3.6.1 \(\int \frac {\sqrt {a+b x^2} (A+B x^2)}{x^{10}} \, dx\)

Optimal. Leaf size=117 \[ \frac {8 b^2 \left (a+b x^2\right )^{3/2} (2 A b-3 a B)}{315 a^4 x^3}-\frac {4 b \left (a+b x^2\right )^{3/2} (2 A b-3 a B)}{105 a^3 x^5}+\frac {\left (a+b x^2\right )^{3/2} (2 A b-3 a B)}{21 a^2 x^7}-\frac {A \left (a+b x^2\right )^{3/2}}{9 a x^9} \]

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Rubi [A]  time = 0.06, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {453, 271, 264} \begin {gather*} \frac {8 b^2 \left (a+b x^2\right )^{3/2} (2 A b-3 a B)}{315 a^4 x^3}-\frac {4 b \left (a+b x^2\right )^{3/2} (2 A b-3 a B)}{105 a^3 x^5}+\frac {\left (a+b x^2\right )^{3/2} (2 A b-3 a B)}{21 a^2 x^7}-\frac {A \left (a+b x^2\right )^{3/2}}{9 a x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[a + b*x^2]*(A + B*x^2))/x^10,x]

[Out]

-(A*(a + b*x^2)^(3/2))/(9*a*x^9) + ((2*A*b - 3*a*B)*(a + b*x^2)^(3/2))/(21*a^2*x^7) - (4*b*(2*A*b - 3*a*B)*(a
+ b*x^2)^(3/2))/(105*a^3*x^5) + (8*b^2*(2*A*b - 3*a*B)*(a + b*x^2)^(3/2))/(315*a^4*x^3)

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]

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 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]

Rubi steps

\begin {align*} \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{x^{10}} \, dx &=-\frac {A \left (a+b x^2\right )^{3/2}}{9 a x^9}-\frac {(6 A b-9 a B) \int \frac {\sqrt {a+b x^2}}{x^8} \, dx}{9 a}\\ &=-\frac {A \left (a+b x^2\right )^{3/2}}{9 a x^9}+\frac {(2 A b-3 a B) \left (a+b x^2\right )^{3/2}}{21 a^2 x^7}+\frac {(4 b (2 A b-3 a B)) \int \frac {\sqrt {a+b x^2}}{x^6} \, dx}{21 a^2}\\ &=-\frac {A \left (a+b x^2\right )^{3/2}}{9 a x^9}+\frac {(2 A b-3 a B) \left (a+b x^2\right )^{3/2}}{21 a^2 x^7}-\frac {4 b (2 A b-3 a B) \left (a+b x^2\right )^{3/2}}{105 a^3 x^5}-\frac {\left (8 b^2 (2 A b-3 a B)\right ) \int \frac {\sqrt {a+b x^2}}{x^4} \, dx}{105 a^3}\\ &=-\frac {A \left (a+b x^2\right )^{3/2}}{9 a x^9}+\frac {(2 A b-3 a B) \left (a+b x^2\right )^{3/2}}{21 a^2 x^7}-\frac {4 b (2 A b-3 a B) \left (a+b x^2\right )^{3/2}}{105 a^3 x^5}+\frac {8 b^2 (2 A b-3 a B) \left (a+b x^2\right )^{3/2}}{315 a^4 x^3}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 81, normalized size = 0.69 \begin {gather*} \frac {\left (a+b x^2\right )^{3/2} \left (-5 a^3 \left (7 A+9 B x^2\right )+6 a^2 b x^2 \left (5 A+6 B x^2\right )-24 a b^2 x^4 \left (A+B x^2\right )+16 A b^3 x^6\right )}{315 a^4 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/x^10,x]

[Out]

((a + b*x^2)^(3/2)*(16*A*b^3*x^6 - 24*a*b^2*x^4*(A + B*x^2) + 6*a^2*b*x^2*(5*A + 6*B*x^2) - 5*a^3*(7*A + 9*B*x
^2)))/(315*a^4*x^9)

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IntegrateAlgebraic [A]  time = 0.18, size = 110, normalized size = 0.94 \begin {gather*} \frac {\sqrt {a+b x^2} \left (-35 a^4 A-45 a^4 B x^2-5 a^3 A b x^2-9 a^3 b B x^4+6 a^2 A b^2 x^4+12 a^2 b^2 B x^6-8 a A b^3 x^6-24 a b^3 B x^8+16 A b^4 x^8\right )}{315 a^4 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(Sqrt[a + b*x^2]*(A + B*x^2))/x^10,x]

[Out]

(Sqrt[a + b*x^2]*(-35*a^4*A - 5*a^3*A*b*x^2 - 45*a^4*B*x^2 + 6*a^2*A*b^2*x^4 - 9*a^3*b*B*x^4 - 8*a*A*b^3*x^6 +
12*a^2*b^2*B*x^6 + 16*A*b^4*x^8 - 24*a*b^3*B*x^8))/(315*a^4*x^9)

