\(\int \frac {1}{(c+d x) \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}} \, dx\) [2540]

Optimal result

Integrand size = 44, antiderivative size = 213 \[ \int \frac {1}{(c+d x) \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}} \, dx=-\frac {2 \sqrt {d} \arctan \left (\frac {\sqrt {d} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {a c-\sqrt {a^2 c^2+b d^2}}}\right )}{\sqrt {a^2 c^2+b d^2} \sqrt {a c-\sqrt {a^2 c^2+b d^2}}}+\frac {2 \sqrt {d} \arctan \left (\frac {\sqrt {d} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {a c+\sqrt {a^2 c^2+b d^2}}}\right )}{\sqrt {a^2 c^2+b d^2} \sqrt {a c+\sqrt {a^2 c^2+b d^2}}} \]

[Out]

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

Rubi [F]

\[ \int \frac {1}{(c+d x) \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}} \, dx=\int \frac {1}{(c+d x) \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(c+d x) \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(c+d x) \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}} \, dx=\frac {2 \sqrt {d} \left (-\frac {\arctan \left (\frac {\sqrt {d} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {a c-\sqrt {a^2 c^2+b d^2}}}\right )}{\sqrt {a c-\sqrt {a^2 c^2+b d^2}}}+\frac {\arctan \left (\frac {\sqrt {d} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {a c+\sqrt {a^2 c^2+b d^2}}}\right )}{\sqrt {a c+\sqrt {a^2 c^2+b d^2}}}\right )}{\sqrt {a^2 c^2+b d^2}} \]

[In]

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

[Out]

(2*Sqrt[d]*(-(ArcTan[(Sqrt[d]*Sqrt[a*x - Sqrt[b + a^2*x^2]])/Sqrt[a*c - Sqrt[a^2*c^2 + b*d^2]]]/Sqrt[a*c - Sqr
t[a^2*c^2 + b*d^2]]) + ArcTan[(Sqrt[d]*Sqrt[a*x - Sqrt[b + a^2*x^2]])/Sqrt[a*c + Sqrt[a^2*c^2 + b*d^2]]]/Sqrt[
a*c + Sqrt[a^2*c^2 + b*d^2]]))/Sqrt[a^2*c^2 + b*d^2]

Maple [F]

\[\int \frac {1}{\left (d x +c \right ) \sqrt {a^{2} x^{2}+b}\, \sqrt {a x -\sqrt {a^{2} x^{2}+b}}}d x\]

[In]

int(1/(d*x+c)/(a^2*x^2+b)^(1/2)/(a*x-(a^2*x^2+b)^(1/2))^(1/2),x)

[Out]

int(1/(d*x+c)/(a^2*x^2+b)^(1/2)/(a*x-(a^2*x^2+b)^(1/2))^(1/2),x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 877 vs. \(2 (177) = 354\).

