3.621 \(\int \frac{(1+a x)^2}{(c+d x) \sqrt{1-a^2 x^2}} \, dx\)

Optimal. Leaf size=107 \[ \frac{(a c-d)^2 \tan ^{-1}\left (\frac{a^2 c x+d}{\sqrt{1-a^2 x^2} \sqrt{a^2 c^2-d^2}}\right )}{d^2 \sqrt{a^2 c^2-d^2}}-\frac{\sqrt{1-a^2 x^2}}{d}-\frac{(a c-2 d) \sin ^{-1}(a x)}{d^2} \]

[Out]

-(Sqrt[1 - a^2*x^2]/d) - ((a*c - 2*d)*ArcSin[a*x])/d^2 + ((a*c - d)^2*ArcTan[(d
+ a^2*c*x)/(Sqrt[a^2*c^2 - d^2]*Sqrt[1 - a^2*x^2])])/(d^2*Sqrt[a^2*c^2 - d^2])

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Rubi [A]  time = 0.353255, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ \frac{(a c-d)^2 \tan ^{-1}\left (\frac{a^2 c x+d}{\sqrt{1-a^2 x^2} \sqrt{a^2 c^2-d^2}}\right )}{d^2 \sqrt{a^2 c^2-d^2}}-\frac{\sqrt{1-a^2 x^2}}{d}-\frac{(a c-2 d) \sin ^{-1}(a x)}{d^2} \]

Antiderivative was successfully verified.

[In]  Int[(1 + a*x)^2/((c + d*x)*Sqrt[1 - a^2*x^2]),x]

[Out]

-(Sqrt[1 - a^2*x^2]/d) - ((a*c - 2*d)*ArcSin[a*x])/d^2 + ((a*c - d)^2*ArcTan[(d
+ a^2*c*x)/(Sqrt[a^2*c^2 - d^2]*Sqrt[1 - a^2*x^2])])/(d^2*Sqrt[a^2*c^2 - d^2])

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Rubi in Sympy [A]  time = 36.6038, size = 92, normalized size = 0.86 \[ - \frac{\sqrt{- a^{2} x^{2} + 1}}{d} + \frac{\operatorname{asin}{\left (a x \right )}}{d} - \frac{\left (- a c + d\right )^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{a^{2} c x + d}{\sqrt{- a c + d} \sqrt{a c + d} \sqrt{- a^{2} x^{2} + 1}} \right )}}{d^{2} \sqrt{a c + d}} + \frac{\left (- a c + d\right ) \operatorname{asin}{\left (a x \right )}}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*x+1)**2/(d*x+c)/(-a**2*x**2+1)**(1/2),x)

[Out]

-sqrt(-a**2*x**2 + 1)/d + asin(a*x)/d - (-a*c + d)**(3/2)*atanh((a**2*c*x + d)/(
sqrt(-a*c + d)*sqrt(a*c + d)*sqrt(-a**2*x**2 + 1)))/(d**2*sqrt(a*c + d)) + (-a*c
+ d)*asin(a*x)/d**2

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Mathematica [C]  time = 0.0776976, size = 148, normalized size = 1.38 \[ -\frac{\frac{i (d-a c)^2 \log \left (\frac{2 d^3 \left (\sqrt{1-a^2 x^2} \sqrt{a^2 c^2-d^2}+i a^2 c x+i d\right )}{(d-a c)^2 \sqrt{a^2 c^2-d^2} (c+d x)}\right )}{\sqrt{a^2 c^2-d^2}}+d \sqrt{1-a^2 x^2}+(a c-2 d) \sin ^{-1}(a x)}{d^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + a*x)^2/((c + d*x)*Sqrt[1 - a^2*x^2]),x]

[Out]

-((d*Sqrt[1 - a^2*x^2] + (a*c - 2*d)*ArcSin[a*x] + (I*(-(a*c) + d)^2*Log[(2*d^3*
(I*d + I*a^2*c*x + Sqrt[a^2*c^2 - d^2]*Sqrt[1 - a^2*x^2]))/((-(a*c) + d)^2*Sqrt[
a^2*c^2 - d^2]*(c + d*x))])/Sqrt[a^2*c^2 - d^2])/d^2)

