3.6.82 \(\int \frac {A+B x^2}{x^7 (a+b x^2)^{3/2}} \, dx\) [582]

Optimal. Leaf size=153 \[ -\frac {5 b^2 (7 A b-6 a B)}{16 a^4 \sqrt {a+b x^2}}-\frac {A}{6 a x^6 \sqrt {a+b x^2}}+\frac {7 A b-6 a B}{24 a^2 x^4 \sqrt {a+b x^2}}-\frac {5 b (7 A b-6 a B)}{48 a^3 x^2 \sqrt {a+b x^2}}+\frac {5 b^2 (7 A b-6 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{9/2}} \]

[Out]

5/16*b^2*(7*A*b-6*B*a)*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(9/2)-5/16*b^2*(7*A*b-6*B*a)/a^4/(b*x^2+a)^(1/2)-1/6
*A/a/x^6/(b*x^2+a)^(1/2)+1/24*(7*A*b-6*B*a)/a^2/x^4/(b*x^2+a)^(1/2)-5/48*b*(7*A*b-6*B*a)/a^3/x^2/(b*x^2+a)^(1/
2)

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Rubi [A]
time = 0.08, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {457, 79, 44, 53, 65, 214} \begin {gather*} \frac {5 b^2 (7 A b-6 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{9/2}}-\frac {5 b^2 (7 A b-6 a B)}{16 a^4 \sqrt {a+b x^2}}-\frac {5 b (7 A b-6 a B)}{48 a^3 x^2 \sqrt {a+b x^2}}+\frac {7 A b-6 a B}{24 a^2 x^4 \sqrt {a+b x^2}}-\frac {A}{6 a x^6 \sqrt {a+b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2)/(x^7*(a + b*x^2)^(3/2)),x]

[Out]

(-5*b^2*(7*A*b - 6*a*B))/(16*a^4*Sqrt[a + b*x^2]) - A/(6*a*x^6*Sqrt[a + b*x^2]) + (7*A*b - 6*a*B)/(24*a^2*x^4*
Sqrt[a + b*x^2]) - (5*b*(7*A*b - 6*a*B))/(48*a^3*x^2*Sqrt[a + b*x^2]) + (5*b^2*(7*A*b - 6*a*B)*ArcTanh[Sqrt[a
+ b*x^2]/Sqrt[a]])/(16*a^(9/2))

Rule 44

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 53

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

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 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 214

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

Rule 457

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 {A+B x^2}{x^7 \left (a+b x^2\right )^{3/2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {A+B x}{x^4 (a+b x)^{3/2}} \, dx,x,x^2\right )\\ &=-\frac {A}{6 a x^6 \sqrt {a+b x^2}}+\frac {\left (-\frac {7 A b}{2}+3 a B\right ) \text {Subst}\left (\int \frac {1}{x^3 (a+b x)^{3/2}} \, dx,x,x^2\right )}{6 a}\\ &=-\frac {A}{6 a x^6 \sqrt {a+b x^2}}-\frac {7 A b-6 a B}{6 a^2 x^4 \sqrt {a+b x^2}}-\frac {(5 (7 A b-6 a B)) \text {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x}} \, dx,x,x^2\right )}{12 a^2}\\ &=-\frac {A}{6 a x^6 \sqrt {a+b x^2}}-\frac {7 A b-6 a B}{6 a^2 x^4 \sqrt {a+b x^2}}+\frac {5 (7 A b-6 a B) \sqrt {a+b x^2}}{24 a^3 x^4}+\frac {(5 b (7 A b-6 a B)) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,x^2\right )}{16 a^3}\\ &=-\frac {A}{6 a x^6 \sqrt {a+b x^2}}-\frac {7 A b-6 a B}{6 a^2 x^4 \sqrt {a+b x^2}}+\frac {5 (7 A b-6 a B) \sqrt {a+b x^2}}{24 a^3 x^4}-\frac {5 b (7 A b-6 a B) \sqrt {a+b x^2}}{16 a^4 x^2}-\frac {\left (5 b^2 (7 A b-6 a B)\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{32 a^4}\\ &=-\frac {A}{6 a x^6 \sqrt {a+b x^2}}-\frac {7 A b-6 a B}{6 a^2 x^4 \sqrt {a+b x^2}}+\frac {5 (7 A b-6 a B) \sqrt {a+b x^2}}{24 a^3 x^4}-\frac {5 b (7 A b-6 a B) \sqrt {a+b x^2}}{16 a^4 x^2}-\frac {(5 b (7 A b-6 a B)) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{16 a^4}\\ &=-\frac {A}{6 a x^6 \sqrt {a+b x^2}}-\frac {7 A b-6 a B}{6 a^2 x^4 \sqrt {a+b x^2}}+\frac {5 (7 A b-6 a B) \sqrt {a+b x^2}}{24 a^3 x^4}-\frac {5 b (7 A b-6 a B) \sqrt {a+b x^2}}{16 a^4 x^2}+\frac {5 b^2 (7 A b-6 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.23, size = 126, normalized size = 0.82 \begin {gather*} \frac {-8 a^3 A+14 a^2 A b x^2-12 a^3 B x^2-35 a A b^2 x^4+30 a^2 b B x^4-105 A b^3 x^6+90 a b^2 B x^6}{48 a^4 x^6 \sqrt {a+b x^2}}-\frac {5 b^2 (-7 A b+6 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2)/(x^7*(a + b*x^2)^(3/2)),x]

