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

Optimal. Leaf size=290 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right ) (2 b d f (4 A b d f-a C (c f+3 d e))+(2 a d f-b c f+b d e) (4 a C d f+b (-4 B d f+c C f+3 C d e)))}{4 b^3 d^{3/2} f^{5/2}}-\frac{2 \sqrt{b c-a d} \left (A b^2-a (b B-a C)\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right )}{b^3 \sqrt{b e-a f}}-\frac{\sqrt{c+d x} \sqrt{e+f x} (4 a C d f+b (-4 B d f+c C f+3 C d e))}{4 b^2 d f^2}+\frac{C (c+d x)^{3/2} \sqrt{e+f x}}{2 b d f} \]

[Out]

-((4*a*C*d*f + b*(3*C*d*e + c*C*f - 4*B*d*f))*Sqrt[c + d*x]*Sqrt[e + f*x])/(4*b^
2*d*f^2) + (C*(c + d*x)^(3/2)*Sqrt[e + f*x])/(2*b*d*f) + ((2*b*d*f*(4*A*b*d*f -
a*C*(3*d*e + c*f)) + (b*d*e - b*c*f + 2*a*d*f)*(4*a*C*d*f + b*(3*C*d*e + c*C*f -
4*B*d*f)))*ArcTanh[(Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[e + f*x])])/(4*b^3*d^(
3/2)*f^(5/2)) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b*e - a
*f]*Sqrt[c + d*x])/(Sqrt[b*c - a*d]*Sqrt[e + f*x])])/(b^3*Sqrt[b*e - a*f])

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Rubi [A]  time = 1.64104, antiderivative size = 290, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right ) (2 b d f (4 A b d f-a C (c f+3 d e))+(2 a d f-b c f+b d e) (4 a C d f+b (-4 B d f+c C f+3 C d e)))}{4 b^3 d^{3/2} f^{5/2}}-\frac{2 \sqrt{b c-a d} \left (A b^2-a (b B-a C)\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right )}{b^3 \sqrt{b e-a f}}-\frac{\sqrt{c+d x} \sqrt{e+f x} (4 a C d f+b (-4 B d f+c C f+3 C d e))}{4 b^2 d f^2}+\frac{C (c+d x)^{3/2} \sqrt{e+f x}}{2 b d f} \]

Antiderivative was successfully verified.

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

[Out]

-((4*a*C*d*f + b*(3*C*d*e + c*C*f - 4*B*d*f))*Sqrt[c + d*x]*Sqrt[e + f*x])/(4*b^
2*d*f^2) + (C*(c + d*x)^(3/2)*Sqrt[e + f*x])/(2*b*d*f) + ((2*b*d*f*(4*A*b*d*f -
a*C*(3*d*e + c*f)) + (b*d*e - b*c*f + 2*a*d*f)*(4*a*C*d*f + b*(3*C*d*e + c*C*f -
4*B*d*f)))*ArcTanh[(Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[e + f*x])])/(4*b^3*d^(
3/2)*f^(5/2)) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b*e - a
*f]*Sqrt[c + d*x])/(Sqrt[b*c - a*d]*Sqrt[e + f*x])])/(b^3*Sqrt[b*e - a*f])

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Rubi in Sympy [A]  time = 106.287, size = 354, normalized size = 1.22 \[ \frac{C \left (c + d x\right )^{\frac{3}{2}} \sqrt{e + f x}}{2 b d f} - \frac{C \sqrt{c + d x} \sqrt{e + f x} \left (c f - d e\right )}{4 b d f^{2}} - \frac{C \left (c f - d e\right )^{2} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{f} \sqrt{c + d x}} \right )}}{4 b d^{\frac{3}{2}} f^{\frac{5}{2}}} + \frac{\sqrt{c + d x} \sqrt{e + f x} \left (B b f - C a f - C b e\right )}{b^{2} f^{2}} + \frac{\left (c f - d e\right ) \left (B b f - C a f - C b e\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{f} \sqrt{c + d x}} \right )}}{b^{2} \sqrt{d} f^{\frac{5}{2}}} + \frac{2 \sqrt{d} \left (A b^{2} - B a b + C a^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{f} \sqrt{c + d x}} \right )}}{b^{3} \sqrt{f}} - \frac{2 \sqrt{a d - b c} \left (A b^{2} - B a b + C a^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x} \sqrt{a f - b e}}{\sqrt{e + f x} \sqrt{a d - b c}} \right )}}{b^{3} \sqrt{a f - b e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

