\(\int \frac {(a^2-b^2 x^2)^{3/2}}{(a+b x)^8} \, dx\) [59]

Optimal result

Integrand size = 24, antiderivative size = 133 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^8} \, dx=-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{11 a b (a+b x)^8}-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{33 a^2 b (a+b x)^7}-\frac {2 \left (a^2-b^2 x^2\right )^{5/2}}{231 a^3 b (a+b x)^6}-\frac {2 \left (a^2-b^2 x^2\right )^{5/2}}{1155 a^4 b (a+b x)^5} \] Output:

-1/11*(-b^2*x^2+a^2)^(5/2)/a/b/(b*x+a)^8-1/33*(-b^2*x^2+a^2)^(5/2)/a^2/b/( 
b*x+a)^7-2/231*(-b^2*x^2+a^2)^(5/2)/a^3/b/(b*x+a)^6-2/1155*(-b^2*x^2+a^2)^ 
(5/2)/a^4/b/(b*x+a)^5
 

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.53 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^8} \, dx=-\frac {(a-b x)^2 \sqrt {a^2-b^2 x^2} \left (152 a^3+61 a^2 b x+16 a b^2 x^2+2 b^3 x^3\right )}{1155 a^4 b (a+b x)^6} \] Input:

Integrate[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^8,x]
 

Output:

-1/1155*((a - b*x)^2*Sqrt[a^2 - b^2*x^2]*(152*a^3 + 61*a^2*b*x + 16*a*b^2* 
x^2 + 2*b^3*x^3))/(a^4*b*(a + b*x)^6)
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {461, 461, 461, 460}

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.

\(\displaystyle \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^8} \, dx\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {3 \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^7}dx}{11 a}-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{11 a b (a+b x)^8}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {3 \left (\frac {2 \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^6}dx}{9 a}-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{9 a b (a+b x)^7}\right )}{11 a}-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{11 a b (a+b x)^8}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {3 \left (\frac {2 \left (\frac {\int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^5}dx}{7 a}-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{7 a b (a+b x)^6}\right )}{9 a}-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{9 a b (a+b x)^7}\right )}{11 a}-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{11 a b (a+b x)^8}\)

\(\Big \downarrow \) 460

\(\displaystyle \frac {3 \left (\frac {2 \left (-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{35 a^2 b (a+b x)^5}-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{7 a b (a+b x)^6}\right )}{9 a}-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{9 a b (a+b x)^7}\right )}{11 a}-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{11 a b (a+b x)^8}\)

Input:

Int[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^8,x]
 

Output:

-1/11*(a^2 - b^2*x^2)^(5/2)/(a*b*(a + b*x)^8) + (3*(-1/9*(a^2 - b^2*x^2)^( 
5/2)/(a*b*(a + b*x)^7) + (2*(-1/7*(a^2 - b^2*x^2)^(5/2)/(a*b*(a + b*x)^6) 
- (a^2 - b^2*x^2)^(5/2)/(35*a^2*b*(a + b*x)^5)))/(9*a)))/(11*a)
 

Defintions of rubi rules used

rule 460

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(-d)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(b*c*n)), x] /; FreeQ[{a, b, c, d, n, 
 p}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + 2*p + 2, 0]
 

rule 461

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(-d)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(2*b*c*(n + p + 1))), x] + Simp[Simpl 
ify[n + 2*p + 2]/(2*c*(n + p + 1))   Int[(c + d*x)^(n + 1)*(a + b*x^2)^p, x 
], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && ILtQ[Simp 
lify[n + 2*p + 2], 0] && (LtQ[n, -1] || GtQ[n + p, 0])
 

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.50

method	result	size
gosper	\(-\frac {\left (-b x +a \right ) \left (2 b^{3} x^{3}+16 a \,b^{2} x^{2}+61 a^{2} b x +152 a^{3}\right ) \left (-b^{2} x^{2}+a^{2}\right )^{\frac {3}{2}}}{1155 \left (b x +a \right )^{7} a^{4} b}\)	\(66\)
orering	\(-\frac {\left (-b x +a \right ) \left (2 b^{3} x^{3}+16 a \,b^{2} x^{2}+61 a^{2} b x +152 a^{3}\right ) \left (-b^{2} x^{2}+a^{2}\right )^{\frac {3}{2}}}{1155 \left (b x +a \right )^{7} a^{4} b}\)	\(66\)
trager	\(-\frac {\left (2 b^{5} x^{5}+12 a \,b^{4} x^{4}+31 a^{2} b^{3} x^{3}+46 a^{3} b^{2} x^{2}-243 a^{4} b x +152 a^{5}\right ) \sqrt {-b^{2} x^{2}+a^{2}}}{1155 a^{4} \left (b x +a \right )^{6} b}\)	\(82\)
default	\(\frac {-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {5}{2}}}{11 a b \left (x +\frac {a}{b}\right )^{8}}+\frac {3 b \left (-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {5}{2}}}{9 a b \left (x +\frac {a}{b}\right )^{7}}+\frac {2 b \left (-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {5}{2}}}{7 a b \left (x +\frac {a}{b}\right )^{6}}-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {5}{2}}}{35 a^{2} \left (x +\frac {a}{b}\right )^{5}}\right )}{9 a}\right )}{11 a}}{b^{8}}\)	\(197\)

