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

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

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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} \begin {gather*} \frac {1}{5} d x^5 (a d+2 b c)+\frac {1}{3} c x^3 (2 a d+b c)+a c^2 x+\frac {1}{7} b d^2 x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)*(c + d*x^2)^2,x]

[Out]

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

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

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a+b x^2\right ) \left (c+d x^2\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 50, normalized size = 1.00 \begin {gather*} \frac {1}{7} x^{7} d^{2} b + \frac {2}{5} x^{5} d c b + \frac {1}{5} x^{5} d^{2} a + \frac {1}{3} x^{3} c^{2} b + \frac {2}{3} x^{3} d c a + x c^{2} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.57, size = 50, normalized size = 1.00 \begin {gather*} \frac {1}{7} \, b d^{2} x^{7} + \frac {2}{5} \, b c d x^{5} + \frac {1}{5} \, a d^{2} x^{5} + \frac {1}{3} \, b c^{2} x^{3} + \frac {2}{3} \, a c d x^{3} + a c^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.35, size = 48, normalized size = 0.96 \begin {gather*} \frac {1}{7} \, b d^{2} x^{7} + \frac {1}{5} \, {\left (2 \, b c d + a d^{2}\right )} x^{5} + a c^{2} x + \frac {1}{3} \, {\left (b c^{2} + 2 \, a c d\right )} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.05, size = 48, normalized size = 0.96 \begin {gather*} x^3\,\left (\frac {b\,c^2}{3}+\frac {2\,a\,d\,c}{3}\right )+x^5\,\left (\frac {a\,d^2}{5}+\frac {2\,b\,c\,d}{5}\right )+\frac {b\,d^2\,x^7}{7}+a\,c^2\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)*(c + d*x^2)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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