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Maple [B]  time = 0.016, size = 524, normalized size = 4.9 \[ -{\frac{1}{d}\sqrt{-{a}^{2}{x}^{2}+1}}+2\,{\frac{a}{d\sqrt{{a}^{2}}}\arctan \left ({\frac{\sqrt{{a}^{2}}x}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }-{\frac{{a}^{2}c}{{d}^{2}}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-{\frac{{a}^{2}{c}^{2}}{{d}^{3}}\ln \left ({1 \left ( -2\,{\frac{{a}^{2}{c}^{2}-{d}^{2}}{{d}^{2}}}+2\,{\frac{{a}^{2}c}{d} \left ( x+{\frac{c}{d}} \right ) }+2\,\sqrt{-{\frac{{a}^{2}{c}^{2}-{d}^{2}}{{d}^{2}}}}\sqrt{- \left ( x+{\frac{c}{d}} \right ) ^{2}{a}^{2}+2\,{\frac{{a}^{2}c}{d} \left ( x+{\frac{c}{d}} \right ) }-{\frac{{a}^{2}{c}^{2}-{d}^{2}}{{d}^{2}}}} \right ) \left ( x+{\frac{c}{d}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{{a}^{2}{c}^{2}-{d}^{2}}{{d}^{2}}}}}}}+2\,{\frac{ac}{{d}^{2}}\ln \left ({1 \left ( -2\,{\frac{{a}^{2}{c}^{2}-{d}^{2}}{{d}^{2}}}+2\,{\frac{{a}^{2}c}{d} \left ( x+{\frac{c}{d}} \right ) }+2\,\sqrt{-{\frac{{a}^{2}{c}^{2}-{d}^{2}}{{d}^{2}}}}\sqrt{- \left ( x+{\frac{c}{d}} \right ) ^{2}{a}^{2}+2\,{\frac{{a}^{2}c}{d} \left ( x+{\frac{c}{d}} \right ) }-{\frac{{a}^{2}{c}^{2}-{d}^{2}}{{d}^{2}}}} \right ) \left ( x+{\frac{c}{d}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{{a}^{2}{c}^{2}-{d}^{2}}{{d}^{2}}}}}}}-{\frac{1}{d}\ln \left ({1 \left ( -2\,{\frac{{a}^{2}{c}^{2}-{d}^{2}}{{d}^{2}}}+2\,{\frac{{a}^{2}c}{d} \left ( x+{\frac{c}{d}} \right ) }+2\,\sqrt{-{\frac{{a}^{2}{c}^{2}-{d}^{2}}{{d}^{2}}}}\sqrt{- \left ( x+{\frac{c}{d}} \right ) ^{2}{a}^{2}+2\,{\frac{{a}^{2}c}{d} \left ( x+{\frac{c}{d}} \right ) }-{\frac{{a}^{2}{c}^{2}-{d}^{2}}{{d}^{2}}}} \right ) \left ( x+{\frac{c}{d}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{{a}^{2}{c}^{2}-{d}^{2}}{{d}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*x+1)^2/(d*x+c)/(-a^2*x^2+1)^(1/2),x)

[Out]

