Optimal. Leaf size=244 \[ \frac{2 b (e+f x)^{5/2} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{5 d^3 f^3}-\frac{2 b^2 (e+f x)^{7/2} (-3 a d f+b c f+2 b d e)}{7 d^2 f^3}+\frac{2 (b c-a d)^3 (d e-c f)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{11/2}}-\frac{2 \sqrt{e+f x} (b c-a d)^3 (d e-c f)}{d^5}-\frac{2 (e+f x)^{3/2} (b c-a d)^3}{3 d^4}+\frac{2 b^3 (e+f x)^{9/2}}{9 d f^3} \]
[Out]
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Rubi [A] time = 0.481759, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{2 b (e+f x)^{5/2} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{5 d^3 f^3}-\frac{2 b^2 (e+f x)^{7/2} (-3 a d f+b c f+2 b d e)}{7 d^2 f^3}+\frac{2 (b c-a d)^3 (d e-c f)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{11/2}}-\frac{2 \sqrt{e+f x} (b c-a d)^3 (d e-c f)}{d^5}-\frac{2 (e+f x)^{3/2} (b c-a d)^3}{3 d^4}+\frac{2 b^3 (e+f x)^{9/2}}{9 d f^3} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)^3*(e + f*x)^(3/2))/(c + d*x),x]
[Out]
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Rubi in Sympy [A] time = 85.6216, size = 250, normalized size = 1.02 \[ \frac{2 b^{3} \left (e + f x\right )^{\frac{9}{2}}}{9 d f^{3}} + \frac{2 b^{2} \left (e + f x\right )^{\frac{7}{2}} \left (3 a d f - b c f - 2 b d e\right )}{7 d^{2} f^{3}} + \frac{2 b \left (e + f x\right )^{\frac{5}{2}} \left (3 a^{2} d^{2} f^{2} - 3 a b c d f^{2} - 3 a b d^{2} e f + b^{2} c^{2} f^{2} + b^{2} c d e f + b^{2} d^{2} e^{2}\right )}{5 d^{3} f^{3}} + \frac{2 \left (e + f x\right )^{\frac{3}{2}} \left (a d - b c\right )^{3}}{3 d^{4}} - \frac{2 \sqrt{e + f x} \left (a d - b c\right )^{3} \left (c f - d e\right )}{d^{5}} + \frac{2 \left (a d - b c\right )^{3} \left (c f - d e\right )^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{c f - d e}} \right )}}{d^{\frac{11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**3*(f*x+e)**(3/2)/(d*x+c),x)
[Out]
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Mathematica [A] time = 0.606743, size = 313, normalized size = 1.28 \[ \frac{2 \sqrt{e+f x} \left (105 a^3 d^3 f^3 (-3 c f+4 d e+d f x)+63 a^2 b d^2 f^2 \left (15 c^2 f^2-5 c d f (4 e+f x)+3 d^2 (e+f x)^2\right )-9 a b^2 d f \left (105 c^3 f^3-35 c^2 d f^2 (4 e+f x)+21 c d^2 f (e+f x)^2+3 d^3 (2 e-5 f x) (e+f x)^2\right )+b^3 \left (315 c^4 f^4-105 c^3 d f^3 (4 e+f x)+63 c^2 d^2 f^2 (e+f x)^2+9 c d^3 f (2 e-5 f x) (e+f x)^2+d^4 (e+f x)^2 \left (8 e^2-20 e f x+35 f^2 x^2\right )\right )\right )}{315 d^5 f^3}+\frac{2 (b c-a d)^3 (d e-c f)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{11/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)^3*(e + f*x)^(3/2))/(c + d*x),x]
[Out]
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Maple [B] time = 0.022, size = 984, normalized size = 4. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^3*(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)^3*(f*x + e)^(3/2)/(d*x + c),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.231866, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^3*(f*x + e)^(3/2)/(d*x + c),x, algorithm="fricas")
[Out]
