Integrand size = 24, antiderivative size = 76 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{a+b x} \, dx=\frac {1}{2} a x \sqrt {a^2-b^2 x^2}+\frac {\left (a^2-b^2 x^2\right )^{3/2}}{3 b}+\frac {a^3 \arctan \left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{2 b} \] Output:
1/2*a*x*(-b^2*x^2+a^2)^(1/2)+1/3*(-b^2*x^2+a^2)^(3/2)/b+1/2*a^3*arctan(b*x /(-b^2*x^2+a^2)^(1/2))/b
Time = 0.15 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.17 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{a+b x} \, dx=\frac {\left (2 a^2+3 a b x-2 b^2 x^2\right ) \sqrt {a^2-b^2 x^2}}{6 b}-\frac {a^3 \log \left (-\sqrt {-b^2} x+\sqrt {a^2-b^2 x^2}\right )}{2 \sqrt {-b^2}} \] Input:
Integrate[(a^2 - b^2*x^2)^(3/2)/(a + b*x),x]
Output:
((2*a^2 + 3*a*b*x - 2*b^2*x^2)*Sqrt[a^2 - b^2*x^2])/(6*b) - (a^3*Log[-(Sqr t[-b^2]*x) + Sqrt[a^2 - b^2*x^2]])/(2*Sqrt[-b^2])
Time = 0.29 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {466, 211, 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.
\(\displaystyle \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{a+b x} \, dx\) |
\(\Big \downarrow \) 466 |
\(\displaystyle a \int \sqrt {a^2-b^2 x^2}dx+\frac {\left (a^2-b^2 x^2\right )^{3/2}}{3 b}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle a \left (\frac {1}{2} a^2 \int \frac {1}{\sqrt {a^2-b^2 x^2}}dx+\frac {1}{2} x \sqrt {a^2-b^2 x^2}\right )+\frac {\left (a^2-b^2 x^2\right )^{3/2}}{3 b}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle a \left (\frac {1}{2} a^2 \int \frac {1}{\frac {b^2 x^2}{a^2-b^2 x^2}+1}d\frac {x}{\sqrt {a^2-b^2 x^2}}+\frac {1}{2} x \sqrt {a^2-b^2 x^2}\right )+\frac {\left (a^2-b^2 x^2\right )^{3/2}}{3 b}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle a \left (\frac {a^2 \arctan \left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{2 b}+\frac {1}{2} x \sqrt {a^2-b^2 x^2}\right )+\frac {\left (a^2-b^2 x^2\right )^{3/2}}{3 b}\) |
Input:
Int[(a^2 - b^2*x^2)^(3/2)/(a + b*x),x]
Output:
(a^2 - b^2*x^2)^(3/2)/(3*b) + a*((x*Sqrt[a^2 - b^2*x^2])/2 + (a^2*ArcTan[( b*x)/Sqrt[a^2 - b^2*x^2]])/(2*b))
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
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_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (c + d*x)^(n + 1)*((a + b*x^2)^p/(d*(n + 2*p + 1))), x] - Simp[2*b*c*(p/(d^ 2*(n + 2*p + 1))) Int[(c + d*x)^(n + 1)*(a + b*x^2)^(p - 1), x], x] /; Fr eeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[p, 0] && (LeQ[-2, n, 0 ] || EqQ[n + p + 1, 0]) && NeQ[n + 2*p + 1, 0] && IntegerQ[2*p]
Time = 0.22 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.95
method | result | size |
risch | \(\frac {\left (-2 b^{2} x^{2}+3 a b x +2 a^{2}\right ) \sqrt {-b^{2} x^{2}+a^{2}}}{6 b}+\frac {a^{3} \arctan \left (\frac {\sqrt {b^{2}}\, x}{\sqrt {-b^{2} x^{2}+a^{2}}}\right )}{2 \sqrt {b^{2}}}\) | \(72\) |
default | \(\frac {\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {3}{2}}}{3}+a b \left (-\frac {\left (-2 b^{2} \left (x +\frac {a}{b}\right )+2 a b \right ) \sqrt {-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )}}{4 b^{2}}+\frac {a^{2} \arctan \left (\frac {\sqrt {b^{2}}\, x}{\sqrt {-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )}}\right )}{2 \sqrt {b^{2}}}\right )}{b}\) | \(136\) |
Input:
int((-b^2*x^2+a^2)^(3/2)/(b*x+a),x,method=_RETURNVERBOSE)
Output:
1/6*(-2*b^2*x^2+3*a*b*x+2*a^2)/b*(-b^2*x^2+a^2)^(1/2)+1/2*a^3/(b^2)^(1/2)* arctan((b^2)^(1/2)*x/(-b^2*x^2+a^2)^(1/2))