C*(c + d*x)**(3/2)*sqrt(e + f*x)/(2*b*d*f) - C*sqrt(c + d*x)*sqrt(e + f*x)*(c*f
- d*e)/(4*b*d*f**2) - C*(c*f - d*e)**2*atanh(sqrt(d)*sqrt(e + f*x)/(sqrt(f)*sqrt
(c + d*x)))/(4*b*d**(3/2)*f**(5/2)) + sqrt(c + d*x)*sqrt(e + f*x)*(B*b*f - C*a*f
- C*b*e)/(b**2*f**2) + (c*f - d*e)*(B*b*f - C*a*f - C*b*e)*atanh(sqrt(d)*sqrt(e
+ f*x)/(sqrt(f)*sqrt(c + d*x)))/(b**2*sqrt(d)*f**(5/2)) + 2*sqrt(d)*(A*b**2 - B
*a*b + C*a**2)*atanh(sqrt(d)*sqrt(e + f*x)/(sqrt(f)*sqrt(c + d*x)))/(b**3*sqrt(f
)) - 2*sqrt(a*d - b*c)*(A*b**2 - B*a*b + C*a**2)*atanh(sqrt(c + d*x)*sqrt(a*f -
b*e)/(sqrt(e + f*x)*sqrt(a*d - b*c)))/(b**3*sqrt(a*f - b*e))

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Mathematica [A]  time = 0.764149, size = 367, normalized size = 1.27 \[ \frac{\frac{\log \left (2 \sqrt{d} \sqrt{f} \sqrt{c+d x} \sqrt{e+f x}+c f+d e+2 d f x\right ) \left (8 a^2 C d^2 f^2-4 a b d f (2 B d f+c C f-C d e)+b^2 \left (4 d f (2 A d f+B c f-B d e)+C \left (-c^2 f^2-2 c d e f+3 d^2 e^2\right )\right )\right )}{d^{3/2} f^{5/2}}+\frac{8 \sqrt{b c-a d} \log (a+b x) \left (a (a C-b B)+A b^2\right )}{\sqrt{b e-a f}}-\frac{8 \sqrt{b c-a d} \left (a (a C-b B)+A b^2\right ) \log \left (2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{b c-a d} \sqrt{b e-a f}-a (c f+d e+2 d f x)+b (2 c e+c f x+d e x)\right )}{\sqrt{b e-a f}}+\frac{2 b \sqrt{c+d x} \sqrt{e+f x} (-4 a C d f+4 b B d f+b C (c f-3 d e+2 d f x))}{d f^2}}{8 b^3} \]

Antiderivative was successfully verified.

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

[Out]

((2*b*Sqrt[c + d*x]*Sqrt[e + f*x]*(4*b*B*d*f - 4*a*C*d*f + b*C*(-3*d*e + c*f + 2
*d*f*x)))/(d*f^2) + (8*(A*b^2 + a*(-(b*B) + a*C))*Sqrt[b*c - a*d]*Log[a + b*x])/
Sqrt[b*e - a*f] + ((8*a^2*C*d^2*f^2 - 4*a*b*d*f*(-(C*d*e) + c*C*f + 2*B*d*f) + b
^2*(4*d*f*(-(B*d*e) + B*c*f + 2*A*d*f) + C*(3*d^2*e^2 - 2*c*d*e*f - c^2*f^2)))*L
og[d*e + c*f + 2*d*f*x + 2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*Sqrt[e + f*x]])/(d^(3/2
)*f^(5/2)) - (8*(A*b^2 + a*(-(b*B) + a*C))*Sqrt[b*c - a*d]*Log[2*Sqrt[b*c - a*d]
*Sqrt[b*e - a*f]*Sqrt[c + d*x]*Sqrt[e + f*x] + b*(2*c*e + d*e*x + c*f*x) - a*(d*
e + c*f + 2*d*f*x)])/Sqrt[b*e - a*f])/(8*b^3)

