Optimal. Leaf size=244 \[ \frac{6 c (d+e x)^{5/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7}-\frac{2 (d+e x)^{3/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{3 e^7}+\frac{6 d \sqrt{d+e x} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7}-\frac{6 c^2 (d+e x)^{7/2} (2 c d-b e)}{7 e^7}-\frac{2 d^3 (c d-b e)^3}{3 e^7 (d+e x)^{3/2}}+\frac{6 d^2 (c d-b e)^2 (2 c d-b e)}{e^7 \sqrt{d+e x}}+\frac{2 c^3 (d+e x)^{9/2}}{9 e^7} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.338831, 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 \[ \frac{6 c (d+e x)^{5/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7}-\frac{2 (d+e x)^{3/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{3 e^7}+\frac{6 d \sqrt{d+e x} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7}-\frac{6 c^2 (d+e x)^{7/2} (2 c d-b e)}{7 e^7}-\frac{2 d^3 (c d-b e)^3}{3 e^7 (d+e x)^{3/2}}+\frac{6 d^2 (c d-b e)^2 (2 c d-b e)}{e^7 \sqrt{d+e x}}+\frac{2 c^3 (d+e x)^{9/2}}{9 e^7} \]
Antiderivative was successfully verified.
[In] Int[(b*x + c*x^2)^3/(d + e*x)^(5/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 54.9224, size = 240, normalized size = 0.98 \[ \frac{2 c^{3} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{7}} + \frac{6 c^{2} \left (d + e x\right )^{\frac{7}{2}} \left (b e - 2 c d\right )}{7 e^{7}} + \frac{6 c \left (d + e x\right )^{\frac{5}{2}} \left (b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{5 e^{7}} + \frac{2 d^{3} \left (b e - c d\right )^{3}}{3 e^{7} \left (d + e x\right )^{\frac{3}{2}}} - \frac{6 d^{2} \left (b e - 2 c d\right ) \left (b e - c d\right )^{2}}{e^{7} \sqrt{d + e x}} - \frac{6 d \sqrt{d + e x} \left (b e - c d\right ) \left (b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{e^{7}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (b e - 2 c d\right ) \left (b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right )}{3 e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x)**3/(e*x+d)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.212441, size = 231, normalized size = 0.95 \[ \sqrt{d+e x} \left (\frac{2 c x^2 \left (63 b^2 e^2-180 b c d e+115 c^2 d^2\right )}{105 e^5}-\frac{2 d \left (840 b^3 e^3-4599 b^2 c d e^2+7110 b c^2 d^2 e-3335 c^3 d^3\right )}{315 e^7}+\frac{2 x \left (105 b^3 e^3-882 b^2 c d e^2+1665 b c^2 d^2 e-880 c^3 d^3\right )}{315 e^6}-\frac{2 c^2 x^3 (26 c d-27 b e)}{63 e^4}-\frac{2 d^3 (c d-b e)^3}{3 e^7 (d+e x)^2}+\frac{6 d^2 (c d-b e)^2 (2 c d-b e)}{e^7 (d+e x)}+\frac{2 c^3 x^4}{9 e^3}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(b*x + c*x^2)^3/(d + e*x)^(5/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.01, size = 286, normalized size = 1.2 \[ -{\frac{-70\,{c}^{3}{x}^{6}{e}^{6}-270\,b{c}^{2}{e}^{6}{x}^{5}+120\,{c}^{3}d{e}^{5}{x}^{5}-378\,{b}^{2}c{e}^{6}{x}^{4}+540\,b{c}^{2}d{e}^{5}{x}^{4}-240\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}-210\,{b}^{3}{e}^{6}{x}^{3}+1008\,{b}^{2}cd{e}^{5}{x}^{3}-1440\,b{c}^{2}{d}^{2}{e}^{4}{x}^{3}+640\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}+1260\,{b}^{3}d{e}^{5}{x}^{2}-6048\,{b}^{2}c{d}^{2}{e}^{4}{x}^{2}+8640\,b{c}^{2}{d}^{3}{e}^{3}{x}^{2}-3840\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}+5040\,{b}^{3}{d}^{2}{e}^{4}x-24192\,{b}^{2}c{d}^{3}{e}^{3}x+34560\,b{c}^{2}{d}^{4}{e}^{2}x-15360\,{c}^{3}{d}^{5}ex+3360\,{b}^{3}{d}^{3}{e}^{3}-16128\,{b}^{2}c{d}^{4}{e}^{2}+23040\,b{c}^{2}{d}^{5}e-10240\,{c}^{3}{d}^{6}}{315\,{e}^{7}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x)^3/(e*x+d)^(5/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.702647, size = 374, normalized size = 1.53 \[ \frac{2 \,{\left (\frac{35 \,{\left (e x + d\right )}^{\frac{9}{2}} c^{3} - 135 \,{\left (2 \, c^{3} d - b c^{2} e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 189 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 105 \,{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} - b^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 945 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )} \sqrt{e x + d}}{e^{6}} - \frac{105 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} - 9 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )}{\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{6}}\right )}}{315 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^3/(e*x + d)^(5/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.22097, size = 378, normalized size = 1.55 \[ \frac{2 \,{\left (35 \, c^{3} e^{6} x^{6} + 5120 \, c^{3} d^{6} - 11520 \, b c^{2} d^{5} e + 8064 \, b^{2} c d^{4} e^{2} - 1680 \, b^{3} d^{3} e^{3} - 15 \,{\left (4 \, c^{3} d e^{5} - 9 \, b c^{2} e^{6}\right )} x^{5} + 3 \,{\left (40 \, c^{3} d^{2} e^{4} - 90 \, b c^{2} d e^{5} + 63 \, b^{2} c e^{6}\right )} x^{4} -{\left (320 \, c^{3} d^{3} e^{3} - 720 \, b c^{2} d^{2} e^{4} + 504 \, b^{2} c d e^{5} - 105 \, b^{3} e^{6}\right )} x^{3} + 6 \,{\left (320 \, c^{3} d^{4} e^{2} - 720 \, b c^{2} d^{3} e^{3} + 504 \, b^{2} c d^{2} e^{4} - 105 \, b^{3} d e^{5}\right )} x^{2} + 24 \,{\left (320 \, c^{3} d^{5} e - 720 \, b c^{2} d^{4} e^{2} + 504 \, b^{2} c d^{3} e^{3} - 105 \, b^{3} d^{2} e^{4}\right )} x\right )}}{315 \,{\left (e^{8} x + d e^{7}\right )} \sqrt{e x + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^3/(e*x + d)^(5/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3} \left (b + c x\right )^{3}}{\left (d + e x\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x)**3/(e*x+d)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.21352, size = 487, normalized size = 2. \[ \frac{2}{315} \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} c^{3} e^{56} - 270 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{3} d e^{56} + 945 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{3} d^{2} e^{56} - 2100 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{3} e^{56} + 4725 \, \sqrt{x e + d} c^{3} d^{4} e^{56} + 135 \,{\left (x e + d\right )}^{\frac{7}{2}} b c^{2} e^{57} - 945 \,{\left (x e + d\right )}^{\frac{5}{2}} b c^{2} d e^{57} + 3150 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{2} d^{2} e^{57} - 9450 \, \sqrt{x e + d} b c^{2} d^{3} e^{57} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{2} c e^{58} - 1260 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} c d e^{58} + 5670 \, \sqrt{x e + d} b^{2} c d^{2} e^{58} + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} e^{59} - 945 \, \sqrt{x e + d} b^{3} d e^{59}\right )} e^{\left (-63\right )} + \frac{2 \,{\left (18 \,{\left (x e + d\right )} c^{3} d^{5} - c^{3} d^{6} - 45 \,{\left (x e + d\right )} b c^{2} d^{4} e + 3 \, b c^{2} d^{5} e + 36 \,{\left (x e + d\right )} b^{2} c d^{3} e^{2} - 3 \, b^{2} c d^{4} e^{2} - 9 \,{\left (x e + d\right )} b^{3} d^{2} e^{3} + b^{3} d^{3} e^{3}\right )} e^{\left (-7\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^3/(e*x + d)^(5/2),x, algorithm="giac")
[Out]