Time = 0.30 (sec) , antiderivative size = 877, normalized size of antiderivative = 4.12 \[ \int \frac {1}{(c+d x) \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}} \, dx=\sqrt {\frac {a c + \frac {a^{2} b c^{2} d + b^{2} d^{3}}{\sqrt {a^{2} b^{2} c^{2} d^{2} + b^{3} d^{4}}}}{a^{2} b c^{2} d + b^{2} d^{3}}} \log \left (2 \, \sqrt {a x - \sqrt {a^{2} x^{2} + b}} d + 2 \, {\left (a^{2} c^{2} + b d^{2} - \frac {a^{3} b c^{3} d + a b^{2} c d^{3}}{\sqrt {a^{2} b^{2} c^{2} d^{2} + b^{3} d^{4}}}\right )} \sqrt {\frac {a c + \frac {a^{2} b c^{2} d + b^{2} d^{3}}{\sqrt {a^{2} b^{2} c^{2} d^{2} + b^{3} d^{4}}}}{a^{2} b c^{2} d + b^{2} d^{3}}}\right ) - \sqrt {\frac {a c + \frac {a^{2} b c^{2} d + b^{2} d^{3}}{\sqrt {a^{2} b^{2} c^{2} d^{2} + b^{3} d^{4}}}}{a^{2} b c^{2} d + b^{2} d^{3}}} \log \left (2 \, \sqrt {a x - \sqrt {a^{2} x^{2} + b}} d - 2 \, {\left (a^{2} c^{2} + b d^{2} - \frac {a^{3} b c^{3} d + a b^{2} c d^{3}}{\sqrt {a^{2} b^{2} c^{2} d^{2} + b^{3} d^{4}}}\right )} \sqrt {\frac {a c + \frac {a^{2} b c^{2} d + b^{2} d^{3}}{\sqrt {a^{2} b^{2} c^{2} d^{2} + b^{3} d^{4}}}}{a^{2} b c^{2} d + b^{2} d^{3}}}\right ) + \sqrt {\frac {a c - \frac {a^{2} b c^{2} d + b^{2} d^{3}}{\sqrt {a^{2} b^{2} c^{2} d^{2} + b^{3} d^{4}}}}{a^{2} b c^{2} d + b^{2} d^{3}}} \log \left (2 \, \sqrt {a x - \sqrt {a^{2} x^{2} + b}} d + 2 \, {\left (a^{2} c^{2} + b d^{2} + \frac {a^{3} b c^{3} d + a b^{2} c d^{3}}{\sqrt {a^{2} b^{2} c^{2} d^{2} + b^{3} d^{4}}}\right )} \sqrt {\frac {a c - \frac {a^{2} b c^{2} d + b^{2} d^{3}}{\sqrt {a^{2} b^{2} c^{2} d^{2} + b^{3} d^{4}}}}{a^{2} b c^{2} d + b^{2} d^{3}}}\right ) - \sqrt {\frac {a c - \frac {a^{2} b c^{2} d + b^{2} d^{3}}{\sqrt {a^{2} b^{2} c^{2} d^{2} + b^{3} d^{4}}}}{a^{2} b c^{2} d + b^{2} d^{3}}} \log \left (2 \, \sqrt {a x - \sqrt {a^{2} x^{2} + b}} d - 2 \, {\left (a^{2} c^{2} + b d^{2} + \frac {a^{3} b c^{3} d + a b^{2} c d^{3}}{\sqrt {a^{2} b^{2} c^{2} d^{2} + b^{3} d^{4}}}\right )} \sqrt {\frac {a c - \frac {a^{2} b c^{2} d + b^{2} d^{3}}{\sqrt {a^{2} b^{2} c^{2} d^{2} + b^{3} d^{4}}}}{a^{2} b c^{2} d + b^{2} d^{3}}}\right ) \]

[In]

integrate(1/(d*x+c)/(a^2*x^2+b)^(1/2)/(a*x-(a^2*x^2+b)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

