Integrand size = 21, antiderivative size = 136 \[ \int x^7 \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2} \, dx=-\frac {2 a^3 \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2} \left (a+b x^2\right )}{b^4}+\frac {6 a^2 \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2} \left (a+b x^2\right )^2}{5 b^4}-\frac {2 a \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2} \left (a+b x^2\right )^3}{3 b^4}+\frac {2 \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2} \left (a+b x^2\right )^4}{13 b^4} \] Output:
-2*a^3*(c/(b*x^2+a)^(1/2))^(3/2)*(b*x^2+a)/b^4+6/5*a^2*(c/(b*x^2+a)^(1/2)) ^(3/2)*(b*x^2+a)^2/b^4-2/3*a*(c/(b*x^2+a)^(1/2))^(3/2)*(b*x^2+a)^3/b^4+2/1 3*(c/(b*x^2+a)^(1/2))^(3/2)*(b*x^2+a)^4/b^4
Time = 0.05 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.43 \[ \int x^7 \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2} \, dx=\frac {2 c^2 \left (-128 a^3+32 a^2 b x^2-20 a b^2 x^4+15 b^3 x^6\right )}{195 b^4 \sqrt {\frac {c}{\sqrt {a+b x^2}}}} \] Input:
Integrate[x^7*(c/Sqrt[a + b*x^2])^(3/2),x]
Output:
(2*c^2*(-128*a^3 + 32*a^2*b*x^2 - 20*a*b^2*x^4 + 15*b^3*x^6))/(195*b^4*Sqr t[c/Sqrt[a + b*x^2]])
Time = 0.48 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2045, 243, 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.
\(\displaystyle \int x^7 \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 2045 |
\(\displaystyle \left (\frac {b x^2}{a}+1\right )^{3/4} \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2} \int \frac {x^7}{\left (\frac {b x^2}{a}+1\right )^{3/4}}dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \left (\frac {b x^2}{a}+1\right )^{3/4} \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2} \int \frac {x^6}{\left (\frac {b x^2}{a}+1\right )^{3/4}}dx^2\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \frac {1}{2} \left (\frac {b x^2}{a}+1\right )^{3/4} \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2} \int \left (\frac {\left (\frac {b x^2}{a}+1\right )^{9/4} a^3}{b^3}-\frac {3 \left (\frac {b x^2}{a}+1\right )^{5/4} a^3}{b^3}+\frac {3 \sqrt [4]{\frac {b x^2}{a}+1} a^3}{b^3}-\frac {a^3}{b^3 \left (\frac {b x^2}{a}+1\right )^{3/4}}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {b x^2}{a}+1\right )^{3/4} \left (\frac {4 a^4 \left (\frac {b x^2}{a}+1\right )^{13/4}}{13 b^4}-\frac {4 a^4 \left (\frac {b x^2}{a}+1\right )^{9/4}}{3 b^4}+\frac {12 a^4 \left (\frac {b x^2}{a}+1\right )^{5/4}}{5 b^4}-\frac {4 a^4 \sqrt [4]{\frac {b x^2}{a}+1}}{b^4}\right ) \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2}\) |
Input:
Int[x^7*(c/Sqrt[a + b*x^2])^(3/2),x]
Output:
((c/Sqrt[a + b*x^2])^(3/2)*(1 + (b*x^2)/a)^(3/4)*((-4*a^4*(1 + (b*x^2)/a)^ (1/4))/b^4 + (12*a^4*(1 + (b*x^2)/a)^(5/4))/(5*b^4) - (4*a^4*(1 + (b*x^2)/ a)^(9/4))/(3*b^4) + (4*a^4*(1 + (b*x^2)/a)^(13/4))/(13*b^4)))/2
