Integrand size = 25, antiderivative size = 95 \[ \int (a+b x) \sqrt {-\frac {a^2 c}{b^2}+c x^2} \, dx=\frac {1}{2} a x \sqrt {-\frac {a^2 c}{b^2}+c x^2}+\frac {b \left (-\frac {a^2 c}{b^2}+c x^2\right )^{3/2}}{3 c}-\frac {a^3 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {-\frac {a^2 c}{b^2}+c x^2}}\right )}{2 b^2} \] Output:
1/2*a*x*(-a^2*c/b^2+c*x^2)^(1/2)+1/3*b*(-a^2*c/b^2+c*x^2)^(3/2)/c-1/2*a^3* c^(1/2)*arctanh(c^(1/2)*x/(-a^2*c/b^2+c*x^2)^(1/2))/b^2
Time = 0.19 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.11 \[ \int (a+b x) \sqrt {-\frac {a^2 c}{b^2}+c x^2} \, dx=\frac {\sqrt {c \left (-\frac {a^2}{b^2}+x^2\right )} \left (b \sqrt {-\frac {a^2}{b^2}+x^2} \left (-2 a^2+3 a b x+2 b^2 x^2\right )+3 a^3 \log \left (-x+\sqrt {-\frac {a^2}{b^2}+x^2}\right )\right )}{6 b^2 \sqrt {-\frac {a^2}{b^2}+x^2}} \] Input:
Integrate[(a + b*x)*Sqrt[-((a^2*c)/b^2) + c*x^2],x]
Output:
(Sqrt[c*(-(a^2/b^2) + x^2)]*(b*Sqrt[-(a^2/b^2) + x^2]*(-2*a^2 + 3*a*b*x + 2*b^2*x^2) + 3*a^3*Log[-x + Sqrt[-(a^2/b^2) + x^2]]))/(6*b^2*Sqrt[-(a^2/b^ 2) + x^2])
Time = 0.33 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {455, 211, 224, 219}
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.
\(\displaystyle \int (a+b x) \sqrt {c x^2-\frac {a^2 c}{b^2}} \, dx\) |
\(\Big \downarrow \) 455 |
\(\displaystyle a \int \sqrt {c x^2-\frac {a^2 c}{b^2}}dx+\frac {b \left (c x^2-\frac {a^2 c}{b^2}\right )^{3/2}}{3 c}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle a \left (\frac {1}{2} x \sqrt {c x^2-\frac {a^2 c}{b^2}}-\frac {a^2 c \int \frac {1}{\sqrt {c x^2-\frac {a^2 c}{b^2}}}dx}{2 b^2}\right )+\frac {b \left (c x^2-\frac {a^2 c}{b^2}\right )^{3/2}}{3 c}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle a \left (\frac {1}{2} x \sqrt {c x^2-\frac {a^2 c}{b^2}}-\frac {a^2 c \int \frac {1}{1-\frac {c x^2}{c x^2-\frac {a^2 c}{b^2}}}d\frac {x}{\sqrt {c x^2-\frac {a^2 c}{b^2}}}}{2 b^2}\right )+\frac {b \left (c x^2-\frac {a^2 c}{b^2}\right )^{3/2}}{3 c}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle a \left (\frac {1}{2} x \sqrt {c x^2-\frac {a^2 c}{b^2}}-\frac {a^2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {c x^2-\frac {a^2 c}{b^2}}}\right )}{2 b^2}\right )+\frac {b \left (c x^2-\frac {a^2 c}{b^2}\right )^{3/2}}{3 c}\) |
Input:
Int[(a + b*x)*Sqrt[-((a^2*c)/b^2) + c*x^2],x]
Output:
(b*(-((a^2*c)/b^2) + c*x^2)^(3/2))/(3*c) + a*((x*Sqrt[-((a^2*c)/b^2) + c*x ^2])/2 - (a^2*Sqrt[c]*ArcTanh[(Sqrt[c]*x)/Sqrt[-((a^2*c)/b^2) + c*x^2]])/( 2*b^2))
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]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[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]
Time = 0.25 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.91
method | result | size |
