\(\int \frac {x^6 (A+B x^2)}{(a+b x^2)^{5/2}} \, dx\) [585]

Optimal result

Integrand size = 22, antiderivative size = 149 \[ \int \frac {x^6 \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {(4 A b-7 a B) x^5}{12 b^2 \left (a+b x^2\right )^{3/2}}+\frac {B x^7}{4 b \left (a+b x^2\right )^{3/2}}-\frac {5 (4 A b-7 a B) x^3}{12 b^3 \sqrt {a+b x^2}}+\frac {5 (4 A b-7 a B) x \sqrt {a+b x^2}}{8 b^4}-\frac {5 a (4 A b-7 a B) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{9/2}} \]

[Out]

-1/12*(4*A*b-7*B*a)*x^5/b^2/(b*x^2+a)^(3/2)+1/4*B*x^7/b/(b*x^2+a)^(3/2)-5/8*a*(4*A*b-7*B*a)*arctanh(x*b^(1/2)/
(b*x^2+a)^(1/2))/b^(9/2)-5/12*(4*A*b-7*B*a)*x^3/b^3/(b*x^2+a)^(1/2)+5/8*(4*A*b-7*B*a)*x*(b*x^2+a)^(1/2)/b^4

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {470, 294, 327, 223, 212} \[ \int \frac {x^6 \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {5 a (4 A b-7 a B) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{9/2}}+\frac {5 x \sqrt {a+b x^2} (4 A b-7 a B)}{8 b^4}-\frac {5 x^3 (4 A b-7 a B)}{12 b^3 \sqrt {a+b x^2}}-\frac {x^5 (4 A b-7 a B)}{12 b^2 \left (a+b x^2\right )^{3/2}}+\frac {B x^7}{4 b \left (a+b x^2\right )^{3/2}} \]

[In]

Int[(x^6*(A + B*x^2))/(a + b*x^2)^(5/2),x]

[Out]

-1/12*((4*A*b - 7*a*B)*x^5)/(b^2*(a + b*x^2)^(3/2)) + (B*x^7)/(4*b*(a + b*x^2)^(3/2)) - (5*(4*A*b - 7*a*B)*x^3
)/(12*b^3*Sqrt[a + b*x^2]) + (5*(4*A*b - 7*a*B)*x*Sqrt[a + b*x^2])/(8*b^4) - (5*a*(4*A*b - 7*a*B)*ArcTanh[(Sqr
t[b]*x)/Sqrt[a + b*x^2]])/(8*b^(9/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 294

