3.1 \(\int (a+b x^2) (c+d x^2)^4 \, dx\)

Optimal. Leaf size=94 \[ \frac {1}{3} c^3 x^3 (4 a d+b c)+\frac {2}{5} c^2 d x^5 (3 a d+2 b c)+\frac {1}{9} d^3 x^9 (a d+4 b c)+\frac {2}{7} c d^2 x^7 (2 a d+3 b c)+a c^4 x+\frac {1}{11} b d^4 x^{11} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {373} \[ \frac {2}{5} c^2 d x^5 (3 a d+2 b c)+\frac {1}{3} c^3 x^3 (4 a d+b c)+\frac {1}{9} d^3 x^9 (a d+4 b c)+\frac {2}{7} c d^2 x^7 (2 a d+3 b c)+a c^4 x+\frac {1}{11} b d^4 x^{11} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)*(c + d*x^2)^4,x]

[Out]

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

Rule 373

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

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Mathematica [A]  time = 0.02, size = 94, normalized size = 1.00 \[ \frac {1}{3} c^3 x^3 (4 a d+b c)+\frac {2}{5} c^2 d x^5 (3 a d+2 b c)+\frac {1}{9} d^3 x^9 (a d+4 b c)+\frac {2}{7} c d^2 x^7 (2 a d+3 b c)+a c^4 x+\frac {1}{11} b d^4 x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.51, size = 98, normalized size = 1.04 \[ \frac {1}{11} x^{11} d^{4} b + \frac {4}{9} x^{9} d^{3} c b + \frac {1}{9} x^{9} d^{4} a + \frac {6}{7} x^{7} d^{2} c^{2} b + \frac {4}{7} x^{7} d^{3} c a + \frac {4}{5} x^{5} d c^{3} b + \frac {6}{5} x^{5} d^{2} c^{2} a + \frac {1}{3} x^{3} c^{4} b + \frac {4}{3} x^{3} d c^{3} a + x c^{4} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*x^11*d^4*b + 4/9*x^9*d^3*c*b + 1/9*x^9*d^4*a + 6/7*x^7*d^2*c^2*b + 4/7*x^7*d^3*c*a + 4/5*x^5*d*c^3*b + 6/
5*x^5*d^2*c^2*a + 1/3*x^3*c^4*b + 4/3*x^3*d*c^3*a + x*c^4*a

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giac [A]  time = 0.57, size = 98, normalized size = 1.04 \[ \frac {1}{11} \, b d^{4} x^{11} + \frac {4}{9} \, b c d^{3} x^{9} + \frac {1}{9} \, a d^{4} x^{9} + \frac {6}{7} \, b c^{2} d^{2} x^{7} + \frac {4}{7} \, a c d^{3} x^{7} + \frac {4}{5} \, b c^{3} d x^{5} + \frac {6}{5} \, a c^{2} d^{2} x^{5} + \frac {1}{3} \, b c^{4} x^{3} + \frac {4}{3} \, a c^{3} d x^{3} + a c^{4} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*b*d^4*x^11 + 4/9*b*c*d^3*x^9 + 1/9*a*d^4*x^9 + 6/7*b*c^2*d^2*x^7 + 4/7*a*c*d^3*x^7 + 4/5*b*c^3*d*x^5 + 6/
5*a*c^2*d^2*x^5 + 1/3*b*c^4*x^3 + 4/3*a*c^3*d*x^3 + a*c^4*x

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maple [A]  time = 0.00, size = 97, normalized size = 1.03 \[ \frac {b \,d^{4} x^{11}}{11}+\frac {\left (a \,d^{4}+4 b c \,d^{3}\right ) x^{9}}{9}+\frac {\left (4 a c \,d^{3}+6 b \,c^{2} d^{2}\right ) x^{7}}{7}+a \,c^{4} x +\frac {\left (6 a \,c^{2} d^{2}+4 b \,c^{3} d \right ) x^{5}}{5}+\frac {\left (4 a \,c^{3} d +b \,c^{4}\right ) x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*b*d^4*x^11+1/9*(a*d^4+4*b*c*d^3)*x^9+1/7*(4*a*c*d^3+6*b*c^2*d^2)*x^7+1/5*(6*a*c^2*d^2+4*b*c^3*d)*x^5+1/3*
(4*a*c^3*d+b*c^4)*x^3+a*c^4*x

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maxima [A]  time = 1.36, size = 96, normalized size = 1.02 \[ \frac {1}{11} \, b d^{4} x^{11} + \frac {1}{9} \, {\left (4 \, b c d^{3} + a d^{4}\right )} x^{9} + \frac {2}{7} \, {\left (3 \, b c^{2} d^{2} + 2 \, a c d^{3}\right )} x^{7} + a c^{4} x + \frac {2}{5} \, {\left (2 \, b c^{3} d + 3 \, a c^{2} d^{2}\right )} x^{5} + \frac {1}{3} \, {\left (b c^{4} + 4 \, a c^{3} d\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 4.77, size = 88, normalized size = 0.94 \[ x^3\,\left (\frac {b\,c^4}{3}+\frac {4\,a\,d\,c^3}{3}\right )+x^9\,\left (\frac {a\,d^4}{9}+\frac {4\,b\,c\,d^3}{9}\right )+\frac {b\,d^4\,x^{11}}{11}+a\,c^4\,x+\frac {2\,c^2\,d\,x^5\,\left (3\,a\,d+2\,b\,c\right )}{5}+\frac {2\,c\,d^2\,x^7\,\left (2\,a\,d+3\,b\,c\right )}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)*(c + d*x^2)^4,x)

[Out]

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

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sympy [A]  time = 0.09, size = 107, normalized size = 1.14 \[ a c^{4} x + \frac {b d^{4} x^{11}}{11} + x^{9} \left (\frac {a d^{4}}{9} + \frac {4 b c d^{3}}{9}\right ) + x^{7} \left (\frac {4 a c d^{3}}{7} + \frac {6 b c^{2} d^{2}}{7}\right ) + x^{5} \left (\frac {6 a c^{2} d^{2}}{5} + \frac {4 b c^{3} d}{5}\right ) + x^{3} \left (\frac {4 a c^{3} d}{3} + \frac {b c^{4}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c**4*x + b*d**4*x**11/11 + x**9*(a*d**4/9 + 4*b*c*d**3/9) + x**7*(4*a*c*d**3/7 + 6*b*c**2*d**2/7) + x**5*(6*
a*c**2*d**2/5 + 4*b*c**3*d/5) + x**3*(4*a*c**3*d/3 + b*c**4/3)

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