d + a c d^{2}\right )} e f + {\left (b c^{3} + 3 \, a c^{2} d\right )} f^{2}\right )} x^{7} + a c^{3} e^{2} x + \frac {1}{5} \, {\left (a c^{3} f^{2} + 3 \, {\left (b c^{2} d + a c d^{2}\right )} e^{2} + 2 \, {\left (b c^{3} + 3 \, a c^{2} d\right )} e f\right )} x^{5} + \frac {1}{3} \, {\left (2 \, a c^{3} e f + {\left (b c^{3} + 3 \, a c^{2} d\right )} e^{2}\right )} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*b*d^3*f^2*x^13 + 1/11*(2*b*d^3*e*f + (3*b*c*d^2 + a*d^3)*f^2)*x^11 + 1/9*(b*d^3*e^2 + 2*(3*b*c*d^2 + a*d^
3)*e*f + 3*(b*c^2*d + a*c*d^2)*f^2)*x^9 + 1/7*((3*b*c*d^2 + a*d^3)*e^2 + 6*(b*c^2*d + a*c*d^2)*e*f + (b*c^3 +
3*a*c^2*d)*f^2)*x^7 + a*c^3*e^2*x + 1/5*(a*c^3*f^2 + 3*(b*c^2*d + a*c*d^2)*e^2 + 2*(b*c^3 + 3*a*c^2*d)*e*f)*x^
5 + 1/3*(2*a*c^3*e*f + (b*c^3 + 3*a*c^2*d)*e^2)*x^3

________________________________________________________________________________________

mupad [B]  time = 0.85, size = 233, normalized size = 1.03 \begin {gather*} x^5\,\left (\frac {2\,b\,c^3\,e\,f}{5}+\frac {a\,c^3\,f^2}{5}+\frac {3\,b\,c^2\,d\,e^2}{5}+\frac {6\,a\,c^2\,d\,e\,f}{5}+\frac {3\,a\,c\,d^2\,e^2}{5}\right )+x^9\,\left (\frac {b\,c^2\,d\,f^2}{3}+\frac {2\,b\,c\,d^2\,e\,f}{3}+\frac {a\,c\,d^2\,f^2}{3}+\frac {b\,d^3\,e^2}{9}+\frac {2\,a\,d^3\,e\,f}{9}\right )+x^7\,\left (\frac {b\,c^3\,f^2}{7}+\frac {6\,b\,c^2\,d\,e\,f}{7}+\frac {3\,a\,c^2\,d\,f^2}{7}+\frac {3\,b\,c\,d^2\,e^2}{7}+\frac {6\,a\,c\,d^2\,e\,f}{7}+\frac {a\,d^3\,e^2}{7}\right )+\frac {b\,d^3\,f^2\,x^{13}}{13}+\frac {c^2\,e\,x^3\,\left (2\,a\,c\,f+3\,a\,d\,e+b\,c\,e\right )}{3}+\frac {d^2\,f\,x^{11}\,\left (a\,d\,f+3\,b\,c\,f+2\,b\,d\,e\right )}{11}+a\,c^3\,e^2\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^5*((a*c^3*f^2)/5 + (2*b*c^3*e*f)/5 + (3*a*c*d^2*e^2)/5 + (3*b*c^2*d*e^2)/5 + (6*a*c^2*d*e*f)/5) + x^9*((b*d^
3*e^2)/9 + (2*a*d^3*e*f)/9 + (a*c*d^2*f^2)/3 + (b*c^2*d*f^2)/3 + (2*b*c*d^2*e*f)/3) + x^7*((a*d^3*e^2)/7 + (b*
c^3*f^2)/7 + (3*a*c^2*d*f^2)/7 + (3*b*c*d^2*e^2)/7 + (6*a*c*d^2*e*f)/7 + (6*b*c^2*d*e*f)/7) + (b*d^3*f^2*x^13)
/13 + (c^2*e*x^3*(2*a*c*f + 3*a*d*e + b*c*e))/3 + (d^2*f*x^11*(a*d*f + 3*b*c*f + 2*b*d*e))/11 + a*c^3*e^2*x

________________________________________________________________________________________

sympy [A]  time = 0.11, size = 304, normalized size = 1.35 \begin {gather*} a c^{3} e^{2} x + \frac {b d^{3} f^{2} x^{13}}{13} + x^{11} \left (\frac {a d^{3} f^{2}}{11} + \frac {3 b c d^{2} f^{2}}{11} + \frac {2 b d^{3} e f}{11}\right ) + x^{9} \left (\frac {a c d^{2} f^{2}}{3} + \frac {2 a d^{3} e f}{9} + \frac {b c^{2} d f^{2}}{3} + \frac {2 b c d^{2} e f}{3} + \frac {b d^{3} e^{2}}{9}\right ) + x^{7} \left (\frac {3 a c^{2} d f^{2}}{7} + \frac {6 a c d^{2} e f}{7} + \frac {a d^{3} e^{2}}{7} + \frac {b c^{3} f^{2}}{7} + \frac {6 b c^{2} d e f}{7} + \frac {3 b c d^{2} e^{2}}{7}\right ) + x^{5} \left (\frac {a c^{3} f^{2}}{5} + \frac {6 a c^{2} d e f}{5} + \frac {3 a c d^{2} e^{2}}{5} + \frac {2 b c^{3} e f}{5} + \frac {3 b c^{2} d e^{2}}{5}\right ) + x^{3} \left (\frac {2 a c^{3} e f}{3} + a c^{2} d e^{2} + \frac {b c^{3} e^{2}}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c**3*e**2*x + b*d**3*f**2*x**13/13 + x**11*(a*d**3*f**2/11 + 3*b*c*d**2*f**2/11 + 2*b*d**3*e*f/11) + x**9*(a
*c*d**2*f**2/3 + 2*a*d**3*e*f/9 + b*c**2*d*f**2/3 + 2*b*c*d**2*e*f/3 + b*d**3*e**2/9) + x**7*(3*a*c**2*d*f**2/
7 + 6*a*c*d**2*e*f/7 + a*d**3*e**2/7 + b*c**3*f**2/7 + 6*b*c**2*d*e*f/7 + 3*b*c*d**2*e**2/7) + x**5*(a*c**3*f*
*2/5 + 6*a*c**2*d*e*f/5 + 3*a*c*d**2*e**2/5 + 2*b*c**3*e*f/5 + 3*b*c**2*d*e**2/5) + x**3*(2*a*c**3*e*f/3 + a*c
**2*d*e**2 + b*c**3*e**2/3)

________________________________________________________________________________________