3.366 \(\int \frac{\left (a+b x^2\right )^3 \left (A+B x^2\right )}{x^{7/2}} \, dx\)

Optimal. Leaf size=81 \[ -\frac{2 a^3 A}{5 x^{5/2}}-\frac{2 a^2 (a B+3 A b)}{\sqrt{x}}+\frac{2}{7} b^2 x^{7/2} (3 a B+A b)+2 a b x^{3/2} (a B+A b)+\frac{2}{11} b^3 B x^{11/2} \]

[Out]

(-2*a^3*A)/(5*x^(5/2)) - (2*a^2*(3*A*b + a*B))/Sqrt[x] + 2*a*b*(A*b + a*B)*x^(3/
2) + (2*b^2*(A*b + 3*a*B)*x^(7/2))/7 + (2*b^3*B*x^(11/2))/11

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Rubi [A]  time = 0.112557, antiderivative size = 81, 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 \[ -\frac{2 a^3 A}{5 x^{5/2}}-\frac{2 a^2 (a B+3 A b)}{\sqrt{x}}+\frac{2}{7} b^2 x^{7/2} (3 a B+A b)+2 a b x^{3/2} (a B+A b)+\frac{2}{11} b^3 B x^{11/2} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^3*(A + B*x^2))/x^(7/2),x]

[Out]

(-2*a^3*A)/(5*x^(5/2)) - (2*a^2*(3*A*b + a*B))/Sqrt[x] + 2*a*b*(A*b + a*B)*x^(3/
2) + (2*b^2*(A*b + 3*a*B)*x^(7/2))/7 + (2*b^3*B*x^(11/2))/11

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Rubi in Sympy [A]  time = 16.6324, size = 82, normalized size = 1.01 \[ - \frac{2 A a^{3}}{5 x^{\frac{5}{2}}} + \frac{2 B b^{3} x^{\frac{11}{2}}}{11} - \frac{2 a^{2} \left (3 A b + B a\right )}{\sqrt{x}} + 2 a b x^{\frac{3}{2}} \left (A b + B a\right ) + \frac{2 b^{2} x^{\frac{7}{2}} \left (A b + 3 B a\right )}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**3*(B*x**2+A)/x**(7/2),x)

[Out]

-2*A*a**3/(5*x**(5/2)) + 2*B*b**3*x**(11/2)/11 - 2*a**2*(3*A*b + B*a)/sqrt(x) +
2*a*b*x**(3/2)*(A*b + B*a) + 2*b**2*x**(7/2)*(A*b + 3*B*a)/7

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Mathematica [A]  time = 0.0374316, size = 78, normalized size = 0.96 \[ \frac{2 \left (-77 a^3 \left (A+5 B x^2\right )+385 a^2 b x^2 \left (B x^2-3 A\right )+55 a b^2 x^4 \left (7 A+3 B x^2\right )+5 b^3 x^6 \left (11 A+7 B x^2\right )\right )}{385 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^3*(A + B*x^2))/x^(7/2),x]

[Out]

(2*(385*a^2*b*x^2*(-3*A + B*x^2) + 55*a*b^2*x^4*(7*A + 3*B*x^2) - 77*a^3*(A + 5*
B*x^2) + 5*b^3*x^6*(11*A + 7*B*x^2)))/(385*x^(5/2))

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Maple [A]  time = 0.008, size = 80, normalized size = 1. \[ -{\frac{-70\,{b}^{3}B{x}^{8}-110\,{x}^{6}{b}^{3}A-330\,{x}^{6}a{b}^{2}B-770\,Aa{b}^{2}{x}^{4}-770\,B{a}^{2}b{x}^{4}+2310\,A{a}^{2}b{x}^{2}+770\,B{a}^{3}{x}^{2}+154\,{a}^{3}A}{385}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^3*(B*x^2+A)/x^(7/2),x)

[Out]