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Maple [B]  time = 0.046, size = 1822, normalized size = 6.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(8*A*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((d*x+c)*(f*x+e))^(1/2)*((a^2*d*f-a*b*
c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b-a*c*f-a*e*d+2*b*c*e)/(b*x+a))*a*b^2*d^2*f^2*(f
*d)^(1/2)-8*A*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((d*x+c)*(f*x+e))^(1/2)*((a^2*d*f
-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b-a*c*f-a*e*d+2*b*c*e)/(b*x+a))*b^3*c*d*f^2
*(f*d)^(1/2)+8*A*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)+c*f+d*e)/
(f*d)^(1/2))*b^3*d^2*f^2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)-8*B*ln((-
2*a*d*f*x+b*c*f*x+b*d*e*x+2*((d*x+c)*(f*x+e))^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^
2*c*e)/b^2)^(1/2)*b-a*c*f-a*e*d+2*b*c*e)/(b*x+a))*a^2*b*d^2*f^2*(f*d)^(1/2)+8*B*
ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((d*x+c)*(f*x+e))^(1/2)*((a^2*d*f-a*b*c*f-a*b*d
*e+b^2*c*e)/b^2)^(1/2)*b-a*c*f-a*e*d+2*b*c*e)/(b*x+a))*a*b^2*c*d*f^2*(f*d)^(1/2)
-8*B*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))
*a*b^2*d^2*f^2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)+4*B*ln(1/2*(2*d*f*x
+2*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*b^3*c*d*f^2*((a^2*d
*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)-4*B*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(
1/2)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*b^3*d^2*e*f*((a^2*d*f-a*b*c*f-a*b*d*e+b^2
*c*e)/b^2)^(1/2)+8*C*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((d*x+c)*(f*x+e))^(1/2)*((
a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b-a*c*f-a*e*d+2*b*c*e)/(b*x+a))*a^3*
d^2*f^2*(f*d)^(1/2)-8*C*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((d*x+c)*(f*x+e))^(1/2)
*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b-a*c*f-a*e*d+2*b*c*e)/(b*x+a))*a
^2*b*c*d*f^2*(f*d)^(1/2)+8*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/
2)+c*f+d*e)/(f*d)^(1/2))*a^2*b*d^2*f^2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(
1/2)-4*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1
/2))*a*b^2*c*d*f^2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)+4*C*ln(1/2*(2*d
*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*a*b^2*d^2*e*f*(
(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)-C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e
))^(1/2)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*b^3*c^2*f^2*((a^2*d*f-a*b*c*f-a*b*d*e
+b^2*c*e)/b^2)^(1/2)-2*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)+c
*f+d*e)/(f*d)^(1/2))*b^3*c*d*e*f*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)+3
*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*b
^3*d^2*e^2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)+4*C*x*b^3*d*f*((a^2*d*f
-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)+8*B*b^3
*d*f*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*(f*d)
^(1/2)-8*C*a*b^2*d*f*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f*x
+e))^(1/2)*(f*d)^(1/2)+2*C*b^3*c*f*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)