Input:

int((-b^2*x^2+a^2)^(3/2)/(b*x+a)^8,x,method=_RETURNVERBOSE)
 

Output:

-1/1155*(-b*x+a)*(2*b^3*x^3+16*a*b^2*x^2+61*a^2*b*x+152*a^3)*(-b^2*x^2+a^2 
)^(3/2)/(b*x+a)^7/a^4/b
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.53 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^8} \, dx=-\frac {152 \, b^{6} x^{6} + 912 \, a b^{5} x^{5} + 2280 \, a^{2} b^{4} x^{4} + 3040 \, a^{3} b^{3} x^{3} + 2280 \, a^{4} b^{2} x^{2} + 912 \, a^{5} b x + 152 \, a^{6} + {\left (2 \, b^{5} x^{5} + 12 \, a b^{4} x^{4} + 31 \, a^{2} b^{3} x^{3} + 46 \, a^{3} b^{2} x^{2} - 243 \, a^{4} b x + 152 \, a^{5}\right )} \sqrt {-b^{2} x^{2} + a^{2}}}{1155 \, {\left (a^{4} b^{7} x^{6} + 6 \, a^{5} b^{6} x^{5} + 15 \, a^{6} b^{5} x^{4} + 20 \, a^{7} b^{4} x^{3} + 15 \, a^{8} b^{3} x^{2} + 6 \, a^{9} b^{2} x + a^{10} b\right )}} \] Input:

integrate((-b^2*x^2+a^2)^(3/2)/(b*x+a)^8,x, algorithm="fricas")
 

Output:

-1/1155*(152*b^6*x^6 + 912*a*b^5*x^5 + 2280*a^2*b^4*x^4 + 3040*a^3*b^3*x^3 
 + 2280*a^4*b^2*x^2 + 912*a^5*b*x + 152*a^6 + (2*b^5*x^5 + 12*a*b^4*x^4 + 
31*a^2*b^3*x^3 + 46*a^3*b^2*x^2 - 243*a^4*b*x + 152*a^5)*sqrt(-b^2*x^2 + a 
^2))/(a^4*b^7*x^6 + 6*a^5*b^6*x^5 + 15*a^6*b^5*x^4 + 20*a^7*b^4*x^3 + 15*a 
^8*b^3*x^2 + 6*a^9*b^2*x + a^10*b)
 

Sympy [F]

\[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^8} \, dx=\int \frac {\left (- \left (- a + b x\right ) \left (a + b x\right )\right )^{\frac {3}{2}}}{\left (a + b x\right )^{8}}\, dx \] Input:

integrate((-b**2*x**2+a**2)**(3/2)/(b*x+a)**8,x)
 

Output:

Integral((-(-a + b*x)*(a + b*x))**(3/2)/(a + b*x)**8, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 440 vs. \(2 (117) = 234\).

Time = 0.04 (sec) , antiderivative size = 440, normalized size of antiderivative = 3.31 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^8} \, dx=-\frac {{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac {3}{2}}}{4 \, {\left (b^{8} x^{7} + 7 \, a b^{7} x^{6} + 21 \, a^{2} b^{6} x^{5} + 35 \, a^{3} b^{5} x^{4} + 35 \, a^{4} b^{4} x^{3} + 21 \, a^{5} b^{3} x^{2} + 7 \, a^{6} b^{2} x + a^{7} b\right )}} + \frac {3 \, \sqrt {-b^{2} x^{2} + a^{2}} a}{22 \, {\left (b^{7} x^{6} + 6 \, a b^{6} x^{5} + 15 \, a^{2} b^{5} x^{4} + 20 \, a^{3} b^{4} x^{3} + 15 \, a^{4} b^{3} x^{2} + 6 \, a^{5} b^{2} x + a^{6} b\right )}} - \frac {\sqrt {-b^{2} x^{2} + a^{2}}}{132 \, {\left (b^{6} x^{5} + 5 \, a b^{5} x^{4} + 10 \, a^{2} b^{4} x^{3} + 10 \, a^{3} b^{3} x^{2} + 5 \, a^{4} b^{2} x + a^{5} b\right )}} - \frac {\sqrt {-b^{2} x^{2} + a^{2}}}{231 \, {\left (a b^{5} x^{4} + 4 \, a^{2} b^{4} x^{3} + 6 \, a^{3} b^{3} x^{2} + 4 \, a^{4} b^{2} x + a^{5} b\right )}} - \frac {\sqrt {-b^{2} x^{2} + a^{2}}}{385 \, {\left (a^{2} b^{4} x^{3} + 3 \, a^{3} b^{3} x^{2} + 3 \, a^{4} b^{2} x + a^{5} b\right )}} - \frac {2 \, \sqrt {-b^{2} x^{2} + a^{2}}}{1155 \, {\left (a^{3} b^{3} x^{2} + 2 \, a^{4} b^{2} x + a^{5} b\right )}} - \frac {2 \, \sqrt {-b^{2} x^{2} + a^{2}}}{1155 \, {\left (a^{4} b^{2} x + a^{5} b\right )}} \] Input:

