Integrand size = 29, antiderivative size = 107 \[ \int \frac {(1+a x)^2}{(c+d x) \sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {1-a^2 x^2}}{d}-\frac {(a c-2 d) \arcsin (a x)}{d^2}+\frac {(a c-d)^2 \arctan \left (\frac {d+a^2 c x}{\sqrt {a^2 c^2-d^2} \sqrt {1-a^2 x^2}}\right )}{d^2 \sqrt {a^2 c^2-d^2}} \]
-(a*c-2*d)*arcsin(a*x)/d^2+(a*c-d)^2*arctan((a^2*c*x+d)/(a^2*c^2-d^2)^(1/2 )/(-a^2*x^2+1)^(1/2))/d^2/(a^2*c^2-d^2)^(1/2)-(-a^2*x^2+1)^(1/2)/d
Time = 0.39 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.21 \[ \int \frac {(1+a x)^2}{(c+d x) \sqrt {1-a^2 x^2}} \, dx=\frac {-d \sqrt {1-a^2 x^2}+(-2 a c+4 d) \arctan \left (\frac {a x}{-1+\sqrt {1-a^2 x^2}}\right )-\frac {2 (a c-d) \sqrt {a^2 c^2-d^2} \arctan \left (\frac {\sqrt {a^2 c^2-d^2} x}{c+d x-c \sqrt {1-a^2 x^2}}\right )}{a c+d}}{d^2} \]
(-(d*Sqrt[1 - a^2*x^2]) + (-2*a*c + 4*d)*ArcTan[(a*x)/(-1 + Sqrt[1 - a^2*x ^2])] - (2*(a*c - d)*Sqrt[a^2*c^2 - d^2]*ArcTan[(Sqrt[a^2*c^2 - d^2]*x)/(c + d*x - c*Sqrt[1 - a^2*x^2])])/(a*c + d))/d^2
Time = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {716, 25, 27, 719, 223, 488, 217}
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.
\(\displaystyle \int \frac {(a x+1)^2}{\sqrt {1-a^2 x^2} (c+d x)} \, dx\) |
\(\Big \downarrow \) 716 |
\(\displaystyle -\frac {\int -\frac {a^2 d (d-a (a c-2 d) x)}{(c+d x) \sqrt {1-a^2 x^2}}dx}{a^2 d^2}-\frac {\sqrt {1-a^2 x^2}}{d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {a^2 d (d-a (a c-2 d) x)}{(c+d x) \sqrt {1-a^2 x^2}}dx}{a^2 d^2}-\frac {\sqrt {1-a^2 x^2}}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {d-a (a c-2 d) x}{(c+d x) \sqrt {1-a^2 x^2}}dx}{d}-\frac {\sqrt {1-a^2 x^2}}{d}\) |
\(\Big \downarrow \) 719 |
\(\displaystyle \frac {\frac {(a c-d)^2 \int \frac {1}{(c+d x) \sqrt {1-a^2 x^2}}dx}{d}-\frac {a (a c-2 d) \int \frac {1}{\sqrt {1-a^2 x^2}}dx}{d}}{d}-\frac {\sqrt {1-a^2 x^2}}{d}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {\frac {(a c-d)^2 \int \frac {1}{(c+d x) \sqrt {1-a^2 x^2}}dx}{d}-\frac {\arcsin (a x) (a c-2 d)}{d}}{d}-\frac {\sqrt {1-a^2 x^2}}{d}\) |
\(\Big \downarrow \) 488 |
\(\displaystyle \frac {-\frac {(a c-d)^2 \int \frac {1}{-a^2 c^2+d^2-\frac {\left (c x a^2+d\right )^2}{1-a^2 x^2}}d\frac {c x a^2+d}{\sqrt {1-a^2 x^2}}}{d}-\frac {\arcsin (a x) (a c-2 d)}{d}}{d}-\frac {\sqrt {1-a^2 x^2}}{d}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {\frac {(a c-d)^2 \arctan \left (\frac {a^2 c x+d}{\sqrt {1-a^2 x^2} \sqrt {a^2 c^2-d^2}}\right )}{d \sqrt {a^2 c^2-d^2}}-\frac {\arcsin (a x) (a c-2 d)}{d}}{d}-\frac {\sqrt {1-a^2 x^2}}{d}\) |
-(Sqrt[1 - a^2*x^2]/d) + (-(((a*c - 2*d)*ArcSin[a*x])/d) + ((a*c - d)^2*Ar cTan[(d + a^2*c*x)/(Sqrt[a^2*c^2 - d^2]*Sqrt[1 - a^2*x^2])])/(d*Sqrt[a^2*c ^2 - d^2]))/d