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Sympy [A] time = 136.518, size = 500, normalized size = 2.05 \[ \frac{2 b^{3} \left (e + f x\right )^{\frac{9}{2}}}{9 d f^{3}} + \frac{\left (e + f x\right )^{\frac{7}{2}} \left (6 a b^{2} d f - 2 b^{3} c f - 4 b^{3} d e\right )}{7 d^{2} f^{3}} + \frac{\left (e + f x\right )^{\frac{5}{2}} \left (6 a^{2} b d^{2} f^{2} - 6 a b^{2} c d f^{2} - 6 a b^{2} d^{2} e f + 2 b^{3} c^{2} f^{2} + 2 b^{3} c d e f + 2 b^{3} d^{2} e^{2}\right )}{5 d^{3} f^{3}} + \frac{\left (e + f x\right )^{\frac{3}{2}} \left (2 a^{3} d^{3} - 6 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - 2 b^{3} c^{3}\right )}{3 d^{4}} + \frac{\sqrt{e + f x} \left (- 2 a^{3} c d^{3} f + 2 a^{3} d^{4} e + 6 a^{2} b c^{2} d^{2} f - 6 a^{2} b c d^{3} e - 6 a b^{2} c^{3} d f + 6 a b^{2} c^{2} d^{2} e + 2 b^{3} c^{4} f - 2 b^{3} c^{3} d e\right )}{d^{5}} + \frac{2 \left (a d - b c\right )^{3} \left (c f - d e\right )^{2} \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{c f - d e}{d}}} \right )}}{d \sqrt{\frac{c f - d e}{d}}} & \text{for}\: \frac{c f - d e}{d} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{- c f + d e}{d}}} \right )}}{d \sqrt{\frac{- c f + d e}{d}}} & \text{for}\: e + f x > \frac{- c f + d e}{d} \wedge \frac{c f - d e}{d} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{- c f + d e}{d}}} \right )}}{d \sqrt{\frac{- c f + d e}{d}}} & \text{for}\: \frac{c f - d e}{d} < 0 \wedge e + f x < \frac{- c f + d e}{d} \end{cases}\right )}{d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**3*(f*x+e)**(3/2)/(d*x+c),x)
[Out]
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GIAC/XCAS [A] time = 0.234999, size = 922, normalized size = 3.78 \[ -\frac{2 \,{\left (b^{3} c^{5} f^{2} - 3 \, a b^{2} c^{4} d f^{2} + 3 \, a^{2} b c^{3} d^{2} f^{2} - a^{3} c^{2} d^{3} f^{2} - 2 \, b^{3} c^{4} d f e + 6 \, a b^{2} c^{3} d^{2} f e - 6 \, a^{2} b c^{2} d^{3} f e + 2 \, a^{3} c d^{4} f e + b^{3} c^{3} d^{2} e^{2} - 3 \, a b^{2} c^{2} d^{3} e^{2} + 3 \, a^{2} b c d^{4} e^{2} - a^{3} d^{5} e^{2}\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{\sqrt{c d f - d^{2} e} d^{5}} + \frac{2 \,{\left (35 \,{\left (f x + e\right )}^{\frac{9}{2}} b^{3} d^{8} f^{24} - 45 \,{\left (f x + e\right )}^{\frac{7}{2}} b^{3} c d^{7} f^{25} + 135 \,{\left (f x + e\right )}^{\frac{7}{2}} a b^{2} d^{8} f^{25} + 63 \,{\left (f x + e\right )}^{\frac{5}{2}} b^{3} c^{2} d^{6} f^{26} - 189 \,{\left (f x + e\right )}^{\frac{5}{2}} a b^{2} c d^{7} f^{26} + 189 \,{\left (f x + e\right )}^{\frac{5}{2}} a^{2} b d^{8} f^{26} - 105 \,{\left (f x + e\right )}^{\frac{3}{2}} b^{3} c^{3} d^{5} f^{27} + 315 \,{\left (f x + e\right )}^{\frac{3}{2}} a b^{2} c^{2} d^{6} f^{27} - 315 \,{\left (f x + e\right )}^{\frac{3}{2}} a^{2} b c d^{7} f^{27} + 105 \,{\left (f x + e\right )}^{\frac{3}{2}} a^{3} d^{8} f^{27} + 315 \, \sqrt{f x + e} b^{3} c^{4} d^{4} f^{28} - 945 \, \sqrt{f x + e} a b^{2} c^{3} d^{5} f^{28} + 945 \, \sqrt{f x + e} a^{2} b c^{2} d^{6} f^{28} - 315 \, \sqrt{f x + e} a^{3} c d^{7} f^{28} - 90 \,{\left (f x + e\right )}^{\frac{7}{2}} b^{3} d^{8} f^{24} e + 63 \,{\left (f x + e\right )}^{\frac{5}{2}} b^{3} c d^{7} f^{25} e - 189 \,{\left (f x + e\right )}^{\frac{5}{2}} a b^{2} d^{8} f^{25} e - 315 \, \sqrt{f x + e} b^{3} c^{3} d^{5} f^{27} e + 945 \, \sqrt{f x + e} a b^{2} c^{2} d^{6} f^{27} e - 945 \, \sqrt{f x + e} a^{2} b c d^{7} f^{27} e + 315 \, \sqrt{f x + e} a^{3} d^{8} f^{27} e + 63 \,{\left (f x + e\right )}^{\frac{5}{2}} b^{3} d^{8} f^{24} e^{2}\right )}}{315 \, d^{9} f^{27}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^3*(f*x + e)^(3/2)/(d*x + c),x, algorithm="giac")
[Out]