Optimal. Leaf size=244 \[ \frac{2 c (d+e x)^{9/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{3 e^7}-\frac{2 (d+e x)^{7/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{7 e^7}+\frac{6 d (d+e x)^{5/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7}-\frac{6 c^2 (d+e x)^{11/2} (2 c d-b e)}{11 e^7}+\frac{2 d^3 \sqrt{d+e x} (c d-b e)^3}{e^7}-\frac{2 d^2 (d+e x)^{3/2} (c d-b e)^2 (2 c d-b e)}{e^7}+\frac{2 c^3 (d+e x)^{13/2}}{13 e^7} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.30202, antiderivative size = 244, 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 \[ \frac{2 c (d+e x)^{9/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{3 e^7}-\frac{2 (d+e x)^{7/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{7 e^7}+\frac{6 d (d+e x)^{5/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7}-\frac{6 c^2 (d+e x)^{11/2} (2 c d-b e)}{11 e^7}+\frac{2 d^3 \sqrt{d+e x} (c d-b e)^3}{e^7}-\frac{2 d^2 (d+e x)^{3/2} (c d-b e)^2 (2 c d-b e)}{e^7}+\frac{2 c^3 (d+e x)^{13/2}}{13 e^7} \]
Antiderivative was successfully verified.
[In] Int[(b*x + c*x^2)^3/Sqrt[d + e*x],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 55.6525, size = 240, normalized size = 0.98 \[ \frac{2 c^{3} \left (d + e x\right )^{\frac{13}{2}}}{13 e^{7}} + \frac{6 c^{2} \left (d + e x\right )^{\frac{11}{2}} \left (b e - 2 c d\right )}{11 e^{7}} + \frac{2 c \left (d + e x\right )^{\frac{9}{2}} \left (b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{3 e^{7}} - \frac{2 d^{3} \sqrt{d + e x} \left (b e - c d\right )^{3}}{e^{7}} + \frac{2 d^{2} \left (d + e x\right )^{\frac{3}{2}} \left (b e - 2 c d\right ) \left (b e - c d\right )^{2}}{e^{7}} - \frac{6 d \left (d + e x\right )^{\frac{5}{2}} \left (b e - c d\right ) \left (b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{5 e^{7}} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (b e - 2 c d\right ) \left (b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right )}{7 e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x)**3/(e*x+d)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.166496, size = 232, normalized size = 0.95 \[ \frac{2 \sqrt{d+e x} \left (429 b^3 e^3 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+143 b^2 c e^2 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )+65 b c^2 e \left (-256 d^5+128 d^4 e x-96 d^3 e^2 x^2+80 d^2 e^3 x^3-70 d e^4 x^4+63 e^5 x^5\right )+5 c^3 \left (1024 d^6-512 d^5 e x+384 d^4 e^2 x^2-320 d^3 e^3 x^3+280 d^2 e^4 x^4-252 d e^5 x^5+231 e^6 x^6\right )\right )}{15015 e^7} \]
Antiderivative was successfully verified.
[In] Integrate[(b*x + c*x^2)^3/Sqrt[d + e*x],x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.01, size = 286, normalized size = 1.2 \[ -{\frac{-2310\,{c}^{3}{x}^{6}{e}^{6}-8190\,b{c}^{2}{e}^{6}{x}^{5}+2520\,{c}^{3}d{e}^{5}{x}^{5}-10010\,{b}^{2}c{e}^{6}{x}^{4}+9100\,b{c}^{2}d{e}^{5}{x}^{4}-2800\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}-4290\,{b}^{3}{e}^{6}{x}^{3}+11440\,{b}^{2}cd{e}^{5}{x}^{3}-10400\,b{c}^{2}{d}^{2}{e}^{4}{x}^{3}+3200\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}+5148\,{b}^{3}d{e}^{5}{x}^{2}-13728\,{b}^{2}c{d}^{2}{e}^{4}{x}^{2}+12480\,b{c}^{2}{d}^{3}{e}^{3}{x}^{2}-3840\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}-6864\,{b}^{3}{d}^{2}{e}^{4}x+18304\,{b}^{2}c{d}^{3}{e}^{3}x-16640\,b{c}^{2}{d}^{4}{e}^{2}x+5120\,{c}^{3}{d}^{5}ex+13728\,{b}^{3}{d}^{3}{e}^{3}-36608\,{b}^{2}c{d}^{4}{e}^{2}+33280\,b{c}^{2}{d}^{5}e-10240\,{c}^{3}{d}^{6}}{15015\,{e}^{7}}\sqrt{ex+d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x)^3/(e*x+d)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.698105, size = 389, normalized size = 1.59 \[ \frac{2 \,{\left (\frac{429 \,{\left (5 \,{\left (e x + d\right )}^{\frac{7}{2}} - 21 \,{\left (e x + d\right )}^{\frac{5}{2}} d + 35 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{2} - 