Integrand size = 15, antiderivative size = 68 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^{19}} \, dx=-\frac {\left (a+c x^4\right )^{5/2}}{18 a x^{18}}+\frac {2 c \left (a+c x^4\right )^{5/2}}{63 a^2 x^{14}}-\frac {4 c^2 \left (a+c x^4\right )^{5/2}}{315 a^3 x^{10}} \] Output:
-1/18*(c*x^4+a)^(5/2)/a/x^18+2/63*c*(c*x^4+a)^(5/2)/a^2/x^14-4/315*c^2*(c* x^4+a)^(5/2)/a^3/x^10
Time = 0.23 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.62 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^{19}} \, dx=\frac {\left (a+c x^4\right )^{5/2} \left (-35 a^2+20 a c x^4-8 c^2 x^8\right )}{630 a^3 x^{18}} \] Input:
Integrate[(a + c*x^4)^(3/2)/x^19,x]
Output:
((a + c*x^4)^(5/2)*(-35*a^2 + 20*a*c*x^4 - 8*c^2*x^8))/(630*a^3*x^18)
Time = 0.32 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {803, 803, 796}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (a+c x^4\right )^{3/2}}{x^{19}} \, dx\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {4 c \int \frac {\left (c x^4+a\right )^{3/2}}{x^{15}}dx}{9 a}-\frac {\left (a+c x^4\right )^{5/2}}{18 a x^{18}}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {4 c \left (-\frac {2 c \int \frac {\left (c x^4+a\right )^{3/2}}{x^{11}}dx}{7 a}-\frac {\left (a+c x^4\right )^{5/2}}{14 a x^{14}}\right )}{9 a}-\frac {\left (a+c x^4\right )^{5/2}}{18 a x^{18}}\) |
\(\Big \downarrow \) 796 |
\(\displaystyle -\frac {4 c \left (\frac {c \left (a+c x^4\right )^{5/2}}{35 a^2 x^{10}}-\frac {\left (a+c x^4\right )^{5/2}}{14 a x^{14}}\right )}{9 a}-\frac {\left (a+c x^4\right )^{5/2}}{18 a x^{18}}\) |
Input:
Int[(a + c*x^4)^(3/2)/x^19,x]
Output:
-1/18*(a + c*x^4)^(5/2)/(a*x^18) - (4*c*(-1/14*(a + c*x^4)^(5/2)/(a*x^14) + (c*(a + c*x^4)^(5/2))/(35*a^2*x^10)))/(9*a)
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]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*(( a + b*x^n)^(p + 1)/(a*(m + 1))), x] - Simp[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] && I LtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]
Time = 1.67 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.57
method | result | size |
gosper | \(-\frac {\left (c \,x^{4}+a \right )^{\frac {5}{2}} \left (8 c^{2} x^{8}-20 a \,x^{4} c +35 a^{2}\right )}{630 x^{18} a^{3}}\) | \(39\) |
pseudoelliptic | \(-\frac {\left (c \,x^{4}+a \right )^{\frac {5}{2}} \left (8 c^{2} x^{8}-20 a \,x^{4} c +35 a^{2}\right )}{630 x^{18} a^{3}}\) | \(39\) |
orering | \(-\frac {\left (c \,x^{4}+a \right )^{\frac {5}{2}} \left (8 c^{2} x^{8}-20 a \,x^{4} c +35 a^{2}\right )}{630 x^{18} a^{3}}\) | \(39\) |
