3.1760 \(\int \frac{(a+b x)^2}{(c+d x) \sqrt{e+f x}} \, dx\)

Optimal. Leaf size=112 \[ -\frac{2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{5/2} \sqrt{d e-c f}}-\frac{2 b \sqrt{e+f x} (-2 a d f+b c f+b d e)}{d^2 f^2}+\frac{2 b^2 (e+f x)^{3/2}}{3 d f^2} \]

[Out]

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

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Rubi [A]  time = 0.226745, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{5/2} \sqrt{d e-c f}}-\frac{2 b \sqrt{e+f x} (-2 a d f+b c f+b d e)}{d^2 f^2}+\frac{2 b^2 (e+f x)^{3/2}}{3 d f^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 31.7234, size = 105, normalized size = 0.94 \[ \frac{2 b^{2} \left (e + f x\right )^{\frac{3}{2}}}{3 d f^{2}} + \frac{2 b \sqrt{e + f x} \left (2 a d f - b c f - b d e\right )}{d^{2} f^{2}} + \frac{2 \left (a d - b c\right )^{2} \operatorname{atan}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{c f - d e}} \right )}}{d^{\frac{5}{2}} \sqrt{c f - d e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**2*(e + f*x)**(3/2)/(3*d*f**2) + 2*b*sqrt(e + f*x)*(2*a*d*f - b*c*f - b*d*e)
/(d**2*f**2) + 2*(a*d - b*c)**2*atan(sqrt(d)*sqrt(e + f*x)/sqrt(c*f - d*e))/(d**
(5/2)*sqrt(c*f - d*e))

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Mathematica [A]  time = 0.155697, size = 99, normalized size = 0.88 \[ \frac{2 b \sqrt{e+f x} (6 a d f+b (-3 c f-2 d e+d f x))}{3 d^2 f^2}-\frac{2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{5/2} \sqrt{d e-c f}} \]

Antiderivative was successfully verified.

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

[Out]

(2*b*Sqrt[e + f*x]*(6*a*d*f + b*(-2*d*e - 3*c*f + d*f*x)))/(3*d^2*f^2) - (2*(b*c
- a*d)^2*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/(d^(5/2)*Sqrt[d*e -
c*f])

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Maple [B]  time = 0.015, size = 201, normalized size = 1.8 \[{\frac{2\,{b}^{2}}{3\,d{f}^{2}} \left ( fx+e \right ) ^{{\frac{3}{2}}}}+4\,{\frac{ab\sqrt{fx+e}}{df}}-2\,{\frac{{b}^{2}c\sqrt{fx+e}}{f{d}^{2}}}-2\,{\frac{{b}^{2}e\sqrt{fx+e}}{d{f}^{2}}}+2\,{\frac{{a}^{2}}{\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-4\,{\frac{abc}{d\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+2\,{\frac{{b}^{2}{c}^{2}}{{d}^{2}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222352, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{2} \log \left (\frac{\sqrt{d^{2} e - c d f}{\left (d f x + 2 \, d e - c f\right )} - 2 \,{\left (d^{2} e - c d f\right )} \sqrt{f x + e}}{d x + c}\right ) + 2 \,{\left (b^{2} d f x - 2 \, b^{2} d e - 3 \,{\left (b^{2} c - 2 \, a b d\right )} f\right )} \sqrt{d^{2} e - c d f} \sqrt{f x + e}}{3 \, \sqrt{d^{2} e - c d f} d^{2} f^{2}}, -\frac{2 \,{\left (3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f^{2} \arctan \left (-\frac{d e - c f}{\sqrt{-d^{2} e + c d f} \sqrt{f x + e}}\right ) -{\left (b^{2} d f x - 2 \, b^{2} d e - 3 \,{\left (b^{2} c - 2 \, a b d\right )} f\right )} \sqrt{-d^{2} e + c d f} \sqrt{f x + e}\right )}}{3 \, \sqrt{-d^{2} e + c d f} d^{2} f^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3*(3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*f^2*log((sqrt(d^2*e - c*d*f)*(d*f*x + 2*
d*e - c*f) - 2*(d^2*e - c*d*f)*sqrt(f*x + e))/(d*x + c)) + 2*(b^2*d*f*x - 2*b^2*
d*e - 3*(b^2*c - 2*a*b*d)*f)*sqrt(d^2*e - c*d*f)*sqrt(f*x + e))/(sqrt(d^2*e - c*
d*f)*d^2*f^2), -2/3*(3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*f^2*arctan(-(d*e - c*f)/(
sqrt(-d^2*e + c*d*f)*sqrt(f*x + e))) - (b^2*d*f*x - 2*b^2*d*e - 3*(b^2*c - 2*a*b
*d)*f)*sqrt(-d^2*e + c*d*f)*sqrt(f*x + e))/(sqrt(-d^2*e + c*d*f)*d^2*f^2)]

