3.852 \(\int \frac{1}{\sqrt{d+e x} \sqrt{f+g x} \left (a+b x+c x^2\right )} \, dx\)

Optimal. Leaf size=287 \[ \frac{4 c \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{f+g x} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \sqrt{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{4 c \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{2 c f-g \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{f+g x} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \sqrt{2 c f-g \left (b-\sqrt{b^2-4 a c}\right )}} \]

[Out]

(-4*c*ArcTanh[(Sqrt[2*c*f - (b - Sqrt[b^2 - 4*a*c])*g]*Sqrt[d + e*x])/(Sqrt[2*c*
d - (b - Sqrt[b^2 - 4*a*c])*e]*Sqrt[f + g*x])])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d -
(b - Sqrt[b^2 - 4*a*c])*e]*Sqrt[2*c*f - (b - Sqrt[b^2 - 4*a*c])*g]) + (4*c*ArcTa
nh[(Sqrt[2*c*f - (b + Sqrt[b^2 - 4*a*c])*g]*Sqrt[d + e*x])/(Sqrt[2*c*d - (b + Sq
rt[b^2 - 4*a*c])*e]*Sqrt[f + g*x])])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b
^2 - 4*a*c])*e]*Sqrt[2*c*f - (b + Sqrt[b^2 - 4*a*c])*g])