integrate((-b^2*x^2+a^2)^(3/2)/(b*x+a)^8,x, algorithm="maxima")
 

Output:

-1/4*(-b^2*x^2 + a^2)^(3/2)/(b^8*x^7 + 7*a*b^7*x^6 + 21*a^2*b^6*x^5 + 35*a 
^3*b^5*x^4 + 35*a^4*b^4*x^3 + 21*a^5*b^3*x^2 + 7*a^6*b^2*x + a^7*b) + 3/22 
*sqrt(-b^2*x^2 + a^2)*a/(b^7*x^6 + 6*a*b^6*x^5 + 15*a^2*b^5*x^4 + 20*a^3*b 
^4*x^3 + 15*a^4*b^3*x^2 + 6*a^5*b^2*x + a^6*b) - 1/132*sqrt(-b^2*x^2 + a^2 
)/(b^6*x^5 + 5*a*b^5*x^4 + 10*a^2*b^4*x^3 + 10*a^3*b^3*x^2 + 5*a^4*b^2*x + 
 a^5*b) - 1/231*sqrt(-b^2*x^2 + a^2)/(a*b^5*x^4 + 4*a^2*b^4*x^3 + 6*a^3*b^ 
3*x^2 + 4*a^4*b^2*x + a^5*b) - 1/385*sqrt(-b^2*x^2 + a^2)/(a^2*b^4*x^3 + 3 
*a^3*b^3*x^2 + 3*a^4*b^2*x + a^5*b) - 2/1155*sqrt(-b^2*x^2 + a^2)/(a^3*b^3 
*x^2 + 2*a^4*b^2*x + a^5*b) - 2/1155*sqrt(-b^2*x^2 + a^2)/(a^4*b^2*x + a^5 
*b)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 351 vs. \(2 (117) = 234\).

Time = 0.14 (sec) , antiderivative size = 351, normalized size of antiderivative = 2.64 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^8} \, dx=\frac {2 \, {\left (\frac {517 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}}{b^{2} x} + \frac {4895 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{2}}{b^{4} x^{2}} + \frac {11220 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{3}}{b^{6} x^{3}} + \frac {27060 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{4}}{b^{8} x^{4}} + \frac {32802 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{5}}{b^{10} x^{5}} + \frac {37422 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{6}}{b^{12} x^{6}} + \frac {23100 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{7}}{b^{14} x^{7}} + \frac {13860 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{8}}{b^{16} x^{8}} + \frac {3465 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{9}}{b^{18} x^{9}} + \frac {1155 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{10}}{b^{20} x^{10}} + 152\right )}}{1155 \, a^{4} {\left (\frac {a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}}{b^{2} x} + 1\right )}^{11} {\left | b \right |}} \] Input:

integrate((-b^2*x^2+a^2)^(3/2)/(b*x+a)^8,x, algorithm="giac")
 

Output:

2/1155*(517*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 4895*(a*b + sqrt 
(-b^2*x^2 + a^2)*abs(b))^2/(b^4*x^2) + 11220*(a*b + sqrt(-b^2*x^2 + a^2)*a 
bs(b))^3/(b^6*x^3) + 27060*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^4/(b^8*x^4) 
 + 32802*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^5/(b^10*x^5) + 37422*(a*b + s 
qrt(-b^2*x^2 + a^2)*abs(b))^6/(b^12*x^6) + 23100*(a*b + sqrt(-b^2*x^2 + a^ 
2)*abs(b))^7/(b^14*x^7) + 13860*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^8/(b^1 
6*x^8) + 3465*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^9/(b^18*x^9) + 1155*(a*b 
 + sqrt(-b^2*x^2 + a^2)*abs(b))^10/(b^20*x^10) + 152)/(a^4*((a*b + sqrt(-b 
^2*x^2 + a^2)*abs(b))/(b^2*x) + 1)^11*abs(b))
 

Mupad [B] (verification not implemented)

