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

Optimal. Leaf size=527 \[ -\frac{2 \sqrt{a d-b c} (d e-c f) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \left (4 a^2 C d f^2+a b f (-5 B d f-c C f+3 C d e)+b^2 (-(5 d f (2 B e-3 A f)-C e (c f+8 d e)))\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{15 b^3 d^{3/2} f^3 \sqrt{c+d x} \sqrt{e+f x}}-\frac{2 \sqrt{e+f x} \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} (3 b d f (a c C f+3 a C d e-5 A b d f+b c C e)-(2 a d f-b c f+2 b d e) (4 a C d f+b (-5 B d f+2 c C f+4 C d e))) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{15 b^3 d^{3/2} f^3 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} (4 a C d f+b (-5 B d f+2 c C f+4 C d e))}{15 b^2 d f^2}+\frac{2 C \sqrt{a+b x} (c+d x)^{3/2} \sqrt{e+f x}}{5 b d f} \]

[Out]

(-2*(4*a*C*d*f + b*(4*C*d*e + 2*c*C*f - 5*B*d*f))*Sqrt[a + b*x]*Sqrt[c + d*x]*Sq
rt[e + f*x])/(15*b^2*d*f^2) + (2*C*Sqrt[a + b*x]*(c + d*x)^(3/2)*Sqrt[e + f*x])/
(5*b*d*f) - (2*Sqrt[-(b*c) + a*d]*(3*b*d*f*(b*c*C*e + 3*a*C*d*e + a*c*C*f - 5*A*
b*d*f) - (2*b*d*e - b*c*f + 2*a*d*f)*(4*a*C*d*f + b*(4*C*d*e + 2*c*C*f - 5*B*d*f
)))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x]*EllipticE[ArcSin[(Sqrt[d]*Sqrt
[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(15*b^3*d^(3/2
)*f^3*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]) - (2*Sqrt[-(b*c) + a*d]*(d*
e - c*f)*(4*a^2*C*d*f^2 + a*b*f*(3*C*d*e - c*C*f - 5*B*d*f) - b^2*(5*d*f*(2*B*e
- 3*A*f) - C*e*(8*d*e + c*f)))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[(b*(e + f*x)
)/(b*e - a*f)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b
*c - a*d)*f)/(d*(b*e - a*f))])/(15*b^3*d^(3/2)*f^3*Sqrt[c + d*x]*Sqrt[e + f*x])

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Rubi [A]  time = 2.72657, antiderivative size = 527, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.184 \[ -\frac{2 \sqrt{a d-b c} (d e-c f) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \left (4 a^2 C d f^2+a b f (-5 B d f-c C f+3 C d e)+b^2 (-(5 d f (2 B e-3 A f)-C e (c f+8 d e)))\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{15 b^3 d^{3/2} f^3 \sqrt{c+d x} \sqrt{e+f x}}-\frac{2 \sqrt{e+f x} \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} (3 b d f (a c C f+3 a C d e-5 A b d f+b c C e)-(2 a d f-b c f+2 b d e) (4 a C d f+b (-5 B d f+2 c C f+4 C d e))) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{15 b^3 d^{3/2} f^3 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} (4 a C d f+b (-5 B d f+2 c C f+4 C d e))}{15 b^2 d f^2}+\frac{2 C \sqrt{a+b x} (c+d x)^{3/2} \sqrt{e+f x}}{5 b d f} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(4*a*C*d*f + b*(4*C*d*e + 2*c*C*f - 5*B*d*f))*Sqrt[a + b*x]*Sqrt[c + d*x]*Sq
rt[e + f*x])/(15*b^2*d*f^2) + (2*C*Sqrt[a + b*x]*(c + d*x)^(3/2)*Sqrt[e + f*x])/
(5*b*d*f) - (2*Sqrt[-(b*c) + a*d]*(3*b*d*f*(b*c*C*e + 3*a*C*d*e + a*c*C*f - 5*A*
b*d*f) - (2*b*d*e - b*c*f + 2*a*d*f)*(4*a*C*d*f + b*(4*C*d*e + 2*c*C*f - 5*B*d*f
)))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x]*EllipticE[ArcSin[(Sqrt[d]*Sqrt
[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(15*b^3*d^(3/2
)*f^3*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]) - (2*Sqrt[-(b*c) + a*d]*(d*
e - c*f)*(4*a^2*C*d*f^2 + a*b*f*(3*C*d*e - c*C*f - 5*B*d*f) - b^2*(5*d*f*(2*B*e
- 3*A*f) - C*e*(8*d*e + c*f)))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[(b*(e + f*x)
)/(b*e - a*f)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b
*c - a*d)*f)/(d*(b*e - a*f))])/(15*b^3*d^(3/2)*f^3*Sqrt[c + d*x]*Sqrt[e + f*x])

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

[Out]

Timed out

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Mathematica [C]  time = 15.6942, size = 5393, normalized size = 10.23 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

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

[Out]

Result too large to show

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Maple [B]  time = 0.043, size = 6049, normalized size = 11.5 \[ \text{output too large to display} \]

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)^(1/2)/(f*x+e)^(1/2),x)

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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