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Rubi [A]  time = 1.05461, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ \frac{4 c \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{f+g x} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \sqrt{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{4 c \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{2 c f-g \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{f+g x} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \sqrt{2 c f-g \left (b-\sqrt{b^2-4 a c}\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(-4*c*ArcTanh[(Sqrt[2*c*f - (b - Sqrt[b^2 - 4*a*c])*g]*Sqrt[d + e*x])/(Sqrt[2*c*
d - (b - Sqrt[b^2 - 4*a*c])*e]*Sqrt[f + g*x])])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d -
(b - Sqrt[b^2 - 4*a*c])*e]*Sqrt[2*c*f - (b - Sqrt[b^2 - 4*a*c])*g]) + (4*c*ArcTa
nh[(Sqrt[2*c*f - (b + Sqrt[b^2 - 4*a*c])*g]*Sqrt[d + e*x])/(Sqrt[2*c*d - (b + Sq
rt[b^2 - 4*a*c])*e]*Sqrt[f + g*x])])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b
^2 - 4*a*c])*e]*Sqrt[2*c*f - (b + Sqrt[b^2 - 4*a*c])*g])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [B]  time = 6.23863, size = 1000, normalized size = 3.48 \[ \frac{\sqrt{2} c \log \left (-b-2 c x+\sqrt{b^2-4 a c}\right )}{\sqrt{b^2-4 a c} \sqrt{e g b^2-c e f b-c d g b-\sqrt{b^2-4 a c} e g b+2 c^2 d f+c \sqrt{b^2-4 a c} e f+c \sqrt{b^2-4 a c} d g-2 a c e g}}-\frac{\sqrt{2} c \log \left (b+2 c x+\sqrt{b^2-4 a c}\right )}{\sqrt{b^2-4 a c} \sqrt{e g b^2-c e f b-c d g b+\sqrt{b^2-4 a c} e g b+2 c^2 d f-c \sqrt{b^2-4 a c} e f-c \sqrt{b^2-4 a c} d g-2 a c e g}}-\frac{\sqrt{2} c \log \left (e f b^2+d g b^2+2 e g x b^2-\sqrt{b^2-4 a c} e f b-\sqrt{b^2-4 a c} d g b-2 \sqrt{b^2-4 a c} e g x b+4 c \sqrt{b^2-4 a c} d f-4 a c e f-4 a c d g+2 c \sqrt{b^2-4 a c} e f x+2 c \sqrt{b^2-4 a c} d g x-8 a c e g x+2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{e g b^2-c e f b-c d g b-\sqrt{b^2-4 a c} e g b+2 c^2 d f+c \sqrt{b^2-4 a c} e f+c \sqrt{b^2-4 a c} d g-2 a c e g} \sqrt{d+e x} \sqrt{f+g x}\right )}{\sqrt{b^2-4 a c} \sqrt{e g b^2-c e f b-c d g b-\sqrt{b^2-4 a c} e g b+2 c^2 d f+c \sqrt{b^2-4 a c} e f+c \sqrt{b^2-4 a c} d g-2 a c e g}}+\frac{\sqrt{2} c \log \left (-e f b^2-d g b^2-2 e g x b^2-\sqrt{b^2-4 a c} e f b-\sqrt{b^2-4 a c} d g b-2 \sqrt{b^2-4 a c} e g x b+4 c \sqrt{b^2-4 a c} d f+4 a c e f+4 a c d g+2 c \sqrt{b^2-4 a c} e f x+2 c \sqrt{b^2-4 a c} d g x+8 a c e g x+2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{e g b^2-c e f b-c d g b+\sqrt{b^2-4 a c} e g b+2 c^2 d f-c \sqrt{b^2-4 a c} e f-c \sqrt{b^2-4 a c} d g-2 a c e g} \sqrt{d+e x} \sqrt{f+g x}\right )}{\sqrt{b^2-4 a c} \sqrt{e g b^2-c e f b-c d g b+\sqrt{b^2-4 a c} e g b+2 c^2 d f-c \sqrt{b^2-4 a c} e f-c \sqrt{b^2-4 a c} d g-2 a c e g}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[2]*c*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2*d*
f - b*c*e*f + c*Sqrt[b^2 - 4*a*c]*e*f - b*c*d*g + c*Sqrt[b^2 - 4*a*c]*d*g + b^2*
e*g - 2*a*c*e*g - b*Sqrt[b^2 - 4*a*c]*e*g]) - (Sqrt[2]*c*Log[b + Sqrt[b^2 - 4*a*
c] + 2*c*x])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2*d*f - b*c*e*f - c*Sqrt[b^2 - 4*a*c]*e
*f - b*c*d*g - c*Sqrt[b^2 - 4*a*c]*d*g + b^2*e*g - 2*a*c*e*g + b*Sqrt[b^2 - 4*a*
c]*e*g]) - (Sqrt[2]*c*Log[4*c*Sqrt[b^2 - 4*a*c]*d*f + b^2*e*f - 4*a*c*e*f - b*Sq
rt[b^2 - 4*a*c]*e*f + b^2*d*g - 4*a*c*d*g - b*Sqrt[b^2 - 4*a*c]*d*g + 2*c*Sqrt[b
^2 - 4*a*c]*e*f*x + 2*c*Sqrt[b^2 - 4*a*c]*d*g*x + 2*b^2*e*g*x - 8*a*c*e*g*x - 2*
b*Sqrt[b^2 - 4*a*c]*e*g*x + 2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2*d*f - b*c*e*f
+ c*Sqrt[b^2 - 4*a*c]*e*f - b*c*d*g + c*Sqrt[b^2 - 4*a*c]*d*g + b^2*e*g - 2*a*c
*e*g - b*Sqrt[b^2 - 4*a*c]*e*g]*Sqrt[d + e*x]*Sqrt[f + g*x]])/(Sqrt[b^2 - 4*a*c]
*Sqrt[2*c^2*d*f - b*c*e*f + c*Sqrt[b^2 - 4*a*c]*e*f - b*c*d*g + c*Sqrt[b^2 - 4*a
*c]*d*g + b^2*e*g - 2*a*c*e*g - b*Sqrt[b^2 - 4*a*c]*e*g]) + (Sqrt[2]*c*Log[4*c*S
qrt[b^2 - 4*a*c]*d*f - b^2*e*f + 4*a*c*e*f - b*Sqrt[b^2 - 4*a*c]*e*f - b^2*d*g +
4*a*c*d*g - b*Sqrt[b^2 - 4*a*c]*d*g + 2*c*Sqrt[b^2 - 4*a*c]*e*f*x + 2*c*Sqrt[b^
2 - 4*a*c]*d*g*x - 2*b^2*e*g*x + 8*a*c*e*g*x - 2*b*Sqrt[b^2 - 4*a*c]*e*g*x + 2*S
qrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2*d*f - b*c*e*f - c*Sqrt[b^2 - 4*a*c]*e*f - b*
c*d*g - c*Sqrt[b^2 - 4*a*c]*d*g + b^2*e*g - 2*a*c*e*g + b*Sqrt[b^2 - 4*a*c]*e*g]
*Sqrt[d + e*x]*Sqrt[f + g*x]])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2*d*f - b*c*e*f - c*S
qrt[b^2 - 4*a*c]*e*f - b*c*d*g - c*Sqrt[b^2 - 4*a*c]*d*g + b^2*e*g - 2*a*c*e*g +
b*Sqrt[b^2 - 4*a*c]*e*g])

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Maple [B]  time = 0.072, size = 5507, normalized size = 19.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x + a\right )} \sqrt{e x + d} \sqrt{g x + f}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(sqrt(d + e*x)*sqrt(f + g*x)*(a + b*x + c*x**2)), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RuntimeError