3.364 \(\int \frac{A+B x}{x^6 \sqrt{a+c x^2}} \, dx\)

Optimal. Leaf size=147 \[ -\frac{8 A c^2 \sqrt{a+c x^2}}{15 a^3 x}+\frac{4 A c \sqrt{a+c x^2}}{15 a^2 x^3}-\frac{3 B c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{8 a^{5/2}}+\frac{3 B c \sqrt{a+c x^2}}{8 a^2 x^2}-\frac{A \sqrt{a+c x^2}}{5 a x^5}-\frac{B \sqrt{a+c x^2}}{4 a x^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.12864, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {835, 807, 266, 63, 208} \[ -\frac{8 A c^2 \sqrt{a+c x^2}}{15 a^3 x}+\frac{4 A c \sqrt{a+c x^2}}{15 a^2 x^3}-\frac{3 B c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{8 a^{5/2}}+\frac{3 B c \sqrt{a+c x^2}}{8 a^2 x^2}-\frac{A \sqrt{a+c x^2}}{5 a x^5}-\frac{B \sqrt{a+c x^2}}{4 a x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 835

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((e*f - d*g)
*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

Rule 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rule 266

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{A+B x}{x^6 \sqrt{a+c x^2}} \, dx &=-\frac{A \sqrt{a+c x^2}}{5 a x^5}-\frac{\int \frac{-5 a B+4 A c x}{x^5 \sqrt{a+c x^2}} \, dx}{5 a}\\ &=-\frac{A \sqrt{a+c x^2}}{5 a x^5}-\frac{B \sqrt{a+c x^2}}{4 a x^4}+\frac{\int \frac{-16 a A c-15 a B c x}{x^4 \sqrt{a+c x^2}} \, dx}{20 a^2}\\ &=-\frac{A \sqrt{a+c x^2}}{5 a x^5}-\frac{B \sqrt{a+c x^2}}{4 a x^4}+\frac{4 A c \sqrt{a+c x^2}}{15 a^2 x^3}-\frac{\int \frac{45 a^2 B c-32 a A c^2 x}{x^3 \sqrt{a+c x^2}} \, dx}{60 a^3}\\ &=-\frac{A \sqrt{a+c x^2}}{5 a x^5}-\frac{B \sqrt{a+c x^2}}{4 a x^4}+\frac{4 A c \sqrt{a+c x^2}}{15 a^2 x^3}+\frac{3 B c \sqrt{a+c x^2}}{8 a^2 x^2}+\frac{\int \frac{64 a^2 A c^2+45 a^2 B c^2 x}{x^2 \sqrt{a+c x^2}} \, dx}{120 a^4}\\ &=-\frac{A \sqrt{a+c x^2}}{5 a x^5}-\frac{B \sqrt{a+c x^2}}{4 a x^4}+\frac{4 A c \sqrt{a+c x^2}}{15 a^2 x^3}+\frac{3 B c \sqrt{a+c x^2}}{8 a^2 x^2}-\frac{8 A c^2 \sqrt{a+c x^2}}{15 a^3 x}+\frac{\left (3 B c^2\right ) \int \frac{1}{x \sqrt{a+c x^2}} \, dx}{8 a^2}\\ &=-\frac{A \sqrt{a+c x^2}}{5 a x^5}-\frac{B \sqrt{a+c x^2}}{4 a x^4}+\frac{4 A c \sqrt{a+c x^2}}{15 a^2 x^3}+\frac{3 B c \sqrt{a+c x^2}}{8 a^2 x^2}-\frac{8 A c^2 \sqrt{a+c x^2}}{15 a^3 x}+\frac{\left (3 B c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^2\right )}{16 a^2}\\ &=-\frac{A \sqrt{a+c x^2}}{5 a x^5}-\frac{B \sqrt{a+c x^2}}{4 a x^4}+\frac{4 A c \sqrt{a+c x^2}}{15 a^2 x^3}+\frac{3 B c \sqrt{a+c x^2}}{8 a^2 x^2}-\frac{8 A c^2 \sqrt{a+c x^2}}{15 a^3 x}+\frac{(3 B c) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^2}\right )}{8 a^2}\\ &=-\frac{A \sqrt{a+c x^2}}{5 a x^5}-\frac{B \sqrt{a+c x^2}}{4 a x^4}+\frac{4 A c \sqrt{a+c x^2}}{15 a^2 x^3}+\frac{3 B c \sqrt{a+c x^2}}{8 a^2 x^2}-\frac{8 A c^2 \sqrt{a+c x^2}}{15 a^3 x}-\frac{3 B c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{8 a^{5/2}}\\ \end{align*}

