3.584 \(\int \frac{x^7 \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=128 \[ \frac{a^3 (A b-a B)}{3 b^5 \left (a+b x^2\right )^{3/2}}-\frac{a^2 (3 A b-4 a B)}{b^5 \sqrt{a+b x^2}}-\frac{3 a \sqrt{a+b x^2} (A b-2 a B)}{b^5}+\frac{\left (a+b x^2\right )^{3/2} (A b-4 a B)}{3 b^5}+\frac{B \left (a+b x^2\right )^{5/2}}{5 b^5} \]

[Out]

(a^3*(A*b - a*B))/(3*b^5*(a + b*x^2)^(3/2)) - (a^2*(3*A*b - 4*a*B))/(b^5*Sqrt[a
+ b*x^2]) - (3*a*(A*b - 2*a*B)*Sqrt[a + b*x^2])/b^5 + ((A*b - 4*a*B)*(a + b*x^2)
^(3/2))/(3*b^5) + (B*(a + b*x^2)^(5/2))/(5*b^5)

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Rubi [A]  time = 0.275654, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{a^3 (A b-a B)}{3 b^5 \left (a+b x^2\right )^{3/2}}-\frac{a^2 (3 A b-4 a B)}{b^5 \sqrt{a+b x^2}}-\frac{3 a \sqrt{a+b x^2} (A b-2 a B)}{b^5}+\frac{\left (a+b x^2\right )^{3/2} (A b-4 a B)}{3 b^5}+\frac{B \left (a+b x^2\right )^{5/2}}{5 b^5} \]

Antiderivative was successfully verified.

[In]  Int[(x^7*(A + B*x^2))/(a + b*x^2)^(5/2),x]

[Out]

(a^3*(A*b - a*B))/(3*b^5*(a + b*x^2)^(3/2)) - (a^2*(3*A*b - 4*a*B))/(b^5*Sqrt[a
+ b*x^2]) - (3*a*(A*b - 2*a*B)*Sqrt[a + b*x^2])/b^5 + ((A*b - 4*a*B)*(a + b*x^2)
^(3/2))/(3*b^5) + (B*(a + b*x^2)^(5/2))/(5*b^5)

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Rubi in Sympy [A]  time = 31.7032, size = 117, normalized size = 0.91 \[ \frac{B \left (a + b x^{2}\right )^{\frac{5}{2}}}{5 b^{5}} + \frac{a^{3} \left (A b - B a\right )}{3 b^{5} \left (a + b x^{2}\right )^{\frac{3}{2}}} - \frac{a^{2} \left (3 A b - 4 B a\right )}{b^{5} \sqrt{a + b x^{2}}} - \frac{3 a \sqrt{a + b x^{2}} \left (A b - 2 B a\right )}{b^{5}} + \frac{\left (a + b x^{2}\right )^{\frac{3}{2}} \left (A b - 4 B a\right )}{3 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7*(B*x**2+A)/(b*x**2+a)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.100143, size = 98, normalized size = 0.77 \[ \frac{128 a^4 B+a^3 \left (192 b B x^2-80 A b\right )+24 a^2 b^2 x^2 \left (2 B x^2-5 A\right )-2 a b^3 x^4 \left (15 A+4 B x^2\right )+b^4 x^6 \left (5 A+3 B x^2\right )}{15 b^5 \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^7*(A + B*x^2))/(a + b*x^2)^(5/2),x]

[Out]

(128*a^4*B + 24*a^2*b^2*x^2*(-5*A + 2*B*x^2) + b^4*x^6*(5*A + 3*B*x^2) - 2*a*b^3
*x^4*(15*A + 4*B*x^2) + a^3*(-80*A*b + 192*b*B*x^2))/(15*b^5*(a + b*x^2)^(3/2))

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Maple [A]  time = 0.009, size = 101, normalized size = 0.8 \[ -{\frac{-3\,{x}^{8}B{b}^{4}-5\,A{b}^{4}{x}^{6}+8\,Ba{b}^{3}{x}^{6}+30\,Aa{b}^{3}{x}^{4}-48\,B{a}^{2}{b}^{2}{x}^{4}+120\,A{a}^{2}{b}^{2}{x}^{2}-192\,B{a}^{3}b{x}^{2}+80\,A{a}^{3}b-128\,B{a}^{4}}{15\,{b}^{5}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(B*x^2+A)/(b*x^2+a)^(5/2),x)

[Out]

-1/15*(-3*B*b^4*x^8-5*A*b^4*x^6+8*B*a*b^3*x^6+30*A*a*b^3*x^4-48*B*a^2*b^2*x^4+12
0*A*a^2*b^2*x^2-192*B*a^3*b*x^2+80*A*a^3*b-128*B*a^4)/(b*x^2+a)^(3/2)/b^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^7/(b*x^2 + a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233815, size = 166, normalized size = 1.3 \[ \frac{{\left (3 \, B b^{4} x^{8} -{\left (8 \, B a b^{3} - 5 \, A b^{4}\right )} x^{6} + 128 \, B a^{4} - 80 \, A a^{3} b + 6 \,{\left (8 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{4} + 24 \,{\left (8 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{15 \,{\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^7/(b*x^2 + a)^(5/2),x, algorithm="fricas")