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Sympy [A]  time = 30.8499, size = 255, normalized size = 2.28 \[ \frac{2 b^{2} \left (e + f x\right )^{\frac{3}{2}}}{3 d f^{2}} + \frac{2 b \sqrt{e + f x} \left (2 a d f - b c f - b d e\right )}{d^{2} f^{2}} - \frac{2 \left (a d - b c\right )^{2} \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{1}{\sqrt{\frac{d}{c f - d e}} \sqrt{e + f x}} \right )}}{\sqrt{\frac{d}{c f - d e}} \left (c f - d e\right )} & \text{for}\: \frac{d}{c f - d e} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{1}{\sqrt{- \frac{d}{c f - d e}} \sqrt{e + f x}} \right )}}{\sqrt{- \frac{d}{c f - d e}} \left (c f - d e\right )} & \text{for}\: \frac{1}{e + f x} > - \frac{d}{c f - d e} \wedge \frac{d}{c f - d e} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{1}{\sqrt{- \frac{d}{c f - d e}} \sqrt{e + f x}} \right )}}{\sqrt{- \frac{d}{c f - d e}} \left (c f - d e\right )} & \text{for}\: \frac{d}{c f - d e} < 0 \wedge \frac{1}{e + f x} < - \frac{d}{c f - d e} \end{cases}\right )}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**2*(e + f*x)**(3/2)/(3*d*f**2) + 2*b*sqrt(e + f*x)*(2*a*d*f - b*c*f - b*d*e)
/(d**2*f**2) - 2*(a*d - b*c)**2*Piecewise((atan(1/(sqrt(d/(c*f - d*e))*sqrt(e +
f*x)))/(sqrt(d/(c*f - d*e))*(c*f - d*e)), d/(c*f - d*e) > 0), (-acoth(1/(sqrt(-d
/(c*f - d*e))*sqrt(e + f*x)))/(sqrt(-d/(c*f - d*e))*(c*f - d*e)), (d/(c*f - d*e)
< 0) & (1/(e + f*x) > -d/(c*f - d*e))), (-atanh(1/(sqrt(-d/(c*f - d*e))*sqrt(e
+ f*x)))/(sqrt(-d/(c*f - d*e))*(c*f - d*e)), (d/(c*f - d*e) < 0) & (1/(e + f*x)
< -d/(c*f - d*e))))/d**2

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GIAC/XCAS [A]  time = 0.214133, size = 203, normalized size = 1.81 \[ \frac{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{\sqrt{c d f - d^{2} e} d^{2}} + \frac{2 \,{\left ({\left (f x + e\right )}^{\frac{3}{2}} b^{2} d^{2} f^{4} - 3 \, \sqrt{f x + e} b^{2} c d f^{5} + 6 \, \sqrt{f x + e} a b d^{2} f^{5} - 3 \, \sqrt{f x + e} b^{2} d^{2} f^{4} e\right )}}{3 \, d^{3} f^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*arctan(sqrt(f*x + e)*d/sqrt(c*d*f - d^2*e))/(s
qrt(c*d*f - d^2*e)*d^2) + 2/3*((f*x + e)^(3/2)*b^2*d^2*f^4 - 3*sqrt(f*x + e)*b^2
*c*d*f^5 + 6*sqrt(f*x + e)*a*b*d^2*f^5 - 3*sqrt(f*x + e)*b^2*d^2*f^4*e)/(d^3*f^6
)