Mathematica [C]  time = 0.0201174, size = 72, normalized size = 0.49 \[ -\frac{\sqrt{a+c x^2} \left (A \left (3 a^2-4 a c x^2+8 c^2 x^4\right )+15 B c^2 x^5 \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};\frac{c x^2}{a}+1\right )\right )}{15 a^3 x^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[a + c*x^2]*(A*(3*a^2 - 4*a*c*x^2 + 8*c^2*x^4) + 15*B*c^2*x^5*Hypergeometric2F1[1/2, 3, 3/2, 1 + (c*x^2)
/a]))/(15*a^3*x^5)

________________________________________________________________________________________

Maple [A]  time = 0.011, size = 129, normalized size = 0.9 \begin{align*} -{\frac{B}{4\,a{x}^{4}}\sqrt{c{x}^{2}+a}}+{\frac{3\,Bc}{8\,{a}^{2}{x}^{2}}\sqrt{c{x}^{2}+a}}-{\frac{3\,B{c}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}-{\frac{A}{5\,a{x}^{5}}\sqrt{c{x}^{2}+a}}+{\frac{4\,Ac}{15\,{a}^{2}{x}^{3}}\sqrt{c{x}^{2}+a}}-{\frac{8\,A{c}^{2}}{15\,{a}^{3}x}\sqrt{c{x}^{2}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^6/(c*x^2+a)^(1/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^6/(c*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.56022, size = 467, normalized size = 3.18 \begin{align*} \left [\frac{45 \, B \sqrt{a} c^{2} x^{5} \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) - 2 \,{\left (64 \, A c^{2} x^{4} - 45 \, B a c x^{3} - 32 \, A a c x^{2} + 30 \, B a^{2} x + 24 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{240 \, a^{3} x^{5}}, \frac{45 \, B \sqrt{-a} c^{2} x^{5} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) -{\left (64 \, A c^{2} x^{4} - 45 \, B a c x^{3} - 32 \, A a c x^{2} + 30 \, B a^{2} x + 24 \, A a^{2}\right )} \sqrt{c x^{2} + a}}{120 \, a^{3} x^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^6/(c*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

[1/240*(45*B*sqrt(a)*c^2*x^5*log(-(c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(64*A*c^2*x^4 - 45*B*a*c*
x^3 - 32*A*a*c*x^2 + 30*B*a^2*x + 24*A*a^2)*sqrt(c*x^2 + a))/(a^3*x^5), 1/120*(45*B*sqrt(-a)*c^2*x^5*arctan(sq
rt(-a)/sqrt(c*x^2 + a)) - (64*A*c^2*x^4 - 45*B*a*c*x^3 - 32*A*a*c*x^2 + 30*B*a^2*x + 24*A*a^2)*sqrt(c*x^2 + a)
)/(a^3*x^5)]

