Integrand size = 15, antiderivative size = 59 \[ \int x^{11} \left (a+c x^4\right )^{3/2} \, dx=\frac {a^2 \left (a+c x^4\right )^{5/2}}{10 c^3}-\frac {a \left (a+c x^4\right )^{7/2}}{7 c^3}+\frac {\left (a+c x^4\right )^{9/2}}{18 c^3} \] Output:
1/10*a^2*(c*x^4+a)^(5/2)/c^3-1/7*a*(c*x^4+a)^(7/2)/c^3+1/18*(c*x^4+a)^(9/2 )/c^3
Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.66 \[ \int x^{11} \left (a+c x^4\right )^{3/2} \, dx=\frac {\left (a+c x^4\right )^{5/2} \left (8 a^2-20 a c x^4+35 c^2 x^8\right )}{630 c^3} \] Input:
Integrate[x^11*(a + c*x^4)^(3/2),x]
Output:
((a + c*x^4)^(5/2)*(8*a^2 - 20*a*c*x^4 + 35*c^2*x^8))/(630*c^3)
Time = 0.30 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {798, 53, 2009}
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.
\(\displaystyle \int x^{11} \left (a+c x^4\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{4} \int x^8 \left (c x^4+a\right )^{3/2}dx^4\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \frac {1}{4} \int \left (\frac {\left (c x^4+a\right )^{7/2}}{c^2}-\frac {2 a \left (c x^4+a\right )^{5/2}}{c^2}+\frac {a^2 \left (c x^4+a\right )^{3/2}}{c^2}\right )dx^4\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} \left (\frac {2 a^2 \left (a+c x^4\right )^{5/2}}{5 c^3}+\frac {2 \left (a+c x^4\right )^{9/2}}{9 c^3}-\frac {4 a \left (a+c x^4\right )^{7/2}}{7 c^3}\right )\) |
Input:
Int[x^11*(a + c*x^4)^(3/2),x]
Output:
((2*a^2*(a + c*x^4)^(5/2))/(5*c^3) - (4*a*(a + c*x^4)^(7/2))/(7*c^3) + (2* (a + c*x^4)^(9/2))/(9*c^3))/4
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.60 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.61
method | result | size |
gosper | \(\frac {\left (c \,x^{4}+a \right )^{\frac {5}{2}} \left (35 c^{2} x^{8}-20 a \,x^{4} c +8 a^{2}\right )}{630 c^{3}}\) | \(36\) |
pseudoelliptic | \(\frac {\left (c \,x^{4}+a \right )^{\frac {5}{2}} \left (35 c^{2} x^{8}-20 a \,x^{4} c +8 a^{2}\right )}{630 c^{3}}\) | \(36\) |
orering | \(\frac {\left (c \,x^{4}+a \right )^{\frac {5}{2}} \left (35 c^{2} x^{8}-20 a \,x^{4} c +8 a^{2}\right )}{630 c^{3}}\) | \(36\) |
default | \(\frac {\sqrt {c \,x^{4}+a}\, \left (35 c^{2} x^{8}-20 a \,x^{4} c +8 a^{2}\right ) \left (c^{2} x^{8}+2 a \,x^{4} c +a^{2}\right )}{630 c^{3}}\) | \(54\) |
elliptic | \(\frac {\sqrt {c \,x^{4}+a}\, \left (35 c^{2} x^{8}-20 a \,x^{4} c +8 a^{2}\right ) \left (c^{2} x^{8}+2 a \,x^{4} c +a^{2}\right )}{630 c^{3}}\) | \(54\) |
trager | \(\frac {\left (35 c^{4} x^{16}+50 c^{3} a \,x^{12}+3 a^{2} x^{8} c^{2}-4 a^{3} c \,x^{4}+8 a^{4}\right ) \sqrt {c \,x^{4}+a}}{630 c^{3}}\) | \(58\) |
risch | \(\frac {\left (35 c^{4} x^{16}+50 c^{3} a \,x^{12}+3 a^{2} x^{8} c^{2}-4 a^{3} c \,x^{4}+8 a^{4}\right ) \sqrt {c \,x^{4}+a}}{630 c^{3}}\) | \(58\) |
Input:
int(x^11*(c*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/630*(c*x^4+a)^(5/2)*(35*c^2*x^8-20*a*c*x^4+8*a^2)/c^3
Time = 0.06 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.97 \[ \int x^{11} \left (a+c x^4\right )^{3/2} \, dx=\frac {{\left (35 \, c^{4} x^{16} + 50 \, a c^{3} x^{12} + 3 \, a^{2} c^{2} x^{8} - 4 \, a^{3} c x^{4} + 8 \, a^{4}\right )} \sqrt {c x^{4} + a}}{630 \, c^{3}} \] Input:
integrate(x^11*(c*x^4+a)^(3/2),x, algorithm="fricas")
Output:
1/630*(35*c^4*x^16 + 50*a*c^3*x^12 + 3*a^2*c^2*x^8 - 4*a^3*c*x^4 + 8*a^4)* sqrt(c*x^4 + a)/c^3
Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (49) = 98\).
