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

Optimal. Leaf size=246 \[ \frac{\sqrt{d+e x} \sqrt{f+g x} \left (2 e g (4 a e g-b (d g+3 e f))+c \left (d^2 g^2+2 d e f g+5 e^2 f^2\right )\right )}{8 e^2 g^3}-\frac{(e f-d g) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right ) \left (2 e g (4 a e g-b (d g+3 e f))+c \left (d^2 g^2+2 d e f g+5 e^2 f^2\right )\right )}{8 e^{5/2} g^{7/2}}-\frac{(d+e x)^{3/2} \sqrt{f+g x} (-6 b e g+7 c d g+5 c e f)}{12 e^2 g^2}+\frac{c (d+e x)^{5/2} \sqrt{f+g x}}{3 e^2 g} \]

[Out]

((c*(5*e^2*f^2 + 2*d*e*f*g + d^2*g^2) + 2*e*g*(4*a*e*g - b*(3*e*f + d*g)))*Sqrt[
d + e*x]*Sqrt[f + g*x])/(8*e^2*g^3) - ((5*c*e*f + 7*c*d*g - 6*b*e*g)*(d + e*x)^(
3/2)*Sqrt[f + g*x])/(12*e^2*g^2) + (c*(d + e*x)^(5/2)*Sqrt[f + g*x])/(3*e^2*g) -
((e*f - d*g)*(c*(5*e^2*f^2 + 2*d*e*f*g + d^2*g^2) + 2*e*g*(4*a*e*g - b*(3*e*f +
d*g)))*ArcTanh[(Sqrt[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g*x])])/(8*e^(5/2)*g^(
7/2))

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Rubi [A]  time = 0.584502, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ \frac{\sqrt{d+e x} \sqrt{f+g x} \left (2 e g (4 a e g-b (d g+3 e f))+c \left (d^2 g^2+2 d e f g+5 e^2 f^2\right )\right )}{8 e^2 g^3}-\frac{(e f-d g) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right ) \left (2 e g (4 a e g-b (d g+3 e f))+c \left (d^2 g^2+2 d e f g+5 e^2 f^2\right )\right )}{8 e^{5/2} g^{7/2}}-\frac{(d+e x)^{3/2} \sqrt{f+g x} (-6 b e g+7 c d g+5 c e f)}{12 e^2 g^2}+\frac{c (d+e x)^{5/2} \sqrt{f+g x}}{3 e^2 g} \]

Antiderivative was successfully verified.

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

[Out]

((c*(5*e^2*f^2 + 2*d*e*f*g + d^2*g^2) + 2*e*g*(4*a*e*g - b*(3*e*f + d*g)))*Sqrt[
d + e*x]*Sqrt[f + g*x])/(8*e^2*g^3) - ((5*c*e*f + 7*c*d*g - 6*b*e*g)*(d + e*x)^(
3/2)*Sqrt[f + g*x])/(12*e^2*g^2) + (c*(d + e*x)^(5/2)*Sqrt[f + g*x])/(3*e^2*g) -
((e*f - d*g)*(c*(5*e^2*f^2 + 2*d*e*f*g + d^2*g^2) + 2*e*g*(4*a*e*g - b*(3*e*f +
d*g)))*ArcTanh[(Sqrt[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g*x])])/(8*e^(5/2)*g^(
7/2))

