3.386 \(\int \frac {(a+b x^2)^{3/2}}{x^{12}} \, dx\)

Optimal. Leaf size=92 \[ \frac {16 b^3 \left (a+b x^2\right )^{5/2}}{1155 a^4 x^5}-\frac {8 b^2 \left (a+b x^2\right )^{5/2}}{231 a^3 x^7}+\frac {2 b \left (a+b x^2\right )^{5/2}}{33 a^2 x^9}-\frac {\left (a+b x^2\right )^{5/2}}{11 a x^{11}} \]

[Out]

-1/11*(b*x^2+a)^(5/2)/a/x^11+2/33*b*(b*x^2+a)^(5/2)/a^2/x^9-8/231*b^2*(b*x^2+a)^(5/2)/a^3/x^7+16/1155*b^3*(b*x
^2+a)^(5/2)/a^4/x^5

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Rubi [A]  time = 0.03, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac {16 b^3 \left (a+b x^2\right )^{5/2}}{1155 a^4 x^5}-\frac {8 b^2 \left (a+b x^2\right )^{5/2}}{231 a^3 x^7}+\frac {2 b \left (a+b x^2\right )^{5/2}}{33 a^2 x^9}-\frac {\left (a+b x^2\right )^{5/2}}{11 a x^{11}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(3/2)/x^12,x]

[Out]

-(a + b*x^2)^(5/2)/(11*a*x^11) + (2*b*(a + b*x^2)^(5/2))/(33*a^2*x^9) - (8*b^2*(a + b*x^2)^(5/2))/(231*a^3*x^7
) + (16*b^3*(a + b*x^2)^(5/2))/(1155*a^4*x^5)

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]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^{3/2}}{x^{12}} \, dx &=-\frac {\left (a+b x^2\right )^{5/2}}{11 a x^{11}}-\frac {(6 b) \int \frac {\left (a+b x^2\right )^{3/2}}{x^{10}} \, dx}{11 a}\\ &=-\frac {\left (a+b x^2\right )^{5/2}}{11 a x^{11}}+\frac {2 b \left (a+b x^2\right )^{5/2}}{33 a^2 x^9}+\frac {\left (8 b^2\right ) \int \frac {\left (a+b x^2\right )^{3/2}}{x^8} \, dx}{33 a^2}\\ &=-\frac {\left (a+b x^2\right )^{5/2}}{11 a x^{11}}+\frac {2 b \left (a+b x^2\right )^{5/2}}{33 a^2 x^9}-\frac {8 b^2 \left (a+b x^2\right )^{5/2}}{231 a^3 x^7}-\frac {\left (16 b^3\right ) \int \frac {\left (a+b x^2\right )^{3/2}}{x^6} \, dx}{231 a^3}\\ &=-\frac {\left (a+b x^2\right )^{5/2}}{11 a x^{11}}+\frac {2 b \left (a+b x^2\right )^{5/2}}{33 a^2 x^9}-\frac {8 b^2 \left (a+b x^2\right )^{5/2}}{231 a^3 x^7}+\frac {16 b^3 \left (a+b x^2\right )^{5/2}}{1155 a^4 x^5}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 53, normalized size = 0.58 \[ \frac {\left (a+b x^2\right )^{5/2} \left (-105 a^3+70 a^2 b x^2-40 a b^2 x^4+16 b^3 x^6\right )}{1155 a^4 x^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(3/2)/x^12,x]

[Out]

((a + b*x^2)^(5/2)*(-105*a^3 + 70*a^2*b*x^2 - 40*a*b^2*x^4 + 16*b^3*x^6))/(1155*a^4*x^11)

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fricas [A]  time = 0.83, size = 71, normalized size = 0.77 \[ \frac {{\left (16 \, b^{5} x^{10} - 8 \, a b^{4} x^{8} + 6 \, a^{2} b^{3} x^{6} - 5 \, a^{3} b^{2} x^{4} - 140 \, a^{4} b x^{2} - 105 \, a^{5}\right )} \sqrt {b x^{2} + a}}{1155 \, a^{4} x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(3/2)/x^12,x, algorithm="fricas")

[Out]

1/1155*(16*b^5*x^10 - 8*a*b^4*x^8 + 6*a^2*b^3*x^6 - 5*a^3*b^2*x^4 - 140*a^4*b*x^2 - 105*a^5)*sqrt(b*x^2 + a)/(
a^4*x^11)