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fricas [A]  time = 0.91, size = 105, normalized size = 0.90 \begin {gather*} -\frac {{\left (8 \, {\left (3 \, B a b^{3} - 2 \, A b^{4}\right )} x^{8} - 4 \, {\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{6} + 35 \, A a^{4} + 3 \, {\left (3 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{4} + 5 \, {\left (9 \, B a^{4} + A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{315 \, a^{4} x^{9}} \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^10,x, algorithm="fricas")

[Out]

-1/315*(8*(3*B*a*b^3 - 2*A*b^4)*x^8 - 4*(3*B*a^2*b^2 - 2*A*a*b^3)*x^6 + 35*A*a^4 + 3*(3*B*a^3*b - 2*A*a^2*b^2)
*x^4 + 5*(9*B*a^4 + A*a^3*b)*x^2)*sqrt(b*x^2 + a)/(a^4*x^9)

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giac [B]  time = 0.42, size = 344, normalized size = 2.94 \begin {gather*} \frac {16 \, {\left (210 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} B b^{\frac {7}{2}} - 315 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} B a b^{\frac {7}{2}} + 630 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} A b^{\frac {9}{2}} + 63 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} B a^{2} b^{\frac {7}{2}} + 378 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} A a b^{\frac {9}{2}} - 42 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a^{3} b^{\frac {7}{2}} + 168 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} A a^{2} b^{\frac {9}{2}} + 108 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{4} b^{\frac {7}{2}} - 72 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a^{3} b^{\frac {9}{2}} - 27 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{5} b^{\frac {7}{2}} + 18 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{4} b^{\frac {9}{2}} + 3 \, B a^{6} b^{\frac {7}{2}} - 2 \, A a^{5} b^{\frac {9}{2}}\right )}}{315 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{9}} \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^10,x, algorithm="giac")

[Out]

16/315*(210*(sqrt(b)*x - sqrt(b*x^2 + a))^12*B*b^(7/2) - 315*(sqrt(b)*x - sqrt(b*x^2 + a))^10*B*a*b^(7/2) + 63
0*(sqrt(b)*x - sqrt(b*x^2 + a))^10*A*b^(9/2) + 63*(sqrt(b)*x - sqrt(b*x^2 + a))^8*B*a^2*b^(7/2) + 378*(sqrt(b)
*x - sqrt(b*x^2 + a))^8*A*a*b^(9/2) - 42*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a^3*b^(7/2) + 168*(sqrt(b)*x - sqrt
(b*x^2 + a))^6*A*a^2*b^(9/2) + 108*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^4*b^(7/2) - 72*(sqrt(b)*x - sqrt(b*x^2
+ a))^4*A*a^3*b^(9/2) - 27*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^5*b^(7/2) + 18*(sqrt(b)*x - sqrt(b*x^2 + a))^2*
A*a^4*b^(9/2) + 3*B*a^6*b^(7/2) - 2*A*a^5*b^(9/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^9

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maple [A]  time = 0.01, size = 83, normalized size = 0.71 \begin {gather*} -\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (-16 A \,b^{3} x^{6}+24 B a \,b^{2} x^{6}+24 x^{4} A a \,b^{2}-36 B \,a^{2} b \,x^{4}-30 A \,a^{2} b \,x^{2}+45 B \,a^{3} x^{2}+35 A \,a^{3}\right )}{315 a^{4} x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)*(b*x^2+a)^(1/2)/x^10,x)

[Out]

-1/315*(b*x^2+a)^(3/2)*(-16*A*b^3*x^6+24*B*a*b^2*x^6+24*A*a*b^2*x^4-36*B*a^2*b*x^4-30*A*a^2*b*x^2+45*B*a^3*x^2
+35*A*a^3)/x^9/a^4

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maxima [A]  time = 0.99, size = 138, normalized size = 1.18 \begin {gather*} -\frac {8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b^{2}}{105 \, a^{3} x^{3}} + \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{3}}{315 \, a^{4} x^{3}} + \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b}{35 \, a^{2} x^{5}} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{2}}{105 \, a^{3} x^{5}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B}{7 \, a x^{7}} + \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b}{21 \, a^{2} x^{7}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A}{9 \, a x^{9}} \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^10,x, algorithm="maxima")

[Out]

-8/105*(b*x^2 + a)^(3/2)*B*b^2/(a^3*x^3) + 16/315*(b*x^2 + a)^(3/2)*A*b^3/(a^4*x^3) + 4/35*(b*x^2 + a)^(3/2)*B
*b/(a^2*x^5) - 8/105*(b*x^2 + a)^(3/2)*A*b^2/(a^3*x^5) - 1/7*(b*x^2 + a)^(3/2)*B/(a*x^7) + 2/21*(b*x^2 + a)^(3
/2)*A*b/(a^2*x^7) - 1/9*(b*x^2 + a)^(3/2)*A/(a*x^9)

