Optimal. Leaf size=242 \[ \frac{6 c (d+e x)^{7/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{7 e^7}-\frac{2 (d+e x)^{5/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{5 e^7}+\frac{2 d (d+e x)^{3/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7}-\frac{2 c^2 (d+e x)^{9/2} (2 c d-b e)}{3 e^7}-\frac{2 d^3 (c d-b e)^3}{e^7 \sqrt{d+e x}}-\frac{6 d^2 \sqrt{d+e x} (c d-b e)^2 (2 c d-b e)}{e^7}+\frac{2 c^3 (d+e x)^{11/2}}{11 e^7} \]
[Out]
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Rubi [A] time = 0.301322, antiderivative size = 242, 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 \[ \frac{6 c (d+e x)^{7/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{7 e^7}-\frac{2 (d+e x)^{5/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{5 e^7}+\frac{2 d (d+e x)^{3/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7}-\frac{2 c^2 (d+e x)^{9/2} (2 c d-b e)}{3 e^7}-\frac{2 d^3 (c d-b e)^3}{e^7 \sqrt{d+e x}}-\frac{6 d^2 \sqrt{d+e x} (c d-b e)^2 (2 c d-b e)}{e^7}+\frac{2 c^3 (d+e x)^{11/2}}{11 e^7} \]
Antiderivative was successfully verified.
[In] Int[(b*x + c*x^2)^3/(d + e*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 55.1187, size = 238, normalized size = 0.98 \[ \frac{2 c^{3} \left (d + e x\right )^{\frac{11}{2}}}{11 e^{7}} + \frac{2 c^{2} \left (d + e x\right )^{\frac{9}{2}} \left (b e - 2 c d\right )}{3 e^{7}} + \frac{6 c \left (d + e x\right )^{\frac{7}{2}} \left (b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{7 e^{7}} + \frac{2 d^{3} \left (b e - c d\right )^{3}}{e^{7} \sqrt{d + e x}} + \frac{6 d^{2} \sqrt{d + e x} \left (b e - 2 c d\right ) \left (b e - c d\right )^{2}}{e^{7}} - \frac{2 d \left (d + e x\right )^{\frac{3}{2}} \left (b e - c d\right ) \left (b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{e^{7}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (b e - 2 c d\right ) \left (b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right )}{5 e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x)**3/(e*x+d)**(3/2),x)
[Out]
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Mathematica [A] time = 0.209297, size = 231, normalized size = 0.95 \[ \frac{2 \left (231 b^3 e^3 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+99 b^2 c e^2 \left (-128 d^4-64 d^3 e x+16 d^2 e^2 x^2-8 d e^3 x^3+5 e^4 x^4\right )+55 b c^2 e \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )-5 c^3 \left (1024 d^6+512 d^5 e x-128 d^4 e^2 x^2+64 d^3 e^3 x^3-40 d^2 e^4 x^4+28 d e^5 x^5-21 e^6 x^6\right )\right )}{1155 e^7 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Integrate[(b*x + c*x^2)^3/(d + e*x)^(3/2),x]
[Out]
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Maple [A] time = 0.01, size = 286, normalized size = 1.2 \[{\frac{210\,{c}^{3}{x}^{6}{e}^{6}+770\,b{c}^{2}{e}^{6}{x}^{5}-280\,{c}^{3}d{e}^{5}{x}^{5}+990\,{b}^{2}c{e}^{6}{x}^{4}-1100\,b{c}^{2}d{e}^{5}{x}^{4}+400\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}+462\,{b}^{3}{e}^{6}{x}^{3}-1584\,{b}^{2}cd{e}^{5}{x}^{3}+1760\,b{c}^{2}{d}^{2}{e}^{4}{x}^{3}-640\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}-924\,{b}^{3}d{e}^{5}{x}^{2}+3168\,{b}^{2}c{d}^{2}{e}^{4}{x}^{2}-3520\,b{c}^{2}{d}^{3}{e}^{3}{x}^{2}+1280\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}+3696\,{b}^{3}{d}^{2}{e}^{4}x-12672\,{b}^{2}c{d}^{3}{e}^{3}x+14080\,b{c}^{2}{d}^{4}{e}^{2}x-5120\,{c}^{3}{d}^{5}ex+7392\,{b}^{3}{d}^{3}{e}^{3}-25344\,{b}^{2}c{d}^{4}{e}^{2}+28160\,b{c}^{2}{d}^{5}e-10240\,{c}^{3}{d}^{6}}{1155\,{e}^{7}}{\frac{1}{\sqrt{ex+d}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x)^3/(e*x+d)^(3/2),x)
