3.6.17 \(\int \frac {\sqrt {a+b x^2} (A+B x^2)}{x^8} \, dx\) [517]

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

[Out]

-1/7*A*(b*x^2+a)^(3/2)/a/x^7+1/35*(4*A*b-7*B*a)*(b*x^2+a)^(3/2)/a^2/x^5-2/105*b*(4*A*b-7*B*a)*(b*x^2+a)^(3/2)/
a^3/x^3

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Rubi [A]
time = 0.02, antiderivative size = 84, normalized size of antiderivative = 1.00, 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 = {464, 277, 270} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/7*(A*(a + b*x^2)^(3/2))/(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 270

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]

Rule 277

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 464

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]

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*}

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Mathematica [A]
time = 0.12, size = 62, normalized size = 0.74 \begin {gather*} \frac {\left (a+b x^2\right )^{3/2} \left (-15 a^2 A+12 a A b x^2-21 a^2 B x^2-8 A b^2 x^4+14 a b B x^4\right )}{105 a^3 x^7} \end {gather*}

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

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Maple [A]
time = 0.09, size = 102, normalized size = 1.21

method	result	size
gosper	\(-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (8 A \,b^{2} x^{4}-14 B a b \,x^{4}-12 a A b \,x^{2}+21 B \,a^{2} x^{2}+15 a^{2} A \right )}{105 a^{3} x^{7}}\)	\(59\)
trager	\(-\frac {\left (8 x^{6} A \,b^{3}-14 x^{6} B a \,b^{2}-4 A a \,b^{2} x^{4}+7 x^{4} B \,a^{2} b +3 x^{2} A \,a^{2} b +21 B \,a^{3} x^{2}+15 A \,a^{3}\right ) \sqrt {b \,x^{2}+a}}{105 a^{3} x^{7}}\)	\(83\)
risch	\(-\frac {\left (8 x^{6} A \,b^{3}-14 x^{6} B a \,b^{2}-4 A a \,b^{2} x^{4}+7 x^{4} B \,a^{2} b +3 x^{2} A \,a^{2} b +21 B \,a^{3} x^{2}+15 A \,a^{3}\right ) \sqrt {b \,x^{2}+a}}{105 a^{3} x^{7}}\)	\(83\)
default	\(B \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{5 a \,x^{5}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )+A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{7 a \,x^{7}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{5 a \,x^{5}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{15 a^{2} x^{3}}\right )}{7 a}\right )\)	\(102\)

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,method=_RETURNVERBOSE)

[Out]

B*(-1/5/a/x^5*(b*x^2+a)^(3/2)+2/15*b/a^2*(b*x^2+a)^(3/2)/x^3)+A*(-1/7/a/x^7*(b*x^2+a)^(3/2)-4/7*b/a*(-1/5/a/x^
5*(b*x^2+a)^(3/2)+2/15*b/a^2*(b*x^2+a)^(3/2)/x^3))

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Maxima [A]
time = 0.32, size = 96, normalized size = 1.14 \begin {gather*} \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b}{15 \, a^{2} x^{3}} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{2}}{105 \, a^{3} x^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B}{5 \, a x^{5}} + \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b}{35 \, a^{2} x^{5}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A}{7 \, a x^{7}} \end {gather*}

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]

2/15*(b*x^2 + a)^(3/2)*B*b/(a^2*x^3) - 8/105*(b*x^2 + a)^(3/2)*A*b^2/(a^3*x^3) - 1/5*(b*x^2 + a)^(3/2)*B/(a*x^
5) + 4/35*(b*x^2 + a)^(3/2)*A*b/(a^2*x^5) - 1/7*(b*x^2 + a)^(3/2)*A/(a*x^7)

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Fricas [A]
time = 1.51, size = 81, normalized size = 0.96 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (78) = 156\).
time = 1.62, size = 442, normalized size = 5.26 \begin {gather*} - \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 288 vs. \(2 (72) = 144\).
time = 0.88, size = 288, normalized size = 3.43 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 0.55, size = 132, normalized size = 1.57 \begin {gather*} \frac {4\,A\,b^2\,\sqrt {b\,x^2+a}}{105\,a^2\,x^3}-\frac {B\,\sqrt {b\,x^2+a}}{5\,x^5}-\frac {A\,b\,\sqrt {b\,x^2+a}}{35\,a\,x^5}-\frac {B\,b\,\sqrt {b\,x^2+a}}{15\,a\,x^3}-\frac {A\,\sqrt {b\,x^2+a}}{7\,x^7}-\frac {8\,A\,b^3\,\sqrt {b\,x^2+a}}{105\,a^3\,x}+\frac {2\,B\,b^2\,\sqrt {b\,x^2+a}}{15\,a^2\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^2)*(a + b*x^2)^(1/2))/x^8,x)

[Out]

(4*A*b^2*(a + b*x^2)^(1/2))/(105*a^2*x^3) - (B*(a + b*x^2)^(1/2))/(5*x^5) - (A*b*(a + b*x^2)^(1/2))/(35*a*x^5)
- (B*b*(a + b*x^2)^(1/2))/(15*a*x^3) - (A*(a + b*x^2)^(1/2))/(7*x^7) - (8*A*b^3*(a + b*x^2)^(1/2))/(105*a^3*x
) + (2*B*b^2*(a + b*x^2)^(1/2))/(15*a^2*x)

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