Integrand size = 24, antiderivative size = 77 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^3} \, dx=-\frac {\sqrt {a^2-b^2 x^2}}{b}-\frac {4 a \sqrt {a^2-b^2 x^2}}{b (a+b x)}-\frac {3 a \arctan \left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{b} \] Output:
-(-b^2*x^2+a^2)^(1/2)/b-4*a*(-b^2*x^2+a^2)^(1/2)/b/(b*x+a)-3*a*arctan(b*x/ (-b^2*x^2+a^2)^(1/2))/b
Time = 0.26 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^3} \, dx=\frac {(-5 a-b x) \sqrt {a^2-b^2 x^2}}{b (a+b x)}+\frac {3 a \log \left (-\sqrt {-b^2} x+\sqrt {a^2-b^2 x^2}\right )}{\sqrt {-b^2}} \] Input:
Integrate[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^3,x]
Output:
((-5*a - b*x)*Sqrt[a^2 - b^2*x^2])/(b*(a + b*x)) + (3*a*Log[-(Sqrt[-b^2]*x ) + Sqrt[a^2 - b^2*x^2]])/Sqrt[-b^2]
Time = 0.31 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {463, 455, 224, 216}
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.
\(\displaystyle \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^3} \, dx\) |
\(\Big \downarrow \) 463 |
\(\displaystyle -\int \frac {3 a-b x}{\sqrt {a^2-b^2 x^2}}dx-\frac {4 a \sqrt {a^2-b^2 x^2}}{b (a+b x)}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle -3 a \int \frac {1}{\sqrt {a^2-b^2 x^2}}dx-\frac {4 a \sqrt {a^2-b^2 x^2}}{b (a+b x)}-\frac {\sqrt {a^2-b^2 x^2}}{b}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle -3 a \int \frac {1}{\frac {b^2 x^2}{a^2-b^2 x^2}+1}d\frac {x}{\sqrt {a^2-b^2 x^2}}-\frac {4 a \sqrt {a^2-b^2 x^2}}{b (a+b x)}-\frac {\sqrt {a^2-b^2 x^2}}{b}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -\frac {3 a \arctan \left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{b}-\frac {4 a \sqrt {a^2-b^2 x^2}}{b (a+b x)}-\frac {\sqrt {a^2-b^2 x^2}}{b}\) |
Input:
Int[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^3,x]
Output:
-(Sqrt[a^2 - b^2*x^2]/b) - (4*a*Sqrt[a^2 - b^2*x^2])/(b*(a + b*x)) - (3*a* ArcTan[(b*x)/Sqrt[a^2 - b^2*x^2]])/b
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (-(-c)^(-n - 2))*d^(2*n + 3)*(Sqrt[a + b*x^2]/(2^(n + 1)*b^(n + 2)*(c + d*x ))), x] - Simp[d^(2*n + 2)/b^(n + 1) Int[(1/Sqrt[a + b*x^2])*ExpandToSum[ (2^(-n - 1)*(-c)^(-n - 1) - (-c + d*x)^(-n - 1))/(c + d*x), x], x], x] /; F reeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && ILtQ[n, 0] && EqQ[n + p, -3/2]
Time = 0.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.22
method | result | size |
risch | \(-\frac {\sqrt {-b^{2} x^{2}+a^{2}}}{b}-\frac {3 a \arctan \left (\frac {\sqrt {b^{2}}\, x}{\sqrt {-b^{2} x^{2}+a^{2}}}\right )}{\sqrt {b^{2}}}-\frac {4 a \sqrt {-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )}}{b^{2} \left (x +\frac {a}{b}\right )}\) | \(94\) |
default | \(\frac {-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {5}{2}}}{a b \left (x +\frac {a}{b}\right )^{3}}-\frac {2 b \left (\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {5}{2}}}{a b \left (x +\frac {a}{b}\right )^{2}}+\frac {3 b \left (\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {3}{2}}}{3}+a b \left (-\frac {\left (-2 b^{2} \left (x +\frac {a}{b}\right )+2 a b \right ) \sqrt {-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )}}{4 b^{2}}+\frac {a^{2} \arctan \left (\frac {\sqrt {b^{2}}\, x}{\sqrt {-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )}}\right )}{2 \sqrt {b^{2}}}\right )\right )}{a}\right )}{a}}{b^{3}}\) | \(239\) |