[Out]

1/15*(3*B*b^4*x^8 - (8*B*a*b^3 - 5*A*b^4)*x^6 + 128*B*a^4 - 80*A*a^3*b + 6*(8*B*
a^2*b^2 - 5*A*a*b^3)*x^4 + 24*(8*B*a^3*b - 5*A*a^2*b^2)*x^2)*sqrt(b*x^2 + a)/(b^
7*x^4 + 2*a*b^6*x^2 + a^2*b^5)

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Sympy [A]  time = 10.0174, size = 437, normalized size = 3.41 \[ \begin{cases} - \frac{80 A a^{3} b}{15 a b^{5} \sqrt{a + b x^{2}} + 15 b^{6} x^{2} \sqrt{a + b x^{2}}} - \frac{120 A a^{2} b^{2} x^{2}}{15 a b^{5} \sqrt{a + b x^{2}} + 15 b^{6} x^{2} \sqrt{a + b x^{2}}} - \frac{30 A a b^{3} x^{4}}{15 a b^{5} \sqrt{a + b x^{2}} + 15 b^{6} x^{2} \sqrt{a + b x^{2}}} + \frac{5 A b^{4} x^{6}}{15 a b^{5} \sqrt{a + b x^{2}} + 15 b^{6} x^{2} \sqrt{a + b x^{2}}} + \frac{128 B a^{4}}{15 a b^{5} \sqrt{a + b x^{2}} + 15 b^{6} x^{2} \sqrt{a + b x^{2}}} + \frac{192 B a^{3} b x^{2}}{15 a b^{5} \sqrt{a + b x^{2}} + 15 b^{6} x^{2} \sqrt{a + b x^{2}}} + \frac{48 B a^{2} b^{2} x^{4}}{15 a b^{5} \sqrt{a + b x^{2}} + 15 b^{6} x^{2} \sqrt{a + b x^{2}}} - \frac{8 B a b^{3} x^{6}}{15 a b^{5} \sqrt{a + b x^{2}} + 15 b^{6} x^{2} \sqrt{a + b x^{2}}} + \frac{3 B b^{4} x^{8}}{15 a b^{5} \sqrt{a + b x^{2}} + 15 b^{6} x^{2} \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{8}}{8} + \frac{B x^{10}}{10}}{a^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7*(B*x**2+A)/(b*x**2+a)**(5/2),x)

[Out]

Piecewise((-80*A*a**3*b/(15*a*b**5*sqrt(a + b*x**2) + 15*b**6*x**2*sqrt(a + b*x*
*2)) - 120*A*a**2*b**2*x**2/(15*a*b**5*sqrt(a + b*x**2) + 15*b**6*x**2*sqrt(a +
b*x**2)) - 30*A*a*b**3*x**4/(15*a*b**5*sqrt(a + b*x**2) + 15*b**6*x**2*sqrt(a +
b*x**2)) + 5*A*b**4*x**6/(15*a*b**5*sqrt(a + b*x**2) + 15*b**6*x**2*sqrt(a + b*x
**2)) + 128*B*a**4/(15*a*b**5*sqrt(a + b*x**2) + 15*b**6*x**2*sqrt(a + b*x**2))
+ 192*B*a**3*b*x**2/(15*a*b**5*sqrt(a + b*x**2) + 15*b**6*x**2*sqrt(a + b*x**2))
+ 48*B*a**2*b**2*x**4/(15*a*b**5*sqrt(a + b*x**2) + 15*b**6*x**2*sqrt(a + b*x**
2)) - 8*B*a*b**3*x**6/(15*a*b**5*sqrt(a + b*x**2) + 15*b**6*x**2*sqrt(a + b*x**2
)) + 3*B*b**4*x**8/(15*a*b**5*sqrt(a + b*x**2) + 15*b**6*x**2*sqrt(a + b*x**2)),
Ne(b, 0)), ((A*x**8/8 + B*x**10/10)/a**(5/2), True))

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GIAC/XCAS [A]  time = 0.230089, size = 167, normalized size = 1.3 \[ \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} B - 20 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a + 90 \, \sqrt{b x^{2} + a} B a^{2} + 5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A b - 45 \, \sqrt{b x^{2} + a} A a b + \frac{5 \,{\left (12 \,{\left (b x^{2} + a\right )} B a^{3} - B a^{4} - 9 \,{\left (b x^{2} + a\right )} A a^{2} b + A a^{3} b\right )}}{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}}{15 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^7/(b*x^2 + a)^(5/2),x, algorithm="giac")

[Out]

1/15*(3*(b*x^2 + a)^(5/2)*B - 20*(b*x^2 + a)^(3/2)*B*a + 90*sqrt(b*x^2 + a)*B*a^
2 + 5*(b*x^2 + a)^(3/2)*A*b - 45*sqrt(b*x^2 + a)*A*a*b + 5*(12*(b*x^2 + a)*B*a^3
- B*a^4 - 9*(b*x^2 + a)*A*a^2*b + A*a^3*b)/(b*x^2 + a)^(3/2))/b^5