3.358 \(\int \frac{\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^{7/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0885723, 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 \[ -\frac{2 a^2 A}{5 x^{5/2}}+\frac{2}{3} b x^{3/2} (2 a B+A b)-\frac{2 a (a B+2 A b)}{\sqrt{x}}+\frac{2}{7} b^2 B x^{7/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.7748, size = 61, normalized size = 1. \[ - \frac{2 A a^{2}}{5 x^{\frac{5}{2}}} + \frac{2 B b^{2} x^{\frac{7}{2}}}{7} - \frac{2 a \left (2 A b + B a\right )}{\sqrt{x}} + \frac{2 b x^{\frac{3}{2}} \left (A b + 2 B a\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2*(B*x**2+A)/x**(7/2),x)

[Out]

-2*A*a**2/(5*x**(5/2)) + 2*B*b**2*x**(7/2)/7 - 2*a*(2*A*b + B*a)/sqrt(x) + 2*b*x
**(3/2)*(A*b + 2*B*a)/3

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Mathematica [A]  time = 0.0289204, size = 57, normalized size = 0.93 \[ \frac{-42 a^2 \left (A+5 B x^2\right )+140 a b x^2 \left (B x^2-3 A\right )+10 b^2 x^4 \left (7 A+3 B x^2\right )}{105 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 56, normalized size = 0.9 \[ -{\frac{-30\,{b}^{2}B{x}^{6}-70\,A{b}^{2}{x}^{4}-140\,{x}^{4}abB+420\,aAb{x}^{2}+210\,B{a}^{2}{x}^{2}+42\,{a}^{2}A}{105}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.32982, size = 72, normalized size = 1.18 \[ \frac{2}{7} \, B b^{2} x^{\frac{7}{2}} + \frac{2}{3} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{3}{2}} - \frac{2 \,{\left (A a^{2} + 5 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}\right )}}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*B*b^2*x^(7/2) + 2/3*(2*B*a*b + A*b^2)*x^(3/2) - 2/5*(A*a^2 + 5*(B*a^2 + 2*A*
a*b)*x^2)/x^(5/2)

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Fricas [A]  time = 0.218293, size = 72, normalized size = 1.18 \[ \frac{2 \,{\left (15 \, B b^{2} x^{6} + 35 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} - 21 \, A a^{2} - 105 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}\right )}}{105 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 15.4564, size = 76, normalized size = 1.25 \[ - \frac{2 A a^{2}}{5 x^{\frac{5}{2}}} - \frac{4 A a b}{\sqrt{x}} + \frac{2 A b^{2} x^{\frac{3}{2}}}{3} - \frac{2 B a^{2}}{\sqrt{x}} + \frac{4 B a b x^{\frac{3}{2}}}{3} + \frac{2 B b^{2} x^{\frac{7}{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2*(B*x**2+A)/x**(7/2),x)

[Out]

-2*A*a**2/(5*x**(5/2)) - 4*A*a*b/sqrt(x) + 2*A*b**2*x**(3/2)/3 - 2*B*a**2/sqrt(x
) + 4*B*a*b*x**(3/2)/3 + 2*B*b**2*x**(7/2)/7

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GIAC/XCAS [A]  time = 0.212193, size = 74, normalized size = 1.21 \[ \frac{2}{7} \, B b^{2} x^{\frac{7}{2}} + \frac{4}{3} \, B a b x^{\frac{3}{2}} + \frac{2}{3} \, A b^{2} x^{\frac{3}{2}} - \frac{2 \,{\left (5 \, B a^{2} x^{2} + 10 \, A a b x^{2} + A a^{2}\right )}}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*B*b^2*x^(7/2) + 4/3*B*a*b*x^(3/2) + 2/3*A*b^2*x^(3/2) - 2/5*(5*B*a^2*x^2 + 1
0*A*a*b*x^2 + A*a^2)/x^(5/2)