Optimal. Leaf size=240 \[ \frac{2 c (d+e x)^{3/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7}-\frac{2 \sqrt{d+e x} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{e^7}-\frac{6 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7 \sqrt{d+e x}}-\frac{6 c^2 (d+e x)^{5/2} (2 c d-b e)}{5 e^7}-\frac{2 d^3 (c d-b e)^3}{5 e^7 (d+e x)^{5/2}}+\frac{2 d^2 (c d-b e)^2 (2 c d-b e)}{e^7 (d+e x)^{3/2}}+\frac{2 c^3 (d+e x)^{7/2}}{7 e^7} \]
[Out]
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Rubi [A] time = 0.330413, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{2 c (d+e x)^{3/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7}-\frac{2 \sqrt{d+e x} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{e^7}-\frac{6 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7 \sqrt{d+e x}}-\frac{6 c^2 (d+e x)^{5/2} (2 c d-b e)}{5 e^7}-\frac{2 d^3 (c d-b e)^3}{5 e^7 (d+e x)^{5/2}}+\frac{2 d^2 (c d-b e)^2 (2 c d-b e)}{e^7 (d+e x)^{3/2}}+\frac{2 c^3 (d+e x)^{7/2}}{7 e^7} \]
Antiderivative was successfully verified.
[In] Int[(b*x + c*x^2)^3/(d + e*x)^(7/2),x]
[Out]
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Rubi in Sympy [A] time = 54.721, size = 236, normalized size = 0.98 \[ \frac{2 c^{3} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{7}} + \frac{6 c^{2} \left (d + e x\right )^{\frac{5}{2}} \left (b e - 2 c d\right )}{5 e^{7}} + \frac{2 c \left (d + e x\right )^{\frac{3}{2}} \left (b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{e^{7}} + \frac{2 d^{3} \left (b e - c d\right )^{3}}{5 e^{7} \left (d + e x\right )^{\frac{5}{2}}} - \frac{2 d^{2} \left (b e - 2 c d\right ) \left (b e - c d\right )^{2}}{e^{7} \left (d + e x\right )^{\frac{3}{2}}} + \frac{6 d \left (b e - c d\right ) \left (b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{e^{7} \sqrt{d + e x}} + \frac{2 \sqrt{d + e x} \left (b e - 2 c d\right ) \left (b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right )}{e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x)**3/(e*x+d)**(7/2),x)
[Out]
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Mathematica [A] time = 0.318642, size = 211, normalized size = 0.88 \[ \frac{2 \sqrt{d+e x} \left (35 b^3 e^3+c e x \left (35 b^2 e^2-133 b c d e+106 c^2 d^2\right )-385 b^2 c d e^2-\frac{105 d \left (-b^3 e^3+6 b^2 c d e^2-10 b c^2 d^2 e+5 c^3 d^3\right )}{d+e x}+896 b c^2 d^2 e-3 c^2 e^2 x^2 (9 c d-7 b e)-\frac{7 d^3 (c d-b e)^3}{(d+e x)^3}+\frac{35 d^2 (c d-b e)^2 (2 c d-b e)}{(d+e x)^2}-562 c^3 d^3+5 c^3 e^3 x^3\right )}{35 e^7} \]
Antiderivative was successfully verified.
