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

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

[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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Rubi [A]  time = 0.064207, antiderivative size = 94, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0194268, size = 94, normalized size = 1. \[ \frac{2}{5} c^2 d x^5 (3 a d+2 b c)+\frac{1}{3} c^3 x^3 (4 a d+b c)+\frac{1}{9} d^3 x^9 (a d+4 b c)+\frac{2}{7} c d^2 x^7 (2 a d+3 b c)+a c^4 x+\frac{1}{11} b d^4 x^{11} \]

Antiderivative was successfully verified.

[In]

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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Maple [A]  time = 0.002, size = 97, normalized size = 1. \begin{align*}{\frac{b{d}^{4}{x}^{11}}{11}}+{\frac{ \left ( a{d}^{4}+4\,bc{d}^{3} \right ){x}^{9}}{9}}+{\frac{ \left ( 4\,ac{d}^{3}+6\,b{c}^{2}{d}^{2} \right ){x}^{7}}{7}}+{\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}}+a{c}^{4}x \end{align*}

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 = 1.13958, size = 130, normalized size = 1.38 \begin{align*} \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} \end{align*}

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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Fricas [A]  time = 1.58752, size = 231, normalized size = 2.46 \begin{align*} \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 \end{align*}

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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Sympy [A]  time = 0.075173, size = 107, normalized size = 1.14 \begin{align*} 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 ) \end{align*}

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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Giac [A]  time = 1.82843, size = 132, normalized size = 1.4 \begin{align*} \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 \end{align*}

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