3.2018 \(\int \frac{(a+b x)^2}{(c+d x)^2 \sqrt{e+f x}} \, dx\)

Optimal. Leaf size=132 \[ \frac{(b c-a d) (-a d f-3 b c f+4 b d e) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{5/2} (d e-c f)^{3/2}}-\frac{\sqrt{e+f x} (b c-a d)^2}{d^2 (c+d x) (d e-c f)}+\frac{2 b^2 \sqrt{e+f x}}{d^2 f} \]

[Out]

(2*b^2*Sqrt[e + f*x])/(d^2*f) - ((b*c - a*d)^2*Sqrt[e + f*x])/(d^2*(d*e - c*f)*(
c + d*x)) + ((b*c - a*d)*(4*b*d*e - 3*b*c*f - a*d*f)*ArcTanh[(Sqrt[d]*Sqrt[e + f
*x])/Sqrt[d*e - c*f]])/(d^(5/2)*(d*e - c*f)^(3/2))

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Rubi [A]  time = 0.434057, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{(b c-a d) (-a d f-3 b c f+4 b d e) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{5/2} (d e-c f)^{3/2}}-\frac{\sqrt{e+f x} (b c-a d)^2}{d^2 (c+d x) (d e-c f)}+\frac{2 b^2 \sqrt{e+f x}}{d^2 f} \]

Antiderivative was successfully verified.

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

[Out]

(2*b^2*Sqrt[e + f*x])/(d^2*f) - ((b*c - a*d)^2*Sqrt[e + f*x])/(d^2*(d*e - c*f)*(
c + d*x)) + ((b*c - a*d)*(4*b*d*e - 3*b*c*f - a*d*f)*ArcTanh[(Sqrt[d]*Sqrt[e + f
*x])/Sqrt[d*e - c*f]])/(d^(5/2)*(d*e - c*f)^(3/2))

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Rubi in Sympy [A]  time = 47.7275, size = 116, normalized size = 0.88 \[ \frac{2 b^{2} \sqrt{e + f x}}{d^{2} f} + \frac{\sqrt{e + f x} \left (a d - b c\right )^{2}}{d^{2} \left (c + d x\right ) \left (c f - d e\right )} + \frac{\left (a d - b c\right ) \left (a d f + 3 b c f - 4 b d e\right ) \operatorname{atan}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{c f - d e}} \right )}}{d^{\frac{5}{2}} \left (c f - d e\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.385783, size = 122, normalized size = 0.92 \[ \frac{\sqrt{e+f x} \left (\frac{2 b^2}{f}-\frac{(b c-a d)^2}{(c+d x) (d e-c f)}\right )}{d^2}-\frac{(b c-a d) (a d f+3 b c f-4 b d e) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{5/2} (d e-c f)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[e + f*x]*((2*b^2)/f - (b*c - a*d)^2/((d*e - c*f)*(c + d*x))))/d^2 - ((b*c
- a*d)*(-4*b*d*e + 3*b*c*f + a*d*f)*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c
*f]])/(d^(5/2)*(d*e - c*f)^(3/2))

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Maple [B]  time = 0.026, size = 387, normalized size = 2.9 \[ 2\,{\frac{{b}^{2}\sqrt{fx+e}}{{d}^{2}f}}+{\frac{f{a}^{2}}{ \left ( cf-de \right ) \left ( dfx+cf \right ) }\sqrt{fx+e}}-2\,{\frac{f\sqrt{fx+e}abc}{ \left ( cf-de \right ) d \left ( dfx+cf \right ) }}+{\frac{{b}^{2}{c}^{2}f}{{d}^{2} \left ( cf-de \right ) \left ( dfx+cf \right ) }\sqrt{fx+e}}+{\frac{f{a}^{2}}{cf-de}\arctan \left ({d\sqrt{fx+e}{\frac{1}{\sqrt{ \left ( cf-de \right ) d}}}} \right ){\frac{1}{\sqrt{ \left ( cf-de \right ) d}}}}+2\,{\frac{bfac}{ \left ( cf-de \right ) d\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-4\,{\frac{abe}{ \left ( cf-de \right ) \sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-3\,{\frac{{b}^{2}{c}^{2}f}{{d}^{2} \left ( cf-de \right ) \sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+4\,{\frac{ce{b}^{2}}{ \left ( cf-de \right ) d\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b^2*(f*x+e)^(1/2)/d^2/f+f/(c*f-d*e)*(f*x+e)^(1/2)/(d*f*x+c*f)*a^2-2*f/d/(c*f-d
*e)*(f*x+e)^(1/2)/(d*f*x+c*f)*a*b*c+f/d^2/(c*f-d*e)*(f*x+e)^(1/2)/(d*f*x+c*f)*b^
2*c^2+f/(c*f-d*e)/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(1/2)
)*a^2+2*f/d/(c*f-d*e)/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)^(
1/2))*a*b*c-4/(c*f-d*e)/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d/((c*f-d*e)*d)
^(1/2))*a*b*e-3*f/d^2/(c*f-d*e)/((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+4/d/(c*f-d*e)/((c*f-d*e)*d)^(1/2)*arctan((f*x+e)^(1/2)*d
/((c*f-d*e)*d)^(1/2))*b^2*c*e

