3.24 \(\int \frac{A+B x}{\left (a+c x^2\right ) \sqrt{d+f x^2}} \, dx\)

Optimal. Leaf size=101 \[ \frac{A \tan ^{-1}\left (\frac{x \sqrt{c d-a f}}{\sqrt{a} \sqrt{d+f x^2}}\right )}{\sqrt{a} \sqrt{c d-a f}}-\frac{B \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+f x^2}}{\sqrt{c d-a f}}\right )}{\sqrt{c} \sqrt{c d-a f}} \]

[Out]

(A*ArcTan[(Sqrt[c*d - a*f]*x)/(Sqrt[a]*Sqrt[d + f*x^2])])/(Sqrt[a]*Sqrt[c*d - a*
f]) - (B*ArcTanh[(Sqrt[c]*Sqrt[d + f*x^2])/Sqrt[c*d - a*f]])/(Sqrt[c]*Sqrt[c*d -
a*f])

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Rubi [A]  time = 0.318022, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{A \tan ^{-1}\left (\frac{x \sqrt{c d-a f}}{\sqrt{a} \sqrt{d+f x^2}}\right )}{\sqrt{a} \sqrt{c d-a f}}-\frac{B \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+f x^2}}{\sqrt{c d-a f}}\right )}{\sqrt{c} \sqrt{c d-a f}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((a + c*x^2)*Sqrt[d + f*x^2]),x]

[Out]

(A*ArcTan[(Sqrt[c*d - a*f]*x)/(Sqrt[a]*Sqrt[d + f*x^2])])/(Sqrt[a]*Sqrt[c*d - a*
f]) - (B*ArcTanh[(Sqrt[c]*Sqrt[d + f*x^2])/Sqrt[c*d - a*f]])/(Sqrt[c]*Sqrt[c*d -
a*f])

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Rubi in Sympy [A]  time = 28.9055, size = 155, normalized size = 1.53 \[ - \frac{\left (A \sqrt{c} - B \sqrt{- a}\right ) \operatorname{atan}{\left (\frac{\sqrt{c} d - f x \sqrt{- a}}{\sqrt{d + f x^{2}} \sqrt{a f - c d}} \right )}}{2 \sqrt{c} \sqrt{- a} \sqrt{a f - c d}} + \frac{\left (A \sqrt{c} + B \sqrt{- a}\right ) \operatorname{atan}{\left (\frac{\sqrt{c} d + f x \sqrt{- a}}{\sqrt{d + f x^{2}} \sqrt{a f - c d}} \right )}}{2 \sqrt{c} \sqrt{- a} \sqrt{a f - c d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(A*sqrt(c) - B*sqrt(-a))*atan((sqrt(c)*d - f*x*sqrt(-a))/(sqrt(d + f*x**2)*sqrt
(a*f - c*d)))/(2*sqrt(c)*sqrt(-a)*sqrt(a*f - c*d)) + (A*sqrt(c) + B*sqrt(-a))*at
an((sqrt(c)*d + f*x*sqrt(-a))/(sqrt(d + f*x**2)*sqrt(a*f - c*d)))/(2*sqrt(c)*sqr
t(-a)*sqrt(a*f - c*d))

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Mathematica [C]  time = 0.773739, size = 282, normalized size = 2.79 \[ \frac{\left (-\sqrt{a} B+i A \sqrt{c}\right ) \log \left (\frac{2 i \sqrt{a} \sqrt{c} \left (\sqrt{d+f x^2} \sqrt{c d-a f}+i \sqrt{a} f x+\sqrt{c} d\right )}{\left (\sqrt{a}+i \sqrt{c} x\right ) \left (\sqrt{a} B-i A \sqrt{c}\right ) \sqrt{c d-a f}}\right )-\left (\sqrt{a} B+i A \sqrt{c}\right ) \log \left (\frac{2 \sqrt{a} \sqrt{c} \left (\sqrt{d+f x^2} \sqrt{c d-a f}-i \sqrt{a} f x+\sqrt{c} d\right )}{\left (\sqrt{c} x+i \sqrt{a}\right ) \left (\sqrt{a} B+i A \sqrt{c}\right ) \sqrt{c d-a f}}\right )}{2 \sqrt{a} \sqrt{c} \sqrt{c d-a f}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((a + c*x^2)*Sqrt[d + f*x^2]),x]

[Out]