Time = 7.49 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^8} \, dx=\frac {8\,\sqrt {a^2-b^2\,x^2}}{33\,b\,{\left (a+b\,x\right )}^5}-\frac {4\,a\,\sqrt {a^2-b^2\,x^2}}{11\,b\,{\left (a+b\,x\right )}^6}-\frac {\sqrt {a^2-b^2\,x^2}}{231\,a\,b\,{\left (a+b\,x\right )}^4}-\frac {\sqrt {a^2-b^2\,x^2}}{385\,a^2\,b\,{\left (a+b\,x\right )}^3}-\frac {2\,\sqrt {a^2-b^2\,x^2}}{1155\,a^3\,b\,{\left (a+b\,x\right )}^2}-\frac {2\,\sqrt {a^2-b^2\,x^2}}{1155\,a^4\,b\,\left (a+b\,x\right )} \] Input:

int((a^2 - b^2*x^2)^(3/2)/(a + b*x)^8,x)
 

Output:

(8*(a^2 - b^2*x^2)^(1/2))/(33*b*(a + b*x)^5) - (4*a*(a^2 - b^2*x^2)^(1/2)) 
/(11*b*(a + b*x)^6) - (a^2 - b^2*x^2)^(1/2)/(231*a*b*(a + b*x)^4) - (a^2 - 
 b^2*x^2)^(1/2)/(385*a^2*b*(a + b*x)^3) - (2*(a^2 - b^2*x^2)^(1/2))/(1155* 
a^3*b*(a + b*x)^2) - (2*(a^2 - b^2*x^2)^(1/2))/(1155*a^4*b*(a + b*x))
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 392, normalized size of antiderivative = 2.95 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^8} \, dx=\frac {210 \sqrt {-b^{2} x^{2}+a^{2}}\, a^{5}+47 \sqrt {-b^{2} x^{2}+a^{2}}\, a^{4} b x +626 \sqrt {-b^{2} x^{2}+a^{2}}\, a^{3} b^{2} x^{2}+611 \sqrt {-b^{2} x^{2}+a^{2}}\, a^{2} b^{3} x^{3}+302 \sqrt {-b^{2} x^{2}+a^{2}}\, a \,b^{4} x^{4}+60 \sqrt {-b^{2} x^{2}+a^{2}}\, b^{5} x^{5}-210 a^{6}+47 a^{5} b x -1159 a^{4} b^{2} x^{2}-1145 a^{3} b^{3} x^{3}-851 a^{2} b^{4} x^{4}-338 a \,b^{5} x^{5}-56 b^{6} x^{6}}{1155 a^{4} b \left (\sqrt {-b^{2} x^{2}+a^{2}}\, a^{5}+5 \sqrt {-b^{2} x^{2}+a^{2}}\, a^{4} b x +10 \sqrt {-b^{2} x^{2}+a^{2}}\, a^{3} b^{2} x^{2}+10 \sqrt {-b^{2} x^{2}+a^{2}}\, a^{2} b^{3} x^{3}+5 \sqrt {-b^{2} x^{2}+a^{2}}\, a \,b^{4} x^{4}+\sqrt {-b^{2} x^{2}+a^{2}}\, b^{5} x^{5}-a^{6}-6 a^{5} b x -15 a^{4} b^{2} x^{2}-20 a^{3} b^{3} x^{3}-15 a^{2} b^{4} x^{4}-6 a \,b^{5} x^{5}-b^{6} x^{6}\right )} \] Input:

int((-b^2*x^2+a^2)^(3/2)/(b*x+a)^8,x)
 

Output:

(210*sqrt(a**2 - b**2*x**2)*a**5 + 47*sqrt(a**2 - b**2*x**2)*a**4*b*x + 62 
6*sqrt(a**2 - b**2*x**2)*a**3*b**2*x**2 + 611*sqrt(a**2 - b**2*x**2)*a**2* 
b**3*x**3 + 302*sqrt(a**2 - b**2*x**2)*a*b**4*x**4 + 60*sqrt(a**2 - b**2*x 
**2)*b**5*x**5 - 210*a**6 + 47*a**5*b*x - 1159*a**4*b**2*x**2 - 1145*a**3* 
b**3*x**3 - 851*a**2*b**4*x**4 - 338*a*b**5*x**5 - 56*b**6*x**6)/(1155*a** 
4*b*(sqrt(a**2 - b**2*x**2)*a**5 + 5*sqrt(a**2 - b**2*x**2)*a**4*b*x + 10* 
sqrt(a**2 - b**2*x**2)*a**3*b**2*x**2 + 10*sqrt(a**2 - b**2*x**2)*a**2*b** 
3*x**3 + 5*sqrt(a**2 - b**2*x**2)*a*b**4*x**4 + sqrt(a**2 - b**2*x**2)*b** 
5*x**5 - a**6 - 6*a**5*b*x - 15*a**4*b**2*x**2 - 20*a**3*b**3*x**3 - 15*a* 
*2*b**4*x**4 - 6*a*b**5*x**5 - b**6*x**6))