Optimal. Leaf size=130 \[ \frac{2 (b c-a d) (d e-c f)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{7/2}}-\frac{2 \sqrt{e+f x} (b c-a d) (d e-c f)}{d^3}-\frac{2 (e+f x)^{3/2} (b c-a d)}{3 d^2}+\frac{2 b (e+f x)^{5/2}}{5 d f} \]
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Rubi [A] time = 0.233401, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 (b c-a d) (d e-c f)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{7/2}}-\frac{2 \sqrt{e+f x} (b c-a d) (d e-c f)}{d^3}-\frac{2 (e+f x)^{3/2} (b c-a d)}{3 d^2}+\frac{2 b (e+f x)^{5/2}}{5 d f} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)*(e + f*x)^(3/2))/(c + d*x),x]
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Rubi in Sympy [A] time = 24.8086, size = 114, normalized size = 0.88 \[ \frac{2 b \left (e + f x\right )^{\frac{5}{2}}}{5 d f} + \frac{2 \left (e + f x\right )^{\frac{3}{2}} \left (a d - b c\right )}{3 d^{2}} - \frac{2 \sqrt{e + f x} \left (a d - b c\right ) \left (c f - d e\right )}{d^{3}} + \frac{2 \left (a d - b c\right ) \left (c f - d e\right )^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{c f - d e}} \right )}}{d^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(f*x+e)**(3/2)/(d*x+c),x)
[Out]
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Mathematica [A] time = 0.203358, size = 129, normalized size = 0.99 \[ \frac{2 \sqrt{e+f x} \left (5 a d f (-3 c f+4 d e+d f x)+b \left (15 c^2 f^2-5 c d f (4 e+f x)+3 d^2 (e+f x)^2\right )\right )}{15 d^3 f}+\frac{2 (b c-a d) (d e-c f)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)*(e + f*x)^(3/2))/(c + d*x),x]
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Maple [B] time = 0.014, size = 370, normalized size = 2.9 \[{\frac{2\,b}{5\,df} \left ( fx+e \right ) ^{{\frac{5}{2}}}}+{\frac{2\,a}{3\,d} \left ( fx+e \right ) ^{{\frac{3}{2}}}}-{\frac{2\,bc}{3\,{d}^{2}} \left ( fx+e \right ) ^{{\frac{3}{2}}}}-2\,{\frac{acf\sqrt{fx+e}}{{d}^{2}}}+2\,{\frac{ae\sqrt{fx+e}}{d}}+2\,{\frac{bf{c}^{2}\sqrt{fx+e}}{{d}^{3}}}-2\,{\frac{bce\sqrt{fx+e}}{{d}^{2}}}+2\,{\frac{{f}^{2}a{c}^{2}}{{d}^{2}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-4\,{\frac{acfe}{d\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+2\,{\frac{a{e}^{2}}{\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-2\,{\frac{b{c}^{3}{f}^{2}}{{d}^{3}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }+4\,{\frac{bf{c}^{2}e}{{d}^{2}\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) }-2\,{\frac{bc{e}^{2}}{d\sqrt{ \left ( cf-de \right ) d}}\arctan \left ({\frac{\sqrt{fx+e}d}{\sqrt{ \left ( cf-de \right ) d}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(f*x+e)^(3/2)/(d*x+c),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)*(f*x + e)^(3/2)/(d*x + c),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.221155, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left ({\left (b c d - a d^{2}\right )} e f -{\left (b c^{2} - a c d\right )} f^{2}\right )} \sqrt{\frac{d e - c f}{d}} \log \left (\frac{d f x + 2 \, d e - c f - 2 \, \sqrt{f x + e} d \sqrt{\frac{d e - c f}{d}}}{d x + c}\right ) - 2 \,{\left (3 \, b d^{2} f^{2} x^{2} + 3 \, b d^{2} e^{2} - 20 \,{\left (b c d - a d^{2}\right )} e f + 15 \,{\left (b c^{2} - a c d\right )} f^{2} +{\left (6 \, b d^{2} e f - 5 \,{\left (b c d - a d^{2}\right )} f^{2}\right )} x\right )} \sqrt{f x + e}}{15 \, d^{3} f}, \frac{2 \,{\left (15 \,{\left ({\left (b c d - a d^{2}\right )} e f -{\left (b c^{2} - a c d\right )} f^{2}\right )} \sqrt{-\frac{d e - c f}{d}} \arctan \left (\frac{\sqrt{f x + e}}{\sqrt{-\frac{d e - c f}{d}}}\right ) +{\left (3 \, b d^{2} f^{2} x^{2} + 3 \, b d^{2} e^{2} - 20 \,{\left (b c d - a d^{2}\right )} e f + 15 \,{\left (b c^{2} - a c d\right )} f^{2} +{\left (6 \, b d^{2} e f - 5 \,{\left (b c d - a d^{2}\right )} f^{2}\right )} x\right )} \sqrt{f x + e}\right )}}{15 \, d^{3} f}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(f*x + e)^(3/2)/(d*x + c),x, algorithm="fricas")
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Sympy [A] time = 46.4003, size = 258, normalized size = 1.98 \[ \frac{2 b \left (e + f x\right )^{\frac{5}{2}}}{5 d f} + \frac{\left (e + f x\right )^{\frac{3}{2}} \left (2 a d - 2 b c\right )}{3 d^{2}} + \frac{\sqrt{e + f x} \left (- 2 a c d f + 2 a d^{2} e + 2 b c^{2} f - 2 b c d e\right )}{d^{3}} + \frac{2 \left (a d - b c\right ) \left (c f - d e\right )^{2} \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{c f - d e}{d}}} \right )}}{d \sqrt{\frac{c f - d e}{d}}} & \text{for}\: \frac{c f - d e}{d} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{- c f + d e}{d}}} \right )}}{d \sqrt{\frac{- c f + d e}{d}}} & \text{for}\: e + f x > \frac{- c f + d e}{d} \wedge \frac{c f - d e}{d} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{- c f + d e}{d}}} \right )}}{d \sqrt{\frac{- c f + d e}{d}}} & \text{for}\: \frac{c f - d e}{d} < 0 \wedge e + f x < \frac{- c f + d e}{d} \end{cases}\right )}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(f*x+e)**(3/2)/(d*x+c),x)
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GIAC/XCAS [A] time = 0.216712, size = 321, normalized size = 2.47 \[ -\frac{2 \,{\left (b c^{3} f^{2} - a c^{2} d f^{2} - 2 \, b c^{2} d f e + 2 \, a c d^{2} f e + b c d^{2} e^{2} - a d^{3} e^{2}\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{\sqrt{c d f - d^{2} e} d^{3}} + \frac{2 \,{\left (3 \,{\left (f x + e\right )}^{\frac{5}{2}} b d^{4} f^{4} - 5 \,{\left (f x + e\right )}^{\frac{3}{2}} b c d^{3} f^{5} + 5 \,{\left (f x + e\right )}^{\frac{3}{2}} a d^{4} f^{5} + 15 \, \sqrt{f x + e} b c^{2} d^{2} f^{6} - 15 \, \sqrt{f x + e} a c d^{3} f^{6} - 15 \, \sqrt{f x + e} b c d^{3} f^{5} e + 15 \, \sqrt{f x + e} a d^{4} f^{5} e\right )}}{15 \, d^{5} f^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(f*x + e)^(3/2)/(d*x + c),x, algorithm="giac")
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