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Rubi in Sympy [A]  time = 51.9142, size = 318, normalized size = 1.29 \[ \frac{b \left (d + e x\right )^{\frac{3}{2}} \sqrt{f + g x}}{2 e g} + \frac{c x \left (d + e x\right )^{\frac{3}{2}} \sqrt{f + g x}}{3 e g} - \frac{c \left (d + e x\right )^{\frac{3}{2}} \sqrt{f + g x} \left (3 d g + 5 e f\right )}{12 e^{2} g^{2}} + \frac{c \sqrt{d + e x} \sqrt{f + g x} \left (d^{2} g^{2} + 2 d e f g + 5 e^{2} f^{2}\right )}{8 e^{2} g^{3}} + \frac{c \left (d g - e f\right ) \left (d^{2} g^{2} + 2 d e f g + 5 e^{2} f^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{f + g x}}{\sqrt{g} \sqrt{d + e x}} \right )}}{8 e^{\frac{5}{2}} g^{\frac{7}{2}}} + \frac{\sqrt{d + e x} \sqrt{f + g x} \left (4 a e g - b d g - 3 b e f\right )}{4 e g^{2}} + \frac{\left (d g - e f\right ) \left (4 a e g - b d g - 3 b e f\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{f + g x}}{\sqrt{g} \sqrt{d + e x}} \right )}}{4 e^{\frac{3}{2}} g^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*(d + e*x)**(3/2)*sqrt(f + g*x)/(2*e*g) + c*x*(d + e*x)**(3/2)*sqrt(f + g*x)/(3
*e*g) - c*(d + e*x)**(3/2)*sqrt(f + g*x)*(3*d*g + 5*e*f)/(12*e**2*g**2) + c*sqrt
(d + e*x)*sqrt(f + g*x)*(d**2*g**2 + 2*d*e*f*g + 5*e**2*f**2)/(8*e**2*g**3) + c*
(d*g - e*f)*(d**2*g**2 + 2*d*e*f*g + 5*e**2*f**2)*atanh(sqrt(e)*sqrt(f + g*x)/(s
qrt(g)*sqrt(d + e*x)))/(8*e**(5/2)*g**(7/2)) + sqrt(d + e*x)*sqrt(f + g*x)*(4*a*
e*g - b*d*g - 3*b*e*f)/(4*e*g**2) + (d*g - e*f)*(4*a*e*g - b*d*g - 3*b*e*f)*atan
h(sqrt(e)*sqrt(f + g*x)/(sqrt(g)*sqrt(d + e*x)))/(4*e**(3/2)*g**(5/2))

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Mathematica [A]  time = 0.297278, size = 212, normalized size = 0.86 \[ \frac{\sqrt{d+e x} \sqrt{f+g x} \left (6 e g (4 a e g+b (d g-3 e f+2 e g x))+c \left (-3 d^2 g^2+2 d e g (g x-2 f)+e^2 \left (15 f^2-10 f g x+8 g^2 x^2\right )\right )\right )}{24 e^2 g^3}-\frac{(e f-d g) \log \left (2 \sqrt{e} \sqrt{g} \sqrt{d+e x} \sqrt{f+g x}+d g+e f+2 e g x\right ) \left (2 e g (4 a e g-b (d g+3 e f))+c \left (d^2 g^2+2 d e f g+5 e^2 f^2\right )\right )}{16 e^{5/2} g^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d + e*x]*Sqrt[f + g*x]*(6*e*g*(4*a*e*g + b*(-3*e*f + d*g + 2*e*g*x)) + c*(
-3*d^2*g^2 + 2*d*e*g*(-2*f + g*x) + e^2*(15*f^2 - 10*f*g*x + 8*g^2*x^2))))/(24*e
^2*g^3) - ((e*f - d*g)*(c*(5*e^2*f^2 + 2*d*e*f*g + d^2*g^2) + 2*e*g*(4*a*e*g - b
*(3*e*f + d*g)))*Log[e*f + d*g + 2*e*g*x + 2*Sqrt[e]*Sqrt[g]*Sqrt[d + e*x]*Sqrt[
f + g*x]])/(16*e^(5/2)*g^(7/2))

