3.15 \(\int (a+b x^2)^3 (c+d x^2)^2 \, dx\)

Optimal. Leaf size=122 \[ \frac{1}{7} b x^7 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{1}{5} a x^5 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{3} a^2 c x^3 (2 a d+3 b c)+a^3 c^2 x+\frac{1}{9} b^2 d x^9 (3 a d+2 b c)+\frac{1}{11} b^3 d^2 x^{11} \]

[Out]

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

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Rubi [A]  time = 0.0700041, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {373} \[ \frac{1}{7} b x^7 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{1}{5} a x^5 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{3} a^2 c x^3 (2 a d+3 b c)+a^3 c^2 x+\frac{1}{9} b^2 d x^9 (3 a d+2 b c)+\frac{1}{11} b^3 d^2 x^{11} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^3*(c + d*x^2)^2,x]

[Out]

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

Mathematica [A]  time = 0.020559, size = 122, normalized size = 1. \[ \frac{1}{7} b x^7 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{1}{5} a x^5 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{3} a^2 c x^3 (2 a d+3 b c)+a^3 c^2 x+\frac{1}{9} b^2 d x^9 (3 a d+2 b c)+\frac{1}{11} b^3 d^2 x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.002, size = 125, normalized size = 1. \begin{align*}{\frac{{b}^{3}{d}^{2}{x}^{11}}{11}}+{\frac{ \left ( 3\,a{b}^{2}{d}^{2}+2\,{b}^{3}cd \right ){x}^{9}}{9}}+{\frac{ \left ( 3\,{a}^{2}b{d}^{2}+6\,a{b}^{2}cd+{b}^{3}{c}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ({a}^{3}{d}^{2}+6\,{a}^{2}bcd+3\,a{b}^{2}{c}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{a}^{3}cd+3\,{a}^{2}b{c}^{2} \right ){x}^{3}}{3}}+{a}^{3}{c}^{2}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.982753, size = 167, normalized size = 1.37 \begin{align*} \frac{1}{11} \, b^{3} d^{2} x^{11} + \frac{1}{9} \,{\left (2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right )} x^{9} + \frac{1}{7} \,{\left (b^{3} c^{2} + 6 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x^{7} + a^{3} c^{2} x + \frac{1}{5} \,{\left (3 \, a b^{2} c^{2} + 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{5} + \frac{1}{3} \,{\left (3 \, a^{2} b c^{2} + 2 \, a^{3} c d\right )} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.47619, size = 296, normalized size = 2.43 \begin{align*} \frac{1}{11} x^{11} d^{2} b^{3} + \frac{2}{9} x^{9} d c b^{3} + \frac{1}{3} x^{9} d^{2} b^{2} a + \frac{1}{7} x^{7} c^{2} b^{3} + \frac{6}{7} x^{7} d c b^{2} a + \frac{3}{7} x^{7} d^{2} b a^{2} + \frac{3}{5} x^{5} c^{2} b^{2} a + \frac{6}{5} x^{5} d c b a^{2} + \frac{1}{5} x^{5} d^{2} a^{3} + x^{3} c^{2} b a^{2} + \frac{2}{3} x^{3} d c a^{3} + x c^{2} a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.080502, size = 136, normalized size = 1.11 \begin{align*} a^{3} c^{2} x + \frac{b^{3} d^{2} x^{11}}{11} + x^{9} \left (\frac{a b^{2} d^{2}}{3} + \frac{2 b^{3} c d}{9}\right ) + x^{7} \left (\frac{3 a^{2} b d^{2}}{7} + \frac{6 a b^{2} c d}{7} + \frac{b^{3} c^{2}}{7}\right ) + x^{5} \left (\frac{a^{3} d^{2}}{5} + \frac{6 a^{2} b c d}{5} + \frac{3 a b^{2} c^{2}}{5}\right ) + x^{3} \left (\frac{2 a^{3} c d}{3} + a^{2} b c^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.08105, size = 177, normalized size = 1.45 \begin{align*} \frac{1}{11} \, b^{3} d^{2} x^{11} + \frac{2}{9} \, b^{3} c d x^{9} + \frac{1}{3} \, a b^{2} d^{2} x^{9} + \frac{1}{7} \, b^{3} c^{2} x^{7} + \frac{6}{7} \, a b^{2} c d x^{7} + \frac{3}{7} \, a^{2} b d^{2} x^{7} + \frac{3}{5} \, a b^{2} c^{2} x^{5} + \frac{6}{5} \, a^{2} b c d x^{5} + \frac{1}{5} \, a^{3} d^{2} x^{5} + a^{2} b c^{2} x^{3} + \frac{2}{3} \, a^{3} c d x^{3} + a^{3} c^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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