Time = 0.55 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.85 \[ \int x^{11} \left (a+c x^4\right )^{3/2} \, dx=\begin {cases} \frac {4 a^{4} \sqrt {a + c x^{4}}}{315 c^{3}} - \frac {2 a^{3} x^{4} \sqrt {a + c x^{4}}}{315 c^{2}} + \frac {a^{2} x^{8} \sqrt {a + c x^{4}}}{210 c} + \frac {5 a x^{12} \sqrt {a + c x^{4}}}{63} + \frac {c x^{16} \sqrt {a + c x^{4}}}{18} & \text {for}\: c \neq 0 \\\frac {a^{\frac {3}{2}} x^{12}}{12} & \text {otherwise} \end {cases} \] Input:
integrate(x**11*(c*x**4+a)**(3/2),x)
Output:
Piecewise((4*a**4*sqrt(a + c*x**4)/(315*c**3) - 2*a**3*x**4*sqrt(a + c*x** 4)/(315*c**2) + a**2*x**8*sqrt(a + c*x**4)/(210*c) + 5*a*x**12*sqrt(a + c* x**4)/63 + c*x**16*sqrt(a + c*x**4)/18, Ne(c, 0)), (a**(3/2)*x**12/12, Tru e))
Time = 0.04 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.80 \[ \int x^{11} \left (a+c x^4\right )^{3/2} \, dx=\frac {{\left (c x^{4} + a\right )}^{\frac {9}{2}}}{18 \, c^{3}} - \frac {{\left (c x^{4} + a\right )}^{\frac {7}{2}} a}{7 \, c^{3}} + \frac {{\left (c x^{4} + a\right )}^{\frac {5}{2}} a^{2}}{10 \, c^{3}} \] Input:
integrate(x^11*(c*x^4+a)^(3/2),x, algorithm="maxima")
Output:
1/18*(c*x^4 + a)^(9/2)/c^3 - 1/7*(c*x^4 + a)^(7/2)*a/c^3 + 1/10*(c*x^4 + a )^(5/2)*a^2/c^3
Time = 0.12 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.73 \[ \int x^{11} \left (a+c x^4\right )^{3/2} \, dx=\frac {35 \, {\left (c x^{4} + a\right )}^{\frac {9}{2}} - 90 \, {\left (c x^{4} + a\right )}^{\frac {7}{2}} a + 63 \, {\left (c x^{4} + a\right )}^{\frac {5}{2}} a^{2}}{630 \, c^{3}} \] Input:
integrate(x^11*(c*x^4+a)^(3/2),x, algorithm="giac")
Output:
1/630*(35*(c*x^4 + a)^(9/2) - 90*(c*x^4 + a)^(7/2)*a + 63*(c*x^4 + a)^(5/2 )*a^2)/c^3
Time = 0.38 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.90 \[ \int x^{11} \left (a+c x^4\right )^{3/2} \, dx=\sqrt {c\,x^4+a}\,\left (\frac {5\,a\,x^{12}}{63}+\frac {c\,x^{16}}{18}+\frac {4\,a^4}{315\,c^3}-\frac {2\,a^3\,x^4}{315\,c^2}+\frac {a^2\,x^8}{210\,c}\right ) \] Input:
int(x^11*(a + c*x^4)^(3/2),x)
Output:
(a + c*x^4)^(1/2)*((5*a*x^12)/63 + (c*x^16)/18 + (4*a^4)/(315*c^3) - (2*a^ 3*x^4)/(315*c^2) + (a^2*x^8)/(210*c))