[Out]
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Maxima [A] time = 0.700121, size = 377, normalized size = 1.56 \[ \frac{2 \,{\left (\frac{105 \,{\left (e x + d\right )}^{\frac{11}{2}} c^{3} - 385 \,{\left (2 \, c^{3} d - b c^{2} e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 495 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 231 \,{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} - b^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 1155 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 3465 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )} \sqrt{e x + d}}{e^{6}} - \frac{1155 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )}}{\sqrt{e x + d} e^{6}}\right )}}{1155 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^3/(e*x + d)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.220448, size = 365, normalized size = 1.51 \[ \frac{2 \,{\left (105 \, c^{3} e^{6} x^{6} - 5120 \, c^{3} d^{6} + 14080 \, b c^{2} d^{5} e - 12672 \, b^{2} c d^{4} e^{2} + 3696 \, b^{3} d^{3} e^{3} - 35 \,{\left (4 \, c^{3} d e^{5} - 11 \, b c^{2} e^{6}\right )} x^{5} + 5 \,{\left (40 \, c^{3} d^{2} e^{4} - 110 \, b c^{2} d e^{5} + 99 \, b^{2} c e^{6}\right )} x^{4} -{\left (320 \, c^{3} d^{3} e^{3} - 880 \, b c^{2} d^{2} e^{4} + 792 \, b^{2} c d e^{5} - 231 \, b^{3} e^{6}\right )} x^{3} + 2 \,{\left (320 \, c^{3} d^{4} e^{2} - 880 \, b c^{2} d^{3} e^{3} + 792 \, b^{2} c d^{2} e^{4} - 231 \, b^{3} d e^{5}\right )} x^{2} - 8 \,{\left (320 \, c^{3} d^{5} e - 880 \, b c^{2} d^{4} e^{2} + 792 \, b^{2} c d^{3} e^{3} - 231 \, b^{3} d^{2} e^{4}\right )} x\right )}}{1155 \, \sqrt{e x + d} e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^3/(e*x + d)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3} \left (b + c x\right )^{3}}{\left (d + e x\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x)**3/(e*x+d)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.211046, size = 501, normalized size = 2.07 \[ \frac{2}{1155} \,{\left (105 \,{\left (x e + d\right )}^{\frac{11}{2}} c^{3} e^{70} - 770 \,{\left (x e + d\right )}^{\frac{9}{2}} c^{3} d e^{70} + 2475 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{3} d^{2} e^{70} - 4620 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{3} d^{3} e^{70} + 5775 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{4} e^{70} - 6930 \, \sqrt{x e + d} c^{3} d^{5} e^{70} + 385 \,{\left (x e + d\right )}^{\frac{9}{2}} b c^{2} e^{71} - 2475 \,{\left (x e + d\right )}^{\frac{7}{2}} b c^{2} d e^{71} + 6930 \,{\left (x e + d\right )}^{\frac{5}{2}} b c^{2} d^{2} e^{71} - 11550 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{2} d^{3} e^{71} + 17325 \, \sqrt{x e + d} b c^{2} d^{4} e^{71} + 495 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{2} c e^{72} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{2} c d e^{72} + 6930 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} c d^{2} e^{72} - 13860 \, \sqrt{x e + d} b^{2} c d^{3} e^{72} + 231 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{3} e^{73} - 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} d e^{73} + 3465 \, \sqrt{x e + d} b^{3} d^{2} e^{73}\right )} e^{\left (-77\right )} - \frac{2 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} e^{\left (-7\right )}}{\sqrt{x e + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^3/(e*x + d)^(3/2),x, algorithm="giac")
[Out]