3.2.56 \(\int \frac {(a+b x^2)^2 (c+d x^2)^2}{x^2} \, dx\)

Optimal. Leaf size=81 \[ \frac {1}{3} x^3 \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac {a^2 c^2}{x}+\frac {2}{5} b d x^5 (a d+b c)+2 a c x (a d+b c)+\frac {1}{7} b^2 d^2 x^7 \]

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Rubi [A]  time = 0.04, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {448} \begin {gather*} \frac {1}{3} x^3 \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac {a^2 c^2}{x}+\frac {2}{5} b d x^5 (a d+b c)+2 a c x (a d+b c)+\frac {1}{7} b^2 d^2 x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 448

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 81, normalized size = 1.00 \begin {gather*} \frac {1}{3} x^3 \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac {a^2 c^2}{x}+\frac {2}{5} b d x^5 (a d+b c)+2 a c x (a d+b c)+\frac {1}{7} b^2 d^2 x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.70, size = 87, normalized size = 1.07 \begin {gather*} \frac {15 \, b^{2} d^{2} x^{8} + 42 \, {\left (b^{2} c d + a b d^{2}\right )} x^{6} + 35 \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} - 105 \, a^{2} c^{2} + 210 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2}}{105 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(15*b^2*d^2*x^8 + 42*(b^2*c*d + a*b*d^2)*x^6 + 35*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4 - 105*a^2*c^2 + 21
0*(a*b*c^2 + a^2*c*d)*x^2)/x

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giac [A]  time = 0.27, size = 90, normalized size = 1.11 \begin {gather*} \frac {1}{7} \, b^{2} d^{2} x^{7} + \frac {2}{5} \, b^{2} c d x^{5} + \frac {2}{5} \, a b d^{2} x^{5} + \frac {1}{3} \, b^{2} c^{2} x^{3} + \frac {4}{3} \, a b c d x^{3} + \frac {1}{3} \, a^{2} d^{2} x^{3} + 2 \, a b c^{2} x + 2 \, a^{2} c d x - \frac {a^{2} c^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^2*d^2*x^7 + 2/5*b^2*c*d*x^5 + 2/5*a*b*d^2*x^5 + 1/3*b^2*c^2*x^3 + 4/3*a*b*c*d*x^3 + 1/3*a^2*d^2*x^3 + 2*
a*b*c^2*x + 2*a^2*c*d*x - a^2*c^2/x

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maple [A]  time = 0.00, size = 91, normalized size = 1.12 \begin {gather*} \frac {b^{2} d^{2} x^{7}}{7}+\frac {2 a b \,d^{2} x^{5}}{5}+\frac {2 b^{2} c d \,x^{5}}{5}+\frac {a^{2} d^{2} x^{3}}{3}+\frac {4 a b c d \,x^{3}}{3}+\frac {b^{2} c^{2} x^{3}}{3}+2 a^{2} c d x +2 a b \,c^{2} x -\frac {a^{2} c^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^2*d^2*x^7+2/5*x^5*a*b*d^2+2/5*x^5*b^2*c*d+1/3*x^3*a^2*d^2+4/3*x^3*a*b*c*d+1/3*x^3*b^2*c^2+2*a^2*c*d*x+2*
a*b*c^2*x-a^2*c^2/x

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maxima [A]  time = 1.10, size = 83, normalized size = 1.02 \begin {gather*} \frac {1}{7} \, b^{2} d^{2} x^{7} + \frac {2}{5} \, {\left (b^{2} c d + a b d^{2}\right )} x^{5} + \frac {1}{3} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{3} - \frac {a^{2} c^{2}}{x} + 2 \, {\left (a b c^{2} + a^{2} c d\right )} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.03, size = 76, normalized size = 0.94 \begin {gather*} x^3\,\left (\frac {a^2\,d^2}{3}+\frac {4\,a\,b\,c\,d}{3}+\frac {b^2\,c^2}{3}\right )-\frac {a^2\,c^2}{x}+\frac {b^2\,d^2\,x^7}{7}+2\,a\,c\,x\,\left (a\,d+b\,c\right )+\frac {2\,b\,d\,x^5\,\left (a\,d+b\,c\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.18, size = 92, normalized size = 1.14 \begin {gather*} - \frac {a^{2} c^{2}}{x} + \frac {b^{2} d^{2} x^{7}}{7} + x^{5} \left (\frac {2 a b d^{2}}{5} + \frac {2 b^{2} c d}{5}\right ) + x^{3} \left (\frac {a^{2} d^{2}}{3} + \frac {4 a b c d}{3} + \frac {b^{2} c^{2}}{3}\right ) + x \left (2 a^{2} c d + 2 a b c^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**2*c**2/x + b**2*d**2*x**7/7 + x**5*(2*a*b*d**2/5 + 2*b**2*c*d/5) + x**3*(a**2*d**2/3 + 4*a*b*c*d/3 + b**2*
c**2/3) + x*(2*a**2*c*d + 2*a*b*c**2)

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