3.398 \(\int \frac{\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x^{7/2}} \, dx\)

Optimal. Leaf size=61 \[ -\frac{2 a^2 c}{5 x^{5/2}}+\frac{2}{3} b x^{3/2} (2 a d+b c)-\frac{2 a (a d+2 b c)}{\sqrt{x}}+\frac{2}{7} b^2 d x^{7/2} \]

[Out]

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

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Rubi [A]  time = 0.0879057, antiderivative size = 61, 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 \[ -\frac{2 a^2 c}{5 x^{5/2}}+\frac{2}{3} b x^{3/2} (2 a d+b c)-\frac{2 a (a d+2 b c)}{\sqrt{x}}+\frac{2}{7} b^2 d x^{7/2} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^2*(c + d*x^2))/x^(7/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 12.1637, size = 61, normalized size = 1. \[ - \frac{2 a^{2} c}{5 x^{\frac{5}{2}}} - \frac{2 a \left (a d + 2 b c\right )}{\sqrt{x}} + \frac{2 b^{2} d x^{\frac{7}{2}}}{7} + \frac{2 b x^{\frac{3}{2}} \left (2 a d + b c\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2*(d*x**2+c)/x**(7/2),x)

[Out]

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

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Mathematica [A]  time = 0.0296077, size = 57, normalized size = 0.93 \[ \frac{-42 a^2 \left (c+5 d x^2\right )+140 a b x^2 \left (d x^2-3 c\right )+10 b^2 x^4 \left (7 c+3 d x^2\right )}{105 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^2*(c + d*x^2))/x^(7/2),x]

[Out]

(140*a*b*x^2*(-3*c + d*x^2) + 10*b^2*x^4*(7*c + 3*d*x^2) - 42*a^2*(c + 5*d*x^2))
/(105*x^(5/2))

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Maple [A]  time = 0.008, size = 56, normalized size = 0.9 \[ -{\frac{-30\,{b}^{2}d{x}^{6}-140\,{x}^{4}abd-70\,{b}^{2}c{x}^{4}+210\,{x}^{2}{a}^{2}d+420\,abc{x}^{2}+42\,{a}^{2}c}{105}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(d*x^2+c)/x^(7/2),x)

[Out]

-2/105*(-15*b^2*d*x^6-70*a*b*d*x^4-35*b^2*c*x^4+105*a^2*d*x^2+210*a*b*c*x^2+21*a
^2*c)/x^(5/2)

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Maxima [A]  time = 1.34331, size = 72, normalized size = 1.18 \[ \frac{2}{7} \, b^{2} d x^{\frac{7}{2}} + \frac{2}{3} \,{\left (b^{2} c + 2 \, a b d\right )} x^{\frac{3}{2}} - \frac{2 \,{\left (a^{2} c + 5 \,{\left (2 \, a b c + a^{2} d\right )} x^{2}\right )}}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)/x^(7/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.221629, size = 72, normalized size = 1.18 \[ \frac{2 \,{\left (15 \, b^{2} d x^{6} + 35 \,{\left (b^{2} c + 2 \, a b d\right )} x^{4} - 21 \, a^{2} c - 105 \,{\left (2 \, a b c + a^{2} d\right )} x^{2}\right )}}{105 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)/x^(7/2),x, algorithm="fricas")

[Out]

2/105*(15*b^2*d*x^6 + 35*(b^2*c + 2*a*b*d)*x^4 - 21*a^2*c - 105*(2*a*b*c + a^2*d
)*x^2)/x^(5/2)

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Sympy [A]  time = 15.5271, size = 76, normalized size = 1.25 \[ - \frac{2 a^{2} c}{5 x^{\frac{5}{2}}} - \frac{2 a^{2} d}{\sqrt{x}} - \frac{4 a b c}{\sqrt{x}} + \frac{4 a b d x^{\frac{3}{2}}}{3} + \frac{2 b^{2} c x^{\frac{3}{2}}}{3} + \frac{2 b^{2} d x^{\frac{7}{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2*(d*x**2+c)/x**(7/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.229806, size = 74, normalized size = 1.21 \[ \frac{2}{7} \, b^{2} d x^{\frac{7}{2}} + \frac{2}{3} \, b^{2} c x^{\frac{3}{2}} + \frac{4}{3} \, a b d x^{\frac{3}{2}} - \frac{2 \,{\left (10 \, a b c x^{2} + 5 \, a^{2} d x^{2} + a^{2} c\right )}}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)/x^(7/2),x, algorithm="giac")

[Out]

2/7*b^2*d*x^(7/2) + 2/3*b^2*c*x^(3/2) + 4/3*a*b*d*x^(3/2) - 2/5*(10*a*b*c*x^2 +
5*a^2*d*x^2 + a^2*c)/x^(5/2)