35 \, \sqrt{e x + d} d^{3}\right )} b^{3}}{e^{3}} + \frac{143 \,{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} - 180 \,{\left (e x + d\right )}^{\frac{7}{2}} d + 378 \,{\left (e x + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{e x + d} d^{4}\right )} b^{2} c}{e^{4}} + \frac{65 \,{\left (63 \,{\left (e x + d\right )}^{\frac{11}{2}} - 385 \,{\left (e x + d\right )}^{\frac{9}{2}} d + 990 \,{\left (e x + d\right )}^{\frac{7}{2}} d^{2} - 1386 \,{\left (e x + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{4} - 693 \, \sqrt{e x + d} d^{5}\right )} b c^{2}}{e^{5}} + \frac{5 \,{\left (231 \,{\left (e x + d\right )}^{\frac{13}{2}} - 1638 \,{\left (e x + d\right )}^{\frac{11}{2}} d + 5005 \,{\left (e x + d\right )}^{\frac{9}{2}} d^{2} - 8580 \,{\left (e x + d\right )}^{\frac{7}{2}} d^{3} + 9009 \,{\left (e x + d\right )}^{\frac{5}{2}} d^{4} - 6006 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{5} + 3003 \, \sqrt{e x + d} d^{6}\right )} c^{3}}{e^{6}}\right )}}{15015 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^3/sqrt(e*x + d),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.220754, size = 365, normalized size = 1.5 \[ \frac{2 \,{\left (1155 \, c^{3} e^{6} x^{6} + 5120 \, c^{3} d^{6} - 16640 \, b c^{2} d^{5} e + 18304 \, b^{2} c d^{4} e^{2} - 6864 \, b^{3} d^{3} e^{3} - 315 \,{\left (4 \, c^{3} d e^{5} - 13 \, b c^{2} e^{6}\right )} x^{5} + 35 \,{\left (40 \, c^{3} d^{2} e^{4} - 130 \, b c^{2} d e^{5} + 143 \, b^{2} c e^{6}\right )} x^{4} - 5 \,{\left (320 \, c^{3} d^{3} e^{3} - 1040 \, b c^{2} d^{2} e^{4} + 1144 \, b^{2} c d e^{5} - 429 \, b^{3} e^{6}\right )} x^{3} + 6 \,{\left (320 \, c^{3} d^{4} e^{2} - 1040 \, b c^{2} d^{3} e^{3} + 1144 \, b^{2} c d^{2} e^{4} - 429 \, b^{3} d e^{5}\right )} x^{2} - 8 \,{\left (320 \, c^{3} d^{5} e - 1040 \, b c^{2} d^{4} e^{2} + 1144 \, b^{2} c d^{3} e^{3} - 429 \, b^{3} d^{2} e^{4}\right )} x\right )} \sqrt{e x + d}}{15015 \, e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^3/sqrt(e*x + d),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 68.2302, size = 745, normalized size = 3.05 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x)**3/(e*x+d)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.209438, size = 471, normalized size = 1.93 \[ \frac{2}{15015} \,{\left (429 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{18} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{18} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{18} - 35 \, \sqrt{x e + d} d^{3} e^{18}\right )} b^{3} e^{\left (-21\right )} + 143 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{32} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{32} + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{32} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{32} + 315 \, \sqrt{x e + d} d^{4} e^{32}\right )} b^{2} c e^{\left (-36\right )} + 65 \,{\left (63 \,{\left (x e + d\right )}^{\frac{11}{2}} e^{50} - 385 \,{\left (x e + d\right )}^{\frac{9}{2}} d e^{50} + 990 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} e^{50} - 1386 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} e^{50} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4} e^{50} - 693 \, \sqrt{x e + d} d^{5} e^{50}\right )} b c^{2} e^{\left (-55\right )} + 5 \,{\left (231 \,{\left (x e + d\right )}^{\frac{13}{2}} e^{72} - 1638 \,{\left (x e + d\right )}^{\frac{11}{2}} d e^{72} + 5005 \,{\left (x e + d\right )}^{\frac{9}{2}} d^{2} e^{72} - 8580 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{3} e^{72} + 9009 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{4} e^{72} - 6006 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{5} e^{72} + 3003 \, \sqrt{x e + d} d^{6} e^{72}\right )} c^{3} e^{\left (-78\right )}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^3/sqrt(e*x + d),x, algorithm="giac")
[Out]