default | \(a \left (\frac {x \sqrt {-\frac {a^{2} c}{b^{2}}+c \,x^{2}}}{2}-\frac {\sqrt {c}\, a^{2} \ln \left (\sqrt {c}\, x +\sqrt {-\frac {a^{2} c}{b^{2}}+c \,x^{2}}\right )}{2 b^{2}}\right )+\frac {b {\left (-\frac {c \left (-b^{2} x^{2}+a^{2}\right )}{b^{2}}\right )}^{\frac {3}{2}}}{3 c}\) | \(86\) |
risch | \(-\frac {\left (-2 b^{2} x^{2}-3 a b x +2 a^{2}\right ) \sqrt {-\frac {c \left (-b^{2} x^{2}+a^{2}\right )}{b^{2}}}\, \sqrt {-c \left (-b^{2} x^{2}+a^{2}\right )}}{6 b \sqrt {c \left (b^{2} x^{2}-a^{2}\right )}}+\frac {a^{3} \ln \left (\frac {b^{2} c x}{\sqrt {b^{2} c}}+\sqrt {b^{2} c \,x^{2}-a^{2} c}\right ) \sqrt {-\frac {c \left (-b^{2} x^{2}+a^{2}\right )}{b^{2}}}\, \sqrt {-c \left (-b^{2} x^{2}+a^{2}\right )}}{2 \sqrt {b^{2} c}\, \left (-b^{2} x^{2}+a^{2}\right )}\) | \(175\) |
Input:
int((b*x+a)*(-a^2*c/b^2+c*x^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
a*(1/2*x*(-a^2*c/b^2+c*x^2)^(1/2)-1/2*c^(1/2)*a^2/b^2*ln(c^(1/2)*x+(-a^2*c /b^2+c*x^2)^(1/2)))+1/3*b/c*(-c*(-b^2*x^2+a^2)/b^2)^(3/2)
Time = 0.09 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.27 \[ \int (a+b x) \sqrt {-\frac {a^2 c}{b^2}+c x^2} \, dx=\left [\frac {3 \, a^{3} \sqrt {c} \log \left (2 \, b^{2} c x^{2} - 2 \, b^{2} \sqrt {c} x \sqrt {\frac {b^{2} c x^{2} - a^{2} c}{b^{2}}} - a^{2} c\right ) + 2 \, {\left (2 \, b^{3} x^{2} + 3 \, a b^{2} x - 2 \, a^{2} b\right )} \sqrt {\frac {b^{2} c x^{2} - a^{2} c}{b^{2}}}}{12 \, b^{2}}, \frac {3 \, a^{3} \sqrt {-c} \arctan \left (\frac {b^{2} \sqrt {-c} x \sqrt {\frac {b^{2} c x^{2} - a^{2} c}{b^{2}}}}{b^{2} c x^{2} - a^{2} c}\right ) + {\left (2 \, b^{3} x^{2} + 3 \, a b^{2} x - 2 \, a^{2} b\right )} \sqrt {\frac {b^{2} c x^{2} - a^{2} c}{b^{2}}}}{6 \, b^{2}}\right ] \] Input:
integrate((b*x+a)*(-a^2*c/b^2+c*x^2)^(1/2),x, algorithm="fricas")
Output:
[1/12*(3*a^3*sqrt(c)*log(2*b^2*c*x^2 - 2*b^2*sqrt(c)*x*sqrt((b^2*c*x^2 - a ^2*c)/b^2) - a^2*c) + 2*(2*b^3*x^2 + 3*a*b^2*x - 2*a^2*b)*sqrt((b^2*c*x^2 - a^2*c)/b^2))/b^2, 1/6*(3*a^3*sqrt(-c)*arctan(b^2*sqrt(-c)*x*sqrt((b^2*c* x^2 - a^2*c)/b^2)/(b^2*c*x^2 - a^2*c)) + (2*b^3*x^2 + 3*a*b^2*x - 2*a^2*b) *sqrt((b^2*c*x^2 - a^2*c)/b^2))/b^2]
Time = 0.44 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.31 \[ \int (a+b x) \sqrt {-\frac {a^2 c}{b^2}+c x^2} \, dx=\begin {cases} - \frac {a^{3} c \left (\begin {cases} \frac {\log {\left (2 \sqrt {c} \sqrt {- \frac {a^{2} c}{b^{2}} + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: \frac {a^{2} c}{b^{2}} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {c x^{2}}} & \text {otherwise} \end {cases}\right )}{2 b^{2}} + \sqrt {- \frac {a^{2} c}{b^{2}} + c x^{2}} \left (- \frac {a^{2}}{3 b} + \frac {a x}{2} + \frac {b x^{2}}{3}\right ) & \text {for}\: c \neq 0 \\\sqrt {- \frac {a^{2} c}{b^{2}}} \left (a x + \frac {b x^{2}}{2}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((b*x+a)*(-a**2*c/b**2+c*x**2)**(1/2),x)
