\(\int \frac {A+B x^2}{x^6 (a+b x^2)^{3/2}} \, dx\) [581]

Optimal result

Integrand size = 22, antiderivative size = 115 \[ \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {A}{5 a x^5 \sqrt {a+b x^2}}+\frac {6 A b-5 a B}{15 a^2 x^3 \sqrt {a+b x^2}}-\frac {4 b (6 A b-5 a B)}{15 a^3 x \sqrt {a+b x^2}}-\frac {8 b^2 (6 A b-5 a B) x}{15 a^4 \sqrt {a+b x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {464, 277, 197} \[ \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {8 b^2 x (6 A b-5 a B)}{15 a^4 \sqrt {a+b x^2}}-\frac {4 b (6 A b-5 a B)}{15 a^3 x \sqrt {a+b x^2}}+\frac {6 A b-5 a B}{15 a^2 x^3 \sqrt {a+b x^2}}-\frac {A}{5 a x^5 \sqrt {a+b x^2}} \]

[In]

Int[(A + B*x^2)/(x^6*(a + b*x^2)^(3/2)),x]

[Out]

-1/5*A/(a*x^5*Sqrt[a + b*x^2]) + (6*A*b - 5*a*B)/(15*a^2*x^3*Sqrt[a + b*x^2]) - (4*b*(6*A*b - 5*a*B))/(15*a^3*
x*Sqrt[a + b*x^2]) - (8*b^2*(6*A*b - 5*a*B)*x)/(15*a^4*Sqrt[a + b*x^2])

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 277

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^(p + 1)/(a*(m + 1))), x]
- Dist[b*((m + n*(p + 1) + 1)/(a*(m + 1))), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 464

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[c*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {A}{5 a x^5 \sqrt {a+b x^2}}-\frac {(6 A b-5 a B) \int \frac {1}{x^4 \left (a+b x^2\right )^{3/2}} \, dx}{5 a} \\ & = -\frac {A}{5 a x^5 \sqrt {a+b x^2}}+\frac {6 A b-5 a B}{15 a^2 x^3 \sqrt {a+b x^2}}+\frac {(4 b (6 A b-5 a B)) \int \frac {1}{x^2 \left (a+b x^2\right )^{3/2}} \, dx}{15 a^2} \\ & = -\frac {A}{5 a x^5 \sqrt {a+b x^2}}+\frac {6 A b-5 a B}{15 a^2 x^3 \sqrt {a+b x^2}}-\frac {4 b (6 A b-5 a B)}{15 a^3 x \sqrt {a+b x^2}}-\frac {\left (8 b^2 (6 A b-5 a B)\right ) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{15 a^3} \\ & = -\frac {A}{5 a x^5 \sqrt {a+b x^2}}+\frac {6 A b-5 a B}{15 a^2 x^3 \sqrt {a+b x^2}}-\frac {4 b (6 A b-5 a B)}{15 a^3 x \sqrt {a+b x^2}}-\frac {8 b^2 (6 A b-5 a B) x}{15 a^4 \sqrt {a+b x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.75 \[ \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )^{3/2}} \, dx=\frac {-3 a^3 A+6 a^2 A b x^2-5 a^3 B x^2-24 a A b^2 x^4+20 a^2 b B x^4-48 A b^3 x^6+40 a b^2 B x^6}{15 a^4 x^5 \sqrt {a+b x^2}} \]

[In]

Integrate[(A + B*x^2)/(x^6*(a + b*x^2)^(3/2)),x]

[Out]

(-3*a^3*A + 6*a^2*A*b*x^2 - 5*a^3*B*x^2 - 24*a*A*b^2*x^4 + 20*a^2*b*B*x^4 - 48*A*b^3*x^6 + 40*a*b^2*B*x^6)/(15
*a^4*x^5*Sqrt[a + b*x^2])

Maple [A] (verified)