default | \(-\frac {\sqrt {c \,x^{4}+a}\, \left (8 c^{2} x^{8}-20 a \,x^{4} c +35 a^{2}\right ) \left (c^{2} x^{8}+2 a \,x^{4} c +a^{2}\right )}{630 a^{3} x^{18}}\) | \(57\) |
elliptic | \(-\frac {\sqrt {c \,x^{4}+a}\, \left (8 c^{2} x^{8}-20 a \,x^{4} c +35 a^{2}\right ) \left (c^{2} x^{8}+2 a \,x^{4} c +a^{2}\right )}{630 a^{3} x^{18}}\) | \(57\) |
trager | \(-\frac {\left (8 c^{4} x^{16}-4 c^{3} a \,x^{12}+3 a^{2} x^{8} c^{2}+50 a^{3} c \,x^{4}+35 a^{4}\right ) \sqrt {c \,x^{4}+a}}{630 x^{18} a^{3}}\) | \(61\) |
risch | \(-\frac {\left (8 c^{4} x^{16}-4 c^{3} a \,x^{12}+3 a^{2} x^{8} c^{2}+50 a^{3} c \,x^{4}+35 a^{4}\right ) \sqrt {c \,x^{4}+a}}{630 x^{18} a^{3}}\) | \(61\) |
Input:
int((c*x^4+a)^(3/2)/x^19,x,method=_RETURNVERBOSE)
Output:
-1/630*(c*x^4+a)^(5/2)*(8*c^2*x^8-20*a*c*x^4+35*a^2)/x^18/a^3
Time = 0.09 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^{19}} \, dx=-\frac {{\left (8 \, c^{4} x^{16} - 4 \, a c^{3} x^{12} + 3 \, a^{2} c^{2} x^{8} + 50 \, a^{3} c x^{4} + 35 \, a^{4}\right )} \sqrt {c x^{4} + a}}{630 \, a^{3} x^{18}} \] Input:
integrate((c*x^4+a)^(3/2)/x^19,x, algorithm="fricas")
Output:
-1/630*(8*c^4*x^16 - 4*a*c^3*x^12 + 3*a^2*c^2*x^8 + 50*a^3*c*x^4 + 35*a^4) *sqrt(c*x^4 + a)/(a^3*x^18)
Leaf count of result is larger than twice the leaf count of optimal. 420 vs. \(2 (61) = 122\).
Time = 1.38 (sec) , antiderivative size = 420, normalized size of antiderivative = 6.18 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^{19}} \, dx=- \frac {35 a^{6} c^{\frac {9}{2}} \sqrt {\frac {a}{c x^{4}} + 1}}{630 a^{5} c^{4} x^{16} + 1260 a^{4} c^{5} x^{20} + 630 a^{3} c^{6} x^{24}} - \frac {120 a^{5} c^{\frac {11}{2}} x^{4} \sqrt {\frac {a}{c x^{4}} + 1}}{630 a^{5} c^{4} x^{16} + 1260 a^{4} c^{5} x^{20} + 630 a^{3} c^{6} x^{24}} - \frac {138 a^{4} c^{\frac {13}{2}} x^{8} \sqrt {\frac {a}{c x^{4}} + 1}}{630 a^{5} c^{4} x^{16} + 1260 a^{4} c^{5} x^{20} + 630 a^{3} c^{6} x^{24}} - \frac {52 a^{3} c^{\frac {15}{2}} x^{12} \sqrt {\frac {a}{c x^{4}} + 1}}{630 a^{5} c^{4} x^{16} + 1260 a^{4} c^{5} x^{20} + 630 a^{3} c^{6} x^{24}} - \frac {3 a^{2} c^{\frac {17}{2}} x^{16} \sqrt {\frac {a}{c x^{4}} + 1}}{630 a^{5} c^{4} x^{16} + 1260 a^{4} c^{5} x^{20} + 630 a^{3} c^{6} x^{24}} - \frac {12 a c^{\frac {19}{2}} x^{20} \sqrt {\frac {a}{c x^{4}} + 1}}{630 a^{5} c^{4} x^{16} + 1260 a^{4} c^{5} x^{20} + 630 a^{3} c^{6} x^{24}} - \frac {8 c^{\frac {21}{2}} x^{24} \sqrt {\frac {a}{c x^{4}} + 1}}{630 a^{5} c^{4} x^{16} + 1260 a^{4} c^{5} x^{20} + 630 a^{3} c^{6} x^{24}} \] Input:
integrate((c*x**4+a)**(3/2)/x**19,x)