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_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(u_.)*((c_.)*((a_) + (b_.)*(x_)^(n_.))^(q_))^(p_), x_Symbol] :> Simp[Si mp[(c*(a + b*x^n)^q)^p/(1 + b*(x^n/a))^(p*q)] Int[u*(1 + b*(x^n/a))^(p*q) , x], x] /; FreeQ[{a, b, c, n, p, q}, x] && !GeQ[a, 0]
Time = 0.14 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.43
method | result | size |
gosper | \(-\frac {2 \left (b \,x^{2}+a \right ) \left (-15 b^{3} x^{6}+20 a \,b^{2} x^{4}-32 a^{2} b \,x^{2}+128 a^{3}\right ) {\left (\frac {c}{\sqrt {b \,x^{2}+a}}\right )}^{\frac {3}{2}}}{195 b^{4}}\) | \(58\) |
orering | \(-\frac {2 \left (b \,x^{2}+a \right ) \left (-15 b^{3} x^{6}+20 a \,b^{2} x^{4}-32 a^{2} b \,x^{2}+128 a^{3}\right ) {\left (\frac {c}{\sqrt {b \,x^{2}+a}}\right )}^{\frac {3}{2}}}{195 b^{4}}\) | \(58\) |
Input:
int(x^7*(c/(b*x^2+a)^(1/2))^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/195*(b*x^2+a)*(-15*b^3*x^6+20*a*b^2*x^4-32*a^2*b*x^2+128*a^3)*(c/(b*x^2 +a)^(1/2))^(3/2)/b^4
Time = 0.14 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.46 \[ \int x^7 \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2} \, dx=\frac {2 \, {\left (15 \, b^{3} c x^{6} - 20 \, a b^{2} c x^{4} + 32 \, a^{2} b c x^{2} - 128 \, a^{3} c\right )} \sqrt {b x^{2} + a} \sqrt {\frac {c}{\sqrt {b x^{2} + a}}}}{195 \, b^{4}} \] Input:
integrate(x^7*(c/(b*x^2+a)^(1/2))^(3/2),x, algorithm="fricas")
Output:
2/195*(15*b^3*c*x^6 - 20*a*b^2*c*x^4 + 32*a^2*b*c*x^2 - 128*a^3*c)*sqrt(b* x^2 + a)*sqrt(c/sqrt(b*x^2 + a))/b^4
Time = 10.73 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.06 \[ \int x^7 \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2} \, dx=\begin {cases} - \frac {256 a^{4} \left (\frac {c}{\sqrt {a + b x^{2}}}\right )^{\frac {3}{2}}}{195 b^{4}} - \frac {64 a^{3} x^{2} \left (\frac {c}{\sqrt {a + b x^{2}}}\right )^{\frac {3}{2}}}{65 b^{3}} + \frac {8 a^{2} x^{4} \left (\frac {c}{\sqrt {a + b x^{2}}}\right )^{\frac {3}{2}}}{65 b^{2}} - \frac {2 a x^{6} \left (\frac {c}{\sqrt {a + b x^{2}}}\right )^{\frac {3}{2}}}{39 b} + \frac {2 x^{8} \left (\frac {c}{\sqrt {a + b x^{2}}}\right )^{\frac {3}{2}}}{13} & \text {for}\: b \neq 0 \\\frac {x^{8} \left (\frac {c}{\sqrt {a}}\right )^{\frac {3}{2}}}{8} & \text {otherwise} \end {cases} \] Input:
integrate(x**7*(c/(b*x**2+a)**(1/2))**(3/2),x)
Output:
Piecewise((-256*a**4*(c/sqrt(a + b*x**2))**(3/2)/(195*b**4) - 64*a**3*x**2 *(c/sqrt(a + b*x**2))**(3/2)/(65*b**3) + 8*a**2*x**4*(c/sqrt(a + b*x**2))* *(3/2)/(65*b**2) - 2*a*x**6*(c/sqrt(a + b*x**2))**(3/2)/(39*b) + 2*x**8*(c /sqrt(a + b*x**2))**(3/2)/13, Ne(b, 0)), (x**8*(c/sqrt(a))**(3/2)/8, True) )
Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.56 \[ \int x^7 \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2} \, dx=-\frac {2 \, {\left (\frac {65 \, a c^{6}}{b x^{2} + a} - \frac {117 \, a^{2} c^{6}}{{\left (b x^{2} + a\right )}^{2}} + \frac {195 \, a^{3} c^{6}}{{\left (b x^{2} + a\right )}^{3}} - 15 \, c^{6}\right )} c^{2}}{195 \, b^{4} \left (\frac {c}{\sqrt {b x^{2} + a}}\right )^{\frac {13}{2}}} \] Input:
integrate(x^7*(c/(b*x^2+a)^(1/2))^(3/2),x, algorithm="maxima")
Output:
-2/195*(65*a*c^6/(b*x^2 + a) - 117*a^2*c^6/(b*x^2 + a)^2 + 195*a^3*c^6/(b* x^2 + a)^3 - 15*c^6)*c^2/(b^4*(c/sqrt(b*x^2 + a))^(13/2))
Time = 0.13 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.51 \[ \int x^7 \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2} \, dx=\frac {2 \, {\left (15 \, {\left (b x^{2} + a\right )}^{\frac {13}{4}} - 65 \, {\left (b x^{2} + a\right )}^{\frac {9}{4}} a + 117 \, {\left (b x^{2} + a\right )}^{\frac {5}{4}} a^{2} - 195 \, {\left (b x^{2} + a\right )}^{\frac {1}{4}} a^{3}\right )} c^{\frac {3}{2}}}{195 \, b^{4} \sqrt {\mathrm {sgn}\left (b x^{2} + a\right )}} \] Input:
integrate(x^7*(c/(b*x^2+a)^(1/2))^(3/2),x, algorithm="giac")
Output:
2/195*(15*(b*x^2 + a)^(13/4) - 65*(b*x^2 + a)^(9/4)*a + 117*(b*x^2 + a)^(5 /4)*a^2 - 195*(b*x^2 + a)^(1/4)*a^3)*c^(3/2)/(b^4*sqrt(sgn(b*x^2 + a)))
Time = 0.55 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.67 \[ \int x^7 \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2} \, dx=\sqrt {\frac {c}{\sqrt {b\,x^2+a}}}\,\left (\frac {2\,c\,x^6\,\sqrt {b\,x^2+a}}{13\,b}-\frac {256\,a^3\,c\,\sqrt {b\,x^2+a}}{195\,b^4}-\frac {8\,a\,c\,x^4\,\sqrt {b\,x^2+a}}{39\,b^2}+\frac {64\,a^2\,c\,x^2\,\sqrt {b\,x^2+a}}{195\,b^3}\right ) \] Input:
int(x^7*(c/(a + b*x^2)^(1/2))^(3/2),x)
Output:
(c/(a + b*x^2)^(1/2))^(1/2)*((2*c*x^6*(a + b*x^2)^(1/2))/(13*b) - (256*a^3 *c*(a + b*x^2)^(1/2))/(195*b^4) - (8*a*c*x^4*(a + b*x^2)^(1/2))/(39*b^2) + (64*a^2*c*x^2*(a + b*x^2)^(1/2))/(195*b^3))
Time = 0.20 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.10 \[ \int x^7 \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2} \, dx=\frac {2 \sqrt {c}\, \sqrt {\sqrt {b}\, \sqrt {b \,x^{2}+a}\, x +a +b \,x^{2}}\, \sqrt {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}\, c \left (-128 \sqrt {b \,x^{2}+a}\, a^{3}+32 \sqrt {b \,x^{2}+a}\, a^{2} b \,x^{2}-20 \sqrt {b \,x^{2}+a}\, a \,b^{2} x^{4}+15 \sqrt {b \,x^{2}+a}\, b^{3} x^{6}+128 \sqrt {b}\, a^{3} x -32 \sqrt {b}\, a^{2} b \,x^{3}+20 \sqrt {b}\, a \,b^{2} x^{5}-15 \sqrt {b}\, b^{3} x^{7}\right )}{195 a \,b^{4}} \] Input:
int(x^7*(c/(b*x^2+a)^(1/2))^(3/2),x)
Output:
(2*sqrt(c)*sqrt(sqrt(b)*sqrt(a + b*x**2)*x + a + b*x**2)*sqrt(sqrt(a + b*x **2) + sqrt(b)*x)*c*( - 128*sqrt(a + b*x**2)*a**3 + 32*sqrt(a + b*x**2)*a* *2*b*x**2 - 20*sqrt(a + b*x**2)*a*b**2*x**4 + 15*sqrt(a + b*x**2)*b**3*x** 6 + 128*sqrt(b)*a**3*x - 32*sqrt(b)*a**2*b*x**3 + 20*sqrt(b)*a*b**2*x**5 - 15*sqrt(b)*b**3*x**7))/(195*a*b**4)