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

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]

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

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Rubi [A]  time = 0.10, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {446, 77} \[ \frac {a^3 (A b-a B)}{3 b^5 \left (a+b x^2\right )^{3/2}}-\frac {a^2 (3 A b-4 a B)}{b^5 \sqrt {a+b x^2}}-\frac {3 a \sqrt {a+b x^2} (A b-2 a B)}{b^5}+\frac {\left (a+b x^2\right )^{3/2} (A b-4 a B)}{3 b^5}+\frac {B \left (a+b x^2\right )^{5/2}}{5 b^5} \]

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)

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {x^7 \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^3 (A+B x)}{(a+b x)^{5/2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a^3 (-A b+a B)}{b^4 (a+b x)^{5/2}}-\frac {a^2 (-3 A b+4 a B)}{b^4 (a+b x)^{3/2}}+\frac {3 a (-A b+2 a B)}{b^4 \sqrt {a+b x}}+\frac {(A b-4 a B) \sqrt {a+b x}}{b^4}+\frac {B (a+b x)^{3/2}}{b^4}\right ) \, dx,x,x^2\right )\\ &=\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 (A b-2 a B) \sqrt {a+b x^2}}{b^5}+\frac {(A b-4 a B) \left (a+b x^2\right )^{3/2}}{3 b^5}+\frac {B \left (a+b x^2\right )^{5/2}}{5 b^5}\\ \end {align*}

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Mathematica [A]  time = 0.07, 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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fricas [A]  time = 0.56, size = 123, normalized size = 0.96 \[ \frac {{\left (3 \, B b^{4} x^{8} - {\left (8 \, B a b^{3} - 5 \, A b^{4}\right )} x^{6} + 128 \, B a^{4} - 80 \, A a^{3} b + 6 \, {\left (8 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{4} + 24 \, {\left (8 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{15 \, {\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(B*x^2+A)/(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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giac [A]  time = 0.36, size = 141, normalized size = 1.10 \[ \frac {12 \, {\left (b x^{2} + a\right )} B a^{3} - B a^{4} - 9 \, {\left (b x^{2} + a\right )} A a^{2} b + A a^{3} b}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{5}} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B b^{20} - 20 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a b^{20} + 90 \, \sqrt {b x^{2} + a} B a^{2} b^{20} + 5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{21} - 45 \, \sqrt {b x^{2} + a} A a b^{21}}{15 \, b^{25}} \]

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, algorithm="giac")

[Out]

1/3*(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) + 1/15*(3*(b*x^2
+ a)^(5/2)*B*b^20 - 20*(b*x^2 + a)^(3/2)*B*a*b^20 + 90*sqrt(b*x^2 + a)*B*a^2*b^20 + 5*(b*x^2 + a)^(3/2)*A*b^21
- 45*sqrt(b*x^2 + a)*A*a*b^21)/b^25

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maple [A]  time = 0.01, size = 101, normalized size = 0.79 \[ -\frac {-3 x^{8} B \,b^{4}-5 A \,b^{4} x^{6}+8 B a \,b^{3} x^{6}+30 A a \,b^{3} x^{4}-48 B \,a^{2} b^{2} x^{4}+120 A \,a^{2} b^{2} x^{2}-192 B \,a^{3} b \,x^{2}+80 A \,a^{3} b -128 B \,a^{4}}{15 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{5}} \]

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+120*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 [A]  time = 1.21, size = 174, normalized size = 1.36 \[ \frac {B x^{8}}{5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} - \frac {8 \, B a x^{6}}{15 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}} + \frac {A x^{6}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {16 \, B a^{2} x^{4}}{5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3}} - \frac {2 \, A a x^{4}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}} + \frac {64 \, B a^{3} x^{2}}{5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{4}} - \frac {8 \, A a^{2} x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3}} + \frac {128 \, B a^{4}}{15 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{5}} - \frac {16 \, A a^{3}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{4}} \]

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, algorithm="maxima")

[Out]

1/5*B*x^8/((b*x^2 + a)^(3/2)*b) - 8/15*B*a*x^6/((b*x^2 + a)^(3/2)*b^2) + 1/3*A*x^6/((b*x^2 + a)^(3/2)*b) + 16/
5*B*a^2*x^4/((b*x^2 + a)^(3/2)*b^3) - 2*A*a*x^4/((b*x^2 + a)^(3/2)*b^2) + 64/5*B*a^3*x^2/((b*x^2 + a)^(3/2)*b^
4) - 8*A*a^2*x^2/((b*x^2 + a)^(3/2)*b^3) + 128/15*B*a^4/((b*x^2 + a)^(3/2)*b^5) - 16/3*A*a^3/((b*x^2 + a)^(3/2
)*b^4)

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mupad [B]  time = 1.00, size = 122, normalized size = 0.95 \[ \frac {3\,B\,{\left (b\,x^2+a\right )}^4-5\,B\,a^4+90\,B\,a^2\,{\left (b\,x^2+a\right )}^2+5\,A\,b\,{\left (b\,x^2+a\right )}^3-20\,B\,a\,{\left (b\,x^2+a\right )}^3+60\,B\,a^3\,\left (b\,x^2+a\right )+5\,A\,a^3\,b-45\,A\,a\,b\,{\left (b\,x^2+a\right )}^2-45\,A\,a^2\,b\,\left (b\,x^2+a\right )}{15\,b^5\,{\left (b\,x^2+a\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*B*(a + b*x^2)^4 - 5*B*a^4 + 90*B*a^2*(a + b*x^2)^2 + 5*A*b*(a + b*x^2)^3 - 20*B*a*(a + b*x^2)^3 + 60*B*a^3*
(a + b*x^2) + 5*A*a^3*b - 45*A*a*b*(a + b*x^2)^2 - 45*A*a^2*b*(a + b*x^2))/(15*b^5*(a + b*x^2)^(3/2))

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sympy [A]  time = 4.21, 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/(1
5*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)) + 12
8*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), Tru
e))

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