Output:
-35*a**6*c**(9/2)*sqrt(a/(c*x**4) + 1)/(630*a**5*c**4*x**16 + 1260*a**4*c* *5*x**20 + 630*a**3*c**6*x**24) - 120*a**5*c**(11/2)*x**4*sqrt(a/(c*x**4) + 1)/(630*a**5*c**4*x**16 + 1260*a**4*c**5*x**20 + 630*a**3*c**6*x**24) - 138*a**4*c**(13/2)*x**8*sqrt(a/(c*x**4) + 1)/(630*a**5*c**4*x**16 + 1260*a **4*c**5*x**20 + 630*a**3*c**6*x**24) - 52*a**3*c**(15/2)*x**12*sqrt(a/(c* x**4) + 1)/(630*a**5*c**4*x**16 + 1260*a**4*c**5*x**20 + 630*a**3*c**6*x** 24) - 3*a**2*c**(17/2)*x**16*sqrt(a/(c*x**4) + 1)/(630*a**5*c**4*x**16 + 1 260*a**4*c**5*x**20 + 630*a**3*c**6*x**24) - 12*a*c**(19/2)*x**20*sqrt(a/( c*x**4) + 1)/(630*a**5*c**4*x**16 + 1260*a**4*c**5*x**20 + 630*a**3*c**6*x **24) - 8*c**(21/2)*x**24*sqrt(a/(c*x**4) + 1)/(630*a**5*c**4*x**16 + 1260 *a**4*c**5*x**20 + 630*a**3*c**6*x**24)
Time = 0.03 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^{19}} \, dx=-\frac {\frac {63 \, {\left (c x^{4} + a\right )}^{\frac {5}{2}} c^{2}}{x^{10}} - \frac {90 \, {\left (c x^{4} + a\right )}^{\frac {7}{2}} c}{x^{14}} + \frac {35 \, {\left (c x^{4} + a\right )}^{\frac {9}{2}}}{x^{18}}}{630 \, a^{3}} \] Input:
integrate((c*x^4+a)^(3/2)/x^19,x, algorithm="maxima")
Output:
-1/630*(63*(c*x^4 + a)^(5/2)*c^2/x^10 - 90*(c*x^4 + a)^(7/2)*c/x^14 + 35*( c*x^4 + a)^(9/2)/x^18)/a^3
Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (56) = 112\).
Time = 0.15 (sec) , antiderivative size = 206, normalized size of antiderivative = 3.03 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^{19}} \, dx=\frac {8 \, {\left (210 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{12} c^{\frac {9}{2}} + 315 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{10} a c^{\frac {9}{2}} + 441 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{8} a^{2} c^{\frac {9}{2}} + 126 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{6} a^{3} c^{\frac {9}{2}} + 36 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{4} a^{4} c^{\frac {9}{2}} - 9 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2} a^{5} c^{\frac {9}{2}} + a^{6} c^{\frac {9}{2}}\right )}}{315 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2} - a\right )}^{9}} \] Input:
integrate((c*x^4+a)^(3/2)/x^19,x, algorithm="giac")
Output:
8/315*(210*(sqrt(c)*x^2 - sqrt(c*x^4 + a))^12*c^(9/2) + 315*(sqrt(c)*x^2 - sqrt(c*x^4 + a))^10*a*c^(9/2) + 441*(sqrt(c)*x^2 - sqrt(c*x^4 + a))^8*a^2 *c^(9/2) + 126*(sqrt(c)*x^2 - sqrt(c*x^4 + a))^6*a^3*c^(9/2) + 36*(sqrt(c) *x^2 - sqrt(c*x^4 + a))^4*a^4*c^(9/2) - 9*(sqrt(c)*x^2 - sqrt(c*x^4 + a))^ 2*a^5*c^(9/2) + a^6*c^(9/2))/((sqrt(c)*x^2 - sqrt(c*x^4 + a))^2 - a)^9