[Out]

(-8*a^3*A + 14*a^2*A*b*x^2 - 12*a^3*B*x^2 - 35*a*A*b^2*x^4 + 30*a^2*b*B*x^4 - 105*A*b^3*x^6 + 90*a*b^2*B*x^6)/
(48*a^4*x^6*Sqrt[a + b*x^2]) - (5*b^2*(-7*A*b + 6*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(16*a^(9/2))

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Maple [A]
time = 0.12, size = 210, normalized size = 1.37

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (57 A \,b^{2} x^{4}-42 B a b \,x^{4}-22 a A b \,x^{2}+12 B \,a^{2} x^{2}+8 a^{2} A \right )}{48 a^{4} x^{6}}-\frac {b^{3} A}{a^{4} \sqrt {b \,x^{2}+a}}+\frac {b^{2} B}{a^{3} \sqrt {b \,x^{2}+a}}+\frac {35 b^{3} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right ) A}{16 a^{\frac {9}{2}}}-\frac {15 b^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right ) B}{8 a^{\frac {7}{2}}}\)	\(159\)
default	\(B \left (-\frac {1}{4 a \,x^{4} \sqrt {b \,x^{2}+a}}-\frac {5 b \left (-\frac {1}{2 a \,x^{2} \sqrt {b \,x^{2}+a}}-\frac {3 b \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{2 a}\right )}{4 a}\right )+A \left (-\frac {1}{6 a \,x^{6} \sqrt {b \,x^{2}+a}}-\frac {7 b \left (-\frac {1}{4 a \,x^{4} \sqrt {b \,x^{2}+a}}-\frac {5 b \left (-\frac {1}{2 a \,x^{2} \sqrt {b \,x^{2}+a}}-\frac {3 b \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )\)	\(210\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)/x^7/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 174, normalized size = 1.14 \begin {gather*} -\frac {15 \, B b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, a^{\frac {7}{2}}} + \frac {35 \, A b^{3} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{16 \, a^{\frac {9}{2}}} + \frac {15 \, B b^{2}}{8 \, \sqrt {b x^{2} + a} a^{3}} - \frac {35 \, A b^{3}}{16 \, \sqrt {b x^{2} + a} a^{4}} + \frac {5 \, B b}{8 \, \sqrt {b x^{2} + a} a^{2} x^{2}} - \frac {35 \, A b^{2}}{48 \, \sqrt {b x^{2} + a} a^{3} x^{2}} - \frac {B}{4 \, \sqrt {b x^{2} + a} a x^{4}} + \frac {7 \, A b}{24 \, \sqrt {b x^{2} + a} a^{2} x^{4}} - \frac {A}{6 \, \sqrt {b x^{2} + a} a x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^7/(b*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