Time = 2.88 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.66

method	result	size
pseudoelliptic	\(\frac {\left (-5 x^{2} B -3 A \right ) a^{3}+6 x^{2} b \left (\frac {10 x^{2} B}{3}+A \right ) a^{2}-24 x^{4} b^{2} \left (-\frac {5 x^{2} B}{3}+A \right ) a -48 x^{6} b^{3} A}{15 \sqrt {b \,x^{2}+a}\, x^{5} a^{4}}\)	\(76\)
gosper	\(-\frac {48 x^{6} b^{3} A -40 x^{6} a \,b^{2} B +24 A a \,b^{2} x^{4}-20 B \,a^{2} b \,x^{4}-6 A \,a^{2} b \,x^{2}+5 B \,a^{3} x^{2}+3 a^{3} A}{15 x^{5} \sqrt {b \,x^{2}+a}\, a^{4}}\)	\(83\)
trager	\(-\frac {48 x^{6} b^{3} A -40 x^{6} a \,b^{2} B +24 A a \,b^{2} x^{4}-20 B \,a^{2} b \,x^{4}-6 A \,a^{2} b \,x^{2}+5 B \,a^{3} x^{2}+3 a^{3} A}{15 x^{5} \sqrt {b \,x^{2}+a}\, a^{4}}\)	\(83\)
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (33 A \,b^{2} x^{4}-25 B a b \,x^{4}-9 a A b \,x^{2}+5 a^{2} B \,x^{2}+3 a^{2} A \right )}{15 a^{4} x^{5}}-\frac {x \,b^{2} \left (A b -B a \right )}{\sqrt {b \,x^{2}+a}\, a^{4}}\)	\(86\)
default	\(B \left (-\frac {1}{3 a \,x^{3} \sqrt {b \,x^{2}+a}}-\frac {4 b \left (-\frac {1}{a x \sqrt {b \,x^{2}+a}}-\frac {2 b x}{a^{2} \sqrt {b \,x^{2}+a}}\right )}{3 a}\right )+A \left (-\frac {1}{5 a \,x^{5} \sqrt {b \,x^{2}+a}}-\frac {6 b \left (-\frac {1}{3 a \,x^{3} \sqrt {b \,x^{2}+a}}-\frac {4 b \left (-\frac {1}{a x \sqrt {b \,x^{2}+a}}-\frac {2 b x}{a^{2} \sqrt {b \,x^{2}+a}}\right )}{3 a}\right )}{5 a}\right )\)	\(146\)

[In]

int((B*x^2+A)/x^6/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/15*((-5*B*x^2-3*A)*a^3+6*x^2*b*(10/3*x^2*B+A)*a^2-24*x^4*b^2*(-5/3*x^2*B+A)*a-48*x^6*b^3*A)/(b*x^2+a)^(1/2)/
x^5/a^4

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.82 \[ \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )^{3/2}} \, dx=\frac {{\left (8 \, {\left (5 \, B a b^{2} - 6 \, A b^{3}\right )} x^{6} + 4 \, {\left (5 \, B a^{2} b - 6 \, A a b^{2}\right )} x^{4} - 3 \, A a^{3} - {\left (5 \, B a^{3} - 6 \, A a^{2} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{15 \, {\left (a^{4} b x^{7} + a^{5} x^{5}\right )}} \]

[In]

integrate((B*x^2+A)/x^6/(b*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 593 vs. \(2 (109) = 218\).

Time = 4.43 (sec) , antiderivative size = 593, normalized size of antiderivative = 5.16 \[ \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )^{3/2}} \, dx=A \left (- \frac {a^{5} b^{\frac {19}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{5 a^{7} b^{9} x^{4} + 15 a^{6} b^{10} x^{6} + 15 a^{5} b^{11} x^{8} + 5 a^{4} b^{12} x^{10}} - \frac {5 a^{3} b^{\frac {23}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{5 a^{7} b^{9} x^{4} + 15 a^{6} b^{10} x^{6} + 15 a^{5} b^{11} x^{8} + 5 a^{4} b^{12} x^{10}} - \frac {30 a^{2} b^{\frac {25}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{5 a^{7} b^{9} x^{4} + 15 a^{6} b^{10} x^{6} + 15 a^{5} b^{11} x^{8} + 5 a^{4} b^{12} x^{10}} - \frac {40 a b^{\frac {27}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{5 a^{7} b^{9} x^{4} + 15 a^{6} b^{10} x^{6} + 15 a^{5} b^{11} x^{8} + 5 a^{4} b^{12} x^{10}} - \frac {16 b^{\frac {29}{2}} x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{5 a^{7} b^{9} x^{4} + 15 a^{6} b^{10} x^{6} + 15 a^{5} b^{11} x^{8} + 5 a^{4} b^{12} x^{10}}\right ) + B \left (- \frac {a^{3} b^{\frac {9}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}} + \frac {3 a^{2} b^{\frac {11}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}} + \frac {12 a b^{\frac {13}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}} + \frac {8 b^{\frac {15}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}}\right ) \]

[In]

integrate((B*x**2+A)/x**6/(b*x**2+a)**(3/2),x)

[Out]