________________________________________________________________________________________

Sympy [B]  time = 7.10817, size = 408, normalized size = 2.78 \begin{align*} - \frac{3 A a^{4} c^{\frac{9}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{15 a^{5} c^{4} x^{4} + 30 a^{4} c^{5} x^{6} + 15 a^{3} c^{6} x^{8}} - \frac{2 A a^{3} c^{\frac{11}{2}} x^{2} \sqrt{\frac{a}{c x^{2}} + 1}}{15 a^{5} c^{4} x^{4} + 30 a^{4} c^{5} x^{6} + 15 a^{3} c^{6} x^{8}} - \frac{3 A a^{2} c^{\frac{13}{2}} x^{4} \sqrt{\frac{a}{c x^{2}} + 1}}{15 a^{5} c^{4} x^{4} + 30 a^{4} c^{5} x^{6} + 15 a^{3} c^{6} x^{8}} - \frac{12 A a c^{\frac{15}{2}} x^{6} \sqrt{\frac{a}{c x^{2}} + 1}}{15 a^{5} c^{4} x^{4} + 30 a^{4} c^{5} x^{6} + 15 a^{3} c^{6} x^{8}} - \frac{8 A c^{\frac{17}{2}} x^{8} \sqrt{\frac{a}{c x^{2}} + 1}}{15 a^{5} c^{4} x^{4} + 30 a^{4} c^{5} x^{6} + 15 a^{3} c^{6} x^{8}} - \frac{B}{4 \sqrt{c} x^{5} \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{B \sqrt{c}}{8 a x^{3} \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{3 B c^{\frac{3}{2}}}{8 a^{2} x \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{3 B c^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{8 a^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x**6/(c*x**2+a)**(1/2),x)

[Out]

-3*A*a**4*c**(9/2)*sqrt(a/(c*x**2) + 1)/(15*a**5*c**4*x**4 + 30*a**4*c**5*x**6 + 15*a**3*c**6*x**8) - 2*A*a**3
*c**(11/2)*x**2*sqrt(a/(c*x**2) + 1)/(15*a**5*c**4*x**4 + 30*a**4*c**5*x**6 + 15*a**3*c**6*x**8) - 3*A*a**2*c*
*(13/2)*x**4*sqrt(a/(c*x**2) + 1)/(15*a**5*c**4*x**4 + 30*a**4*c**5*x**6 + 15*a**3*c**6*x**8) - 12*A*a*c**(15/
2)*x**6*sqrt(a/(c*x**2) + 1)/(15*a**5*c**4*x**4 + 30*a**4*c**5*x**6 + 15*a**3*c**6*x**8) - 8*A*c**(17/2)*x**8*
sqrt(a/(c*x**2) + 1)/(15*a**5*c**4*x**4 + 30*a**4*c**5*x**6 + 15*a**3*c**6*x**8) - B/(4*sqrt(c)*x**5*sqrt(a/(c
*x**2) + 1)) + B*sqrt(c)/(8*a*x**3*sqrt(a/(c*x**2) + 1)) + 3*B*c**(3/2)/(8*a**2*x*sqrt(a/(c*x**2) + 1)) - 3*B*
c**2*asinh(sqrt(a)/(sqrt(c)*x))/(8*a**(5/2))

________________________________________________________________________________________

Giac [B]  time = 1.20971, size = 325, normalized size = 2.21 \begin{align*} \frac{3 \, B c^{2} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{4 \, \sqrt{-a} a^{2}} - \frac{45 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{9} B c^{2} - 210 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{7} B a c^{2} - 640 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} A a^{2} c^{\frac{5}{2}} + 210 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} B a^{3} c^{2} + 320 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} A a^{3} c^{\frac{5}{2}} - 45 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} B a^{4} c^{2} - 64 \, A a^{4} c^{\frac{5}{2}}}{60 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{5} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^6/(c*x^2+a)^(1/2),x, algorithm="giac")

[Out]

3/4*B*c^2*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/(sqrt(-a)*a^2) - 1/60*(45*(sqrt(c)*x - sqrt(c*x^2 +
a))^9*B*c^2 - 210*(sqrt(c)*x - sqrt(c*x^2 + a))^7*B*a*c^2 - 640*(sqrt(c)*x - sqrt(c*x^2 + a))^4*A*a^2*c^(5/2)
+ 210*(sqrt(c)*x - sqrt(c*x^2 + a))^3*B*a^3*c^2 + 320*(sqrt(c)*x - sqrt(c*x^2 + a))^2*A*a^3*c^(5/2) - 45*(sqrt
(c)*x - sqrt(c*x^2 + a))*B*a^4*c^2 - 64*A*a^4*c^(5/2))/(((sqrt(c)*x - sqrt(c*x^2 + a))^2 - a)^5*a^2)