Output:
Piecewise((-a**3*c*Piecewise((log(2*sqrt(c)*sqrt(-a**2*c/b**2 + c*x**2) + 2*c*x)/sqrt(c), Ne(a**2*c/b**2, 0)), (x*log(x)/sqrt(c*x**2), True))/(2*b** 2) + sqrt(-a**2*c/b**2 + c*x**2)*(-a**2/(3*b) + a*x/2 + b*x**2/3), Ne(c, 0 )), (sqrt(-a**2*c/b**2)*(a*x + b*x**2/2), True))
Time = 0.03 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.88 \[ \int (a+b x) \sqrt {-\frac {a^2 c}{b^2}+c x^2} \, dx=\frac {1}{2} \, \sqrt {c x^{2} - \frac {a^{2} c}{b^{2}}} a x - \frac {a^{3} \sqrt {c} \log \left (2 \, c x + 2 \, \sqrt {c x^{2} - \frac {a^{2} c}{b^{2}}} \sqrt {c}\right )}{2 \, b^{2}} + \frac {{\left (c x^{2} - \frac {a^{2} c}{b^{2}}\right )}^{\frac {3}{2}} b}{3 \, c} \] Input:
integrate((b*x+a)*(-a^2*c/b^2+c*x^2)^(1/2),x, algorithm="maxima")
Output:
1/2*sqrt(c*x^2 - a^2*c/b^2)*a*x - 1/2*a^3*sqrt(c)*log(2*c*x + 2*sqrt(c*x^2 - a^2*c/b^2)*sqrt(c))/b^2 + 1/3*(c*x^2 - a^2*c/b^2)^(3/2)*b/c
Time = 0.13 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.92 \[ \int (a+b x) \sqrt {-\frac {a^2 c}{b^2}+c x^2} \, dx=\frac {{\left (\frac {3 \, a^{3} \sqrt {c} \log \left ({\left | -\sqrt {b^{2} c} x + \sqrt {b^{2} c x^{2} - a^{2} c} \right |}\right )}{{\left | b \right |}} + \sqrt {b^{2} c x^{2} - a^{2} c} {\left ({\left (2 \, b x + 3 \, a\right )} x - \frac {2 \, a^{2}}{b}\right )}\right )} {\left | b \right |}}{6 \, b^{2}} \] Input:
integrate((b*x+a)*(-a^2*c/b^2+c*x^2)^(1/2),x, algorithm="giac")
Output:
1/6*(3*a^3*sqrt(c)*log(abs(-sqrt(b^2*c)*x + sqrt(b^2*c*x^2 - a^2*c)))/abs( b) + sqrt(b^2*c*x^2 - a^2*c)*((2*b*x + 3*a)*x - 2*a^2/b))*abs(b)/b^2
Time = 7.35 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.96 \[ \int (a+b x) \sqrt {-\frac {a^2 c}{b^2}+c x^2} \, dx=\frac {a\,x\,\sqrt {c\,x^2-\frac {a^2\,c}{b^2}}}{2}-\frac {\left (a^2-b^2\,x^2\right )\,\sqrt {c\,x^2-\frac {a^2\,c}{b^2}}}{3\,b}-\frac {a^3\,\sqrt {c}\,\ln \left (\sqrt {c}\,x+\sqrt {c\,x^2-\frac {a^2\,c}{b^2}}\right )}{2\,b^2} \] Input:
int((c*x^2 - (a^2*c)/b^2)^(1/2)*(a + b*x),x)
Output:
(a*x*(c*x^2 - (a^2*c)/b^2)^(1/2))/2 - ((a^2 - b^2*x^2)*(c*x^2 - (a^2*c)/b^ 2)^(1/2))/(3*b) - (a^3*c^(1/2)*log(c^(1/2)*x + (c*x^2 - (a^2*c)/b^2)^(1/2) ))/(2*b^2)
Time = 0.22 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.01 \[ \int (a+b x) \sqrt {-\frac {a^2 c}{b^2}+c x^2} \, dx=\frac {\sqrt {c}\, \left (-2 \sqrt {b^{2} x^{2}-a^{2}}\, a^{2}+3 \sqrt {b^{2} x^{2}-a^{2}}\, a b x +2 \sqrt {b^{2} x^{2}-a^{2}}\, b^{2} x^{2}-3 \,\mathrm {log}\left (\frac {\sqrt {b^{2} x^{2}-a^{2}}+b x}{a}\right ) a^{3}\right )}{6 b^{2}} \] Input:
int((b*x+a)*(-a^2*c/b^2+c*x^2)^(1/2),x)
Output:
(sqrt(c)*( - 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 - 3*log((sqrt( - a**2 + b* *2*x**2) + b*x)/a)*a**3))/(6*b**2)