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

Rule 327

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

Rule 470

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

Rubi steps \begin{align*} \text {integral}& = \frac {B x^7}{4 b \left (a+b x^2\right )^{3/2}}-\frac {(-4 A b+7 a B) \int \frac {x^6}{\left (a+b x^2\right )^{5/2}} \, dx}{4 b} \\ & = -\frac {(4 A b-7 a B) x^5}{12 b^2 \left (a+b x^2\right )^{3/2}}+\frac {B x^7}{4 b \left (a+b x^2\right )^{3/2}}+\frac {(5 (4 A b-7 a B)) \int \frac {x^4}{\left (a+b x^2\right )^{3/2}} \, dx}{12 b^2} \\ & = -\frac {(4 A b-7 a B) x^5}{12 b^2 \left (a+b x^2\right )^{3/2}}+\frac {B x^7}{4 b \left (a+b x^2\right )^{3/2}}-\frac {5 (4 A b-7 a B) x^3}{12 b^3 \sqrt {a+b x^2}}+\frac {(5 (4 A b-7 a B)) \int \frac {x^2}{\sqrt {a+b x^2}} \, dx}{4 b^3} \\ & = -\frac {(4 A b-7 a B) x^5}{12 b^2 \left (a+b x^2\right )^{3/2}}+\frac {B x^7}{4 b \left (a+b x^2\right )^{3/2}}-\frac {5 (4 A b-7 a B) x^3}{12 b^3 \sqrt {a+b x^2}}+\frac {5 (4 A b-7 a B) x \sqrt {a+b x^2}}{8 b^4}-\frac {(5 a (4 A b-7 a B)) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{8 b^4} \\ & = -\frac {(4 A b-7 a B) x^5}{12 b^2 \left (a+b x^2\right )^{3/2}}+\frac {B x^7}{4 b \left (a+b x^2\right )^{3/2}}-\frac {5 (4 A b-7 a B) x^3}{12 b^3 \sqrt {a+b x^2}}+\frac {5 (4 A b-7 a B) x \sqrt {a+b x^2}}{8 b^4}-\frac {(5 a (4 A b-7 a B)) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{8 b^4} \\ & = -\frac {(4 A b-7 a B) x^5}{12 b^2 \left (a+b x^2\right )^{3/2}}+\frac {B x^7}{4 b \left (a+b x^2\right )^{3/2}}-\frac {5 (4 A b-7 a B) x^3}{12 b^3 \sqrt {a+b x^2}}+\frac {5 (4 A b-7 a B) x \sqrt {a+b x^2}}{8 b^4}-\frac {5 a (4 A b-7 a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.85 \[ \int \frac {x^6 \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {-105 a^3 B x+a b^2 x^3 \left (80 A-21 B x^2\right )+20 a^2 b x \left (3 A-7 B x^2\right )+6 b^3 x^5 \left (2 A+B x^2\right )}{24 b^4 \left (a+b x^2\right )^{3/2}}+\frac {5 a (-4 A b+7 a B) \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{4 b^{9/2}} \]

[In]

Integrate[(x^6*(A + B*x^2))/(a + b*x^2)^(5/2),x]

[Out]

(-105*a^3*B*x + a*b^2*x^3*(80*A - 21*B*x^2) + 20*a^2*b*x*(3*A - 7*B*x^2) + 6*b^3*x^5*(2*A + B*x^2))/(24*b^4*(a
+ b*x^2)^(3/2)) + (5*a*(-4*A*b + 7*a*B)*ArcTanh[(Sqrt[b]*x)/(-Sqrt[a] + Sqrt[a + b*x^2])])/(4*b^(9/2))

Maple [A] (verified)

Time = 2.96 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.77

method	result	size
pseudoelliptic	\(-\frac {5 \left (-x \left (-\frac {7 x^{2} B}{3}+A \right ) a^{2} b^{\frac {3}{2}}-\frac {4 x^{3} \left (-\frac {21 x^{2} B}{80}+A \right ) a \,b^{\frac {5}{2}}}{3}-\frac {x^{5} \left (\frac {x^{2} B}{2}+A \right ) b^{\frac {7}{2}}}{5}+\left (\frac {7 B \sqrt {b}\, a^{2} x}{4}+\left (A b -\frac {7 B a}{4}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right ) \left (b \,x^{2}+a \right )^{\frac {3}{2}}\right ) a \right )}{2 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{\frac {9}{2}}}\)	\(114\)
default	\(B \left (\frac {x^{7}}{4 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {7 a \left (\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {5 a \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )}{2 b}\right )}{4 b}\right )+A \left (\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {5 a \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )}{2 b}\right )\)	\(194\)
risch	\(\frac {x \left (2 b B \,x^{2}+4 A b -11 B a \right ) \sqrt {b \,x^{2}+a}}{8 b^{4}}-\frac {a \left (20 A \sqrt {b}\, \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )-\frac {35 B a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}-\frac {2 a \left (A b -B a \right ) \left (\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (x +\frac {\sqrt {-a b}}{b}\right )}\right )}{b}-\frac {2 a \left (A b -B a \right ) \left (-\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (x -\frac {\sqrt {-a b}}{b}\right )}\right )}{b}-\frac {2 \left (5 A b -7 B a \right ) \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{b \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {2 \left (5 A b -7 B a \right ) \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{b \left (x +\frac {\sqrt {-a b}}{b}\right )}\right )}{8 b^{4}}\)	\(496\)

