3.165 \(\int \frac{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{x^2} \, dx\)

Optimal. Leaf size=120 \[ \frac{1}{5} d x^5 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{3} c x^3 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac{a^2 c^3}{x}+a c^2 x (3 a d+2 b c)+\frac{1}{7} b d^2 x^7 (2 a d+3 b c)+\frac{1}{9} b^2 d^3 x^9 \]

[Out]

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

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Rubi [A]  time = 0.155721, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{1}{5} d x^5 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{3} c x^3 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac{a^2 c^3}{x}+a c^2 x (3 a d+2 b c)+\frac{1}{7} b d^2 x^7 (2 a d+3 b c)+\frac{1}{9} b^2 d^3 x^9 \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^2*(c + d*x^2)^3)/x^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 34.8776, size = 117, normalized size = 0.98 \[ - \frac{a^{2} c^{3}}{x} + a c^{2} x \left (3 a d + 2 b c\right ) + \frac{b^{2} d^{3} x^{9}}{9} + \frac{b d^{2} x^{7} \left (2 a d + 3 b c\right )}{7} + \frac{c x^{3} \left (3 a^{2} d^{2} + 6 a b c d + b^{2} c^{2}\right )}{3} + \frac{d x^{5} \left (a^{2} d^{2} + 6 a b c d + 3 b^{2} c^{2}\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2*(d*x**2+c)**3/x**2,x)

[Out]

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

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Mathematica [A]  time = 0.0719165, size = 120, normalized size = 1. \[ \frac{1}{5} d x^5 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{3} c x^3 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac{a^2 c^3}{x}+a c^2 x (3 a d+2 b c)+\frac{1}{7} b d^2 x^7 (2 a d+3 b c)+\frac{1}{9} b^2 d^3 x^9 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 131, normalized size = 1.1 \[{\frac{{b}^{2}{d}^{3}{x}^{9}}{9}}+{\frac{2\,{x}^{7}ab{d}^{3}}{7}}+{\frac{3\,{x}^{7}{b}^{2}c{d}^{2}}{7}}+{\frac{{x}^{5}{a}^{2}{d}^{3}}{5}}+{\frac{6\,{x}^{5}abc{d}^{2}}{5}}+{\frac{3\,{x}^{5}{b}^{2}{c}^{2}d}{5}}+{x}^{3}{a}^{2}c{d}^{2}+2\,{x}^{3}ab{c}^{2}d+{\frac{{x}^{3}{b}^{2}{c}^{3}}{3}}+3\,x{a}^{2}{c}^{2}d+2\,xab{c}^{3}-{\frac{{a}^{2}{c}^{3}}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(d*x^2+c)^3/x^2,x)

[Out]

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

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Maxima [A]  time = 1.3534, size = 167, normalized size = 1.39 \[ \frac{1}{9} \, b^{2} d^{3} x^{9} + \frac{1}{7} \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{7} + \frac{1}{5} \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{5} - \frac{a^{2} c^{3}}{x} + \frac{1}{3} \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{3} +{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.225515, size = 174, normalized size = 1.45 \[ \frac{35 \, b^{2} d^{3} x^{10} + 45 \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{8} + 63 \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} - 315 \, a^{2} c^{3} + 105 \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} + 315 \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}}{315 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/315*(35*b^2*d^3*x^10 + 45*(3*b^2*c*d^2 + 2*a*b*d^3)*x^8 + 63*(3*b^2*c^2*d + 6*
a*b*c*d^2 + a^2*d^3)*x^6 - 315*a^2*c^3 + 105*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^
2)*x^4 + 315*(2*a*b*c^3 + 3*a^2*c^2*d)*x^2)/x

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Sympy [A]  time = 1.52237, size = 131, normalized size = 1.09 \[ - \frac{a^{2} c^{3}}{x} + \frac{b^{2} d^{3} x^{9}}{9} + x^{7} \left (\frac{2 a b d^{3}}{7} + \frac{3 b^{2} c d^{2}}{7}\right ) + x^{5} \left (\frac{a^{2} d^{3}}{5} + \frac{6 a b c d^{2}}{5} + \frac{3 b^{2} c^{2} d}{5}\right ) + x^{3} \left (a^{2} c d^{2} + 2 a b c^{2} d + \frac{b^{2} c^{3}}{3}\right ) + x \left (3 a^{2} c^{2} d + 2 a b c^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2*(d*x**2+c)**3/x**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.219856, size = 176, normalized size = 1.47 \[ \frac{1}{9} \, b^{2} d^{3} x^{9} + \frac{3}{7} \, b^{2} c d^{2} x^{7} + \frac{2}{7} \, a b d^{3} x^{7} + \frac{3}{5} \, b^{2} c^{2} d x^{5} + \frac{6}{5} \, a b c d^{2} x^{5} + \frac{1}{5} \, a^{2} d^{3} x^{5} + \frac{1}{3} \, b^{2} c^{3} x^{3} + 2 \, a b c^{2} d x^{3} + a^{2} c d^{2} x^{3} + 2 \, a b c^{3} x + 3 \, a^{2} c^{2} d x - \frac{a^{2} c^{3}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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