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

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

[Out]

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

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Rubi [A]  time = 0.580217, 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 \[ \frac{\sqrt{c+d x} \sqrt{e+f x} \left (2 d f (4 A d f-B (c f+3 d e))+C \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )}{8 d^2 f^3}-\frac{(d e-c f) \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right ) \left (2 d f (4 A d f-B (c f+3 d e))+C \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )}{8 d^{5/2} f^{7/2}}-\frac{(c+d x)^{3/2} \sqrt{e+f x} (-6 B d f+7 c C f+5 C d e)}{12 d^2 f^2}+\frac{C (c+d x)^{5/2} \sqrt{e+f x}}{3 d^2 f} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 48.632, size = 318, normalized size = 1.29 \[ \frac{B \left (c + d x\right )^{\frac{3}{2}} \sqrt{e + f x}}{2 d f} + \frac{C x \left (c + d x\right )^{\frac{3}{2}} \sqrt{e + f x}}{3 d f} - \frac{C \left (c + d x\right )^{\frac{3}{2}} \sqrt{e + f x} \left (3 c f + 5 d e\right )}{12 d^{2} f^{2}} + \frac{C \sqrt{c + d x} \sqrt{e + f x} \left (c^{2} f^{2} + 2 c d e f + 5 d^{2} e^{2}\right )}{8 d^{2} f^{3}} + \frac{C \left (c f - d e\right ) \left (c^{2} f^{2} + 2 c d e f + 5 d^{2} e^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{f} \sqrt{c + d x}} \right )}}{8 d^{\frac{5}{2}} f^{\frac{7}{2}}} + \frac{\sqrt{c + d x} \sqrt{e + f x} \left (4 A d f - B c f - 3 B d e\right )}{4 d f^{2}} + \frac{\left (c f - d e\right ) \left (4 A d f - B c f - 3 B d e\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{f} \sqrt{c + d x}} \right )}}{4 d^{\frac{3}{2}} f^{\frac{5}{2}}} \]

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

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Mathematica [A]  time = 0.306874, size = 212, normalized size = 0.86 \[ \frac{\sqrt{c+d x} \sqrt{e+f x} \left (6 d f (4 A d f+B (c f-3 d e+2 d f x))+C \left (-3 c^2 f^2+2 c d f (f x-2 e)+d^2 \left (15 e^2-10 e f x+8 f^2 x^2\right )\right )\right )}{24 d^2 f^3}-\frac{(d e-c f) \log \left (2 \sqrt{d} \sqrt{f} \sqrt{c+d x} \sqrt{e+f x}+c f+d e+2 d f x\right ) \left (2 d f (4 A d f-B (c f+3 d e))+C \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )}{16 d^{5/2} f^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c + d*x]*Sqrt[e + f*x]*(6*d*f*(4*A*d*f + B*(-3*d*e + c*f + 2*d*f*x)) + C*(
-3*c^2*f^2 + 2*c*d*f*(-2*e + f*x) + d^2*(15*e^2 - 10*e*f*x + 8*f^2*x^2))))/(24*d
^2*f^3) - ((d*e - c*f)*(C*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2) + 2*d*f*(4*A*d*f - B
*(3*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]])/(16*d^(5/2)*f^(7/2))

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Maple [B]  time = 0.031, size = 763, normalized size = 3.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(16*C*x^2*d^2*f^2*(f*d)^(1/2)*((d*x+c)*(f*x+e))
^(1/2)+24*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*A*f^3*d^2-24*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^3*e*A*f^2-6*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^2*f^3*d-12*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*B*e*f^2*d^2+18*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^3*e^2*B*f+24*B*((d*x+c)*(f*
x+e))^(1/2)*x*f^2*d^2*(f*d)^(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))*c^3*f^3+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))*c^2*e*f^2*d+9*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*C*e^2*f*d^2-15*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^3*e^3*C+4*C*(
(d*x+c)*(f*x+e))^(1/2)*x*c*f^2*d*(f*d)^(1/2)-20*C*((d*x+c)*(f*x+e))^(1/2)*x*e*f*
d^2*(f*d)^(1/2)+48*((d*x+c)*(f*x+e))^(1/2)*A*f^2*d^2*(f*d)^(1/2)+12*B*((d*x+c)*(
f*x+e))^(1/2)*c*f^2*d*(f*d)^(1/2)-36*((d*x+c)*(f*x+e))^(1/2)*B*e*f*d^2*(f*d)^(1/
2)-6*C*((d*x+c)*(f*x+e))^(1/2)*c^2*f^2*(f*d)^(1/2)-8*C*((d*x+c)*(f*x+e))^(1/2)*c
*e*f*d*(f*d)^(1/2)+30*((d*x+c)*(f*x+e))^(1/2)*C*e^2*d^2*(f*d)^(1/2))/((d*x+c)*(f
*x+e))^(1/2)/f^3/d^2/(f*d)^(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 = 1.2951, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (8 \, C d^{2} f^{2} x^{2} + 15 \, C d^{2} e^{2} - 2 \,{\left (2 \, C c d + 9 \, B d^{2}\right )} e f - 3 \,{\left (C c^{2} - 2 \, B c d - 8 \, A d^{2}\right )} f^{2} - 2 \,{\left (5 \, C d^{2} e f -{\left (C c d + 6 \, B d^{2}\right )} f^{2}\right )} x\right )} \sqrt{d f} \sqrt{d x + c} \sqrt{f x + e} - 3 \,{\left (5 \, C d^{3} e^{3} - 3 \,{\left (C c d^{2} + 2 \, B d^{3}\right )} e^{2} f -{\left (C c^{2} d - 4 \, B c d^{2} - 8 \, A d^{3}\right )} e f^{2} -{\left (C c^{3} - 2 \, B c^{2} d + 8 \, A c d^{2}\right )} f^{3}\right )} \log \left (4 \,{\left (2 \, d^{2} f^{2} x + d^{2} e f + c d f^{2}\right )} \sqrt{d x + c} \sqrt{f x + e} +{\left (8 \, d^{2} f^{2} x^{2} + d^{2} e^{2} + 6 \, c d e f + c^{2} f^{2} + 8 \,{\left (d^{2} e f + c d f^{2}\right )} x\right )} \sqrt{d f}\right )}{96 \, \sqrt{d f} d^{2} f^{3}}, \frac{2 \,{\left (8 \, C d^{2} f^{2} x^{2} + 15 \, C d^{2} e^{2} - 2 \,{\left (2 \, C c d + 9 \, B d^{2}\right )} e f - 3 \,{\left (C c^{2} - 2 \, B c d - 8 \, A d^{2}\right )} f^{2} - 2 \,{\left (5 \, C d^{2} e f -{\left (C c d + 6 \, B d^{2}\right )} f^{2}\right )} x\right )} \sqrt{-d f} \sqrt{d x + c} \sqrt{f x + e} - 3 \,{\left (5 \, C d^{3} e^{3} - 3 \,{\left (C c d^{2} + 2 \, B d^{3}\right )} e^{2} f -{\left (C c^{2} d - 4 \, B c d^{2} - 8 \, A d^{3}\right )} e f^{2} -{\left (C c^{3} - 2 \, B c^{2} d + 8 \, A c d^{2}\right )} f^{3}\right )} \arctan \left (\frac{{\left (2 \, d f x + d e + c f\right )} \sqrt{-d f}}{2 \, \sqrt{d x + c} \sqrt{f x + e} d f}\right )}{48 \, \sqrt{-d f} d^{2} f^{3}}\right ] \]