[In]

int(x^6*(B*x^2+A)/(b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-5/2/(b*x^2+a)^(3/2)*(-x*(-7/3*x^2*B+A)*a^2*b^(3/2)-4/3*x^3*(-21/80*x^2*B+A)*a*b^(5/2)-1/5*x^5*(1/2*x^2*B+A)*b
^(7/2)+(7/4*B*b^(1/2)*a^2*x+(A*b-7/4*B*a)*arctanh((b*x^2+a)^(1/2)/x/b^(1/2))*(b*x^2+a)^(3/2))*a)/b^(9/2)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 392, normalized size of antiderivative = 2.63 \[ \int \frac {x^6 \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\left [-\frac {15 \, {\left (7 \, B a^{4} - 4 \, A a^{3} b + {\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{4} + 2 \, {\left (7 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (6 \, B b^{4} x^{7} - 3 \, {\left (7 \, B a b^{3} - 4 \, A b^{4}\right )} x^{5} - 20 \, {\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3} - 15 \, {\left (7 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{48 \, {\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\right )}}, -\frac {15 \, {\left (7 \, B a^{4} - 4 \, A a^{3} b + {\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{4} + 2 \, {\left (7 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (6 \, B b^{4} x^{7} - 3 \, {\left (7 \, B a b^{3} - 4 \, A b^{4}\right )} x^{5} - 20 \, {\left (7 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3} - 15 \, {\left (7 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{24 \, {\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\right )}}\right ] \]

[In]

integrate(x^6*(B*x^2+A)/(b*x^2+a)^(5/2),x, algorithm="fricas")

[Out]

[-1/48*(15*(7*B*a^4 - 4*A*a^3*b + (7*B*a^2*b^2 - 4*A*a*b^3)*x^4 + 2*(7*B*a^3*b - 4*A*a^2*b^2)*x^2)*sqrt(b)*log
(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(6*B*b^4*x^7 - 3*(7*B*a*b^3 - 4*A*b^4)*x^5 - 20*(7*B*a^2*b^2
- 4*A*a*b^3)*x^3 - 15*(7*B*a^3*b - 4*A*a^2*b^2)*x)*sqrt(b*x^2 + a))/(b^7*x^4 + 2*a*b^6*x^2 + a^2*b^5), -1/24*(
15*(7*B*a^4 - 4*A*a^3*b + (7*B*a^2*b^2 - 4*A*a*b^3)*x^4 + 2*(7*B*a^3*b - 4*A*a^2*b^2)*x^2)*sqrt(-b)*arctan(sqr
t(-b)*x/sqrt(b*x^2 + a)) - (6*B*b^4*x^7 - 3*(7*B*a*b^3 - 4*A*b^4)*x^5 - 20*(7*B*a^2*b^2 - 4*A*a*b^3)*x^3 - 15*
(7*B*a^3*b - 4*A*a^2*b^2)*x)*sqrt(b*x^2 + a))/(b^7*x^4 + 2*a*b^6*x^2 + a^2*b^5)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 804 vs. \(2 (144) = 288\).

