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

Optimal. Leaf size=213 \[ \frac {2 \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a x-\sqrt {a^2 x^2+b}}}{\sqrt {\sqrt {a^2 c^2+b d^2}+a c}}\right )}{\sqrt {a^2 c^2+b d^2} \sqrt {\sqrt {a^2 c^2+b d^2}+a c}}-\frac {2 \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a x-\sqrt {a^2 x^2+b}}}{\sqrt {a c-\sqrt {a^2 c^2+b d^2}}}\right )}{\sqrt {a^2 c^2+b d^2} \sqrt {a c-\sqrt {a^2 c^2+b d^2}}} \]

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Rubi [F]  time = 0.88, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{(c+d x) \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[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*} \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\\ \end {align*}

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Mathematica [A]  time = 0.74, size = 317, normalized size = 1.49 \begin {gather*} \frac {2 \left (a x \left (a x-\sqrt {a^2 x^2+b}\right )+b\right ) \left (\left (\sqrt {a^2 c^2+b d^2}+a c\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d}}{\sqrt {a x-\sqrt {a^2 x^2+b}} \sqrt {-\sqrt {a^2 c^2+b d^2}-a c}}\right )+\sqrt {-\sqrt {a^2 c^2+b d^2}-a c} \sqrt {\sqrt {a^2 c^2+b d^2}-a c} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d}}{\sqrt {a x-\sqrt {a^2 x^2+b}} \sqrt {\sqrt {a^2 c^2+b d^2}-a c}}\right )\right )}{\sqrt {b} \sqrt {d} \sqrt {a^2 x^2+b} \left (a x-\sqrt {a^2 x^2+b}\right ) \sqrt {a^2 c^2+b d^2} \sqrt {-\sqrt {a^2 c^2+b d^2}-a c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b + a*x*(a*x - Sqrt[b + a^2*x^2]))*((a*c + Sqrt[a^2*c^2 + b*d^2])*ArcTan[(Sqrt[b]*Sqrt[d])/(Sqrt[-(a*c) -
Sqrt[a^2*c^2 + b*d^2]]*Sqrt[a*x - Sqrt[b + a^2*x^2]])] + Sqrt[-(a*c) - Sqrt[a^2*c^2 + b*d^2]]*Sqrt[-(a*c) + Sq
rt[a^2*c^2 + b*d^2]]*ArcTan[(Sqrt[b]*Sqrt[d])/(Sqrt[-(a*c) + Sqrt[a^2*c^2 + b*d^2]]*Sqrt[a*x - Sqrt[b + a^2*x^
2]])]))/(Sqrt[b]*Sqrt[d]*Sqrt[a^2*c^2 + b*d^2]*Sqrt[-(a*c) - Sqrt[a^2*c^2 + b*d^2]]*Sqrt[b + a^2*x^2]*(a*x - S
qrt[b + a^2*x^2]))

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IntegrateAlgebraic [A]  time = 0.57, size = 213, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {d} \tan ^{-1}\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} \tan ^{-1}\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}}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[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^2*c^2 +
b*d^2]*Sqrt[a*c - Sqrt[a^2*c^2 + b*d^2]]) + (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^2*c^2 + b*d^2]*Sqrt[a*c + Sqrt[a^2*c^2 + b*d^2]])

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fricas [B]  time = 0.79, size = 877, normalized size = 4.12 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (d x +c \right ) \sqrt {a^{2} x^{2}+b}\, \sqrt {a x -\sqrt {a^{2} x^{2}+b}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (c + d x\right ) \sqrt {a x - \sqrt {a^{2} x^{2} + b}} \sqrt {a^{2} x^{2} + b}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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