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

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

[Out]

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

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Rubi [A]  time = 0.214069, antiderivative size = 226, normalized size of antiderivative = 1., 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} \[ \frac{1}{9} f x^9 \left (a d f (2 c f+3 d e)+b \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )\right )+\frac{1}{7} x^7 \left (a f \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )+b e \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )\right )+\frac{1}{5} e x^5 \left (a \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )+b c e (3 c f+2 d e)\right )+\frac{1}{3} c e^2 x^3 (3 a c f+2 a d e+b c e)+\frac{1}{11} d f^2 x^{11} (a d f+2 b c f+3 b d e)+a c^2 e^3 x+\frac{1}{13} b d^2 f^3 x^{13} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.081365, size = 226, normalized size = 1. \[ \frac{1}{9} f x^9 \left (a d f (2 c f+3 d e)+b \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )\right )+\frac{1}{7} x^7 \left (a f \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )+b e \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )\right )+\frac{1}{5} e x^5 \left (a \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )+b c e (3 c f+2 d e)\right )+\frac{1}{3} c e^2 x^3 (3 a c f+2 a d e+b c e)+\frac{1}{11} d f^2 x^{11} (a d f+2 b c f+3 b d e)+a c^2 e^3 x+\frac{1}{13} b d^2 f^3 x^{13} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 237, normalized size = 1.1 \begin{align*}{\frac{b{d}^{2}{f}^{3}{x}^{13}}{13}}+{\frac{ \left ( \left ( a{d}^{2}+2\,bcd \right ){f}^{3}+3\,b{d}^{2}e{f}^{2} \right ){x}^{11}}{11}}+{\frac{ \left ( \left ( 2\,acd+b{c}^{2} \right ){f}^{3}+3\, \left ( a{d}^{2}+2\,bcd \right ) e{f}^{2}+3\,b{d}^{2}{e}^{2}f \right ){x}^{9}}{9}}+{\frac{ \left ( a{c}^{2}{f}^{3}+3\, \left ( 2\,acd+b{c}^{2} \right ) e{f}^{2}+3\, \left ( a{d}^{2}+2\,bcd \right ){e}^{2}f+b{d}^{2}{e}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,a{c}^{2}e{f}^{2}+3\, \left ( 2\,acd+b{c}^{2} \right ){e}^{2}f+ \left ( a{d}^{2}+2\,bcd \right ){e}^{3} \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,a{c}^{2}{e}^{2}f+ \left ( 2\,acd+b{c}^{2} \right ){e}^{3} \right ){x}^{3}}{3}}+a{c}^{2}{e}^{3}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.03594, size = 319, normalized size = 1.41 \begin{align*} \frac{1}{13} \, b d^{2} f^{3} x^{13} + \frac{1}{11} \,{\left (3 \, b d^{2} e f^{2} +{\left (2 \, b c d + a d^{2}\right )} f^{3}\right )} x^{11} + \frac{1}{9} \,{\left (3 \, b d^{2} e^{2} f + 3 \,{\left (2 \, b c d + a d^{2}\right )} e f^{2} +{\left (b c^{2} + 2 \, a c d\right )} f^{3}\right )} x^{9} + \frac{1}{7} \,{\left (b d^{2} e^{3} + a c^{2} f^{3} + 3 \,{\left (2 \, b c d + a d^{2}\right )} e^{2} f + 3 \,{\left (b c^{2} + 2 \, a c d\right )} e f^{2}\right )} x^{7} + a c^{2} e^{3} x + \frac{1}{5} \,{\left (3 \, a c^{2} e f^{2} +{\left (2 \, b c d + a d^{2}\right )} e^{3} + 3 \,{\left (b c^{2} + 2 \, a c d\right )} e^{2} f\right )} x^{5} + \frac{1}{3} \,{\left (3 \, a c^{2} e^{2} f +{\left (b c^{2} + 2 \, a c d\right )} e^{3}\right )} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.2578, size = 676, normalized size = 2.99 \begin{align*} \frac{1}{13} x^{13} f^{3} d^{2} b + \frac{3}{11} x^{11} f^{2} e d^{2} b + \frac{2}{11} x^{11} f^{3} d c b + \frac{1}{11} x^{11} f^{3} d^{2} a + \frac{1}{3} x^{9} f e^{2} d^{2} b + \frac{2}{3} x^{9} f^{2} e d c b + \frac{1}{9} x^{9} f^{3} c^{2} b + \frac{1}{3} x^{9} f^{2} e d^{2} a + \frac{2}{9} x^{9} f^{3} d c a + \frac{1}{7} x^{7} e^{3} d^{2} b + \frac{6}{7} x^{7} f e^{2} d c b + \frac{3}{7} x^{7} f^{2} e c^{2} b + \frac{3}{7} x^{7} f e^{2} d^{2} a + \frac{6}{7} x^{7} f^{2} e d c a + \frac{1}{7} x^{7} f^{3} c^{2} a + \frac{2}{5} x^{5} e^{3} d c b + \frac{3}{5} x^{5} f e^{2} c^{2} b + \frac{1}{5} x^{5} e^{3} d^{2} a + \frac{6}{5} x^{5} f e^{2} d c a + \frac{3}{5} x^{5} f^{2} e c^{2} a + \frac{1}{3} x^{3} e^{3} c^{2} b + \frac{2}{3} x^{3} e^{3} d c a + x^{3} f e^{2} c^{2} a + x e^{3} c^{2} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.098201, size = 304, normalized size = 1.35 \begin{align*} a c^{2} e^{3} x + \frac{b d^{2} f^{3} x^{13}}{13} + x^{11} \left (\frac{a d^{2} f^{3}}{11} + \frac{2 b c d f^{3}}{11} + \frac{3 b d^{2} e f^{2}}{11}\right ) + x^{9} \left (\frac{2 a c d f^{3}}{9} + \frac{a d^{2} e f^{2}}{3} + \frac{b c^{2} f^{3}}{9} + \frac{2 b c d e f^{2}}{3} + \frac{b d^{2} e^{2} f}{3}\right ) + x^{7} \left (\frac{a c^{2} f^{3}}{7} + \frac{6 a c d e f^{2}}{7} + \frac{3 a d^{2} e^{2} f}{7} + \frac{3 b c^{2} e f^{2}}{7} + \frac{6 b c d e^{2} f}{7} + \frac{b d^{2} e^{3}}{7}\right ) + x^{5} \left (\frac{3 a c^{2} e f^{2}}{5} + \frac{6 a c d e^{2} f}{5} + \frac{a d^{2} e^{3}}{5} + \frac{3 b c^{2} e^{2} f}{5} + \frac{2 b c d e^{3}}{5}\right ) + x^{3} \left (a c^{2} e^{2} f + \frac{2 a c d e^{3}}{3} + \frac{b c^{2} e^{3}}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.17603, size = 382, normalized size = 1.69 \begin{align*} \frac{1}{13} \, b d^{2} f^{3} x^{13} + \frac{2}{11} \, b c d f^{3} x^{11} + \frac{1}{11} \, a d^{2} f^{3} x^{11} + \frac{3}{11} \, b d^{2} f^{2} x^{11} e + \frac{1}{9} \, b c^{2} f^{3} x^{9} + \frac{2}{9} \, a c d f^{3} x^{9} + \frac{2}{3} \, b c d f^{2} x^{9} e + \frac{1}{3} \, a d^{2} f^{2} x^{9} e + \frac{1}{3} \, b d^{2} f x^{9} e^{2} + \frac{1}{7} \, a c^{2} f^{3} x^{7} + \frac{3}{7} \, b c^{2} f^{2} x^{7} e + \frac{6}{7} \, a c d f^{2} x^{7} e + \frac{6}{7} \, b c d f x^{7} e^{2} + \frac{3}{7} \, a d^{2} f x^{7} e^{2} + \frac{1}{7} \, b d^{2} x^{7} e^{3} + \frac{3}{5} \, a c^{2} f^{2} x^{5} e + \frac{3}{5} \, b c^{2} f x^{5} e^{2} + \frac{6}{5} \, a c d f x^{5} e^{2} + \frac{2}{5} \, b c d x^{5} e^{3} + \frac{1}{5} \, a d^{2} x^{5} e^{3} + a c^{2} f x^{3} e^{2} + \frac{1}{3} \, b c^{2} x^{3} e^{3} + \frac{2}{3} \, a c d x^{3} e^{3} + a c^{2} x e^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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