Time = 0.22 (sec) , antiderivative size = 421, normalized size of antiderivative = 7.14 \[ \int x^{11} \left (a+c x^4\right )^{3/2} \, dx=\frac {72 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, a^{8} x^{2}+924 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, a^{7} c \,x^{6}+3003 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, a^{6} c^{2} x^{10}+3690 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, a^{5} c^{3} x^{14}+7355 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, a^{4} c^{4} x^{18}+26504 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, a^{3} c^{5} x^{22}+44688 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, a^{2} c^{6} x^{26}+32960 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, a \,c^{7} x^{30}+8960 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, c^{8} x^{34}+8 a^{9}+324 a^{8} c \,x^{4}+2079 a^{7} c^{2} x^{8}+4557 a^{6} c^{3} x^{12}+5805 a^{5} c^{4} x^{16}+16731 a^{4} c^{5} x^{20}+45288 a^{3} c^{6} x^{24}+60048 a^{2} c^{7} x^{28}+37440 a \,c^{8} x^{32}+8960 c^{9} x^{36}}{630 c^{3} \left (\sqrt {c \,x^{4}+a}\, a^{4}+40 \sqrt {c \,x^{4}+a}\, a^{3} c \,x^{4}+240 \sqrt {c \,x^{4}+a}\, a^{2} c^{2} x^{8}+448 \sqrt {c \,x^{4}+a}\, a \,c^{3} x^{12}+256 \sqrt {c \,x^{4}+a}\, c^{4} x^{16}+9 \sqrt {c}\, a^{4} x^{2}+120 \sqrt {c}\, a^{3} c \,x^{6}+432 \sqrt {c}\, a^{2} c^{2} x^{10}+576 \sqrt {c}\, a \,c^{3} x^{14}+256 \sqrt {c}\, c^{4} x^{18}\right )} \] Input:
int(x^11*(c*x^4+a)^(3/2),x)
Output:
(72*sqrt(c)*sqrt(a + c*x**4)*a**8*x**2 + 924*sqrt(c)*sqrt(a + c*x**4)*a**7 *c*x**6 + 3003*sqrt(c)*sqrt(a + c*x**4)*a**6*c**2*x**10 + 3690*sqrt(c)*sqr t(a + c*x**4)*a**5*c**3*x**14 + 7355*sqrt(c)*sqrt(a + c*x**4)*a**4*c**4*x* *18 + 26504*sqrt(c)*sqrt(a + c*x**4)*a**3*c**5*x**22 + 44688*sqrt(c)*sqrt( a + c*x**4)*a**2*c**6*x**26 + 32960*sqrt(c)*sqrt(a + c*x**4)*a*c**7*x**30 + 8960*sqrt(c)*sqrt(a + c*x**4)*c**8*x**34 + 8*a**9 + 324*a**8*c*x**4 + 20 79*a**7*c**2*x**8 + 4557*a**6*c**3*x**12 + 5805*a**5*c**4*x**16 + 16731*a* *4*c**5*x**20 + 45288*a**3*c**6*x**24 + 60048*a**2*c**7*x**28 + 37440*a*c* *8*x**32 + 8960*c**9*x**36)/(630*c**3*(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))