sqrt((a*c + (a^2*b*c^2*d + b^2*d^3)/sqrt(a^2*b^2*c^2*d^2 + b^3*d^4))/(a^2*b*c^2*d + b^2*d^3))*log(2*sqrt(a*x -
sqrt(a^2*x^2 + b))*d + 2*(a^2*c^2 + b*d^2 - (a^3*b*c^3*d + a*b^2*c*d^3)/sqrt(a^2*b^2*c^2*d^2 + b^3*d^4))*sqrt
((a*c + (a^2*b*c^2*d + b^2*d^3)/sqrt(a^2*b^2*c^2*d^2 + b^3*d^4))/(a^2*b*c^2*d + b^2*d^3))) - sqrt((a*c + (a^2*
b*c^2*d + b^2*d^3)/sqrt(a^2*b^2*c^2*d^2 + b^3*d^4))/(a^2*b*c^2*d + b^2*d^3))*log(2*sqrt(a*x - sqrt(a^2*x^2 + b
))*d - 2*(a^2*c^2 + b*d^2 - (a^3*b*c^3*d + a*b^2*c*d^3)/sqrt(a^2*b^2*c^2*d^2 + b^3*d^4))*sqrt((a*c + (a^2*b*c^
2*d + b^2*d^3)/sqrt(a^2*b^2*c^2*d^2 + b^3*d^4))/(a^2*b*c^2*d + b^2*d^3))) + sqrt((a*c - (a^2*b*c^2*d + b^2*d^3
)/sqrt(a^2*b^2*c^2*d^2 + b^3*d^4))/(a^2*b*c^2*d + b^2*d^3))*log(2*sqrt(a*x - sqrt(a^2*x^2 + b))*d + 2*(a^2*c^2
+ b*d^2 + (a^3*b*c^3*d + a*b^2*c*d^3)/sqrt(a^2*b^2*c^2*d^2 + b^3*d^4))*sqrt((a*c - (a^2*b*c^2*d + b^2*d^3)/sq
rt(a^2*b^2*c^2*d^2 + b^3*d^4))/(a^2*b*c^2*d + b^2*d^3))) - sqrt((a*c - (a^2*b*c^2*d + b^2*d^3)/sqrt(a^2*b^2*c^
2*d^2 + b^3*d^4))/(a^2*b*c^2*d + b^2*d^3))*log(2*sqrt(a*x - sqrt(a^2*x^2 + b))*d - 2*(a^2*c^2 + b*d^2 + (a^3*b
*c^3*d + a*b^2*c*d^3)/sqrt(a^2*b^2*c^2*d^2 + b^3*d^4))*sqrt((a*c - (a^2*b*c^2*d + b^2*d^3)/sqrt(a^2*b^2*c^2*d^
2 + b^3*d^4))/(a^2*b*c^2*d + b^2*d^3)))

Sympy [F]

\[ \int \frac {1}{(c+d x) \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}} \, dx=\int \frac {1}{\left (c + d x\right ) \sqrt {a x - \sqrt {a^{2} x^{2} + b}} \sqrt {a^{2} x^{2} + b}}\, dx \]

[In]

integrate(1/(d*x+c)/(a**2*x**2+b)**(1/2)/(a*x-(a**2*x**2+b)**(1/2))**(1/2),x)

[Out]

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

Maxima [F]

\[ \int \frac {1}{(c+d x) \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}} \, dx=\int { \frac {1}{\sqrt {a^{2} x^{2} + b} \sqrt {a x - \sqrt {a^{2} x^{2} + b}} {\left (d x + c\right )}} \,d x } \]

[In]

integrate(1/(d*x+c)/(a^2*x^2+b)^(1/2)/(a*x-(a^2*x^2+b)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(a^2*x^2 + b)*sqrt(a*x - sqrt(a^2*x^2 + b))*(d*x + c)), x)

Giac [F]

\[ \int \frac {1}{(c+d x) \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}} \, dx=\int { \frac {1}{\sqrt {a^{2} x^{2} + b} \sqrt {a x - \sqrt {a^{2} x^{2} + b}} {\left (d x + c\right )}} \,d x } \]

[In]

integrate(1/(d*x+c)/(a^2*x^2+b)^(1/2)/(a*x-(a^2*x^2+b)^(1/2))^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(a^2*x^2 + b)*sqrt(a*x - sqrt(a^2*x^2 + b))*(d*x + c)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(c+d x) \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}} \, dx=\int \frac {1}{\sqrt {a\,x-\sqrt {a^2\,x^2+b}}\,\sqrt {a^2\,x^2+b}\,\left (c+d\,x\right )} \,d x \]

[In]

int(1/((a*x - (b + a^2*x^2)^(1/2))^(1/2)*(b + a^2*x^2)^(1/2)*(c + d*x)),x)

[Out]

int(1/((a*x - (b + a^2*x^2)^(1/2))^(1/2)*(b + a^2*x^2)^(1/2)*(c + d*x)), x)