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Maple [B]  time = 0.03, size = 763, normalized size = 3.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(e*x+d)^(1/2)*(g*x+f)^(1/2)*(16*x^2*c*e^2*g^2*((e*x+d)*(g*x+f))^(1/2)*(e*g)
^(1/2)+24*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(
1/2))*d*a*g^3*e^2-24*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+d*g+e
*f)/(e*g)^(1/2))*e^3*f*a*g^2-6*b*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)
^(1/2)+d*g+e*f)/(e*g)^(1/2))*d^2*g^3*e-12*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1
/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*d*b*f*g^2*e^2+18*ln(1/2*(2*e*g*x+2*((e*x+d
)*(g*x+f))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*e^3*f^2*b*g+3*c*ln(1/2*(2*e*g
*x+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*d^3*g^3+3*c*ln(1/
2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*d^2*f*g^2
*e+9*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))
*d*c*f^2*g*e^2-15*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+d*g+e*f)
/(e*g)^(1/2))*e^3*f^3*c+24*b*((e*x+d)*(g*x+f))^(1/2)*x*g^2*e^2*(e*g)^(1/2)+4*c*(
(e*x+d)*(g*x+f))^(1/2)*x*d*g^2*e*(e*g)^(1/2)-20*c*((e*x+d)*(g*x+f))^(1/2)*x*f*g*
e^2*(e*g)^(1/2)+48*((e*x+d)*(g*x+f))^(1/2)*a*g^2*e^2*(e*g)^(1/2)+12*b*((e*x+d)*(
g*x+f))^(1/2)*d*g^2*e*(e*g)^(1/2)-36*((e*x+d)*(g*x+f))^(1/2)*b*f*g*e^2*(e*g)^(1/
2)-6*c*((e*x+d)*(g*x+f))^(1/2)*d^2*g^2*(e*g)^(1/2)-8*c*((e*x+d)*(g*x+f))^(1/2)*d
*f*g*e*(e*g)^(1/2)+30*((e*x+d)*(g*x+f))^(1/2)*c*f^2*e^2*(e*g)^(1/2))/((e*x+d)*(g
*x+f))^(1/2)/g^3/e^2/(e*g)^(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((c*x^2 + b*x + a)*sqrt(e*x + d)/sqrt(g*x + f),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.37137, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (8 \, c e^{2} g^{2} x^{2} + 15 \, c e^{2} f^{2} - 2 \,{\left (2 \, c d e + 9 \, b e^{2}\right )} f g - 3 \,{\left (c d^{2} - 2 \, b d e - 8 \, a e^{2}\right )} g^{2} - 2 \,{\left (5 \, c e^{2} f g -{\left (c d e + 6 \, b e^{2}\right )} g^{2}\right )} x\right )} \sqrt{e g} \sqrt{e x + d} \sqrt{g x + f} - 3 \,{\left (5 \, c e^{3} f^{3} - 3 \,{\left (c d e^{2} + 2 \, b e^{3}\right )} f^{2} g -{\left (c d^{2} e - 4 \, b d e^{2} - 8 \, a e^{3}\right )} f g^{2} -{\left (c d^{3} - 2 \, b d^{2} e + 8 \, a d e^{2}\right )} g^{3}\right )} \log \left (4 \,{\left (2 \, e^{2} g^{2} x + e^{2} f g + d e g^{2}\right )} \sqrt{e x + d} \sqrt{g x + f} +{\left (8 \, e^{2} g^{2} x^{2} + e^{2} f^{2} + 6 \, d e f g + d^{2} g^{2} + 8 \,{\left (e^{2} f g + d e g^{2}\right )} x\right )} \sqrt{e g}\right )}{96 \, \sqrt{e g} e^{2} g^{3}}, \frac{2 \,{\left (8 \, c e^{2} g^{2} x^{2} + 15 \, c e^{2} f^{2} - 2 \,{\left (2 \, c d e + 9 \, b e^{2}\right )} f g - 3 \,{\left (c d^{2} - 2 \, b d e - 8 \, a e^{2}\right )} g^{2} - 2 \,{\left (5 \, c e^{2} f g -{\left (c d e + 6 \, b e^{2}\right )} g^{2}\right )} x\right )} \sqrt{-e g} \sqrt{e x + d} \sqrt{g x + f} - 3 \,{\left (5 \, c e^{3} f^{3} - 3 \,{\left (c d e^{2} + 2 \, b e^{3}\right )} f^{2} g -{\left (c d^{2} e - 4 \, b d e^{2} - 8 \, a e^{3}\right )} f g^{2} -{\left (c d^{3} - 2 \, b d^{2} e + 8 \, a d e^{2}\right )} g^{3}\right )} \arctan \left (\frac{{\left (2 \, e g x + e f + d g\right )} \sqrt{-e g}}{2 \, \sqrt{e x + d} \sqrt{g x + f} e g}\right )}{48 \, \sqrt{-e g} e^{2} g^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(4*(8*c*e^2*g^2*x^2 + 15*c*e^2*f^2 - 2*(2*c*d*e + 9*b*e^2)*f*g - 3*(c*d^2
- 2*b*d*e - 8*a*e^2)*g^2 - 2*(5*c*e^2*f*g - (c*d*e + 6*b*e^2)*g^2)*x)*sqrt(e*g)*
sqrt(e*x + d)*sqrt(g*x + f) - 3*(5*c*e^3*f^3 - 3*(c*d*e^2 + 2*b*e^3)*f^2*g - (c*
d^2*e - 4*b*d*e^2 - 8*a*e^3)*f*g^2 - (c*d^3 - 2*b*d^2*e + 8*a*d*e^2)*g^3)*log(4*
(2*e^2*g^2*x + e^2*f*g + d*e*g^2)*sqrt(e*x + d)*sqrt(g*x + f) + (8*e^2*g^2*x^2 +
e^2*f^2 + 6*d*e*f*g + d^2*g^2 + 8*(e^2*f*g + d*e*g^2)*x)*sqrt(e*g)))/(sqrt(e*g)
*e^2*g^3), 1/48*(2*(8*c*e^2*g^2*x^2 + 15*c*e^2*f^2 - 2*(2*c*d*e + 9*b*e^2)*f*g -
3*(c*d^2 - 2*b*d*e - 8*a*e^2)*g^2 - 2*(5*c*e^2*f*g - (c*d*e + 6*b*e^2)*g^2)*x)*
sqrt(-e*g)*sqrt(e*x + d)*sqrt(g*x + f) - 3*(5*c*e^3*f^3 - 3*(c*d*e^2 + 2*b*e^3)*
f^2*g - (c*d^2*e - 4*b*d*e^2 - 8*a*e^3)*f*g^2 - (c*d^3 - 2*b*d^2*e + 8*a*d*e^2)*
g^3)*arctan(1/2*(2*e*g*x + e*f + d*g)*sqrt(-e*g)/(sqrt(e*x + d)*sqrt(g*x + f)*e*
g)))/(sqrt(-e*g)*e^2*g^3)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.290715, size = 393, normalized size = 1.6 \[ \frac{1}{24} \, \sqrt{{\left (x e + d\right )} g e - d g e + f e^{2}}{\left (2 \,{\left (x e + d\right )}{\left (\frac{4 \,{\left (x e + d\right )} c e^{\left (-3\right )}}{g} - \frac{{\left (7 \, c d g^{4} e^{6} + 5 \, c f g^{3} e^{7} - 6 \, b g^{4} e^{7}\right )} e^{\left (-9\right )}}{g^{5}}\right )} + \frac{3 \,{\left (c d^{2} g^{4} e^{6} + 2 \, c d f g^{3} e^{7} - 2 \, b d g^{4} e^{7} + 5 \, c f^{2} g^{2} e^{8} - 6 \, b f g^{3} e^{8} + 8 \, a g^{4} e^{8}\right )} e^{\left (-9\right )}}{g^{5}}\right )} \sqrt{x e + d} - \frac{{\left (c d^{3} g^{3} + c d^{2} f g^{2} e - 2 \, b d^{2} g^{3} e + 3 \, c d f^{2} g e^{2} - 4 \, b d f g^{2} e^{2} + 8 \, a d g^{3} e^{2} - 5 \, c f^{3} e^{3} + 6 \, b f^{2} g e^{3} - 8 \, a f g^{2} e^{3}\right )} e^{\left (-\frac{5}{2}\right )}{\rm ln}\left ({\left | -\sqrt{x e + d} \sqrt{g} e^{\frac{1}{2}} + \sqrt{{\left (x e + d\right )} g e - d g e + f e^{2}} \right |}\right )}{8 \, g^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*sqrt((x*e + d)*g*e - d*g*e + f*e^2)*(2*(x*e + d)*(4*(x*e + d)*c*e^(-3)/g -
(7*c*d*g^4*e^6 + 5*c*f*g^3*e^7 - 6*b*g^4*e^7)*e^(-9)/g^5) + 3*(c*d^2*g^4*e^6 + 2
*c*d*f*g^3*e^7 - 2*b*d*g^4*e^7 + 5*c*f^2*g^2*e^8 - 6*b*f*g^3*e^8 + 8*a*g^4*e^8)*
e^(-9)/g^5)*sqrt(x*e + d) - 1/8*(c*d^3*g^3 + c*d^2*f*g^2*e - 2*b*d^2*g^3*e + 3*c
*d*f^2*g*e^2 - 4*b*d*f*g^2*e^2 + 8*a*d*g^3*e^2 - 5*c*f^3*e^3 + 6*b*f^2*g*e^3 - 8
*a*f*g^2*e^3)*e^(-5/2)*ln(abs(-sqrt(x*e + d)*sqrt(g)*e^(1/2) + sqrt((x*e + d)*g*
e - d*g*e + f*e^2)))/g^(7/2)