Optimal. Leaf size=184 \[ \frac{2 b \sqrt{e+f x} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{d^3 f^3}-\frac{2 b^2 (e+f x)^{3/2} (-3 a d f+b c f+2 b d e)}{3 d^2 f^3}+\frac{2 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{7/2} \sqrt{d e-c f}}+\frac{2 b^3 (e+f x)^{5/2}}{5 d f^3} \]
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Rubi [A] time = 0.347774, antiderivative size = 184, 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 \[ \frac{2 b \sqrt{e+f x} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{d^3 f^3}-\frac{2 b^2 (e+f x)^{3/2} (-3 a d f+b c f+2 b d e)}{3 d^2 f^3}+\frac{2 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{7/2} \sqrt{d e-c f}}+\frac{2 b^3 (e+f x)^{5/2}}{5 d f^3} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^3/((c + d*x)*Sqrt[e + f*x]),x]
[Out]
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Rubi in Sympy [A] time = 70.5545, size = 196, normalized size = 1.07 \[ \frac{2 b^{3} \left (e + f x\right )^{\frac{5}{2}}}{5 d f^{3}} + \frac{2 b^{2} \left (e + f x\right )^{\frac{3}{2}} \left (3 a d f - b c f - 2 b d e\right )}{3 d^{2} f^{3}} + \frac{2 b \sqrt{e + f x} \left (3 a^{2} d^{2} f^{2} - 3 a b c d f^{2} - 3 a b d^{2} e f + b^{2} c^{2} f^{2} + b^{2} c d e f + b^{2} d^{2} e^{2}\right )}{d^{3} f^{3}} + \frac{2 \left (a d - b c\right )^{3} \operatorname{atan}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{c f - d e}} \right )}}{d^{\frac{7}{2}} \sqrt{c f - d e}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**3/(d*x+c)/(f*x+e)**(1/2),x)
[Out]
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Mathematica [A] time = 0.268279, size = 157, normalized size = 0.85 \[ \frac{2 b \sqrt{e+f x} \left (45 a^2 d^2 f^2+15 a b d f (-3 c f-2 d e+d f x)+b^2 \left (15 c^2 f^2-5 c d f (f x-2 e)+d^2 \left (8 e^2-4 e f x+3 f^2 x^2\right )\right )\right )}{15 d^3 f^3}+\frac{2 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{7/2} \sqrt{d e-c f}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^3/((c + d*x)*Sqrt[e + f*x]),x]
[Out]
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Maple [B] time = 0.016, size = 372, normalized size = 2. \[{\frac{2\,{b}^{3}}{5\,d{f}^{3}} \left ( fx+e \right ) ^{{\frac{5}{2}}}}+2\,{\frac{{b}^{2} \left ( fx+e \right ) ^{3/2}a}{d{f}^{2}}}-{\frac{2\,{b}^{3}c}{3\,{d}^{2}{f}^{2}} \left ( fx+e \right ) ^{{\frac{3}{2}}}}-{\frac{4\,{b}^{3}e}{3\,d{f}^{3}} \left ( fx+e \right ) ^{{\frac{3}{2}}}}+6\,{\frac{{a}^{2}b\sqrt{fx+e}}{df}}-6\,{\frac{a{b}^{2}c\sqrt{fx+e}}{{d}^{2}f}}-6\,{\frac{a{b}^{2}e\sqrt{fx+e}}{d{f}^{2}}}+2\,{\frac{{b}^{3}{c}^{2}\sqrt{fx+e}}{f{d}^{3}}}+2\,{\frac{ce{b}^{3}\sqrt{fx+e}}{{d}^{2}{f}^{2}}}+2\,{\frac{{b}^{3}{e}^{2}\sqrt{fx+e}}{d{f}^{3}}}+2\,{\frac{{a}^{3}}{\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-6\,{\frac{{a}^{2}bc}{d\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+6\,{\frac{a{b}^{2}{c}^{2}}{{d}^{2}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-2\,{\frac{{b}^{3}{c}^{3}}{{d}^{3}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^3/(d*x+c)/(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)^3/((d*x + c)*sqrt(f*x + e)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.220908, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{3} \log \left (\frac{\sqrt{d^{2} e - c d f}{\left (d f x + 2 \, d e - c f\right )} - 2 \,{\left (d^{2} e - c d f\right )} \sqrt{f x + e}}{d x + c}\right ) - 2 \,{\left (3 \, b^{3} d^{2} f^{2} x^{2} + 8 \, b^{3} d^{2} e^{2} + 10 \,{\left (b^{3} c d - 3 \, a b^{2} d^{2}\right )} e f + 15 \,{\left (b^{3} c^{2} - 3 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} f^{2} -{\left (4 \, b^{3} d^{2} e f + 5 \,{\left (b^{3} c d - 3 \, a b^{2} d^{2}\right )} f^{2}\right )} x\right )} \sqrt{d^{2} e - c d f} \sqrt{f x + e}}{15 \, \sqrt{d^{2} e - c d f} d^{3} f^{3}}, \frac{2 \,{\left (15 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{3} \arctan \left (-\frac{d e - c f}{\sqrt{-d^{2} e + c d f} \sqrt{f x + e}}\right ) +{\left (3 \, b^{3} d^{2} f^{2} x^{2} + 8 \, b^{3} d^{2} e^{2} + 10 \,{\left (b^{3} c d - 3 \, a b^{2} d^{2}\right )} e f + 15 \,{\left (b^{3} c^{2} - 3 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} f^{2} -{\left (4 \, b^{3} d^{2} e f + 5 \,{\left (b^{3} c d - 3 \, a b^{2} d^{2}\right )} f^{2}\right )} x\right )} \sqrt{-d^{2} e + c d f} \sqrt{f x + e}\right )}}{15 \, \sqrt{-d^{2} e + c d f} d^{3} f^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^3/((d*x + c)*sqrt(f*x + e)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 75.7796, size = 345, normalized size = 1.88 \[ \frac{2 b^{3} \left (e + f x\right )^{\frac{5}{2}}}{5 d f^{3}} + \frac{2 b^{2} \left (e + f x\right )^{\frac{3}{2}} \left (3 a d f - b c f - 2 b d e\right )}{3 d^{2} f^{3}} + \frac{2 b \sqrt{e + f x} \left (3 a^{2} d^{2} f^{2} - 3 a b c d f^{2} - 3 a b d^{2} e f + b^{2} c^{2} f^{2} + b^{2} c d e f + b^{2} d^{2} e^{2}\right )}{d^{3} f^{3}} - \frac{2 \left (a d - b c\right )^{3} \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{1}{\sqrt{\frac{d}{c f - d e}} \sqrt{e + f x}} \right )}}{\sqrt{\frac{d}{c f - d e}} \left (c f - d e\right )} & \text{for}\: \frac{d}{c f - d e} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{1}{\sqrt{- \frac{d}{c f - d e}} \sqrt{e + f x}} \right )}}{\sqrt{- \frac{d}{c f - d e}} \left (c f - d e\right )} & \text{for}\: \frac{1}{e + f x} > - \frac{d}{c f - d e} \wedge \frac{d}{c f - d e} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{1}{\sqrt{- \frac{d}{c f - d e}} \sqrt{e + f x}} \right )}}{\sqrt{- \frac{d}{c f - d e}} \left (c f - d e\right )} & \text{for}\: \frac{d}{c f - d e} < 0 \wedge \frac{1}{e + f x} < - \frac{d}{c f - d e} \end{cases}\right )}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**3/(d*x+c)/(f*x+e)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.216274, size = 402, normalized size = 2.18 \[ -\frac{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{\sqrt{c d f - d^{2} e} d^{3}} + \frac{2 \,{\left (3 \,{\left (f x + e\right )}^{\frac{5}{2}} b^{3} d^{4} f^{12} - 5 \,{\left (f x + e\right )}^{\frac{3}{2}} b^{3} c d^{3} f^{13} + 15 \,{\left (f x + e\right )}^{\frac{3}{2}} a b^{2} d^{4} f^{13} + 15 \, \sqrt{f x + e} b^{3} c^{2} d^{2} f^{14} - 45 \, \sqrt{f x + e} a b^{2} c d^{3} f^{14} + 45 \, \sqrt{f x + e} a^{2} b d^{4} f^{14} - 10 \,{\left (f x + e\right )}^{\frac{3}{2}} b^{3} d^{4} f^{12} e + 15 \, \sqrt{f x + e} b^{3} c d^{3} f^{13} e - 45 \, \sqrt{f x + e} a b^{2} d^{4} f^{13} e + 15 \, \sqrt{f x + e} b^{3} d^{4} f^{12} e^{2}\right )}}{15 \, d^{5} f^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^3/((d*x + c)*sqrt(f*x + e)),x, algorithm="giac")
[Out]