Time = 1.40 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.34 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^{19}} \, dx=\frac {2\,c^3\,\sqrt {c\,x^4+a}}{315\,a^2\,x^6}-\frac {5\,c\,\sqrt {c\,x^4+a}}{63\,x^{14}}-\frac {4\,c^4\,\sqrt {c\,x^4+a}}{315\,a^3\,x^2}-\frac {a\,\sqrt {c\,x^4+a}}{18\,x^{18}}-\frac {c^2\,\sqrt {c\,x^4+a}}{210\,a\,x^{10}} \] Input:
int((a + c*x^4)^(3/2)/x^19,x)
Output:
(2*c^3*(a + c*x^4)^(1/2))/(315*a^2*x^6) - (5*c*(a + c*x^4)^(1/2))/(63*x^14 ) - (4*c^4*(a + c*x^4)^(1/2))/(315*a^3*x^2) - (a*(a + c*x^4)^(1/2))/(18*x^ 18) - (c^2*(a + c*x^4)^(1/2))/(210*a*x^10)
Time = 0.21 (sec) , antiderivative size = 325, normalized size of antiderivative = 4.78 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^{19}} \, dx=\frac {-315 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, a^{5} x^{2}-4650 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, a^{4} c \,x^{6}-21147 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, a^{3} c^{2} x^{10}-42084 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, a^{2} c^{3} x^{14}-38640 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, a \,c^{4} x^{18}-13440 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, c^{5} x^{22}-35 a^{6}-1485 a^{5} c \,x^{4}-11853 a^{4} c^{2} x^{8}-38199 a^{3} c^{3} x^{12}-59724 a^{2} c^{4} x^{16}-45360 a \,c^{5} x^{20}-13440 c^{6} x^{24}}{630 x^{18} \left (\sqrt {c \,x^{4}+a}\, a^{4}+40 \sqrt {c \,x^{4}+a}\, a^{3} c \,x^{4}+240 \sqrt {c \,x^{4}+a}\, a^{2} c^{2} x^{8}+448 \sqrt {c \,x^{4}+a}\, a \,c^{3} x^{12}+256 \sqrt {c \,x^{4}+a}\, c^{4} x^{16}+9 \sqrt {c}\, a^{4} x^{2}+120 \sqrt {c}\, a^{3} c \,x^{6}+432 \sqrt {c}\, a^{2} c^{2} x^{10}+576 \sqrt {c}\, a \,c^{3} x^{14}+256 \sqrt {c}\, c^{4} x^{18}\right )} \] Input:
int((c*x^4+a)^(3/2)/x^19,x)
Output:
( - 315*sqrt(c)*sqrt(a + c*x**4)*a**5*x**2 - 4650*sqrt(c)*sqrt(a + c*x**4) *a**4*c*x**6 - 21147*sqrt(c)*sqrt(a + c*x**4)*a**3*c**2*x**10 - 42084*sqrt (c)*sqrt(a + c*x**4)*a**2*c**3*x**14 - 38640*sqrt(c)*sqrt(a + c*x**4)*a*c* *4*x**18 - 13440*sqrt(c)*sqrt(a + c*x**4)*c**5*x**22 - 35*a**6 - 1485*a**5 *c*x**4 - 11853*a**4*c**2*x**8 - 38199*a**3*c**3*x**12 - 59724*a**2*c**4*x **16 - 45360*a*c**5*x**20 - 13440*c**6*x**24)/(630*x**18*(sqrt(a + c*x**4) *a**4 + 40*sqrt(a + c*x**4)*a**3*c*x**4 + 240*sqrt(a + c*x**4)*a**2*c**2*x **8 + 448*sqrt(a + c*x**4)*a*c**3*x**12 + 256*sqrt(a + c*x**4)*c**4*x**16 + 9*sqrt(c)*a**4*x**2 + 120*sqrt(c)*a**3*c*x**6 + 432*sqrt(c)*a**2*c**2*x* *10 + 576*sqrt(c)*a*c**3*x**14 + 256*sqrt(c)*c**4*x**18))