3.517 \(\int \frac{\sqrt{a+b x^2} (A+B x^2)}{x^8} \, dx\)

Optimal. Leaf size=84 \[ -\frac{2 b \left (a+b x^2\right )^{3/2} (4 A b-7 a B)}{105 a^3 x^3}+\frac{\left (a+b x^2\right )^{3/2} (4 A b-7 a B)}{35 a^2 x^5}-\frac{A \left (a+b x^2\right )^{3/2}}{7 a x^7} \]

[Out]

-(A*(a + b*x^2)^(3/2))/(7*a*x^7) + ((4*A*b - 7*a*B)*(a + b*x^2)^(3/2))/(35*a^2*x^5) - (2*b*(4*A*b - 7*a*B)*(a
+ b*x^2)^(3/2))/(105*a^3*x^3)

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Rubi [A]  time = 0.0336299, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {453, 271, 264} \[ -\frac{2 b \left (a+b x^2\right )^{3/2} (4 A b-7 a B)}{105 a^3 x^3}+\frac{\left (a+b x^2\right )^{3/2} (4 A b-7 a B)}{35 a^2 x^5}-\frac{A \left (a+b x^2\right )^{3/2}}{7 a x^7} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[a + b*x^2]*(A + B*x^2))/x^8,x]

[Out]

-(A*(a + b*x^2)^(3/2))/(7*a*x^7) + ((4*A*b - 7*a*B)*(a + b*x^2)^(3/2))/(35*a^2*x^5) - (2*b*(4*A*b - 7*a*B)*(a
+ b*x^2)^(3/2))/(105*a^3*x^3)

Rule 453

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

Rule 271

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

Rule 264

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

Rubi steps

\begin{align*} \int \frac{\sqrt{a+b x^2} \left (A+B x^2\right )}{x^8} \, dx &=-\frac{A \left (a+b x^2\right )^{3/2}}{7 a x^7}-\frac{(4 A b-7 a B) \int \frac{\sqrt{a+b x^2}}{x^6} \, dx}{7 a}\\ &=-\frac{A \left (a+b x^2\right )^{3/2}}{7 a x^7}+\frac{(4 A b-7 a B) \left (a+b x^2\right )^{3/2}}{35 a^2 x^5}+\frac{(2 b (4 A b-7 a B)) \int \frac{\sqrt{a+b x^2}}{x^4} \, dx}{35 a^2}\\ &=-\frac{A \left (a+b x^2\right )^{3/2}}{7 a x^7}+\frac{(4 A b-7 a B) \left (a+b x^2\right )^{3/2}}{35 a^2 x^5}-\frac{2 b (4 A b-7 a B) \left (a+b x^2\right )^{3/2}}{105 a^3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0299985, size = 63, normalized size = 0.75 \[ \frac{\left (a+b x^2\right )^{3/2} \left (-3 a^2 \left (5 A+7 B x^2\right )+2 a b x^2 \left (6 A+7 B x^2\right )-8 A b^2 x^4\right )}{105 a^3 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/x^8,x]

[Out]

((a + b*x^2)^(3/2)*(-8*A*b^2*x^4 - 3*a^2*(5*A + 7*B*x^2) + 2*a*b*x^2*(6*A + 7*B*x^2)))/(105*a^3*x^7)

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Maple [A]  time = 0.005, size = 59, normalized size = 0.7 \begin{align*} -{\frac{8\,A{b}^{2}{x}^{4}-14\,B{x}^{4}ab-12\,aAb{x}^{2}+21\,B{x}^{2}{a}^{2}+15\,A{a}^{2}}{105\,{x}^{7}{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}} \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^8,x)

[Out]

-1/105*(b*x^2+a)^(3/2)*(8*A*b^2*x^4-14*B*a*b*x^4-12*A*a*b*x^2+21*B*a^2*x^2+15*A*a^2)/x^7/a^3