(-((Sqrt[a]*B + I*A*Sqrt[c])*Log[(2*Sqrt[a]*Sqrt[c]*(Sqrt[c]*d - I*Sqrt[a]*f*x +
Sqrt[c*d - a*f]*Sqrt[d + f*x^2]))/((Sqrt[a]*B + I*A*Sqrt[c])*Sqrt[c*d - a*f]*(I
*Sqrt[a] + Sqrt[c]*x))]) + (-(Sqrt[a]*B) + I*A*Sqrt[c])*Log[((2*I)*Sqrt[a]*Sqrt[
c]*(Sqrt[c]*d + I*Sqrt[a]*f*x + Sqrt[c*d - a*f]*Sqrt[d + f*x^2]))/((Sqrt[a]*B -
I*A*Sqrt[c])*Sqrt[c*d - a*f]*(Sqrt[a] + I*Sqrt[c]*x))])/(2*Sqrt[a]*Sqrt[c]*Sqrt[
c*d - a*f])

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Maple [B]  time = 0.044, size = 608, normalized size = 6. \[{\frac{A}{2}\ln \left ({1 \left ( -2\,{\frac{fa-cd}{c}}-2\,{\frac{f\sqrt{-ac}}{c} \left ( x+{\frac{\sqrt{-ac}}{c}} \right ) }+2\,\sqrt{-{\frac{fa-cd}{c}}}\sqrt{ \left ( x+{\frac{\sqrt{-ac}}{c}} \right ) ^{2}f-2\,{\frac{f\sqrt{-ac}}{c} \left ( x+{\frac{\sqrt{-ac}}{c}} \right ) }-{\frac{fa-cd}{c}}} \right ) \left ( x+{\frac{1}{c}\sqrt{-ac}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ac}}}{\frac{1}{\sqrt{-{\frac{fa-cd}{c}}}}}}-{\frac{B}{2\,c}\ln \left ({1 \left ( -2\,{\frac{fa-cd}{c}}-2\,{\frac{f\sqrt{-ac}}{c} \left ( x+{\frac{\sqrt{-ac}}{c}} \right ) }+2\,\sqrt{-{\frac{fa-cd}{c}}}\sqrt{ \left ( x+{\frac{\sqrt{-ac}}{c}} \right ) ^{2}f-2\,{\frac{f\sqrt{-ac}}{c} \left ( x+{\frac{\sqrt{-ac}}{c}} \right ) }-{\frac{fa-cd}{c}}} \right ) \left ( x+{\frac{1}{c}\sqrt{-ac}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{fa-cd}{c}}}}}}-{\frac{A}{2}\ln \left ({1 \left ( -2\,{\frac{fa-cd}{c}}+2\,{\frac{f\sqrt{-ac}}{c} \left ( x-{\frac{\sqrt{-ac}}{c}} \right ) }+2\,\sqrt{-{\frac{fa-cd}{c}}}\sqrt{ \left ( x-{\frac{\sqrt{-ac}}{c}} \right ) ^{2}f+2\,{\frac{f\sqrt{-ac}}{c} \left ( x-{\frac{\sqrt{-ac}}{c}} \right ) }-{\frac{fa-cd}{c}}} \right ) \left ( x-{\frac{1}{c}\sqrt{-ac}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ac}}}{\frac{1}{\sqrt{-{\frac{fa-cd}{c}}}}}}-{\frac{B}{2\,c}\ln \left ({1 \left ( -2\,{\frac{fa-cd}{c}}+2\,{\frac{f\sqrt{-ac}}{c} \left ( x-{\frac{\sqrt{-ac}}{c}} \right ) }+2\,\sqrt{-{\frac{fa-cd}{c}}}\sqrt{ \left ( x-{\frac{\sqrt{-ac}}{c}} \right ) ^{2}f+2\,{\frac{f\sqrt{-ac}}{c} \left ( x-{\frac{\sqrt{-ac}}{c}} \right ) }-{\frac{fa-cd}{c}}} \right ) \left ( x-{\frac{1}{c}\sqrt{-ac}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{fa-cd}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2/(-a*c)^(1/2)/(-(a*f-c*d)/c)^(1/2)*ln((-2*(a*f-c*d)/c-2*f*(-a*c)^(1/2)/c*(x+(
-a*c)^(1/2)/c)+2*(-(a*f-c*d)/c)^(1/2)*((x+(-a*c)^(1/2)/c)^2*f-2*f*(-a*c)^(1/2)/c
*(x+(-a*c)^(1/2)/c)-(a*f-c*d)/c)^(1/2))/(x+(-a*c)^(1/2)/c))*A-1/2/c/(-(a*f-c*d)/
c)^(1/2)*ln((-2*(a*f-c*d)/c-2*f*(-a*c)^(1/2)/c*(x+(-a*c)^(1/2)/c)+2*(-(a*f-c*d)/
c)^(1/2)*((x+(-a*c)^(1/2)/c)^2*f-2*f*(-a*c)^(1/2)/c*(x+(-a*c)^(1/2)/c)-(a*f-c*d)
/c)^(1/2))/(x+(-a*c)^(1/2)/c))*B-1/2/(-a*c)^(1/2)/(-(a*f-c*d)/c)^(1/2)*ln((-2*(a
*f-c*d)/c+2*f*(-a*c)^(1/2)/c*(x-(-a*c)^(1/2)/c)+2*(-(a*f-c*d)/c)^(1/2)*((x-(-a*c
)^(1/2)/c)^2*f+2*f*(-a*c)^(1/2)/c*(x-(-a*c)^(1/2)/c)-(a*f-c*d)/c)^(1/2))/(x-(-a*
c)^(1/2)/c))*A-1/2/c/(-(a*f-c*d)/c)^(1/2)*ln((-2*(a*f-c*d)/c+2*f*(-a*c)^(1/2)/c*
(x-(-a*c)^(1/2)/c)+2*(-(a*f-c*d)/c)^(1/2)*((x-(-a*c)^(1/2)/c)^2*f+2*f*(-a*c)^(1/
2)/c*(x-(-a*c)^(1/2)/c)-(a*f-c*d)/c)^(1/2))/(x-(-a*c)^(1/2)/c))*B

