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

Optimal. Leaf size=50 \[ a^2 c x+\frac {1}{5} b x^5 (2 a d+b c)+\frac {1}{3} a x^3 (a d+2 b c)+\frac {1}{7} b^2 d x^7 \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 50, 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} \[ a^2 c x+\frac {1}{5} b x^5 (2 a d+b c)+\frac {1}{3} a x^3 (a d+2 b c)+\frac {1}{7} b^2 d x^7 \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2*(c + d*x^2),x]

[Out]

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

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

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Mathematica [A]  time = 0.01, size = 50, normalized size = 1.00 \[ a^2 c x+\frac {1}{5} b x^5 (2 a d+b c)+\frac {1}{3} a x^3 (a d+2 b c)+\frac {1}{7} b^2 d x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.38, size = 50, normalized size = 1.00 \[ \frac {1}{7} x^{7} d b^{2} + \frac {1}{5} x^{5} c b^{2} + \frac {2}{5} x^{5} d b a + \frac {2}{3} x^{3} c b a + \frac {1}{3} x^{3} d a^{2} + x c a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.24, size = 50, normalized size = 1.00 \[ \frac {1}{7} \, b^{2} d x^{7} + \frac {1}{5} \, b^{2} c x^{5} + \frac {2}{5} \, a b d x^{5} + \frac {2}{3} \, a b c x^{3} + \frac {1}{3} \, a^{2} d x^{3} + a^{2} c x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 49, normalized size = 0.98 \[ \frac {b^{2} d \,x^{7}}{7}+\frac {\left (2 a b d +b^{2} c \right ) x^{5}}{5}+a^{2} c x +\frac {\left (a^{2} d +2 a b c \right ) x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.05, size = 48, normalized size = 0.96 \[ \frac {1}{7} \, b^{2} d x^{7} + \frac {1}{5} \, {\left (b^{2} c + 2 \, a b d\right )} x^{5} + a^{2} c x + \frac {1}{3} \, {\left (2 \, a b c + a^{2} d\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.04, size = 48, normalized size = 0.96 \[ x^3\,\left (\frac {d\,a^2}{3}+\frac {2\,b\,c\,a}{3}\right )+x^5\,\left (\frac {c\,b^2}{5}+\frac {2\,a\,d\,b}{5}\right )+\frac {b^2\,d\,x^7}{7}+a^2\,c\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^2*(c + d*x^2),x)

[Out]

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

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sympy [A]  time = 0.07, size = 53, normalized size = 1.06 \[ a^{2} c x + \frac {b^{2} d x^{7}}{7} + x^{5} \left (\frac {2 a b d}{5} + \frac {b^{2} c}{5}\right ) + x^{3} \left (\frac {a^{2} d}{3} + \frac {2 a b c}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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