3.2282 \(\int \frac{\sqrt{a+b x}}{\sqrt{c+d x} (e+f x)} \, dx\)

Optimal. Leaf size=119 \[ \frac{2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{d} f}-\frac{2 \sqrt{b e-a f} \tanh ^{-1}\left (\frac{\sqrt{a+b x} \sqrt{d e-c f}}{\sqrt{c+d x} \sqrt{b e-a f}}\right )}{f \sqrt{d e-c f}} \]

[Out]

(2*Sqrt[b]*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(Sqrt[d]*f)
- (2*Sqrt[b*e - a*f]*ArcTanh[(Sqrt[d*e - c*f]*Sqrt[a + b*x])/(Sqrt[b*e - a*f]*S
qrt[c + d*x])])/(f*Sqrt[d*e - c*f])

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Rubi [A]  time = 0.317495, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{d} f}-\frac{2 \sqrt{b e-a f} \tanh ^{-1}\left (\frac{\sqrt{a+b x} \sqrt{d e-c f}}{\sqrt{c+d x} \sqrt{b e-a f}}\right )}{f \sqrt{d e-c f}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x]/(Sqrt[c + d*x]*(e + f*x)),x]

[Out]

(2*Sqrt[b]*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(Sqrt[d]*f)
- (2*Sqrt[b*e - a*f]*ArcTanh[(Sqrt[d*e - c*f]*Sqrt[a + b*x])/(Sqrt[b*e - a*f]*S
qrt[c + d*x])])/(f*Sqrt[d*e - c*f])

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Rubi in Sympy [A]  time = 26.9616, size = 104, normalized size = 0.87 \[ \frac{2 \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{\sqrt{d} f} - \frac{2 \sqrt{a f - b e} \operatorname{atanh}{\left (\frac{\sqrt{a + b x} \sqrt{c f - d e}}{\sqrt{c + d x} \sqrt{a f - b e}} \right )}}{f \sqrt{c f - d e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(b)*atanh(sqrt(b)*sqrt(c + d*x)/(sqrt(d)*sqrt(a + b*x)))/(sqrt(d)*f) - 2*s
qrt(a*f - b*e)*atanh(sqrt(a + b*x)*sqrt(c*f - d*e)/(sqrt(c + d*x)*sqrt(a*f - b*e
)))/(f*sqrt(c*f - d*e))

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Mathematica [A]  time = 0.39837, size = 193, normalized size = 1.62 \[ \frac{-\frac{\sqrt{b e-a f} \log (e+f x)}{\sqrt{d e-c f}}+\frac{\sqrt{b e-a f} \log \left (2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{b e-a f} \sqrt{d e-c f}+2 a c f-a d e+a d f x-b c e+b c f x-2 b d e x\right )}{\sqrt{d e-c f}}+\frac{\sqrt{b} \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{\sqrt{d}}}{f} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x]/(Sqrt[c + d*x]*(e + f*x)),x]

[Out]

(-((Sqrt[b*e - a*f]*Log[e + f*x])/Sqrt[d*e - c*f]) + (Sqrt[b]*Log[b*c + a*d + 2*
b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/Sqrt[d] + (Sqrt[b*e - a*
f]*Log[-(b*c*e) - a*d*e + 2*a*c*f - 2*b*d*e*x + b*c*f*x + a*d*f*x + 2*Sqrt[b*e -
a*f]*Sqrt[d*e - c*f]*Sqrt[a + b*x]*Sqrt[c + d*x]])/Sqrt[d*e - c*f])/f

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Maple [B]  time = 0.064, size = 300, normalized size = 2.5 \[{\frac{1}{{f}^{2}} \left ( \ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) bf\sqrt{{\frac{ \left ( cf-de \right ) \left ( af-be \right ) }{{f}^{2}}}}-\ln \left ({\frac{1}{fx+e} \left ( adfx+bcfx-2\,bdex+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{{\frac{ \left ( cf-de \right ) \left ( af-be \right ) }{{f}^{2}}}}f+2\,acf-ade-bce \right ) } \right ) af\sqrt{bd}+\ln \left ({\frac{1}{fx+e} \left ( adfx+bcfx-2\,bdex+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{{\frac{ \left ( cf-de \right ) \left ( af-be \right ) }{{f}^{2}}}}f+2\,acf-ade-bce \right ) } \right ) be\sqrt{bd} \right ) \sqrt{dx+c}\sqrt{bx+a}{\frac{1}{\sqrt{{\frac{ \left ( cf-de \right ) \left ( af-be \right ) }{{f}^{2}}}}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*b*f
*((c*f-d*e)*(a*f-b*e)/f^2)^(1/2)-ln((a*d*f*x+b*c*f*x-2*b*d*e*x+2*((b*x+a)*(d*x+c
))^(1/2)*((c*f-d*e)*(a*f-b*e)/f^2)^(1/2)*f+2*a*c*f-a*d*e-b*c*e)/(f*x+e))*a*f*(b*
d)^(1/2)+ln((a*d*f*x+b*c*f*x-2*b*d*e*x+2*((b*x+a)*(d*x+c))^(1/2)*((c*f-d*e)*(a*f
-b*e)/f^2)^(1/2)*f+2*a*c*f-a*d*e-b*c*e)/(f*x+e))*b*e*(b*d)^(1/2))*(d*x+c)^(1/2)*
(b*x+a)^(1/2)/((c*f-d*e)*(a*f-b*e)/f^2)^(1/2)/f^2/((b*x+a)*(d*x+c))^(1/2)/(b*d)^
(1/2)

