3.516 \(\int \frac{\sqrt{a+b x^2} (A+B x^2)}{x^7} \, dx\)

Optimal. Leaf size=120 \[ -\frac{b^2 (A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{16 a^{5/2}}+\frac{b \sqrt{a+b x^2} (A b-2 a B)}{16 a^2 x^2}+\frac{\sqrt{a+b x^2} (A b-2 a B)}{8 a x^4}-\frac{A \left (a+b x^2\right )^{3/2}}{6 a x^6} \]

[Out]

((A*b - 2*a*B)*Sqrt[a + b*x^2])/(8*a*x^4) + (b*(A*b - 2*a*B)*Sqrt[a + b*x^2])/(16*a^2*x^2) - (A*(a + b*x^2)^(3
/2))/(6*a*x^6) - (b^2*(A*b - 2*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(16*a^(5/2))

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Rubi [A]  time = 0.0936198, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {446, 78, 47, 51, 63, 208} \[ -\frac{b^2 (A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{16 a^{5/2}}+\frac{b \sqrt{a+b x^2} (A b-2 a B)}{16 a^2 x^2}+\frac{\sqrt{a+b x^2} (A b-2 a B)}{8 a x^4}-\frac{A \left (a+b x^2\right )^{3/2}}{6 a x^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((A*b - 2*a*B)*Sqrt[a + b*x^2])/(8*a*x^4) + (b*(A*b - 2*a*B)*Sqrt[a + b*x^2])/(16*a^2*x^2) - (A*(a + b*x^2)^(3
/2))/(6*a*x^6) - (b^2*(A*b - 2*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(16*a^(5/2))

Rule 446

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]]

Rule 78

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] || LtQ
[p, n]))))

Rule 47

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

Rule 51

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 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{\sqrt{a+b x^2} \left (A+B x^2\right )}{x^7} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x} (A+B x)}{x^4} \, dx,x,x^2\right )\\ &=-\frac{A \left (a+b x^2\right )^{3/2}}{6 a x^6}+\frac{\left (-\frac{3 A b}{2}+3 a B\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^3} \, dx,x,x^2\right )}{6 a}\\ &=\frac{(A b-2 a B) \sqrt{a+b x^2}}{8 a x^4}-\frac{A \left (a+b x^2\right )^{3/2}}{6 a x^6}-\frac{(b (A b-2 a B)) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,x^2\right )}{16 a}\\ &=\frac{(A b-2 a B) \sqrt{a+b x^2}}{8 a x^4}+\frac{b (A b-2 a B) \sqrt{a+b x^2}}{16 a^2 x^2}-\frac{A \left (a+b x^2\right )^{3/2}}{6 a x^6}+\frac{\left (b^2 (A b-2 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^2\right )}{32 a^2}\\ &=\frac{(A b-2 a B) \sqrt{a+b x^2}}{8 a x^4}+\frac{b (A b-2 a B) \sqrt{a+b x^2}}{16 a^2 x^2}-\frac{A \left (a+b x^2\right )^{3/2}}{6 a x^6}+\frac{(b (A b-2 a B)) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )}{16 a^2}\\ &=\frac{(A b-2 a B) \sqrt{a+b x^2}}{8 a x^4}+\frac{b (A b-2 a B) \sqrt{a+b x^2}}{16 a^2 x^2}-\frac{A \left (a+b x^2\right )^{3/2}}{6 a x^6}-\frac{b^2 (A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{16 a^{5/2}}\\ \end{align*}

Mathematica [C]  time = 0.0211102, size = 61, normalized size = 0.51 \[ -\frac{\left (a+b x^2\right )^{3/2} \left (a^3 A+b^2 x^6 (2 a B-A b) \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};\frac{b x^2}{a}+1\right )\right )}{6 a^4 x^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a + b*x^2)^(3/2)*(a^3*A + b^2*(-(A*b) + 2*a*B)*x^6*Hypergeometric2F1[3/2, 3, 5/2, 1 + (b*x^2)/a]))/(6*a^4*x
^6)

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Maple [A]  time = 0.013, size = 197, normalized size = 1.6 \begin{align*} -{\frac{A}{6\,a{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{Ab}{8\,{a}^{2}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{A{b}^{2}}{16\,{a}^{3}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{A{b}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{A{b}^{3}}{16\,{a}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{B}{4\,a{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{Bb}{8\,{a}^{2}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{B{b}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{B{b}^{2}}{8\,{a}^{2}}\sqrt{b{x}^{2}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)*(b*x^2+a)^(1/2)/x^7,x)