Input:
int((-b^2*x^2+a^2)^(3/2)/(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
-(-b^2*x^2+a^2)^(1/2)/b-3*a/(b^2)^(1/2)*arctan((b^2)^(1/2)*x/(-b^2*x^2+a^2 )^(1/2))-4*a/b^2/(x+a/b)*(-b^2*(x+a/b)^2+2*a*b*(x+a/b))^(1/2)
Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^3} \, dx=-\frac {5 \, a b x + 5 \, a^{2} - 6 \, {\left (a b x + a^{2}\right )} \arctan \left (-\frac {a - \sqrt {-b^{2} x^{2} + a^{2}}}{b x}\right ) + \sqrt {-b^{2} x^{2} + a^{2}} {\left (b x + 5 \, a\right )}}{b^{2} x + a b} \] Input:
integrate((-b^2*x^2+a^2)^(3/2)/(b*x+a)^3,x, algorithm="fricas")
Output:
-(5*a*b*x + 5*a^2 - 6*(a*b*x + a^2)*arctan(-(a - sqrt(-b^2*x^2 + a^2))/(b* x)) + sqrt(-b^2*x^2 + a^2)*(b*x + 5*a))/(b^2*x + a*b)
\[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^3} \, dx=\int \frac {\left (- \left (- a + b x\right ) \left (a + b x\right )\right )^{\frac {3}{2}}}{\left (a + b x\right )^{3}}\, dx \] Input:
integrate((-b**2*x**2+a**2)**(3/2)/(b*x+a)**3,x)
Output:
Integral((-(-a + b*x)*(a + b*x))**(3/2)/(a + b*x)**3, x)
Time = 0.11 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^3} \, dx=-\frac {3 \, a \arcsin \left (\frac {b x}{a}\right )}{b} + \frac {{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac {3}{2}}}{b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b} - \frac {6 \, \sqrt {-b^{2} x^{2} + a^{2}} a}{b^{2} x + a b} \] Input:
integrate((-b^2*x^2+a^2)^(3/2)/(b*x+a)^3,x, algorithm="maxima")
Output:
-3*a*arcsin(b*x/a)/b + (-b^2*x^2 + a^2)^(3/2)/(b^3*x^2 + 2*a*b^2*x + a^2*b ) - 6*sqrt(-b^2*x^2 + a^2)*a/(b^2*x + a*b)
Time = 0.14 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^3} \, dx=-\frac {3 \, a \arcsin \left (\frac {b x}{a}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (b\right )}{{\left | b \right |}} - \frac {\sqrt {-b^{2} x^{2} + a^{2}}}{b} + \frac {8 \, a}{{\left (\frac {a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}}{b^{2} x} + 1\right )} {\left | b \right |}} \] Input:
integrate((-b^2*x^2+a^2)^(3/2)/(b*x+a)^3,x, algorithm="giac")
Output:
-3*a*arcsin(b*x/a)*sgn(a)*sgn(b)/abs(b) - sqrt(-b^2*x^2 + a^2)/b + 8*a/((( a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 1)*abs(b))
Timed out. \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^3} \, dx=\int \frac {{\left (a^2-b^2\,x^2\right )}^{3/2}}{{\left (a+b\,x\right )}^3} \,d x \] Input:
int((a^2 - b^2*x^2)^(3/2)/(a + b*x)^3,x)
Output:
int((a^2 - b^2*x^2)^(3/2)/(a + b*x)^3, x)
Time = 0.22 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.60 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^3} \, dx=\frac {-3 \sqrt {-b^{2} x^{2}+a^{2}}\, \mathit {asin} \left (\frac {b x}{a}\right ) a +3 \mathit {asin} \left (\frac {b x}{a}\right ) a^{2}+3 \mathit {asin} \left (\frac {b x}{a}\right ) a b x +8 \sqrt {-b^{2} x^{2}+a^{2}}\, a +\sqrt {-b^{2} x^{2}+a^{2}}\, b x -8 a^{2}+a b x +b^{2} x^{2}}{b \left (\sqrt {-b^{2} x^{2}+a^{2}}-a -b x \right )} \] Input:
int((-b^2*x^2+a^2)^(3/2)/(b*x+a)^3,x)
Output:
( - 3*sqrt(a**2 - b**2*x**2)*asin((b*x)/a)*a + 3*asin((b*x)/a)*a**2 + 3*as in((b*x)/a)*a*b*x + 8*sqrt(a**2 - b**2*x**2)*a + sqrt(a**2 - b**2*x**2)*b* x - 8*a**2 + a*b*x + b**2*x**2)/(b*(sqrt(a**2 - b**2*x**2) - a - b*x))