3.606 \(\int \frac{\sqrt{d+e x} \sqrt{f+g x}}{a+c x^2} \, dx\)

Optimal. Leaf size=342 \[ \frac{\left (-\sqrt{-a} \sqrt{c} (d g+e f)-a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{\sqrt{-a} c \sqrt{\sqrt{c} d-\sqrt{-a} e} \sqrt{\sqrt{c} f-\sqrt{-a} g}}-\frac{\left (\sqrt{-a} \sqrt{c} (d g+e f)-a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{-a} c \sqrt{\sqrt{-a} e+\sqrt{c} d} \sqrt{\sqrt{-a} g+\sqrt{c} f}}+\frac{2 \sqrt{e} \sqrt{g} \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{c} \]

[Out]

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

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Rubi [A]  time = 3.86892, antiderivative size = 342, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{\left (-\sqrt{-a} \sqrt{c} (d g+e f)-a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{\sqrt{-a} c \sqrt{\sqrt{c} d-\sqrt{-a} e} \sqrt{\sqrt{c} f-\sqrt{-a} g}}-\frac{\left (\sqrt{-a} \sqrt{c} (d g+e f)-a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{-a} c \sqrt{\sqrt{-a} e+\sqrt{c} d} \sqrt{\sqrt{-a} g+\sqrt{c} f}}+\frac{2 \sqrt{e} \sqrt{g} \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{c} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[e]*Sqrt[g]*ArcTanh[(Sqrt[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g*x])])/c +
((c*d*f - a*e*g - Sqrt[-a]*Sqrt[c]*(e*f + d*g))*ArcTanh[(Sqrt[Sqrt[c]*f - Sqrt[
-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*Sqrt[f + g*x])])/(Sqrt[-a]*c
*Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*Sqrt[Sqrt[c]*f - Sqrt[-a]*g]) - ((c*d*f - a*e*g +
Sqrt[-a]*Sqrt[c]*(e*f + d*g))*ArcTanh[(Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*Sqrt[d + e*x
])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[f + g*x])])/(Sqrt[-a]*c*Sqrt[Sqrt[c]*d + S
qrt[-a]*e]*Sqrt[Sqrt[c]*f + Sqrt[-a]*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((e*x+d)**(1/2)*(g*x+f)**(1/2)/(c*x**2+a),x)

[Out]

Timed out

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Mathematica [C]  time = 3.72561, size = 524, normalized size = 1.53 \[ \frac{i \sqrt{\sqrt{c} d+i \sqrt{a} e} \sqrt{\sqrt{c} f+i \sqrt{a} g} \log \left (\frac{-a c (d g+e (f+2 g x))+i \sqrt{a} \left (c^{3/2} (2 d f+d g x+e f x)+2 c \sqrt{d+e x} \sqrt{f+g x} \sqrt{\sqrt{c} d+i \sqrt{a} e} \sqrt{\sqrt{c} f+i \sqrt{a} g}\right )}{\left (\sqrt{c} x-i \sqrt{a}\right ) \left (\sqrt{c} d+i \sqrt{a} e\right )^{3/2} \left (\sqrt{c} f+i \sqrt{a} g\right )^{3/2}}\right )-i \sqrt{\sqrt{c} d-i \sqrt{a} e} \sqrt{\sqrt{c} f-i \sqrt{a} g} \log \left (-\frac{\sqrt{a} c \left (2 i \sqrt{d+e x} \sqrt{f+g x} \sqrt{\sqrt{c} d-i \sqrt{a} e} \sqrt{\sqrt{c} f-i \sqrt{a} g}+\sqrt{a} (d g+e f+2 e g x)+i \sqrt{c} (2 d f+d g x+e f x)\right )}{\left (\sqrt{c} x+i \sqrt{a}\right ) \left (\sqrt{c} d-i \sqrt{a} e\right )^{3/2} \left (\sqrt{c} f-i \sqrt{a} g\right )^{3/2}}\right )+2 \sqrt{a} \sqrt{e} \sqrt{g} \log \left (2 \sqrt{e} \sqrt{g} \sqrt{d+e x} \sqrt{f+g x}+d g+e f+2 e g x\right )}{2 \sqrt{a} c} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[a]*Sqrt[e]*Sqrt[g]*Log[e*f + d*g + 2*e*g*x + 2*Sqrt[e]*Sqrt[g]*Sqrt[d +
e*x]*Sqrt[f + g*x]] - I*Sqrt[Sqrt[c]*d - I*Sqrt[a]*e]*Sqrt[Sqrt[c]*f - I*Sqrt[a]
*g]*Log[-((Sqrt[a]*c*((2*I)*Sqrt[Sqrt[c]*d - I*Sqrt[a]*e]*Sqrt[Sqrt[c]*f - I*Sqr
t[a]*g]*Sqrt[d + e*x]*Sqrt[f + g*x] + I*Sqrt[c]*(2*d*f + e*f*x + d*g*x) + Sqrt[a
]*(e*f + d*g + 2*e*g*x)))/((Sqrt[c]*d - I*Sqrt[a]*e)^(3/2)*(Sqrt[c]*f - I*Sqrt[a
]*g)^(3/2)*(I*Sqrt[a] + Sqrt[c]*x)))] + I*Sqrt[Sqrt[c]*d + I*Sqrt[a]*e]*Sqrt[Sqr
t[c]*f + I*Sqrt[a]*g]*Log[(-(a*c*(d*g + e*(f + 2*g*x))) + I*Sqrt[a]*(2*c*Sqrt[Sq
rt[c]*d + I*Sqrt[a]*e]*Sqrt[Sqrt[c]*f + I*Sqrt[a]*g]*Sqrt[d + e*x]*Sqrt[f + g*x]
+ c^(3/2)*(2*d*f + e*f*x + d*g*x)))/((Sqrt[c]*d + I*Sqrt[a]*e)^(3/2)*(Sqrt[c]*f
+ I*Sqrt[a]*g)^(3/2)*((-I)*Sqrt[a] + Sqrt[c]*x))])/(2*Sqrt[a]*c)

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Maple [B]  time = 0.036, size = 1569, normalized size = 4.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 3.64717, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x