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giac [B]  time = 0.69, size = 220, normalized size = 2.39 \[ \frac {32 \, {\left (1155 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} b^{\frac {11}{2}} + 2079 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a b^{\frac {11}{2}} + 2541 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{2} b^{\frac {11}{2}} + 825 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{3} b^{\frac {11}{2}} + 165 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{4} b^{\frac {11}{2}} - 55 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{5} b^{\frac {11}{2}} + 11 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{6} b^{\frac {11}{2}} - a^{7} b^{\frac {11}{2}}\right )}}{1155 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(3/2)/x^12,x, algorithm="giac")

[Out]

32/1155*(1155*(sqrt(b)*x - sqrt(b*x^2 + a))^14*b^(11/2) + 2079*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a*b^(11/2) + 2
541*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^2*b^(11/2) + 825*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^3*b^(11/2) + 165*(sq
rt(b)*x - sqrt(b*x^2 + a))^6*a^4*b^(11/2) - 55*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^5*b^(11/2) + 11*(sqrt(b)*x -
sqrt(b*x^2 + a))^2*a^6*b^(11/2) - a^7*b^(11/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^11

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maple [A]  time = 0.01, size = 50, normalized size = 0.54 \[ -\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (-16 b^{3} x^{6}+40 a \,b^{2} x^{4}-70 a^{2} b \,x^{2}+105 a^{3}\right )}{1155 a^{4} x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(3/2)/x^12,x)

[Out]

-1/1155*(b*x^2+a)^(5/2)*(-16*b^3*x^6+40*a*b^2*x^4-70*a^2*b*x^2+105*a^3)/x^11/a^4

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maxima [A]  time = 1.42, size = 76, normalized size = 0.83 \[ \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}}{1155 \, a^{4} x^{5}} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2}}{231 \, a^{3} x^{7}} + \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b}{33 \, a^{2} x^{9}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{11 \, a x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(3/2)/x^12,x, algorithm="maxima")

[Out]

16/1155*(b*x^2 + a)^(5/2)*b^3/(a^4*x^5) - 8/231*(b*x^2 + a)^(5/2)*b^2/(a^3*x^7) + 2/33*(b*x^2 + a)^(5/2)*b/(a^
2*x^9) - 1/11*(b*x^2 + a)^(5/2)/(a*x^11)

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mupad [B]  time = 5.78, size = 111, normalized size = 1.21 \[ \frac {2\,b^3\,\sqrt {b\,x^2+a}}{385\,a^2\,x^5}-\frac {4\,b\,\sqrt {b\,x^2+a}}{33\,x^9}-\frac {b^2\,\sqrt {b\,x^2+a}}{231\,a\,x^7}-\frac {a\,\sqrt {b\,x^2+a}}{11\,x^{11}}-\frac {8\,b^4\,\sqrt {b\,x^2+a}}{1155\,a^3\,x^3}+\frac {16\,b^5\,\sqrt {b\,x^2+a}}{1155\,a^4\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^(3/2)/x^12,x)

[Out]

(2*b^3*(a + b*x^2)^(1/2))/(385*a^2*x^5) - (4*b*(a + b*x^2)^(1/2))/(33*x^9) - (b^2*(a + b*x^2)^(1/2))/(231*a*x^
7) - (a*(a + b*x^2)^(1/2))/(11*x^11) - (8*b^4*(a + b*x^2)^(1/2))/(1155*a^3*x^3) + (16*b^5*(a + b*x^2)^(1/2))/(
1155*a^4*x)

