3.90 \(\int \frac{A+B x+C x^2}{\sqrt{c+d x} \sqrt{e+f x}} \, dx\)

Optimal. Leaf size=164 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right ) \left (4 d f (2 A d f-B (c f+d e))+C \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )}{4 d^{5/2} f^{5/2}}-\frac{\sqrt{c+d x} \sqrt{e+f x} (-4 B d f+5 c C f+3 C d e)}{4 d^2 f^2}+\frac{C (c+d x)^{3/2} \sqrt{e+f x}}{2 d^2 f} \]

[Out]

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

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Rubi [A]  time = 0.385665, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right ) \left (4 d f (2 A d f-B (c f+d e))+C \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )}{4 d^{5/2} f^{5/2}}-\frac{\sqrt{c+d x} \sqrt{e+f x} (-4 B d f+5 c C f+3 C d e)}{4 d^2 f^2}+\frac{C (c+d x)^{3/2} \sqrt{e+f x}}{2 d^2 f} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 35.9556, size = 199, normalized size = 1.21 \[ \frac{B \sqrt{c + d x} \sqrt{e + f x}}{d f} + \frac{C x \sqrt{c + d x} \sqrt{e + f x}}{2 d f} - \frac{3 C \sqrt{c + d x} \sqrt{e + f x} \left (c f + d e\right )}{4 d^{2} f^{2}} - \frac{C \left (4 c d e f - 3 \left (c f + d e\right )^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{f} \sqrt{c + d x}} \right )}}{4 d^{\frac{5}{2}} f^{\frac{5}{2}}} - \frac{2 \left (- A d f + \frac{B \left (c f + d e\right )}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{f} \sqrt{c + d x}} \right )}}{d^{\frac{3}{2}} f^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.192523, size = 154, normalized size = 0.94 \[ \frac{\log \left (2 \sqrt{d} \sqrt{f} \sqrt{c+d x} \sqrt{e+f x}+c f+d e+2 d f x\right ) \left (4 d f (2 A d f-B (c f+d e))+C \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )}{8 d^{5/2} f^{5/2}}+\frac{\sqrt{c+d x} \sqrt{e+f x} (4 B d f+C (-3 c f-3 d e+2 d f x))}{4 d^2 f^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.032, size = 425, normalized size = 2.6 \[{\frac{1}{8\,{d}^{2}{f}^{2}} \left ( 8\,A\ln \left ( 1/2\,{\frac{2\,dfx+2\,\sqrt{ \left ( dx+c \right ) \left ( fx+e \right ) }\sqrt{fd}+cf+de}{\sqrt{fd}}} \right ){d}^{2}{f}^{2}-4\,B\ln \left ( 1/2\,{\frac{2\,dfx+2\,\sqrt{ \left ( dx+c \right ) \left ( fx+e \right ) }\sqrt{fd}+cf+de}{\sqrt{fd}}} \right ) cd{f}^{2}-4\,B\ln \left ( 1/2\,{\frac{2\,dfx+2\,\sqrt{ \left ( dx+c \right ) \left ( fx+e \right ) }\sqrt{fd}+cf+de}{\sqrt{fd}}} \right ){d}^{2}ef+4\,C\sqrt{ \left ( dx+c \right ) \left ( fx+e \right ) }xfd\sqrt{fd}+3\,C\ln \left ( 1/2\,{\frac{2\,dfx+2\,\sqrt{ \left ( dx+c \right ) \left ( fx+e \right ) }\sqrt{fd}+cf+de}{\sqrt{fd}}} \right ){c}^{2}{f}^{2}+2\,C\ln \left ( 1/2\,{\frac{2\,dfx+2\,\sqrt{ \left ( dx+c \right ) \left ( fx+e \right ) }\sqrt{fd}+cf+de}{\sqrt{fd}}} \right ) cefd+3\,C\ln \left ( 1/2\,{\frac{2\,dfx+2\,\sqrt{ \left ( dx+c \right ) \left ( fx+e \right ) }\sqrt{fd}+cf+de}{\sqrt{fd}}} \right ){d}^{2}{e}^{2}+8\,B\sqrt{ \left ( dx+c \right ) \left ( fx+e \right ) }\sqrt{fd}df-6\,C\sqrt{ \left ( dx+c \right ) \left ( fx+e \right ) }\sqrt{fd}cf-6\,C\sqrt{ \left ( dx+c \right ) \left ( fx+e \right ) }\sqrt{fd}de \right ) \sqrt{dx+c}\sqrt{fx+e}{\frac{1}{\sqrt{ \left ( dx+c \right ) \left ( fx+e \right ) }}}{\frac{1}{\sqrt{fd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.43053, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (2 \, C d f x - 3 \, C d e -{\left (3 \, C c - 4 \, B d\right )} f\right )} \sqrt{d f} \sqrt{d x + c} \sqrt{f x + e} +{\left (3 \, C d^{2} e^{2} + 2 \,{\left (C c d - 2 \, B d^{2}\right )} e f +{\left (3 \, C c^{2} - 4 \, B c d + 8 \, A d^{2}\right )} f^{2}\right )} \log \left (4 \,{\left (2 \, d^{2} f^{2} x + d^{2} e f + c d f^{2}\right )} \sqrt{d x + c} \sqrt{f x + e} +{\left (8 \, d^{2} f^{2} x^{2} + d^{2} e^{2} + 6 \, c d e f + c^{2} f^{2} + 8 \,{\left (d^{2} e f + c d f^{2}\right )} x\right )} \sqrt{d f}\right )}{16 \, \sqrt{d f} d^{2} f^{2}}, \frac{2 \,{\left (2 \, C d f x - 3 \, C d e -{\left (3 \, C c - 4 \, B d\right )} f\right )} \sqrt{-d f} \sqrt{d x + c} \sqrt{f x + e} +{\left (3 \, C d^{2} e^{2} + 2 \,{\left (C c d - 2 \, B d^{2}\right )} e f +{\left (3 \, C c^{2} - 4 \, B c d + 8 \, A d^{2}\right )} f^{2}\right )} \arctan \left (\frac{{\left (2 \, d f x + d e + c f\right )} \sqrt{-d f}}{2 \, \sqrt{d x + c} \sqrt{f x + e} d f}\right )}{8 \, \sqrt{-d f} d^{2} f^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(4*(2*C*d*f*x - 3*C*d*e - (3*C*c - 4*B*d)*f)*sqrt(d*f)*sqrt(d*x + c)*sqrt(
f*x + e) + (3*C*d^2*e^2 + 2*(C*c*d - 2*B*d^2)*e*f + (3*C*c^2 - 4*B*c*d + 8*A*d^2
)*f^2)*log(4*(2*d^2*f^2*x + d^2*e*f + c*d*f^2)*sqrt(d*x + c)*sqrt(f*x + e) + (8*
d^2*f^2*x^2 + d^2*e^2 + 6*c*d*e*f + c^2*f^2 + 8*(d^2*e*f + c*d*f^2)*x)*sqrt(d*f)
))/(sqrt(d*f)*d^2*f^2), 1/8*(2*(2*C*d*f*x - 3*C*d*e - (3*C*c - 4*B*d)*f)*sqrt(-d
*f)*sqrt(d*x + c)*sqrt(f*x + e) + (3*C*d^2*e^2 + 2*(C*c*d - 2*B*d^2)*e*f + (3*C*
c^2 - 4*B*c*d + 8*A*d^2)*f^2)*arctan(1/2*(2*d*f*x + d*e + c*f)*sqrt(-d*f)/(sqrt(
d*x + c)*sqrt(f*x + e)*d*f)))/(sqrt(-d*f)*d^2*f^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.301845, size = 262, normalized size = 1.6 \[ \frac{{\left (\sqrt{{\left (d x + c\right )} d f - c d f + d^{2} e} \sqrt{d x + c}{\left (\frac{2 \,{\left (d x + c\right )} C}{d^{3} f} - \frac{5 \, C c d^{5} f^{2} - 4 \, B d^{6} f^{2} + 3 \, C d^{6} f e}{d^{8} f^{3}}\right )} - \frac{{\left (3 \, C c^{2} f^{2} - 4 \, B c d f^{2} + 8 \, A d^{2} f^{2} + 2 \, C c d f e - 4 \, B d^{2} f e + 3 \, C d^{2} e^{2}\right )}{\rm ln}\left ({\left | -\sqrt{d f} \sqrt{d x + c} + \sqrt{{\left (d x + c\right )} d f - c d f + d^{2} e} \right |}\right )}{\sqrt{d f} d^{2} f^{2}}\right )} d}{4 \,{\left | d \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)/(sqrt(d*x + c)*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/(d^3*f) -
(5*C*c*d^5*f^2 - 4*B*d^6*f^2 + 3*C*d^6*f*e)/(d^8*f^3)) - (3*C*c^2*f^2 - 4*B*c*d*
f^2 + 8*A*d^2*f^2 + 2*C*c*d*f*e - 4*B*d^2*f*e + 3*C*d^2*e^2)*ln(abs(-sqrt(d*f)*s
qrt(d*x + c) + sqrt((d*x + c)*d*f - c*d*f + d^2*e)))/(sqrt(d*f)*d^2*f^2))*d/abs(
d)