Time = 15.31 (sec) , antiderivative size = 804, normalized size of antiderivative = 5.40 \[ \int \frac {x^6 \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=A \left (- \frac {15 a^{\frac {81}{2}} b^{22} \sqrt {1 + \frac {b x^{2}}{a}} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{6 a^{\frac {79}{2}} b^{\frac {51}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 6 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {15 a^{\frac {79}{2}} b^{23} x^{2} \sqrt {1 + \frac {b x^{2}}{a}} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{6 a^{\frac {79}{2}} b^{\frac {51}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 6 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {15 a^{40} b^{\frac {45}{2}} x}{6 a^{\frac {79}{2}} b^{\frac {51}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 6 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {20 a^{39} b^{\frac {47}{2}} x^{3}}{6 a^{\frac {79}{2}} b^{\frac {51}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 6 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 a^{38} b^{\frac {49}{2}} x^{5}}{6 a^{\frac {79}{2}} b^{\frac {51}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 6 a^{\frac {77}{2}} b^{\frac {53}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + B \left (\frac {105 a^{\frac {157}{2}} b^{41} \sqrt {1 + \frac {b x^{2}}{a}} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{24 a^{\frac {153}{2}} b^{\frac {91}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 24 a^{\frac {151}{2}} b^{\frac {93}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {105 a^{\frac {155}{2}} b^{42} x^{2} \sqrt {1 + \frac {b x^{2}}{a}} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{24 a^{\frac {153}{2}} b^{\frac {91}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 24 a^{\frac {151}{2}} b^{\frac {93}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {105 a^{78} b^{\frac {83}{2}} x}{24 a^{\frac {153}{2}} b^{\frac {91}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 24 a^{\frac {151}{2}} b^{\frac {93}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {140 a^{77} b^{\frac {85}{2}} x^{3}}{24 a^{\frac {153}{2}} b^{\frac {91}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 24 a^{\frac {151}{2}} b^{\frac {93}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {21 a^{76} b^{\frac {87}{2}} x^{5}}{24 a^{\frac {153}{2}} b^{\frac {91}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 24 a^{\frac {151}{2}} b^{\frac {93}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {6 a^{75} b^{\frac {89}{2}} x^{7}}{24 a^{\frac {153}{2}} b^{\frac {91}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 24 a^{\frac {151}{2}} b^{\frac {93}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}}\right ) \]

[In]

integrate(x**6*(B*x**2+A)/(b*x**2+a)**(5/2),x)

[Out]

A*(-15*a**(81/2)*b**22*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(6*a**(79/2)*b**(51/2)*sqrt(1 + b*x**2/a) +
6*a**(77/2)*b**(53/2)*x**2*sqrt(1 + b*x**2/a)) - 15*a**(79/2)*b**23*x**2*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/s
qrt(a))/(6*a**(79/2)*b**(51/2)*sqrt(1 + b*x**2/a) + 6*a**(77/2)*b**(53/2)*x**2*sqrt(1 + b*x**2/a)) + 15*a**40*
b**(45/2)*x/(6*a**(79/2)*b**(51/2)*sqrt(1 + b*x**2/a) + 6*a**(77/2)*b**(53/2)*x**2*sqrt(1 + b*x**2/a)) + 20*a*
*39*b**(47/2)*x**3/(6*a**(79/2)*b**(51/2)*sqrt(1 + b*x**2/a) + 6*a**(77/2)*b**(53/2)*x**2*sqrt(1 + b*x**2/a))
+ 3*a**38*b**(49/2)*x**5/(6*a**(79/2)*b**(51/2)*sqrt(1 + b*x**2/a) + 6*a**(77/2)*b**(53/2)*x**2*sqrt(1 + b*x**
2/a))) + B*(105*a**(157/2)*b**41*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(24*a**(153/2)*b**(91/2)*sqrt(1 +
b*x**2/a) + 24*a**(151/2)*b**(93/2)*x**2*sqrt(1 + b*x**2/a)) + 105*a**(155/2)*b**42*x**2*sqrt(1 + b*x**2/a)*a
sinh(sqrt(b)*x/sqrt(a))/(24*a**(153/2)*b**(91/2)*sqrt(1 + b*x**2/a) + 24*a**(151/2)*b**(93/2)*x**2*sqrt(1 + b*
x**2/a)) - 105*a**78*b**(83/2)*x/(24*a**(153/2)*b**(91/2)*sqrt(1 + b*x**2/a) + 24*a**(151/2)*b**(93/2)*x**2*sq
rt(1 + b*x**2/a)) - 140*a**77*b**(85/2)*x**3/(24*a**(153/2)*b**(91/2)*sqrt(1 + b*x**2/a) + 24*a**(151/2)*b**(9
3/2)*x**2*sqrt(1 + b*x**2/a)) - 21*a**76*b**(87/2)*x**5/(24*a**(153/2)*b**(91/2)*sqrt(1 + b*x**2/a) + 24*a**(1
51/2)*b**(93/2)*x**2*sqrt(1 + b*x**2/a)) + 6*a**75*b**(89/2)*x**7/(24*a**(153/2)*b**(91/2)*sqrt(1 + b*x**2/a)
+ 24*a**(151/2)*b**(93/2)*x**2*sqrt(1 + b*x**2/a)))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.41 \[ \int \frac {x^6 \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {B x^{7}}{4 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} - \frac {7 \, B a x^{5}}{8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}} + \frac {A x^{5}}{2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} - \frac {35 \, B a^{2} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )}}{24 \, b^{2}} + \frac {5 \, A a x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )}}{6 \, b} - \frac {35 \, B a^{2} x}{24 \, \sqrt {b x^{2} + a} b^{4}} + \frac {5 \, A a x}{6 \, \sqrt {b x^{2} + a} b^{3}} + \frac {35 \, B a^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {9}{2}}} - \frac {5 \, A a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {7}{2}}} \]

