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

Optimal. Leaf size=246 \[ -\frac{\left (a^2-b^2 x^2\right ) \left (2 \left (2 a^2 C f^2-b^2 \left (C e^2-3 f (A f+B e)\right )\right )-b^2 f x (C e-3 B f)\right )}{6 b^4 f \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{\sqrt{a^2 c-b^2 c x^2} \tan ^{-1}\left (\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right ) \left (a^2 (B f+C e)+2 A b^2 e\right )}{2 b^3 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{C \left (a^2-b^2 x^2\right ) (e+f x)^2}{3 b^2 f \sqrt{a+b x} \sqrt{a c-b c x}} \]

[Out]

-(C*(e + f*x)^2*(a^2 - b^2*x^2))/(3*b^2*f*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) - ((2
*(2*a^2*C*f^2 - b^2*(C*e^2 - 3*f*(B*e + A*f))) - b^2*f*(C*e - 3*B*f)*x)*(a^2 - b
^2*x^2))/(6*b^4*f*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) + ((2*A*b^2*e + a^2*(C*e + B*
f))*Sqrt[a^2*c - b^2*c*x^2]*ArcTan[(b*Sqrt[c]*x)/Sqrt[a^2*c - b^2*c*x^2]])/(2*b^
3*Sqrt[c]*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])

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Rubi [A]  time = 0.817934, antiderivative size = 249, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 5, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.132 \[ -\frac{\left (a^2-b^2 x^2\right ) \left (2 \left (2 a^2 C f^2-\frac{1}{2} b^2 \left (2 C e^2-6 f (A f+B e)\right )\right )-b^2 f x (C e-3 B f)\right )}{6 b^4 f \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{\sqrt{a^2 c-b^2 c x^2} \tan ^{-1}\left (\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right ) \left (a^2 (B f+C e)+2 A b^2 e\right )}{2 b^3 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{C \left (a^2-b^2 x^2\right ) (e+f x)^2}{3 b^2 f \sqrt{a+b x} \sqrt{a c-b c x}} \]

Antiderivative was successfully verified.

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

[Out]

-(C*(e + f*x)^2*(a^2 - b^2*x^2))/(3*b^2*f*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) - ((2
*(2*a^2*C*f^2 - (b^2*(2*C*e^2 - 6*f*(B*e + A*f)))/2) - b^2*f*(C*e - 3*B*f)*x)*(a
^2 - b^2*x^2))/(6*b^4*f*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) + ((2*A*b^2*e + a^2*(C*
e + B*f))*Sqrt[a^2*c - b^2*c*x^2]*ArcTan[(b*Sqrt[c]*x)/Sqrt[a^2*c - b^2*c*x^2]])
/(2*b^3*Sqrt[c]*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])

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Rubi in Sympy [A]  time = 85.4478, size = 211, normalized size = 0.86 \[ - \frac{C \sqrt{a + b x} \left (e + f x\right )^{2} \sqrt{a c - b c x}}{3 b^{2} c f} + \frac{\sqrt{a + b x} \sqrt{a c - b c x} \left (2 A b^{2} e + B a^{2} f + C a^{2} e\right ) \operatorname{atan}{\left (\frac{b \sqrt{c} x}{\sqrt{a^{2} c - b^{2} c x^{2}}} \right )}}{2 b^{3} \sqrt{c} \sqrt{a^{2} c - b^{2} c x^{2}}} - \frac{\sqrt{a + b x} \sqrt{a c - b c x} \left (2 b^{2} e \left (3 B f - C e\right ) + b^{2} f x \left (3 B f - C e\right ) + 2 f^{2} \left (3 A b^{2} + 2 C a^{2}\right )\right )}{6 b^{4} c f} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-C*sqrt(a + b*x)*(e + f*x)**2*sqrt(a*c - b*c*x)/(3*b**2*c*f) + sqrt(a + b*x)*sqr
t(a*c - b*c*x)*(2*A*b**2*e + B*a**2*f + C*a**2*e)*atan(b*sqrt(c)*x/sqrt(a**2*c -
b**2*c*x**2))/(2*b**3*sqrt(c)*sqrt(a**2*c - b**2*c*x**2)) - sqrt(a + b*x)*sqrt(
a*c - b*c*x)*(2*b**2*e*(3*B*f - C*e) + b**2*f*x*(3*B*f - C*e) + 2*f**2*(3*A*b**2
+ 2*C*a**2))/(6*b**4*c*f)

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Mathematica [A]  time = 0.372752, size = 130, normalized size = 0.53 \[ \frac{3 b \sqrt{a-b x} \tan ^{-1}\left (\frac{b x}{\sqrt{a-b x} \sqrt{a+b x}}\right ) \left (a^2 (B f+C e)+2 A b^2 e\right )-(a-b x) \sqrt{a+b x} \left (4 a^2 C f+b^2 \left (6 A f+6 B e+3 B f x+3 C e x+2 C f x^2\right )\right )}{6 b^4 \sqrt{c (a-b x)}} \]

Antiderivative was successfully verified.