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sympy [B]  time = 2.12, size = 648, normalized size = 7.04 \[ - \frac {105 a^{8} b^{\frac {19}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{1155 a^{7} b^{9} x^{10} + 3465 a^{6} b^{10} x^{12} + 3465 a^{5} b^{11} x^{14} + 1155 a^{4} b^{12} x^{16}} - \frac {455 a^{7} b^{\frac {21}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{1155 a^{7} b^{9} x^{10} + 3465 a^{6} b^{10} x^{12} + 3465 a^{5} b^{11} x^{14} + 1155 a^{4} b^{12} x^{16}} - \frac {740 a^{6} b^{\frac {23}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{1155 a^{7} b^{9} x^{10} + 3465 a^{6} b^{10} x^{12} + 3465 a^{5} b^{11} x^{14} + 1155 a^{4} b^{12} x^{16}} - \frac {534 a^{5} b^{\frac {25}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{1155 a^{7} b^{9} x^{10} + 3465 a^{6} b^{10} x^{12} + 3465 a^{5} b^{11} x^{14} + 1155 a^{4} b^{12} x^{16}} - \frac {145 a^{4} b^{\frac {27}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{1155 a^{7} b^{9} x^{10} + 3465 a^{6} b^{10} x^{12} + 3465 a^{5} b^{11} x^{14} + 1155 a^{4} b^{12} x^{16}} + \frac {5 a^{3} b^{\frac {29}{2}} x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{1155 a^{7} b^{9} x^{10} + 3465 a^{6} b^{10} x^{12} + 3465 a^{5} b^{11} x^{14} + 1155 a^{4} b^{12} x^{16}} + \frac {30 a^{2} b^{\frac {31}{2}} x^{12} \sqrt {\frac {a}{b x^{2}} + 1}}{1155 a^{7} b^{9} x^{10} + 3465 a^{6} b^{10} x^{12} + 3465 a^{5} b^{11} x^{14} + 1155 a^{4} b^{12} x^{16}} + \frac {40 a b^{\frac {33}{2}} x^{14} \sqrt {\frac {a}{b x^{2}} + 1}}{1155 a^{7} b^{9} x^{10} + 3465 a^{6} b^{10} x^{12} + 3465 a^{5} b^{11} x^{14} + 1155 a^{4} b^{12} x^{16}} + \frac {16 b^{\frac {35}{2}} x^{16} \sqrt {\frac {a}{b x^{2}} + 1}}{1155 a^{7} b^{9} x^{10} + 3465 a^{6} b^{10} x^{12} + 3465 a^{5} b^{11} x^{14} + 1155 a^{4} b^{12} x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(3/2)/x**12,x)

[Out]

-105*a**8*b**(19/2)*sqrt(a/(b*x**2) + 1)/(1155*a**7*b**9*x**10 + 3465*a**6*b**10*x**12 + 3465*a**5*b**11*x**14
+ 1155*a**4*b**12*x**16) - 455*a**7*b**(21/2)*x**2*sqrt(a/(b*x**2) + 1)/(1155*a**7*b**9*x**10 + 3465*a**6*b**
10*x**12 + 3465*a**5*b**11*x**14 + 1155*a**4*b**12*x**16) - 740*a**6*b**(23/2)*x**4*sqrt(a/(b*x**2) + 1)/(1155
*a**7*b**9*x**10 + 3465*a**6*b**10*x**12 + 3465*a**5*b**11*x**14 + 1155*a**4*b**12*x**16) - 534*a**5*b**(25/2)
*x**6*sqrt(a/(b*x**2) + 1)/(1155*a**7*b**9*x**10 + 3465*a**6*b**10*x**12 + 3465*a**5*b**11*x**14 + 1155*a**4*b
**12*x**16) - 145*a**4*b**(27/2)*x**8*sqrt(a/(b*x**2) + 1)/(1155*a**7*b**9*x**10 + 3465*a**6*b**10*x**12 + 346
5*a**5*b**11*x**14 + 1155*a**4*b**12*x**16) + 5*a**3*b**(29/2)*x**10*sqrt(a/(b*x**2) + 1)/(1155*a**7*b**9*x**1
0 + 3465*a**6*b**10*x**12 + 3465*a**5*b**11*x**14 + 1155*a**4*b**12*x**16) + 30*a**2*b**(31/2)*x**12*sqrt(a/(b
*x**2) + 1)/(1155*a**7*b**9*x**10 + 3465*a**6*b**10*x**12 + 3465*a**5*b**11*x**14 + 1155*a**4*b**12*x**16) + 4
0*a*b**(33/2)*x**14*sqrt(a/(b*x**2) + 1)/(1155*a**7*b**9*x**10 + 3465*a**6*b**10*x**12 + 3465*a**5*b**11*x**14
+ 1155*a**4*b**12*x**16) + 16*b**(35/2)*x**16*sqrt(a/(b*x**2) + 1)/(1155*a**7*b**9*x**10 + 3465*a**6*b**10*x*
*12 + 3465*a**5*b**11*x**14 + 1155*a**4*b**12*x**16)

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