Time = 0.08 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{a+b x} \, dx=-\frac {6 \, a^{3} \arctan \left (-\frac {a - \sqrt {-b^{2} x^{2} + a^{2}}}{b x}\right ) + {\left (2 \, b^{2} x^{2} - 3 \, a b x - 2 \, a^{2}\right )} \sqrt {-b^{2} x^{2} + a^{2}}}{6 \, b} \] Input:
integrate((-b^2*x^2+a^2)^(3/2)/(b*x+a),x, algorithm="fricas")
Output:
-1/6*(6*a^3*arctan(-(a - sqrt(-b^2*x^2 + a^2))/(b*x)) + (2*b^2*x^2 - 3*a*b *x - 2*a^2)*sqrt(-b^2*x^2 + a^2))/b
Time = 1.36 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.76 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{a+b x} \, dx=a \left (\begin {cases} \frac {a^{2} \left (\begin {cases} \frac {\log {\left (- 2 b^{2} x + 2 \sqrt {- b^{2}} \sqrt {a^{2} - b^{2} x^{2}} \right )}}{\sqrt {- b^{2}}} & \text {for}\: a^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- b^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {a^{2} - b^{2} x^{2}}}{2} & \text {for}\: b^{2} \neq 0 \\x \sqrt {a^{2}} & \text {otherwise} \end {cases}\right ) - b \left (\begin {cases} \sqrt {a^{2} - b^{2} x^{2}} \left (- \frac {a^{2}}{3 b^{2}} + \frac {x^{2}}{3}\right ) & \text {for}\: b^{2} \neq 0 \\\frac {x^{2} \sqrt {a^{2}}}{2} & \text {otherwise} \end {cases}\right ) \] Input:
integrate((-b**2*x**2+a**2)**(3/2)/(b*x+a),x)
Output:
a*Piecewise((a**2*Piecewise((log(-2*b**2*x + 2*sqrt(-b**2)*sqrt(a**2 - b** 2*x**2))/sqrt(-b**2), Ne(a**2, 0)), (x*log(x)/sqrt(-b**2*x**2), True))/2 + x*sqrt(a**2 - b**2*x**2)/2, Ne(b**2, 0)), (x*sqrt(a**2), True)) - b*Piece wise((sqrt(a**2 - b**2*x**2)*(-a**2/(3*b**2) + x**2/3), Ne(b**2, 0)), (x** 2*sqrt(a**2)/2, True))
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.16 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{a+b x} \, dx=-\frac {i \, a^{3} \arcsin \left (\frac {b x}{a} + 2\right )}{2 \, b} + \frac {1}{2} \, \sqrt {b^{2} x^{2} + 4 \, a b x + 3 \, a^{2}} a x + \frac {\sqrt {b^{2} x^{2} + 4 \, a b x + 3 \, a^{2}} a^{2}}{b} + \frac {{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac {3}{2}}}{3 \, b} \] Input:
integrate((-b^2*x^2+a^2)^(3/2)/(b*x+a),x, algorithm="maxima")
Output:
-1/2*I*a^3*arcsin(b*x/a + 2)/b + 1/2*sqrt(b^2*x^2 + 4*a*b*x + 3*a^2)*a*x + sqrt(b^2*x^2 + 4*a*b*x + 3*a^2)*a^2/b + 1/3*(-b^2*x^2 + a^2)^(3/2)/b
Time = 0.14 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.74 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{a+b x} \, dx=\frac {a^{3} \arcsin \left (\frac {b x}{a}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (b\right )}{2 \, {\left | b \right |}} - \frac {1}{6} \, \sqrt {-b^{2} x^{2} + a^{2}} {\left ({\left (2 \, b x - 3 \, a\right )} x - \frac {2 \, a^{2}}{b}\right )} \] Input:
integrate((-b^2*x^2+a^2)^(3/2)/(b*x+a),x, algorithm="giac")
Output:
1/2*a^3*arcsin(b*x/a)*sgn(a)*sgn(b)/abs(b) - 1/6*sqrt(-b^2*x^2 + a^2)*((2* b*x - 3*a)*x - 2*a^2/b)
Timed out. \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{a+b x} \, dx=\int \frac {{\left (a^2-b^2\,x^2\right )}^{3/2}}{a+b\,x} \,d x \] Input:
int((a^2 - b^2*x^2)^(3/2)/(a + b*x),x)
Output:
int((a^2 - b^2*x^2)^(3/2)/(a + b*x), x)
Time = 0.21 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{a+b x} \, dx=\frac {3 \mathit {asin} \left (\frac {b x}{a}\right ) a^{3}+2 \sqrt {-b^{2} x^{2}+a^{2}}\, a^{2}+3 \sqrt {-b^{2} x^{2}+a^{2}}\, a b x -2 \sqrt {-b^{2} x^{2}+a^{2}}\, b^{2} x^{2}-2 a^{3}}{6 b} \] Input:
int((-b^2*x^2+a^2)^(3/2)/(b*x+a),x)
Output:
(3*asin((b*x)/a)*a**3 + 2*sqrt(a**2 - b**2*x**2)*a**2 + 3*sqrt(a**2 - b**2 *x**2)*a*b*x - 2*sqrt(a**2 - b**2*x**2)*b**2*x**2 - 2*a**3)/(6*b)