Optimal. Leaf size=107 \[ \frac {(a c-d)^2 \tan ^{-1}\left (\frac {a^2 c x+d}{\sqrt {1-a^2 x^2} \sqrt {a^2 c^2-d^2}}\right )}{d^2 \sqrt {a^2 c^2-d^2}}-\frac {\sqrt {1-a^2 x^2}}{d}-\frac {(a c-2 d) \sin ^{-1}(a x)}{d^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.18, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1654, 844, 216, 725, 204} \begin {gather*} \frac {(a c-d)^2 \tan ^{-1}\left (\frac {a^2 c x+d}{\sqrt {1-a^2 x^2} \sqrt {a^2 c^2-d^2}}\right )}{d^2 \sqrt {a^2 c^2-d^2}}-\frac {\sqrt {1-a^2 x^2}}{d}-\frac {(a c-2 d) \sin ^{-1}(a x)}{d^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 216
Rule 725
Rule 844
Rule 1654
Rubi steps
\begin {align*} \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 {\int \frac {-a^2 d^2+a^3 (a c-2 d) d x}{(c+d x) \sqrt {1-a^2 x^2}} \, dx}{a^2 d^2}\\ &=-\frac {\sqrt {1-a^2 x^2}}{d}-\frac {(a (a c-2 d)) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{d^2}+\frac {(a c-d)^2 \int \frac {1}{(c+d x) \sqrt {1-a^2 x^2}} \, dx}{d^2}\\ &=-\frac {\sqrt {1-a^2 x^2}}{d}-\frac {(a c-2 d) \sin ^{-1}(a x)}{d^2}-\frac {(a c-d)^2 \operatorname {Subst}\left (\int \frac {1}{-a^2 c^2+d^2-x^2} \, dx,x,\frac {d+a^2 c x}{\sqrt {1-a^2 x^2}}\right )}{d^2}\\ &=-\frac {\sqrt {1-a^2 x^2}}{d}-\frac {(a c-2 d) \sin ^{-1}(a x)}{d^2}+\frac {(a c-d)^2 \tan ^{-1}\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}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 120, normalized size = 1.12 \begin {gather*} \frac {(a c-d) \sqrt {a^2 c^2-d^2} \tan ^{-1}\left (\frac {a^2 c x+d}{\sqrt {1-a^2 x^2} \sqrt {a^2 c^2-d^2}}\right )}{d^2 (a c+d)}-\frac {\sqrt {1-a^2 x^2}}{d}-\frac {(a c-d) \sin ^{-1}(a x)}{d^2}+\frac {\sin ^{-1}(a x)}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [B] time = 1.78, size = 971, normalized size = 9.07 \begin {gather*} \frac {\left (a^3 \sqrt {2 a^2 c^2-2 a \sqrt {a^2 c^2-d^2} c-d^2} c^3-a^2 d \sqrt {2 a^2 c^2-2 a \sqrt {a^2 c^2-d^2} c-d^2} c^2+a^2 \sqrt {a^2 c^2-d^2} \sqrt {2 a^2 c^2-2 a \sqrt {a^2 c^2-d^2} c-d^2} c^2-a d^2 \sqrt {2 a^2 c^2-2 a \sqrt {a^2 c^2-d^2} c-d^2} c-a d \sqrt {a^2 c^2-d^2} \sqrt {2 a^2 c^2-2 a \sqrt {a^2 c^2-d^2} c-d^2} c+d^3 \sqrt {2 a^2 c^2-2 a \sqrt {a^2 c^2-d^2} c-d^2}\right ) \tan ^{-1}\left (\frac {\sqrt {-a^2} d x-d \sqrt {1-a^2 x^2}}{\sqrt {2 a^2 c^2-2 a \sqrt {a^2 c^2-d^2} c-d^2}}\right )}{d^4 (a c+d)}+\frac {\left (a^3 \sqrt {2 a^2 c^2+2 a \sqrt {a^2 c^2-d^2} c-d^2} c^3-a^2 d \sqrt {2 a^2 c^2+2 a \sqrt {a^2 c^2-d^2} c-d^2} c^2-a^2 \sqrt {a^2 c^2-d^2} \sqrt {2 a^2 c^2+2 a \sqrt {a^2 c^2-d^2} c-d^2} c^2-a d^2 \sqrt {2 a^2 c^2+2 a \sqrt {a^2 c^2-d^2} c-d^2} c+a d \sqrt {a^2 c^2-d^2} \sqrt {2 a^2 c^2+2 a \sqrt {a^2 c^2-d^2} c-d^2} c+d^3 \sqrt {2 a^2 c^2+2 a \sqrt {a^2 c^2-d^2} c-d^2}\right ) \tan ^{-1}\left (\frac {\sqrt {-a^2} d x-d \sqrt {1-a^2 x^2}}{\sqrt {2 a^2 c^2+2 a \sqrt {a^2 c^2-d^2} c-d^2}}\right )}{d^4 (a c+d)}-\frac {\sqrt {-a^2} (a c-d) \sqrt {d^2-a^2 c^2} \tan ^{-1}\left (\frac {a^2 c^2-a^2 d^2 x^2-\sqrt {-a^2} d^2 x \sqrt {1-a^2 x^2}}{a c \sqrt {d^2-a^2 c^2}}\right )}{a d^2 (a c+d)}-\frac {\sqrt {-a^2} (a c-2 d) \log \left (\sqrt {1-a^2 x^2}-\sqrt {-a^2} x\right )}{a d^2}-\frac {\sqrt {1-a^2 x^2}}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.56, size = 318, normalized size = 2.97 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.46, size = 131, normalized size = 1.22 \begin {gather*} -\frac {{\left (a^{2} c - 2 \, a d\right )} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{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 |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.02, size = 524, normalized size = 4.90 \begin {gather*} -\frac {a^{2} c^{2} \ln \left (\frac {\frac {2 \left (x +\frac {c}{d}\right ) a^{2} c}{d}-\frac {2 \left (a^{2} c^{2}-d^{2}\right )}{d^{2}}+2 \sqrt {-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}\, \sqrt {\frac {2 \left (x +\frac {c}{d}\right ) a^{2} c}{d}-\left (x +\frac {c}{d}\right )^{2} a^{2}-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\sqrt {-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}\, d^{3}}-\frac {a^{2} c \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}\, d^{2}}+\frac {2 a c \ln \left (\frac {\frac {2 \left (x +\frac {c}{d}\right ) a^{2} c}{d}-\frac {2 \left (a^{2} c^{2}-d^{2}\right )}{d^{2}}+2 \sqrt {-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}\, \sqrt {\frac {2 \left (x +\frac {c}{d}\right ) a^{2} c}{d}-\left (x +\frac {c}{d}\right )^{2} a^{2}-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\sqrt {-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}\, d^{2}}+\frac {2 a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}\, d}-\frac {\ln \left (\frac {\frac {2 \left (x +\frac {c}{d}\right ) a^{2} c}{d}-\frac {2 \left (a^{2} c^{2}-d^{2}\right )}{d^{2}}+2 \sqrt {-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}\, \sqrt {\frac {2 \left (x +\frac {c}{d}\right ) a^{2} c}{d}-\left (x +\frac {c}{d}\right )^{2} a^{2}-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\sqrt {-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}\, d}-\frac {\sqrt {-a^{2} x^{2}+1}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.12, size = 148, normalized size = 1.38 \begin {gather*} -\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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________