-15/8*B*b^2*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(7/2) + 35/16*A*b^3*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(9/2) + 15/8*B
*b^2/(sqrt(b*x^2 + a)*a^3) - 35/16*A*b^3/(sqrt(b*x^2 + a)*a^4) + 5/8*B*b/(sqrt(b*x^2 + a)*a^2*x^2) - 35/48*A*b
^2/(sqrt(b*x^2 + a)*a^3*x^2) - 1/4*B/(sqrt(b*x^2 + a)*a*x^4) + 7/24*A*b/(sqrt(b*x^2 + a)*a^2*x^4) - 1/6*A/(sqr
t(b*x^2 + a)*a*x^6)

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Fricas [A]
time = 2.34, size = 341, normalized size = 2.23 \begin {gather*} \left [-\frac {15 \, {\left ({\left (6 \, B a b^{3} - 7 \, A b^{4}\right )} x^{8} + {\left (6 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{6}\right )} \sqrt {a} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (15 \, {\left (6 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{6} - 8 \, A a^{4} + 5 \, {\left (6 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{4} - 2 \, {\left (6 \, B a^{4} - 7 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{96 \, {\left (a^{5} b x^{8} + a^{6} x^{6}\right )}}, \frac {15 \, {\left ({\left (6 \, B a b^{3} - 7 \, A b^{4}\right )} x^{8} + {\left (6 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{6}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (15 \, {\left (6 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{6} - 8 \, A a^{4} + 5 \, {\left (6 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{4} - 2 \, {\left (6 \, B a^{4} - 7 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{48 \, {\left (a^{5} b x^{8} + a^{6} x^{6}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^7/(b*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

[-1/96*(15*((6*B*a*b^3 - 7*A*b^4)*x^8 + (6*B*a^2*b^2 - 7*A*a*b^3)*x^6)*sqrt(a)*log(-(b*x^2 + 2*sqrt(b*x^2 + a)
*sqrt(a) + 2*a)/x^2) - 2*(15*(6*B*a^2*b^2 - 7*A*a*b^3)*x^6 - 8*A*a^4 + 5*(6*B*a^3*b - 7*A*a^2*b^2)*x^4 - 2*(6*
B*a^4 - 7*A*a^3*b)*x^2)*sqrt(b*x^2 + a))/(a^5*b*x^8 + a^6*x^6), 1/48*(15*((6*B*a*b^3 - 7*A*b^4)*x^8 + (6*B*a^2
*b^2 - 7*A*a*b^3)*x^6)*sqrt(-a)*arctan(sqrt(-a)/sqrt(b*x^2 + a)) + (15*(6*B*a^2*b^2 - 7*A*a*b^3)*x^6 - 8*A*a^4
+ 5*(6*B*a^3*b - 7*A*a^2*b^2)*x^4 - 2*(6*B*a^4 - 7*A*a^3*b)*x^2)*sqrt(b*x^2 + a))/(a^5*b*x^8 + a^6*x^6)]

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Sympy [A]
time = 60.83, size = 236, normalized size = 1.54 \begin {gather*} A \left (- \frac {1}{6 a \sqrt {b} x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {7 \sqrt {b}}{24 a^{2} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {35 b^{\frac {3}{2}}}{48 a^{3} x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {35 b^{\frac {5}{2}}}{16 a^{4} x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {35 b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{16 a^{\frac {9}{2}}}\right ) + B \left (- \frac {1}{4 a \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {5 \sqrt {b}}{8 a^{2} x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {15 b^{\frac {3}{2}}}{8 a^{3} x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {15 b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8 a^{\frac {7}{2}}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)/x**7/(b*x**2+a)**(3/2),x)

[Out]