[Out]

-1/6*A*(b*x^2+a)^(3/2)/a/x^6+1/8*A*b/a^2/x^4*(b*x^2+a)^(3/2)-1/16*A*b^2/a^3/x^2*(b*x^2+a)^(3/2)-1/16*A*b^3/a^(
5/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)+1/16*A*b^3/a^3*(b*x^2+a)^(1/2)-1/4*B/a/x^4*(b*x^2+a)^(3/2)+1/8*B*b/
a^2/x^2*(b*x^2+a)^(3/2)+1/8*B*b^2/a^(3/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)-1/8*B*b^2/a^2*(b*x^2+a)^(1/2)

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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^2+A)*(b*x^2+a)^(1/2)/x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.76611, size = 500, normalized size = 4.17 \begin{align*} \left [-\frac{3 \,{\left (2 \, B a b^{2} - A b^{3}\right )} \sqrt{a} x^{6} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (3 \,{\left (2 \, B a^{2} b - A a b^{2}\right )} x^{4} + 8 \, A a^{3} + 2 \,{\left (6 \, B a^{3} + A a^{2} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{96 \, a^{3} x^{6}}, -\frac{3 \,{\left (2 \, B a b^{2} - A b^{3}\right )} \sqrt{-a} x^{6} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) +{\left (3 \,{\left (2 \, B a^{2} b - A a b^{2}\right )} x^{4} + 8 \, A a^{3} + 2 \,{\left (6 \, B a^{3} + A a^{2} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{48 \, a^{3} x^{6}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)*(b*x^2+a)^(1/2)/x^7,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 59.6569, size = 226, normalized size = 1.88 \begin{align*} - \frac{A a}{6 \sqrt{b} x^{7} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{5 A \sqrt{b}}{24 x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{A b^{\frac{3}{2}}}{48 a x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{A b^{\frac{5}{2}}}{16 a^{2} x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{A b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{16 a^{\frac{5}{2}}} - \frac{B a}{4 \sqrt{b} x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 B \sqrt{b}}{8 x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{B b^{\frac{3}{2}}}{8 a x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{B b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{8 a^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)*(b*x**2+a)**(1/2)/x**7,x)

[Out]

-A*a/(6*sqrt(b)*x**7*sqrt(a/(b*x**2) + 1)) - 5*A*sqrt(b)/(24*x**5*sqrt(a/(b*x**2) + 1)) + A*b**(3/2)/(48*a*x**
3*sqrt(a/(b*x**2) + 1)) + A*b**(5/2)/(16*a**2*x*sqrt(a/(b*x**2) + 1)) - A*b**3*asinh(sqrt(a)/(sqrt(b)*x))/(16*
a**(5/2)) - B*a/(4*sqrt(b)*x**5*sqrt(a/(b*x**2) + 1)) - 3*B*sqrt(b)/(8*x**3*sqrt(a/(b*x**2) + 1)) - B*b**(3/2)
/(8*a*x*sqrt(a/(b*x**2) + 1)) + B*b**2*asinh(sqrt(a)/(sqrt(b)*x))/(8*a**(3/2))

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Giac [A]  time = 1.11098, size = 189, normalized size = 1.58 \begin{align*} -\frac{\frac{3 \,{\left (2 \, B a b^{3} - A b^{4}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{6 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} B a b^{3} - 6 \, \sqrt{b x^{2} + a} B a^{3} b^{3} - 3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} A b^{4} + 8 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A a b^{4} + 3 \, \sqrt{b x^{2} + a} A a^{2} b^{4}}{a^{2} b^{3} x^{6}}}{48 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)*(b*x^2+a)^(1/2)/x^7,x, algorithm="giac")

[Out]

-1/48*(3*(2*B*a*b^3 - A*b^4)*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^2) + (6*(b*x^2 + a)^(5/2)*B*a*b^3 -
6*sqrt(b*x^2 + a)*B*a^3*b^3 - 3*(b*x^2 + a)^(5/2)*A*b^4 + 8*(b*x^2 + a)^(3/2)*A*a*b^4 + 3*sqrt(b*x^2 + a)*A*a^
2*b^4)/(a^2*b^3*x^6))/b