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

[Out]

(-((a - b*x)*Sqrt[a + b*x]*(4*a^2*C*f + b^2*(6*B*e + 6*A*f + 3*C*e*x + 3*B*f*x +
2*C*f*x^2))) + 3*b*(2*A*b^2*e + a^2*(C*e + B*f))*Sqrt[a - b*x]*ArcTan[(b*x)/(Sq
rt[a - b*x]*Sqrt[a + b*x])])/(6*b^4*Sqrt[c*(a - b*x)])

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Maple [A]  time = 0.03, size = 365, normalized size = 1.5 \[{\frac{1}{6\,{b}^{4}c}\sqrt{bx+a}\sqrt{-c \left ( bx-a \right ) } \left ( 6\,c\arctan \left ({\frac{\sqrt{{b}^{2}c}x}{\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }}} \right ) Ae{b}^{4}+3\,Bfc{a}^{2}\arctan \left ({\frac{\sqrt{{b}^{2}c}x}{\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }}} \right ){b}^{2}+3\,Cec{a}^{2}\arctan \left ({\frac{\sqrt{{b}^{2}c}x}{\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }}} \right ){b}^{2}-2\,C{x}^{2}{b}^{2}f\sqrt{{b}^{2}c}\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }-3\,Bfx\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }{b}^{2}\sqrt{{b}^{2}c}-3\,Cex\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }{b}^{2}\sqrt{{b}^{2}c}-6\,\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }Af{b}^{2}\sqrt{{b}^{2}c}-6\,\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }Be{b}^{2}\sqrt{{b}^{2}c}-4\,{a}^{2}\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }Cf\sqrt{{b}^{2}c} \right ){\frac{1}{\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }}}{\frac{1}{\sqrt{{b}^{2}c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.258428, size = 1, normalized size = 0. \[ \left [-\frac{2 \,{\left (2 \, C b^{2} f x^{2} + 6 \, B b^{2} e + 2 \,{\left (2 \, C a^{2} + 3 \, A b^{2}\right )} f + 3 \,{\left (C b^{2} e + B b^{2} f\right )} x\right )} \sqrt{-b c x + a c} \sqrt{b x + a} \sqrt{-c} - 3 \,{\left (B a^{2} b c f +{\left (C a^{2} b + 2 \, A b^{3}\right )} c e\right )} \log \left (2 \, \sqrt{-b c x + a c} \sqrt{b x + a} b x +{\left (2 \, b^{2} x^{2} - a^{2}\right )} \sqrt{-c}\right )}{12 \, b^{4} \sqrt{-c} c}, -\frac{{\left (2 \, C b^{2} f x^{2} + 6 \, B b^{2} e + 2 \,{\left (2 \, C a^{2} + 3 \, A b^{2}\right )} f + 3 \,{\left (C b^{2} e + B b^{2} f\right )} x\right )} \sqrt{-b c x + a c} \sqrt{b x + a} \sqrt{c} - 3 \,{\left (B a^{2} b c f +{\left (C a^{2} b + 2 \, A b^{3}\right )} c e\right )} \arctan \left (\frac{b \sqrt{c} x}{\sqrt{-b c x + a c} \sqrt{b x + a}}\right )}{6 \, b^{4} c^{\frac{3}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/12*(2*(2*C*b^2*f*x^2 + 6*B*b^2*e + 2*(2*C*a^2 + 3*A*b^2)*f + 3*(C*b^2*e + B*
b^2*f)*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*sqrt(-c) - 3*(B*a^2*b*c*f + (C*a^2*b
+ 2*A*b^3)*c*e)*log(2*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*b*x + (2*b^2*x^2 - a^2)*s
qrt(-c)))/(b^4*sqrt(-c)*c), -1/6*((2*C*b^2*f*x^2 + 6*B*b^2*e + 2*(2*C*a^2 + 3*A*
b^2)*f + 3*(C*b^2*e + B*b^2*f)*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*sqrt(c) - 3*(
B*a^2*b*c*f + (C*a^2*b + 2*A*b^3)*c*e)*arctan(b*sqrt(c)*x/(sqrt(-b*c*x + a*c)*sq
rt(b*x + a))))/(b^4*c^(3/2))]

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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((f*x+e)*(C*x**2+B*x+A)/(b*x+a)**(1/2)/(-b*c*x+a*c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [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)*(f*x + e)/(sqrt(-b*c*x + a*c)*sqrt(b*x + a)),x, algorithm="giac")

[Out]

Timed out