[In] Integrate[(b*x + c*x^2)^3/(d + e*x)^(7/2),x]
[Out]
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Maple [A] time = 0.011, size = 286, normalized size = 1.2 \[{\frac{10\,{c}^{3}{x}^{6}{e}^{6}+42\,b{c}^{2}{e}^{6}{x}^{5}-24\,{c}^{3}d{e}^{5}{x}^{5}+70\,{b}^{2}c{e}^{6}{x}^{4}-140\,b{c}^{2}d{e}^{5}{x}^{4}+80\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}+70\,{b}^{3}{e}^{6}{x}^{3}-560\,{b}^{2}cd{e}^{5}{x}^{3}+1120\,b{c}^{2}{d}^{2}{e}^{4}{x}^{3}-640\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}+420\,{b}^{3}d{e}^{5}{x}^{2}-3360\,{b}^{2}c{d}^{2}{e}^{4}{x}^{2}+6720\,b{c}^{2}{d}^{3}{e}^{3}{x}^{2}-3840\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}+560\,{b}^{3}{d}^{2}{e}^{4}x-4480\,{b}^{2}c{d}^{3}{e}^{3}x+8960\,b{c}^{2}{d}^{4}{e}^{2}x-5120\,{c}^{3}{d}^{5}ex+224\,{b}^{3}{d}^{3}{e}^{3}-1792\,{b}^{2}c{d}^{4}{e}^{2}+3584\,b{c}^{2}{d}^{5}e-2048\,{c}^{3}{d}^{6}}{35\,{e}^{7}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x)^3/(e*x+d)^(7/2),x)
[Out]
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Maxima [A] time = 0.704831, size = 374, normalized size = 1.56 \[ \frac{2 \,{\left (\frac{5 \,{\left (e x + d\right )}^{\frac{7}{2}} c^{3} - 21 \,{\left (2 \, c^{3} d - b c^{2} e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 35 \,{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \sqrt{e x + d}}{e^{6}} - \frac{7 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} + 15 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )}{\left (e x + d\right )}^{2} - 5 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )}{\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac{5}{2}} e^{6}}\right )}}{35 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^3/(e*x + d)^(7/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.212086, size = 393, normalized size = 1.64 \[ \frac{2 \,{\left (5 \, c^{3} e^{6} x^{6} - 1024 \, c^{3} d^{6} + 1792 \, b c^{2} d^{5} e - 896 \, b^{2} c d^{4} e^{2} + 112 \, b^{3} d^{3} e^{3} - 3 \,{\left (4 \, c^{3} d e^{5} - 7 \, b c^{2} e^{6}\right )} x^{5} + 5 \,{\left (8 \, c^{3} d^{2} e^{4} - 14 \, b c^{2} d e^{5} + 7 \, b^{2} c e^{6}\right )} x^{4} - 5 \,{\left (64 \, c^{3} d^{3} e^{3} - 112 \, b c^{2} d^{2} e^{4} + 56 \, b^{2} c d e^{5} - 7 \, b^{3} e^{6}\right )} x^{3} - 30 \,{\left (64 \, c^{3} d^{4} e^{2} - 112 \, b c^{2} d^{3} e^{3} + 56 \, b^{2} c d^{2} e^{4} - 7 \, b^{3} d e^{5}\right )} x^{2} - 40 \,{\left (64 \, c^{3} d^{5} e - 112 \, b c^{2} d^{4} e^{2} + 56 \, b^{2} c d^{3} e^{3} - 7 \, b^{3} d^{2} e^{4}\right )} x\right )}}{35 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )} \sqrt{e x + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^3/(e*x + d)^(7/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3} \left (b + c x\right )^{3}}{\left (d + e x\right )^{\frac{7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x)**3/(e*x+d)**(7/2),x)
[Out]
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GIAC/XCAS [A] time = 0.212373, size = 485, normalized size = 2.02 \[ \frac{2}{35} \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{3} e^{42} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{3} d e^{42} + 175 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{2} e^{42} - 700 \, \sqrt{x e + d} c^{3} d^{3} e^{42} + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} b c^{2} e^{43} - 175 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{2} d e^{43} + 1050 \, \sqrt{x e + d} b c^{2} d^{2} e^{43} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} c e^{44} - 420 \, \sqrt{x e + d} b^{2} c d e^{44} + 35 \, \sqrt{x e + d} b^{3} e^{45}\right )} e^{\left (-49\right )} - \frac{2 \,{\left (75 \,{\left (x e + d\right )}^{2} c^{3} d^{4} - 10 \,{\left (x e + d\right )} c^{3} d^{5} + c^{3} d^{6} - 150 \,{\left (x e + d\right )}^{2} b c^{2} d^{3} e + 25 \,{\left (x e + d\right )} b c^{2} d^{4} e - 3 \, b c^{2} d^{5} e + 90 \,{\left (x e + d\right )}^{2} b^{2} c d^{2} e^{2} - 20 \,{\left (x e + d\right )} b^{2} c d^{3} e^{2} + 3 \, b^{2} c d^{4} e^{2} - 15 \,{\left (x e + d\right )}^{2} b^{3} d e^{3} + 5 \,{\left (x e + d\right )} b^{3} d^{2} e^{3} - b^{3} d^{3} e^{3}\right )} e^{\left (-7\right )}}{5 \,{\left (x e + d\right )}^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^3/(e*x + d)^(7/2),x, algorithm="giac")
[Out]