3.7.21.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_) ^2)^(p_), x_Symbol] :> Simp[g^n*(d + e*x)^(m + n - 1)*((a + c*x^2)^(p + 1)/ (c*e^(n - 1)*(m + n + 2*p + 1))), x] + Simp[1/(c*e^n*(m + n + 2*p + 1)) I nt[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^n*(m + n + 2*p + 1)*(f + g*x)^ n - c*g^n*(m + n + 2*p + 1)*(d + e*x)^n - g^n*(d + e*x)^(n - 2)*(a*e^2*(m + n - 1) - c*d^2*(m + n + 2*p + 1) - 2*c*d*e*(m + n + p)*x), x], x], x] /; F reeQ[{a, c, d, e, f, g, m, p}, x] && IGtQ[n, 1] && IntegerQ[m] && NeQ[m + n + 2*p + 1, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(232\) vs. \(2(99)=198\).
Time = 0.47 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.18
method | result | size |
risch | \(\frac {a^{2} x^{2}-1}{d \sqrt {-a^{2} x^{2}+1}}-\frac {\frac {a \left (a c -2 d \right ) \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{d \sqrt {a^{2}}}-\frac {\left (-a^{2} c^{2}+2 a c d -d^{2}\right ) \ln \left (\frac {-\frac {2 \left (a^{2} c^{2}-d^{2}\right )}{d^{2}}+\frac {2 a^{2} c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}\, \sqrt {-a^{2} \left (x +\frac {c}{d}\right )^{2}+\frac {2 a^{2} c \left (x +\frac {c}{d}\right )}{d}-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}}}{d}\) | \(233\) |
default | \(-\frac {a \left (-\frac {2 d \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\frac {a c \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\frac {d \sqrt {-a^{2} x^{2}+1}}{a}\right )}{d^{2}}-\frac {\left (a^{2} c^{2}-2 a c d +d^{2}\right ) \ln \left (\frac {-\frac {2 \left (a^{2} c^{2}-d^{2}\right )}{d^{2}}+\frac {2 a^{2} c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}\, \sqrt {-a^{2} \left (x +\frac {c}{d}\right )^{2}+\frac {2 a^{2} c \left (x +\frac {c}{d}\right )}{d}-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{3} \sqrt {-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}}\) | \(242\) |
1/d*(a^2*x^2-1)/(-a^2*x^2+1)^(1/2)-1/d*(a*(a*c-2*d)/d/(a^2)^(1/2)*arctan(( a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))-(-a^2*c^2+2*a*c*d-d^2)/d^2/(-(a^2*c^2-d^2 )/d^2)^(1/2)*ln((-2*(a^2*c^2-d^2)/d^2+2*a^2*c/d*(x+c/d)+2*(-(a^2*c^2-d^2)/ d^2)^(1/2)*(-a^2*(x+c/d)^2+2*a^2*c/d*(x+c/d)-(a^2*c^2-d^2)/d^2)^(1/2))/(x+ c/d)))
Time = 0.39 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.97 \[ \int \frac {(1+a x)^2}{(c+d x) \sqrt {1-a^2 x^2}} \, dx=\left [-\frac {{\left (a c - d\right )} \sqrt {-\frac {a c - d}{a c + d}} \log \left (\frac {a^{2} c d x + d^{2} - {\left (a^{2} c^{2} - d^{2}\right )} \sqrt {-a^{2} x^{2} + 1} - {\left (a c d + d^{2} + {\left (a^{3} c^{2} + a^{2} c d\right )} x + \sqrt {-a^{2} x^{2} + 1} {\left (a c d + d^{2}\right )}\right )} \sqrt {-\frac {a c - d}{a c + d}}}{d x + c}\right ) - 2 \, {\left (a c - 2 \, d\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt {-a^{2} x^{2} + 1} d}{d^{2}}, \frac {2 \, {\left (a c - d\right )} \sqrt {\frac {a c - d}{a c + d}} \arctan \left (\frac {{\left (d x - \sqrt {-a^{2} x^{2} + 1} c + c\right )} \sqrt {\frac {a c - d}{a c + d}}}{{\left (a c - d\right )} x}\right ) + 2 \, {\left (a c - 2 \, d\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - \sqrt {-a^{2} x^{2} + 1} d}{d^{2}}\right ] \]
[-((a*c - d)*sqrt(-(a*c - d)/(a*c + d))*log((a^2*c*d*x + d^2 - (a^2*c^2 - d^2)*sqrt(-a^2*x^2 + 1) - (a*c*d + d^2 + (a^3*c^2 + a^2*c*d)*x + sqrt(-a^2 *x^2 + 1)*(a*c*d + d^2))*sqrt(-(a*c - d)/(a*c + d)))/(d*x + c)) - 2*(a*c - 2*d)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + sqrt(-a^2*x^2 + 1)*d)/d^2, (2*(a*c - d)*sqrt((a*c - d)/(a*c + d))*arctan((d*x - sqrt(-a^2*x^2 + 1)*c + c)*sqrt((a*c - d)/(a*c + d))/((a*c - d)*x)) + 2*(a*c - 2*d)*arctan((sqrt (-a^2*x^2 + 1) - 1)/(a*x)) - sqrt(-a^2*x^2 + 1)*d)/d^2]
\[ \int \frac {(1+a x)^2}{(c+d x) \sqrt {1-a^2 x^2}} \, dx=\int \frac {\left (a x + 1\right )^{2}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (c + d x\right )}\, dx \]
Exception generated. \[ \int \frac {(1+a x)^2}{(c+d x) \sqrt {1-a^2 x^2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(d-a*c>0)', see `assume?` for mor e details)
Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.22 \[ \int \frac {(1+a x)^2}{(c+d x) \sqrt {1-a^2 x^2}} \, dx=-\frac {{\left (a^{2} c - 2 \, a d\right )} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{d^{2} {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{d} - \frac {2 \, {\left (a^{3} c^{2} - 2 \, a^{2} c d + a d^{2}\right )} \arctan \left (\frac {d + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c}{a x}}{\sqrt {a^{2} c^{2} - d^{2}}}\right )}{\sqrt {a^{2} c^{2} - d^{2}} d^{2} {\left | a \right |}} \]
-(a^2*c - 2*a*d)*arcsin(a*x)*sgn(a)/(d^2*abs(a)) - sqrt(-a^2*x^2 + 1)/d - 2*(a^3*c^2 - 2*a^2*c*d + a*d^2)*arctan((d + (sqrt(-a^2*x^2 + 1)*abs(a) + a )*c/(a*x))/sqrt(a^2*c^2 - d^2))/(sqrt(a^2*c^2 - d^2)*d^2*abs(a))
Time = 0.13 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.38 \[ \int \frac {(1+a x)^2}{(c+d x) \sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {1-a^2\,x^2}}{d}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )\,\left (2\,a\,\sqrt {-a^2}-\frac {a^2\,c\,\sqrt {-a^2}}{d}\right )}{a^2\,d}-\frac {\left (\ln \left (\sqrt {1-\frac {a^2\,c^2}{d^2}}\,\sqrt {1-a^2\,x^2}+\frac {a^2\,c\,x}{d}+1\right )-\ln \left (c+d\,x\right )\right )\,\left (a^2\,c^2-2\,a\,c\,d+d^2\right )}{d^3\,\sqrt {1-\frac {a^2\,c^2}{d^2}}} \]