A*(-a**5*b**(19/2)*sqrt(a/(b*x**2) + 1)/(5*a**7*b**9*x**4 + 15*a**6*b**10*x**6 + 15*a**5*b**11*x**8 + 5*a**4*b
**12*x**10) - 5*a**3*b**(23/2)*x**4*sqrt(a/(b*x**2) + 1)/(5*a**7*b**9*x**4 + 15*a**6*b**10*x**6 + 15*a**5*b**1
1*x**8 + 5*a**4*b**12*x**10) - 30*a**2*b**(25/2)*x**6*sqrt(a/(b*x**2) + 1)/(5*a**7*b**9*x**4 + 15*a**6*b**10*x
**6 + 15*a**5*b**11*x**8 + 5*a**4*b**12*x**10) - 40*a*b**(27/2)*x**8*sqrt(a/(b*x**2) + 1)/(5*a**7*b**9*x**4 +
15*a**6*b**10*x**6 + 15*a**5*b**11*x**8 + 5*a**4*b**12*x**10) - 16*b**(29/2)*x**10*sqrt(a/(b*x**2) + 1)/(5*a**
7*b**9*x**4 + 15*a**6*b**10*x**6 + 15*a**5*b**11*x**8 + 5*a**4*b**12*x**10)) + B*(-a**3*b**(9/2)*sqrt(a/(b*x**
2) + 1)/(3*a**5*b**4*x**2 + 6*a**4*b**5*x**4 + 3*a**3*b**6*x**6) + 3*a**2*b**(11/2)*x**2*sqrt(a/(b*x**2) + 1)/
(3*a**5*b**4*x**2 + 6*a**4*b**5*x**4 + 3*a**3*b**6*x**6) + 12*a*b**(13/2)*x**4*sqrt(a/(b*x**2) + 1)/(3*a**5*b*
*4*x**2 + 6*a**4*b**5*x**4 + 3*a**3*b**6*x**6) + 8*b**(15/2)*x**6*sqrt(a/(b*x**2) + 1)/(3*a**5*b**4*x**2 + 6*a
**4*b**5*x**4 + 3*a**3*b**6*x**6))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.17 \[ \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )^{3/2}} \, dx=\frac {8 \, B b^{2} x}{3 \, \sqrt {b x^{2} + a} a^{3}} - \frac {16 \, A b^{3} x}{5 \, \sqrt {b x^{2} + a} a^{4}} + \frac {4 \, B b}{3 \, \sqrt {b x^{2} + a} a^{2} x} - \frac {8 \, A b^{2}}{5 \, \sqrt {b x^{2} + a} a^{3} x} - \frac {B}{3 \, \sqrt {b x^{2} + a} a x^{3}} + \frac {2 \, A b}{5 \, \sqrt {b x^{2} + a} a^{2} x^{3}} - \frac {A}{5 \, \sqrt {b x^{2} + a} a x^{5}} \]

[In]

integrate((B*x^2+A)/x^6/(b*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

8/3*B*b^2*x/(sqrt(b*x^2 + a)*a^3) - 16/5*A*b^3*x/(sqrt(b*x^2 + a)*a^4) + 4/3*B*b/(sqrt(b*x^2 + a)*a^2*x) - 8/5
*A*b^2/(sqrt(b*x^2 + a)*a^3*x) - 1/3*B/(sqrt(b*x^2 + a)*a*x^3) + 2/5*A*b/(sqrt(b*x^2 + a)*a^2*x^3) - 1/5*A/(sq
rt(b*x^2 + a)*a*x^5)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (99) = 198\).

Time = 0.31 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.56 \[ \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )^{3/2}} \, dx=\frac {{\left (B a b^{2} - A b^{3}\right )} x}{\sqrt {b x^{2} + a} a^{4}} - \frac {2 \, {\left (15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} B a b^{\frac {3}{2}} - 15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} A b^{\frac {5}{2}} - 90 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a^{2} b^{\frac {3}{2}} + 90 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} A a b^{\frac {5}{2}} + 160 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{3} b^{\frac {3}{2}} - 240 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a^{2} b^{\frac {5}{2}} - 110 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{4} b^{\frac {3}{2}} + 150 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{3} b^{\frac {5}{2}} + 25 \, B a^{5} b^{\frac {3}{2}} - 33 \, A a^{4} b^{\frac {5}{2}}\right )}}{15 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{5} a^{3}} \]

[In]

integrate((B*x^2+A)/x^6/(b*x^2+a)^(3/2),x, algorithm="giac")

[Out]

(B*a*b^2 - A*b^3)*x/(sqrt(b*x^2 + a)*a^4) - 2/15*(15*(sqrt(b)*x - sqrt(b*x^2 + a))^8*B*a*b^(3/2) - 15*(sqrt(b)
*x - sqrt(b*x^2 + a))^8*A*b^(5/2) - 90*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a^2*b^(3/2) + 90*(sqrt(b)*x - sqrt(b*
x^2 + a))^6*A*a*b^(5/2) + 160*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^3*b^(3/2) - 240*(sqrt(b)*x - sqrt(b*x^2 + a)
)^4*A*a^2*b^(5/2) - 110*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^4*b^(3/2) + 150*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*
a^3*b^(5/2) + 25*B*a^5*b^(3/2) - 33*A*a^4*b^(5/2))/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^5*a^3)

Mupad [B] (verification not implemented)

Time = 5.33 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.71 \[ \int \frac {A+B x^2}{x^6 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {5\,B\,a^3\,x^2+3\,A\,a^3-20\,B\,a^2\,b\,x^4-6\,A\,a^2\,b\,x^2-40\,B\,a\,b^2\,x^6+24\,A\,a\,b^2\,x^4+48\,A\,b^3\,x^6}{15\,a^4\,x^5\,\sqrt {b\,x^2+a}} \]

[In]

int((A + B*x^2)/(x^6*(a + b*x^2)^(3/2)),x)

[Out]

-(3*A*a^3 + 5*B*a^3*x^2 + 48*A*b^3*x^6 - 6*A*a^2*b*x^2 + 24*A*a*b^2*x^4 - 20*B*a^2*b*x^4 - 40*B*a*b^2*x^6)/(15
*a^4*x^5*(a + b*x^2)^(1/2))