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