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mupad [B]  time = 1.46, size = 174, normalized size = 1.49 \begin {gather*} \frac {2\,A\,b^2\,\sqrt {b\,x^2+a}}{105\,a^2\,x^5}-\frac {B\,\sqrt {b\,x^2+a}}{7\,x^7}-\frac {A\,b\,\sqrt {b\,x^2+a}}{63\,a\,x^7}-\frac {B\,b\,\sqrt {b\,x^2+a}}{35\,a\,x^5}-\frac {A\,\sqrt {b\,x^2+a}}{9\,x^9}-\frac {8\,A\,b^3\,\sqrt {b\,x^2+a}}{315\,a^3\,x^3}+\frac {16\,A\,b^4\,\sqrt {b\,x^2+a}}{315\,a^4\,x}+\frac {4\,B\,b^2\,\sqrt {b\,x^2+a}}{105\,a^2\,x^3}-\frac {8\,B\,b^3\,\sqrt {b\,x^2+a}}{105\,a^3\,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^10,x)

[Out]

(2*A*b^2*(a + b*x^2)^(1/2))/(105*a^2*x^5) - (B*(a + b*x^2)^(1/2))/(7*x^7) - (A*b*(a + b*x^2)^(1/2))/(63*a*x^7)
- (B*b*(a + b*x^2)^(1/2))/(35*a*x^5) - (A*(a + b*x^2)^(1/2))/(9*x^9) - (8*A*b^3*(a + b*x^2)^(1/2))/(315*a^3*x
^3) + (16*A*b^4*(a + b*x^2)^(1/2))/(315*a^4*x) + (4*B*b^2*(a + b*x^2)^(1/2))/(105*a^2*x^3) - (8*B*b^3*(a + b*x
^2)^(1/2))/(105*a^3*x)

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sympy [B]  time = 5.24, size = 957, normalized size = 8.18 \begin {gather*} - \frac {35 A a^{7} b^{\frac {19}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{315 a^{7} b^{9} x^{8} + 945 a^{6} b^{10} x^{10} + 945 a^{5} b^{11} x^{12} + 315 a^{4} b^{12} x^{14}} - \frac {110 A a^{6} b^{\frac {21}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{315 a^{7} b^{9} x^{8} + 945 a^{6} b^{10} x^{10} + 945 a^{5} b^{11} x^{12} + 315 a^{4} b^{12} x^{14}} - \frac {114 A a^{5} b^{\frac {23}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{315 a^{7} b^{9} x^{8} + 945 a^{6} b^{10} x^{10} + 945 a^{5} b^{11} x^{12} + 315 a^{4} b^{12} x^{14}} - \frac {40 A a^{4} b^{\frac {25}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{315 a^{7} b^{9} x^{8} + 945 a^{6} b^{10} x^{10} + 945 a^{5} b^{11} x^{12} + 315 a^{4} b^{12} x^{14}} + \frac {5 A a^{3} b^{\frac {27}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{315 a^{7} b^{9} x^{8} + 945 a^{6} b^{10} x^{10} + 945 a^{5} b^{11} x^{12} + 315 a^{4} b^{12} x^{14}} + \frac {30 A a^{2} b^{\frac {29}{2}} x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{315 a^{7} b^{9} x^{8} + 945 a^{6} b^{10} x^{10} + 945 a^{5} b^{11} x^{12} + 315 a^{4} b^{12} x^{14}} + \frac {40 A a b^{\frac {31}{2}} x^{12} \sqrt {\frac {a}{b x^{2}} + 1}}{315 a^{7} b^{9} x^{8} + 945 a^{6} b^{10} x^{10} + 945 a^{5} b^{11} x^{12} + 315 a^{4} b^{12} x^{14}} + \frac {16 A b^{\frac {33}{2}} x^{14} \sqrt {\frac {a}{b x^{2}} + 1}}{315 a^{7} b^{9} x^{8} + 945 a^{6} b^{10} x^{10} + 945 a^{5} b^{11} x^{12} + 315 a^{4} b^{12} x^{14}} - \frac {15 B 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 B 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 B 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 B 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 B 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 B 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}} \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**10,x)

[Out]

-35*A*a**7*b**(19/2)*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*x**10 + 945*a**5*b**11*x**12 +
315*a**4*b**12*x**14) - 110*A*a**6*b**(21/2)*x**2*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*x*
*10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14) - 114*A*a**5*b**(23/2)*x**4*sqrt(a/(b*x**2) + 1)/(315*a**7*
b**9*x**8 + 945*a**6*b**10*x**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14) - 40*A*a**4*b**(25/2)*x**6*sqr
t(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*x**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14) +
5*A*a**3*b**(27/2)*x**8*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*x**10 + 945*a**5*b**11*x**12
+ 315*a**4*b**12*x**14) + 30*A*a**2*b**(29/2)*x**10*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10
*x**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14) + 40*A*a*b**(31/2)*x**12*sqrt(a/(b*x**2) + 1)/(315*a**7*
b**9*x**8 + 945*a**6*b**10*x**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14) + 16*A*b**(33/2)*x**14*sqrt(a/
(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*x**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14) - 15*B
*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*B*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) - 17*B*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*B*
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*
B*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*B*
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)

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