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]

[1/96*(4*(8*C*d^2*f^2*x^2 + 15*C*d^2*e^2 - 2*(2*C*c*d + 9*B*d^2)*e*f - 3*(C*c^2
- 2*B*c*d - 8*A*d^2)*f^2 - 2*(5*C*d^2*e*f - (C*c*d + 6*B*d^2)*f^2)*x)*sqrt(d*f)*
sqrt(d*x + c)*sqrt(f*x + e) - 3*(5*C*d^3*e^3 - 3*(C*c*d^2 + 2*B*d^3)*e^2*f - (C*
c^2*d - 4*B*c*d^2 - 8*A*d^3)*e*f^2 - (C*c^3 - 2*B*c^2*d + 8*A*c*d^2)*f^3)*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^3), 1/48*(2*(8*C*d^2*f^2*x^2 + 15*C*d^2*e^2 - 2*(2*C*c*d + 9*B*d^2)*e*f -
3*(C*c^2 - 2*B*c*d - 8*A*d^2)*f^2 - 2*(5*C*d^2*e*f - (C*c*d + 6*B*d^2)*f^2)*x)*
sqrt(-d*f)*sqrt(d*x + c)*sqrt(f*x + e) - 3*(5*C*d^3*e^3 - 3*(C*c*d^2 + 2*B*d^3)*
e^2*f - (C*c^2*d - 4*B*c*d^2 - 8*A*d^3)*e*f^2 - (C*c^3 - 2*B*c^2*d + 8*A*c*d^2)*
f^3)*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^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((C*x**2+B*x+A)*(d*x+c)**(1/2)/(f*x+e)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.261266, size = 425, normalized size = 1.73 \[ \frac{{\left (\sqrt{{\left (d x + c\right )} d f - c d f + d^{2} e} \sqrt{d x + c}{\left (2 \,{\left (d x + c\right )}{\left (\frac{4 \,{\left (d x + c\right )} C}{d^{3} f} - \frac{7 \, C c d^{6} f^{4} - 6 \, B d^{7} f^{4} + 5 \, C d^{7} f^{3} e}{d^{9} f^{5}}\right )} + \frac{3 \,{\left (C c^{2} d^{6} f^{4} - 2 \, B c d^{7} f^{4} + 8 \, A d^{8} f^{4} + 2 \, C c d^{7} f^{3} e - 6 \, B d^{8} f^{3} e + 5 \, C d^{8} f^{2} e^{2}\right )}}{d^{9} f^{5}}\right )} - \frac{3 \,{\left (C c^{3} f^{3} - 2 \, B c^{2} d f^{3} + 8 \, A c d^{2} f^{3} + C c^{2} d f^{2} e - 4 \, B c d^{2} f^{2} e - 8 \, A d^{3} f^{2} e + 3 \, C c d^{2} f e^{2} + 6 \, B d^{3} f e^{2} - 5 \, C d^{3} e^{3}\right )}{\rm ln}\left ({\left | -\sqrt{d f} \sqrt{d x + c} + \sqrt{{\left (d x + c\right )} d f - c d f + d^{2} e} \right |}\right )}{\sqrt{d f} d^{2} f^{3}}\right )} d}{24 \,{\left | d \right |}} \]

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]

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