-2/385*(-35*B*b^3*x^8-55*A*b^3*x^6-165*B*a*b^2*x^6-385*A*a*b^2*x^4-385*B*a^2*b*x
^4+1155*A*a^2*b*x^2+385*B*a^3*x^2+77*A*a^3)/x^(5/2)

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Maxima [A]  time = 1.33754, size = 101, normalized size = 1.25 \[ \frac{2}{11} \, B b^{3} x^{\frac{11}{2}} + \frac{2}{7} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac{7}{2}} + 2 \,{\left (B a^{2} b + A a b^{2}\right )} x^{\frac{3}{2}} - \frac{2 \,{\left (A a^{3} + 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2}\right )}}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^3/x^(7/2),x, algorithm="maxima")

[Out]

2/11*B*b^3*x^(11/2) + 2/7*(3*B*a*b^2 + A*b^3)*x^(7/2) + 2*(B*a^2*b + A*a*b^2)*x^
(3/2) - 2/5*(A*a^3 + 5*(B*a^3 + 3*A*a^2*b)*x^2)/x^(5/2)

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Fricas [A]  time = 0.223549, size = 101, normalized size = 1.25 \[ \frac{2 \,{\left (35 \, B b^{3} x^{8} + 55 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{6} + 385 \,{\left (B a^{2} b + A a b^{2}\right )} x^{4} - 77 \, A a^{3} - 385 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2}\right )}}{385 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^3/x^(7/2),x, algorithm="fricas")

[Out]

2/385*(35*B*b^3*x^8 + 55*(3*B*a*b^2 + A*b^3)*x^6 + 385*(B*a^2*b + A*a*b^2)*x^4 -
77*A*a^3 - 385*(B*a^3 + 3*A*a^2*b)*x^2)/x^(5/2)

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Sympy [A]  time = 33.2381, size = 107, normalized size = 1.32 \[ - \frac{2 A a^{3}}{5 x^{\frac{5}{2}}} - \frac{6 A a^{2} b}{\sqrt{x}} + 2 A a b^{2} x^{\frac{3}{2}} + \frac{2 A b^{3} x^{\frac{7}{2}}}{7} - \frac{2 B a^{3}}{\sqrt{x}} + 2 B a^{2} b x^{\frac{3}{2}} + \frac{6 B a b^{2} x^{\frac{7}{2}}}{7} + \frac{2 B b^{3} x^{\frac{11}{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**3*(B*x**2+A)/x**(7/2),x)

[Out]

-2*A*a**3/(5*x**(5/2)) - 6*A*a**2*b/sqrt(x) + 2*A*a*b**2*x**(3/2) + 2*A*b**3*x**
(7/2)/7 - 2*B*a**3/sqrt(x) + 2*B*a**2*b*x**(3/2) + 6*B*a*b**2*x**(7/2)/7 + 2*B*b
**3*x**(11/2)/11

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GIAC/XCAS [A]  time = 0.223142, size = 107, normalized size = 1.32 \[ \frac{2}{11} \, B b^{3} x^{\frac{11}{2}} + \frac{6}{7} \, B a b^{2} x^{\frac{7}{2}} + \frac{2}{7} \, A b^{3} x^{\frac{7}{2}} + 2 \, B a^{2} b x^{\frac{3}{2}} + 2 \, A a b^{2} x^{\frac{3}{2}} - \frac{2 \,{\left (5 \, B a^{3} x^{2} + 15 \, A a^{2} b x^{2} + A a^{3}\right )}}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^3/x^(7/2),x, algorithm="giac")

[Out]

2/11*B*b^3*x^(11/2) + 6/7*B*a*b^2*x^(7/2) + 2/7*A*b^3*x^(7/2) + 2*B*a^2*b*x^(3/2
) + 2*A*a*b^2*x^(3/2) - 2/5*(5*B*a^3*x^2 + 15*A*a^2*b*x^2 + A*a^3)/x^(5/2)