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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((B*x + A)/((c*x^2 + a)*sqrt(f*x^2 + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.399553, size = 2045, normalized size = 20.25 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*sqrt((B^2*a - A^2*c + 2*(a*c^2*d - a^2*c*f)*sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^
2*c^2*d*f + a^3*c*f^2)))/(a*c^2*d - a^2*c*f))*log(((A*B^3*a + A^3*B*c)*f*x + (A^
2*B*c^2*d - A^2*B*a*c*f + (B*a*c^3*d^2 - 2*B*a^2*c^2*d*f + B*a^3*c*f^2)*sqrt(-A^
2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)))*sqrt(f*x^2 + d)*sqrt((B^2*a - A^
2*c + 2*(a*c^2*d - a^2*c*f)*sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2
)))/(a*c^2*d - a^2*c*f)) + sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)
)*((B^2*a*c^2 + A^2*c^3)*d^2 - (B^2*a^2*c + A^2*a*c^2)*d*f))/x) + 1/4*sqrt((B^2*
a - A^2*c + 2*(a*c^2*d - a^2*c*f)*sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3
*c*f^2)))/(a*c^2*d - a^2*c*f))*log(((A*B^3*a + A^3*B*c)*f*x - (A^2*B*c^2*d - A^2
*B*a*c*f + (B*a*c^3*d^2 - 2*B*a^2*c^2*d*f + B*a^3*c*f^2)*sqrt(-A^2*B^2/(a*c^3*d^
2 - 2*a^2*c^2*d*f + a^3*c*f^2)))*sqrt(f*x^2 + d)*sqrt((B^2*a - A^2*c + 2*(a*c^2*
d - a^2*c*f)*sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)))/(a*c^2*d -
a^2*c*f)) + sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2))*((B^2*a*c^2 +
A^2*c^3)*d^2 - (B^2*a^2*c + A^2*a*c^2)*d*f))/x) - 1/4*sqrt((B^2*a - A^2*c - 2*(
a*c^2*d - a^2*c*f)*sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)))/(a*c^
2*d - a^2*c*f))*log(((A*B^3*a + A^3*B*c)*f*x + (A^2*B*c^2*d - A^2*B*a*c*f - (B*a
*c^3*d^2 - 2*B*a^2*c^2*d*f + B*a^3*c*f^2)*sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d
*f + a^3*c*f^2)))*sqrt(f*x^2 + d)*sqrt((B^2*a - A^2*c - 2*(a*c^2*d - a^2*c*f)*sq
rt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)))/(a*c^2*d - a^2*c*f)) - sqr
t(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2))*((B^2*a*c^2 + A^2*c^3)*d^2 -
(B^2*a^2*c + A^2*a*c^2)*d*f))/x) + 1/4*sqrt((B^2*a - A^2*c - 2*(a*c^2*d - a^2*c
*f)*sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)))/(a*c^2*d - a^2*c*f))
*log(((A*B^3*a + A^3*B*c)*f*x - (A^2*B*c^2*d - A^2*B*a*c*f - (B*a*c^3*d^2 - 2*B*
a^2*c^2*d*f + B*a^3*c*f^2)*sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)
))*sqrt(f*x^2 + d)*sqrt((B^2*a - A^2*c - 2*(a*c^2*d - a^2*c*f)*sqrt(-A^2*B^2/(a*
c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)))/(a*c^2*d - a^2*c*f)) - sqrt(-A^2*B^2/(a*c
^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2))*((B^2*a*c^2 + A^2*c^3)*d^2 - (B^2*a^2*c + A
^2*a*c^2)*d*f))/x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((A + B*x)/((a + c*x**2)*sqrt(d + f*x**2)), x)

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GIAC/XCAS [A]  time = 1.28648, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done