A*(-1/(6*a*sqrt(b)*x**7*sqrt(a/(b*x**2) + 1)) + 7*sqrt(b)/(24*a**2*x**5*sqrt(a/(b*x**2) + 1)) - 35*b**(3/2)/(4
8*a**3*x**3*sqrt(a/(b*x**2) + 1)) - 35*b**(5/2)/(16*a**4*x*sqrt(a/(b*x**2) + 1)) + 35*b**3*asinh(sqrt(a)/(sqrt
(b)*x))/(16*a**(9/2))) + B*(-1/(4*a*sqrt(b)*x**5*sqrt(a/(b*x**2) + 1)) + 5*sqrt(b)/(8*a**2*x**3*sqrt(a/(b*x**2
) + 1)) + 15*b**(3/2)/(8*a**3*x*sqrt(a/(b*x**2) + 1)) - 15*b**2*asinh(sqrt(a)/(sqrt(b)*x))/(8*a**(7/2)))

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Giac [A]
time = 0.67, size = 180, normalized size = 1.18 \begin {gather*} \frac {5 \, {\left (6 \, B a b^{2} - 7 \, A b^{3}\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{16 \, \sqrt {-a} a^{4}} + \frac {B a b^{2} - A b^{3}}{\sqrt {b x^{2} + a} a^{4}} + \frac {42 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B a b^{2} - 96 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{2} b^{2} + 54 \, \sqrt {b x^{2} + a} B a^{3} b^{2} - 57 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{3} + 136 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A a b^{3} - 87 \, \sqrt {b x^{2} + a} A a^{2} b^{3}}{48 \, a^{4} b^{3} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^7/(b*x^2+a)^(3/2),x, algorithm="giac")

[Out]

5/16*(6*B*a*b^2 - 7*A*b^3)*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^4) + (B*a*b^2 - A*b^3)/(sqrt(b*x^2 + a
)*a^4) + 1/48*(42*(b*x^2 + a)^(5/2)*B*a*b^2 - 96*(b*x^2 + a)^(3/2)*B*a^2*b^2 + 54*sqrt(b*x^2 + a)*B*a^3*b^2 -
57*(b*x^2 + a)^(5/2)*A*b^3 + 136*(b*x^2 + a)^(3/2)*A*a*b^3 - 87*sqrt(b*x^2 + a)*A*a^2*b^3)/(a^4*b^3*x^6)

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Mupad [B]
time = 1.08, size = 178, normalized size = 1.16 \begin {gather*} \frac {35\,A\,b^3\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{16\,a^{9/2}}-\frac {15\,B\,b^2\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{8\,a^{7/2}}-\frac {35\,A\,b^3}{16\,a^4\,\sqrt {b\,x^2+a}}+\frac {15\,B\,b^2}{8\,a^3\,\sqrt {b\,x^2+a}}-\frac {A}{6\,a\,x^6\,\sqrt {b\,x^2+a}}-\frac {B}{4\,a\,x^4\,\sqrt {b\,x^2+a}}+\frac {7\,A\,b}{24\,a^2\,x^4\,\sqrt {b\,x^2+a}}+\frac {5\,B\,b}{8\,a^2\,x^2\,\sqrt {b\,x^2+a}}-\frac {35\,A\,b^2}{48\,a^3\,x^2\,\sqrt {b\,x^2+a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x^2)/(x^7*(a + b*x^2)^(3/2)),x)

[Out]

(35*A*b^3*atanh((a + b*x^2)^(1/2)/a^(1/2)))/(16*a^(9/2)) - (15*B*b^2*atanh((a + b*x^2)^(1/2)/a^(1/2)))/(8*a^(7
/2)) - (35*A*b^3)/(16*a^4*(a + b*x^2)^(1/2)) + (15*B*b^2)/(8*a^3*(a + b*x^2)^(1/2)) - A/(6*a*x^6*(a + b*x^2)^(
1/2)) - B/(4*a*x^4*(a + b*x^2)^(1/2)) + (7*A*b)/(24*a^2*x^4*(a + b*x^2)^(1/2)) + (5*B*b)/(8*a^2*x^2*(a + b*x^2
)^(1/2)) - (35*A*b^2)/(48*a^3*x^2*(a + b*x^2)^(1/2))

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