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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^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.73242, size = 178, normalized size = 2.12 \begin{align*} \frac{{\left (2 \,{\left (7 \, B a b^{2} - 4 \, A b^{3}\right )} x^{6} -{\left (7 \, B a^{2} b - 4 \, A a b^{2}\right )} x^{4} - 15 \, A a^{3} - 3 \,{\left (7 \, B a^{3} + A a^{2} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{105 \, a^{3} x^{7}} \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^8,x, algorithm="fricas")

[Out]

1/105*(2*(7*B*a*b^2 - 4*A*b^3)*x^6 - (7*B*a^2*b - 4*A*a*b^2)*x^4 - 15*A*a^3 - 3*(7*B*a^3 + A*a^2*b)*x^2)*sqrt(
b*x^2 + a)/(a^3*x^7)

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Sympy [B]  time = 3.0506, size = 442, normalized size = 5.26 \begin{align*} - \frac{15 A a^{5} b^{\frac{9}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac{33 A a^{4} b^{\frac{11}{2}} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac{17 A a^{3} b^{\frac{13}{2}} x^{4} \sqrt{\frac{a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac{3 A a^{2} b^{\frac{15}{2}} x^{6} \sqrt{\frac{a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac{12 A a b^{\frac{17}{2}} x^{8} \sqrt{\frac{a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac{8 A b^{\frac{19}{2}} x^{10} \sqrt{\frac{a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac{B \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{5 x^{4}} - \frac{B b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a x^{2}} + \frac{2 B b^{\frac{5}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a^{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**8,x)

[Out]

-15*A*a**5*b**(9/2)*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) - 33*
A*a**4*b**(11/2)*x**2*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) - 1
7*A*a**3*b**(13/2)*x**4*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) -
3*A*a**2*b**(15/2)*x**6*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10)
- 12*A*a*b**(17/2)*x**8*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) -
8*A*b**(19/2)*x**10*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) - B*
sqrt(b)*sqrt(a/(b*x**2) + 1)/(5*x**4) - B*b**(3/2)*sqrt(a/(b*x**2) + 1)/(15*a*x**2) + 2*B*b**(5/2)*sqrt(a/(b*x
**2) + 1)/(15*a**2)

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Giac [B]  time = 1.14164, size = 389, normalized size = 4.63 \begin{align*} \frac{4 \,{\left (105 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} B b^{\frac{5}{2}} - 175 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} B a b^{\frac{5}{2}} + 280 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} A b^{\frac{7}{2}} + 70 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} B a^{2} b^{\frac{5}{2}} + 140 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} A a b^{\frac{7}{2}} - 42 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a^{3} b^{\frac{5}{2}} + 84 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A a^{2} b^{\frac{7}{2}} + 49 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{4} b^{\frac{5}{2}} - 28 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A a^{3} b^{\frac{7}{2}} - 7 \, B a^{5} b^{\frac{5}{2}} + 4 \, A a^{4} b^{\frac{7}{2}}\right )}}{105 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{7}} \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^8,x, algorithm="giac")

[Out]

4/105*(105*(sqrt(b)*x - sqrt(b*x^2 + a))^10*B*b^(5/2) - 175*(sqrt(b)*x - sqrt(b*x^2 + a))^8*B*a*b^(5/2) + 280*
(sqrt(b)*x - sqrt(b*x^2 + a))^8*A*b^(7/2) + 70*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a^2*b^(5/2) + 140*(sqrt(b)*x
- sqrt(b*x^2 + a))^6*A*a*b^(7/2) - 42*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^3*b^(5/2) + 84*(sqrt(b)*x - sqrt(b*x
^2 + a))^4*A*a^2*b^(7/2) + 49*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^4*b^(5/2) - 28*(sqrt(b)*x - sqrt(b*x^2 + a))
^2*A*a^3*b^(7/2) - 7*B*a^5*b^(5/2) + 4*A*a^4*b^(7/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^7