*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)-6*C*b^3*d*e*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c
*e)/b^2)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2))*(f*x+e)^(1/2)*(d*x+c)^(1/2)/
d/f^2/(f*d)^(1/2)/((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)/b^4/((d*x+c)*(f*
x+e))^(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(d*x + c)/((b*x + a)*sqrt(f*x + e)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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((C*x^2 + B*x + A)*sqrt(d*x + c)/((b*x + a)*sqrt(f*x + e)),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.363483, size = 797, normalized size = 2.75 \[ \frac{1}{4} \, \sqrt{{\left (d x + c\right )} d f - c d f + d^{2} e} \sqrt{d x + c}{\left (\frac{2 \,{\left (d x + c\right )} C}{b d f{\left | d \right |}} - \frac{C b^{5} c d^{3} f^{2} + 4 \, C a b^{4} d^{4} f^{2} - 4 \, B b^{5} d^{4} f^{2} + 3 \, C b^{5} d^{4} f e}{b^{6} d^{4} f^{3}{\left | d \right |}}\right )} - \frac{2 \,{\left (\sqrt{d f} C a^{2} b c d^{2} - \sqrt{d f} B a b^{2} c d^{2} + \sqrt{d f} A b^{3} c d^{2} - \sqrt{d f} C a^{3} d^{3} + \sqrt{d f} B a^{2} b d^{3} - \sqrt{d f} A a b^{2} d^{3}\right )} \arctan \left (-\frac{b c d f - 2 \, a d^{2} f + b d^{2} e -{\left (\sqrt{d f} \sqrt{d x + c} - \sqrt{{\left (d x + c\right )} d f - c d f + d^{2} e}\right )}^{2} b}{2 \, \sqrt{a b c d f^{2} - a^{2} d^{2} f^{2} - b^{2} c d f e + a b d^{2} f e} d}\right )}{\sqrt{a b c d f^{2} - a^{2} d^{2} f^{2} - b^{2} c d f e + a b d^{2} f e} b^{3} d{\left | d \right |}} + \frac{{\left (\sqrt{d f} C b^{2} c^{2} f^{2} + 4 \, \sqrt{d f} C a b c d f^{2} - 4 \, \sqrt{d f} B b^{2} c d f^{2} - 8 \, \sqrt{d f} C a^{2} d^{2} f^{2} + 8 \, \sqrt{d f} B a b d^{2} f^{2} - 8 \, \sqrt{d f} A b^{2} d^{2} f^{2} + 2 \, \sqrt{d f} C b^{2} c d f e - 4 \, \sqrt{d f} C a b d^{2} f e + 4 \, \sqrt{d f} B b^{2} d^{2} f e - 3 \, \sqrt{d f} C b^{2} d^{2} e^{2}\right )}{\rm ln}\left ({\left (\sqrt{d f} \sqrt{d x + c} - \sqrt{{\left (d x + c\right )} d f - c d f + d^{2} e}\right )}^{2}\right )}{8 \, b^{3} d f^{3}{\left | d \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt((d*x + c)*d*f - c*d*f + d^2*e)*sqrt(d*x + c)*(2*(d*x + c)*C/(b*d*f*abs(
d)) - (C*b^5*c*d^3*f^2 + 4*C*a*b^4*d^4*f^2 - 4*B*b^5*d^4*f^2 + 3*C*b^5*d^4*f*e)/
(b^6*d^4*f^3*abs(d))) - 2*(sqrt(d*f)*C*a^2*b*c*d^2 - sqrt(d*f)*B*a*b^2*c*d^2 + s
qrt(d*f)*A*b^3*c*d^2 - sqrt(d*f)*C*a^3*d^3 + sqrt(d*f)*B*a^2*b*d^3 - sqrt(d*f)*A
*a*b^2*d^3)*arctan(-1/2*(b*c*d*f - 2*a*d^2*f + b*d^2*e - (sqrt(d*f)*sqrt(d*x + c
) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*b)/(sqrt(a*b*c*d*f^2 - a^2*d^2*f^2 -
b^2*c*d*f*e + a*b*d^2*f*e)*d))/(sqrt(a*b*c*d*f^2 - a^2*d^2*f^2 - b^2*c*d*f*e + a
*b*d^2*f*e)*b^3*d*abs(d)) + 1/8*(sqrt(d*f)*C*b^2*c^2*f^2 + 4*sqrt(d*f)*C*a*b*c*d
*f^2 - 4*sqrt(d*f)*B*b^2*c*d*f^2 - 8*sqrt(d*f)*C*a^2*d^2*f^2 + 8*sqrt(d*f)*B*a*b
*d^2*f^2 - 8*sqrt(d*f)*A*b^2*d^2*f^2 + 2*sqrt(d*f)*C*b^2*c*d*f*e - 4*sqrt(d*f)*C
*a*b*d^2*f*e + 4*sqrt(d*f)*B*b^2*d^2*f*e - 3*sqrt(d*f)*C*b^2*d^2*e^2)*ln((sqrt(d
*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2)/(b^3*d*f^3*abs(d))