[In]

integrate(x^6*(B*x^2+A)/(b*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

1/4*B*x^7/((b*x^2 + a)^(3/2)*b) - 7/8*B*a*x^5/((b*x^2 + a)^(3/2)*b^2) + 1/2*A*x^5/((b*x^2 + a)^(3/2)*b) - 35/2
4*B*a^2*x*(3*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/((b*x^2 + a)^(3/2)*b^2))/b^2 + 5/6*A*a*x*(3*x^2/((b*x^2 + a)^(3/2
)*b) + 2*a/((b*x^2 + a)^(3/2)*b^2))/b - 35/24*B*a^2*x/(sqrt(b*x^2 + a)*b^4) + 5/6*A*a*x/(sqrt(b*x^2 + a)*b^3)
+ 35/8*B*a^2*arcsinh(b*x/sqrt(a*b))/b^(9/2) - 5/2*A*a*arcsinh(b*x/sqrt(a*b))/b^(7/2)

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.99 \[ \int \frac {x^6 \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {{\left ({\left (3 \, {\left (\frac {2 \, B x^{2}}{b} - \frac {7 \, B a^{2} b^{5} - 4 \, A a b^{6}}{a b^{7}}\right )} x^{2} - \frac {20 \, {\left (7 \, B a^{3} b^{4} - 4 \, A a^{2} b^{5}\right )}}{a b^{7}}\right )} x^{2} - \frac {15 \, {\left (7 \, B a^{4} b^{3} - 4 \, A a^{3} b^{4}\right )}}{a b^{7}}\right )} x}{24 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} - \frac {5 \, {\left (7 \, B a^{2} - 4 \, A a b\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{8 \, b^{\frac {9}{2}}} \]

[In]

integrate(x^6*(B*x^2+A)/(b*x^2+a)^(5/2),x, algorithm="giac")

[Out]

1/24*((3*(2*B*x^2/b - (7*B*a^2*b^5 - 4*A*a*b^6)/(a*b^7))*x^2 - 20*(7*B*a^3*b^4 - 4*A*a^2*b^5)/(a*b^7))*x^2 - 1
5*(7*B*a^4*b^3 - 4*A*a^3*b^4)/(a*b^7))*x/(b*x^2 + a)^(3/2) - 5/8*(7*B*a^2 - 4*A*a*b)*log(abs(-sqrt(b)*x + sqrt
(b*x^2 + a)))/b^(9/2)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6 \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {x^6\,\left (B\,x^2+A\right )}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \]

[In]

int((x^6*(A + B*x^2))/(a + b*x^2)^(5/2),x)

[Out]

int((x^6*(A + B*x^2))/(a + b*x^2)^(5/2), x)