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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(sqrt(b*x + a)/(sqrt(d*x + c)*(f*x + e)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 5.75318, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)/(sqrt(d*x + c)*(f*x + e)),x, algorithm="fricas")

[Out]

[1/2*(sqrt(b/d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d^2*x
+ b*c*d + a*d^2)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(b/d) + 8*(b^2*c*d + a*b*d^2)*
x) + sqrt((b*e - a*f)/(d*e - c*f))*log((8*a^2*c^2*f^2 + (b^2*c^2 + 6*a*b*c*d + a
^2*d^2)*e^2 - 8*(a*b*c^2 + a^2*c*d)*e*f + (8*b^2*d^2*e^2 - 8*(b^2*c*d + a*b*d^2)
*e*f + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*f^2)*x^2 - 4*(2*a*c^2*f^2 + (b*c*d + a*d^
2)*e^2 - (b*c^2 + 3*a*c*d)*e*f + (2*b*d^2*e^2 - (3*b*c*d + a*d^2)*e*f + (b*c^2 +
a*c*d)*f^2)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt((b*e - a*f)/(d*e - c*f)) + 2*(4
*(b^2*c*d + a*b*d^2)*e^2 - (3*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*e*f + 4*(a*b*c^2
+ a^2*c*d)*f^2)*x)/(f^2*x^2 + 2*e*f*x + e^2)))/f, 1/2*(2*sqrt(-b/d)*arctan(1/2*
(2*b*d*x + b*c + a*d)/(sqrt(b*x + a)*sqrt(d*x + c)*d*sqrt(-b/d))) + sqrt((b*e -
a*f)/(d*e - c*f))*log((8*a^2*c^2*f^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*e^2 - 8*(
a*b*c^2 + a^2*c*d)*e*f + (8*b^2*d^2*e^2 - 8*(b^2*c*d + a*b*d^2)*e*f + (b^2*c^2 +
6*a*b*c*d + a^2*d^2)*f^2)*x^2 - 4*(2*a*c^2*f^2 + (b*c*d + a*d^2)*e^2 - (b*c^2 +
3*a*c*d)*e*f + (2*b*d^2*e^2 - (3*b*c*d + a*d^2)*e*f + (b*c^2 + a*c*d)*f^2)*x)*s
qrt(b*x + a)*sqrt(d*x + c)*sqrt((b*e - a*f)/(d*e - c*f)) + 2*(4*(b^2*c*d + a*b*d
^2)*e^2 - (3*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*e*f + 4*(a*b*c^2 + a^2*c*d)*f^2)*
x)/(f^2*x^2 + 2*e*f*x + e^2)))/f, -1/2*(2*sqrt(-(b*e - a*f)/(d*e - c*f))*arctan(
-1/2*(2*a*c*f - (b*c + a*d)*e - (2*b*d*e - (b*c + a*d)*f)*x)/((d*e - c*f)*sqrt(b
*x + a)*sqrt(d*x + c)*sqrt(-(b*e - a*f)/(d*e - c*f)))) - sqrt(b/d)*log(8*b^2*d^2
*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d^2*x + b*c*d + a*d^2)*sqrt(b*x +
a)*sqrt(d*x + c)*sqrt(b/d) + 8*(b^2*c*d + a*b*d^2)*x))/f, -(sqrt(-(b*e - a*f)/(d
*e - c*f))*arctan(-1/2*(2*a*c*f - (b*c + a*d)*e - (2*b*d*e - (b*c + a*d)*f)*x)/(
(d*e - c*f)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-(b*e - a*f)/(d*e - c*f)))) - sqrt(
-b/d)*arctan(1/2*(2*b*d*x + b*c + a*d)/(sqrt(b*x + a)*sqrt(d*x + c)*d*sqrt(-b/d)
)))/f]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(a + b*x)/(sqrt(c + d*x)*(e + f*x)), x)

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GIAC/XCAS [A]  time = 0.283105, size = 328, normalized size = 2.76 \[ -\frac{\sqrt{b d} b{\rm ln}\left ({\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{d f{\left | b \right |}} - \frac{2 \,{\left (\sqrt{b d} a b^{2} f - \sqrt{b d} b^{3} e\right )} \arctan \left (-\frac{b^{2} c f + a b d f - 2 \, b^{2} d e -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} f}{2 \, \sqrt{-a b c d f^{2} + b^{2} c d f e + a b d^{2} f e - b^{2} d^{2} e^{2}} b}\right )}{\sqrt{-a b c d f^{2} + b^{2} c d f e + a b d^{2} f e - b^{2} d^{2} e^{2}} b f{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)/(sqrt(d*x + c)*(f*x + e)),x, algorithm="giac")

[Out]

-sqrt(b*d)*b*ln((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^
2)/(d*f*abs(b)) - 2*(sqrt(b*d)*a*b^2*f - sqrt(b*d)*b^3*e)*arctan(-1/2*(b^2*c*f +
a*b*d*f - 2*b^2*d*e - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a
*b*d))^2*f)/(sqrt(-a*b*c*d*f^2 + b^2*c*d*f*e + a*b*d^2*f*e - b^2*d^2*e^2)*b))/(s
qrt(-a*b*c*d*f^2 + b^2*c*d*f*e + a*b*d^2*f*e - b^2*d^2*e^2)*b*f*abs(b))