-(-a^2*x^2+1)^(1/2)/d+2*a/d/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))
-a^2/d^2*c/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))-1/d^3/(-(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)*(-(x+c/d)^2*a^2+2*a^2*c/d*(x+c/d)-(a^2*c^2-d^2)/d^2)^(1/2))/(x+c/d))*a^
2*c^2+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)*(-(x+c/d)^2*a^2+2*a^2*c/d*(x+c/d)-(a^2*c^2-d^2)/d
^2)^(1/2))/(x+c/d))*a*c-1/d/(-(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)*(-(x+c/d)^2*a^2+2*a^2*c/d*(x+c/d)
-(a^2*c^2-d^2)/d^2)^(1/2))/(x+c/d))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + 1)^2/(sqrt(-a^2*x^2 + 1)*(d*x + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.418532, size = 1, normalized size = 0.01 \[ \left [\frac{a^{2} d x^{2} +{\left (a c - \sqrt{-a^{2} x^{2} + 1}{\left (a c - d\right )} - d\right )} \sqrt{-\frac{a c - d}{a c + d}} \log \left (\frac{c d x -{\left (a^{2} c^{2} - d^{2}\right )} x^{2} + c^{2} - \sqrt{-a^{2} x^{2} + 1}{\left (c d x + c^{2}\right )} -{\left ({\left (a c d + d^{2}\right )} x^{2} - \sqrt{-a^{2} x^{2} + 1}{\left (a c^{2} + c d\right )} x +{\left (a c^{2} + c d\right )} x\right )} \sqrt{-\frac{a c - d}{a c + d}}}{d x - \sqrt{-a^{2} x^{2} + 1}{\left (d x + c\right )} + c}\right ) - 2 \,{\left (a c - \sqrt{-a^{2} x^{2} + 1}{\left (a c - 2 \, d\right )} - 2 \, d\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right )}{\sqrt{-a^{2} x^{2} + 1} d^{2} - d^{2}}, \frac{a^{2} d x^{2} + 2 \,{\left (a c - \sqrt{-a^{2} x^{2} + 1}{\left (a c - d\right )} - d\right )} \sqrt{\frac{a c - d}{a c + d}} \arctan \left (-\frac{d x - \sqrt{-a^{2} x^{2} + 1} c + c}{{\left (a c + d\right )} x \sqrt{\frac{a c - d}{a c + d}}}\right ) - 2 \,{\left (a c - \sqrt{-a^{2} x^{2} + 1}{\left (a c - 2 \, d\right )} - 2 \, d\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right )}{\sqrt{-a^{2} x^{2} + 1} d^{2} - d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + 1)^2/(sqrt(-a^2*x^2 + 1)*(d*x + c)),x, algorithm="fricas")

[Out]

[(a^2*d*x^2 + (a*c - sqrt(-a^2*x^2 + 1)*(a*c - d) - d)*sqrt(-(a*c - d)/(a*c + d)
)*log((c*d*x - (a^2*c^2 - d^2)*x^2 + c^2 - sqrt(-a^2*x^2 + 1)*(c*d*x + c^2) - ((
a*c*d + d^2)*x^2 - sqrt(-a^2*x^2 + 1)*(a*c^2 + c*d)*x + (a*c^2 + c*d)*x)*sqrt(-(
a*c - d)/(a*c + d)))/(d*x - sqrt(-a^2*x^2 + 1)*(d*x + c) + c)) - 2*(a*c - sqrt(-
a^2*x^2 + 1)*(a*c - 2*d) - 2*d)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)))/(sqrt(-a
^2*x^2 + 1)*d^2 - d^2), (a^2*d*x^2 + 2*(a*c - sqrt(-a^2*x^2 + 1)*(a*c - d) - d)*
sqrt((a*c - d)/(a*c + d))*arctan(-(d*x - sqrt(-a^2*x^2 + 1)*c + c)/((a*c + d)*x*
sqrt((a*c - d)/(a*c + d)))) - 2*(a*c - sqrt(-a^2*x^2 + 1)*(a*c - 2*d) - 2*d)*arc
tan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)))/(sqrt(-a^2*x^2 + 1)*d^2 - d^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a x + 1\right )^{2}}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )} \left (c + d x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x+1)**2/(d*x+c)/(-a**2*x**2+1)**(1/2),x)

[Out]

Integral((a*x + 1)**2/(sqrt(-(a*x - 1)*(a*x + 1))*(c + d*x)), x)

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GIAC/XCAS [A]  time = 0.310594, size = 177, normalized size = 1.65 \[ -\frac{{\left (a^{2} c - 2 \, a d\right )} \arcsin \left (a x\right ){\rm sign}\left (a\right )}{d^{2}{\left | a \right |}} - \frac{\sqrt{-a^{2} x^{2} + 1}}{d} - \frac{2 \,{\left (a^{3} c^{2} - 2 \, a^{2} c d + a d^{2}\right )} \arctan \left (\frac{d + \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} c}{a x}}{\sqrt{a^{2} c^{2} - d^{2}}}\right )}{\sqrt{a^{2} c^{2} - d^{2}} d^{2}{\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + 1)^2/(sqrt(-a^2*x^2 + 1)*(d*x + c)),x, algorithm="giac")

[Out]

-(a^2*c - 2*a*d)*arcsin(a*x)*sign(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))