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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)^2*sqrt(f*x + e)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236838, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \, b^{2} c d e -{\left (3 \, b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f + 2 \,{\left (b^{2} d^{2} e - b^{2} c d f\right )} x\right )} \sqrt{d^{2} e - c d f} \sqrt{f x + e} -{\left (4 \,{\left (b^{2} c^{2} d - a b c d^{2}\right )} e f -{\left (3 \, b^{2} c^{3} - 2 \, a b c^{2} d - a^{2} c d^{2}\right )} f^{2} +{\left (4 \,{\left (b^{2} c d^{2} - a b d^{3}\right )} e f -{\left (3 \, b^{2} c^{2} d - 2 \, a b c d^{2} - a^{2} d^{3}\right )} f^{2}\right )} x\right )} \log \left (\frac{\sqrt{d^{2} e - c d f}{\left (d f x + 2 \, d e - c f\right )} - 2 \,{\left (d^{2} e - c d f\right )} \sqrt{f x + e}}{d x + c}\right )}{2 \,{\left (c d^{3} e f - c^{2} d^{2} f^{2} +{\left (d^{4} e f - c d^{3} f^{2}\right )} x\right )} \sqrt{d^{2} e - c d f}}, \frac{{\left (2 \, b^{2} c d e -{\left (3 \, b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f + 2 \,{\left (b^{2} d^{2} e - b^{2} c d f\right )} x\right )} \sqrt{-d^{2} e + c d f} \sqrt{f x + e} +{\left (4 \,{\left (b^{2} c^{2} d - a b c d^{2}\right )} e f -{\left (3 \, b^{2} c^{3} - 2 \, a b c^{2} d - a^{2} c d^{2}\right )} f^{2} +{\left (4 \,{\left (b^{2} c d^{2} - a b d^{3}\right )} e f -{\left (3 \, b^{2} c^{2} d - 2 \, a b c d^{2} - a^{2} d^{3}\right )} f^{2}\right )} x\right )} \arctan \left (-\frac{d e - c f}{\sqrt{-d^{2} e + c d f} \sqrt{f x + e}}\right )}{{\left (c d^{3} e f - c^{2} d^{2} f^{2} +{\left (d^{4} e f - c d^{3} f^{2}\right )} x\right )} \sqrt{-d^{2} e + c d f}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(2*(2*b^2*c*d*e - (3*b^2*c^2 - 2*a*b*c*d + a^2*d^2)*f + 2*(b^2*d^2*e - b^2*
c*d*f)*x)*sqrt(d^2*e - c*d*f)*sqrt(f*x + e) - (4*(b^2*c^2*d - a*b*c*d^2)*e*f - (
3*b^2*c^3 - 2*a*b*c^2*d - a^2*c*d^2)*f^2 + (4*(b^2*c*d^2 - a*b*d^3)*e*f - (3*b^2
*c^2*d - 2*a*b*c*d^2 - a^2*d^3)*f^2)*x)*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)))/((c*d^3*e*f - c^2*d^2*f^2
+ (d^4*e*f - c*d^3*f^2)*x)*sqrt(d^2*e - c*d*f)), ((2*b^2*c*d*e - (3*b^2*c^2 - 2*
a*b*c*d + a^2*d^2)*f + 2*(b^2*d^2*e - b^2*c*d*f)*x)*sqrt(-d^2*e + c*d*f)*sqrt(f*
x + e) + (4*(b^2*c^2*d - a*b*c*d^2)*e*f - (3*b^2*c^3 - 2*a*b*c^2*d - a^2*c*d^2)*
f^2 + (4*(b^2*c*d^2 - a*b*d^3)*e*f - (3*b^2*c^2*d - 2*a*b*c*d^2 - a^2*d^3)*f^2)*
x)*arctan(-(d*e - c*f)/(sqrt(-d^2*e + c*d*f)*sqrt(f*x + e))))/((c*d^3*e*f - c^2*
d^2*f^2 + (d^4*e*f - c*d^3*f^2)*x)*sqrt(-d^2*e + c*d*f))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.215906, size = 277, normalized size = 2.1 \[ -\frac{{\left (3 \, b^{2} c^{2} f - 2 \, a b c d f - a^{2} d^{2} f - 4 \, b^{2} c d e + 4 \, a b d^{2} e\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{{\left (c d^{2} f - d^{3} e\right )} \sqrt{c d f - d^{2} e}} + \frac{2 \, \sqrt{f x + e} b^{2}}{d^{2} f} + \frac{\sqrt{f x + e} b^{2} c^{2} f - 2 \, \sqrt{f x + e} a b c d f + \sqrt{f x + e} a^{2} d^{2} f}{{\left (c d^{2} f - d^{3} e\right )}{\left ({\left (f x + e\right )} d + c f - d e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(3*b^2*c^2*f - 2*a*b*c*d*f - a^2*d^2*f - 4*b^2*c*d*e + 4*a*b*d^2*e)*arctan(sqrt
(f*x + e)*d/sqrt(c*d*f - d^2*e))/((c*d^2*f - d^3*e)*sqrt(c*d*f - d^2*e)) + 2*sqr
t(f*x + e)*b^2/(d^2*f) + (sqrt(f*x + e)*b^2*c^2*f - 2*sqrt(f*x + e)*a*b*c*d*f +
sqrt(f*x + e)*a^2*d^2*f)/((c*d^2*f - d^3*e)*((f*x + e)*d + c*f - d*e))