\(\int \frac {(a+b x^2)^{3/2} (A+B x^2)}{x^6} \, dx\) [532]

Optimal result

Integrand size = 22, antiderivative size = 86 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^6} \, dx=-\frac {b B \sqrt {a+b x^2}}{x}-\frac {B \left (a+b x^2\right )^{3/2}}{3 x^3}-\frac {A \left (a+b x^2\right )^{5/2}}{5 a x^5}+b^{3/2} B \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {462, 283, 223, 212} \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^6} \, dx=-\frac {A \left (a+b x^2\right )^{5/2}}{5 a x^5}+b^{3/2} B \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {b B \sqrt {a+b x^2}}{x}-\frac {B \left (a+b x^2\right )^{3/2}}{3 x^3} \]

[In]

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

[Out]

-((b*B*Sqrt[a + b*x^2])/x) - (B*(a + b*x^2)^(3/2))/(3*x^3) - (A*(a + b*x^2)^(5/2))/(5*a*x^5) + b^(3/2)*B*ArcTa
nh[(Sqrt[b]*x)/Sqrt[a + b*x^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 283

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

Rule 462

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[d/e^n, Int[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a,
b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n*(p + 1) + 1, 0] && (IntegerQ[n] || GtQ[e, 0]) && (
(GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1]))

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^6} \, dx=\frac {\sqrt {a+b x^2} \left (-3 a^2 A-6 a A b x^2-5 a^2 B x^2-3 A b^2 x^4-20 a b B x^4\right )}{15 a x^5}-b^{3/2} B \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.84 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.94

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (3 A \,b^{2} x^{4}+20 B a b \,x^{4}+6 a A b \,x^{2}+5 a^{2} B \,x^{2}+3 a^{2} A \right )}{15 x^{5} a}+B \,b^{\frac {3}{2}} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )\)	\(81\)
pseudoelliptic	\(\frac {5 a \,b^{\frac {3}{2}} B \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right ) x^{5}-\left (\left (\frac {5 x^{2} B}{3}+A \right ) a^{2}+2 \left (\frac {10 x^{2} B}{3}+A \right ) x^{2} b a +A \,b^{2} x^{4}\right ) \sqrt {b \,x^{2}+a}}{5 x^{5} a}\)	\(84\)
default	\(B \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{3 a \,x^{3}}+\frac {2 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{a x}+\frac {4 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{a}\right )}{3 a}\right )-\frac {A \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5 a \,x^{5}}\)	\(121\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.14 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^6} \, dx=\left [\frac {15 \, B a b^{\frac {3}{2}} x^{5} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left ({\left (20 \, B a b + 3 \, A b^{2}\right )} x^{4} + 3 \, A a^{2} + {\left (5 \, B a^{2} + 6 \, A a b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{30 \, a x^{5}}, -\frac {15 \, B a \sqrt {-b} b x^{5} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left ({\left (20 \, B a b + 3 \, A b^{2}\right )} x^{4} + 3 \, A a^{2} + {\left (5 \, B a^{2} + 6 \, A a b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{15 \, a x^{5}}\right ] \]

[In]

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

[Out]

[1/30*(15*B*a*b^(3/2)*x^5*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*((20*B*a*b + 3*A*b^2)*x^4 + 3*A*
a^2 + (5*B*a^2 + 6*A*a*b)*x^2)*sqrt(b*x^2 + a))/(a*x^5), -1/15*(15*B*a*sqrt(-b)*b*x^5*arctan(sqrt(-b)*x/sqrt(b
*x^2 + a)) + ((20*B*a*b + 3*A*b^2)*x^4 + 3*A*a^2 + (5*B*a^2 + 6*A*a*b)*x^2)*sqrt(b*x^2 + a))/(a*x^5)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (75) = 150\).

Time = 2.17 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.14 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^6} \, dx=- \frac {A a \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{5 x^{4}} - \frac {2 A b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{5 x^{2}} - \frac {A b^{\frac {5}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{5 a} - \frac {B \sqrt {a} b}{x \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {B a \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{3 x^{2}} - \frac {B b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3} + B b^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )} - \frac {B b^{2} x}{\sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^6} \, dx=\frac {\sqrt {b x^{2} + a} B b^{2} x}{a} + B b^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b}{3 \, a x} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B}{3 \, a x^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A}{5 \, a x^{5}} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (70) = 140\).

Time = 0.32 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.74 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^6} \, dx=-\frac {1}{2} \, B b^{\frac {3}{2}} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right ) + \frac {2 \, {\left (30 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} B a b^{\frac {3}{2}} + 15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} A b^{\frac {5}{2}} - 90 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a^{2} b^{\frac {3}{2}} + 110 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{3} b^{\frac {3}{2}} + 30 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a^{2} b^{\frac {5}{2}} - 70 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{4} b^{\frac {3}{2}} + 20 \, B a^{5} b^{\frac {3}{2}} + 3 \, A a^{4} b^{\frac {5}{2}}\right )}}{15 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{5}} \]

[In]

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

[Out]

-1/2*B*b^(3/2)*log((sqrt(b)*x - sqrt(b*x^2 + a))^2) + 2/15*(30*(sqrt(b)*x - sqrt(b*x^2 + a))^8*B*a*b^(3/2) + 1
5*(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) + 110*(sqrt(b)*
x - sqrt(b*x^2 + a))^4*B*a^3*b^(3/2) + 30*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*a^2*b^(5/2) - 70*(sqrt(b)*x - sqrt
(b*x^2 + a))^2*B*a^4*b^(3/2) + 20*B*a^5*b^(3/2) + 3*A*a^4*b^(5/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^5

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^6} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (b\,x^2+a\right )}^{3/2}}{x^6} \,d x \]

[In]

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

[Out]

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