Optimal. Leaf size=100 \[ -\frac{6 b^2 (d+e x)^{7/2} (b d-a e)}{7 e^4}+\frac{6 b (d+e x)^{5/2} (b d-a e)^2}{5 e^4}-\frac{2 (d+e x)^{3/2} (b d-a e)^3}{3 e^4}+\frac{2 b^3 (d+e x)^{9/2}}{9 e^4} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0842205, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{6 b^2 (d+e x)^{7/2} (b d-a e)}{7 e^4}+\frac{6 b (d+e x)^{5/2} (b d-a e)^2}{5 e^4}-\frac{2 (d+e x)^{3/2} (b d-a e)^3}{3 e^4}+\frac{2 b^3 (d+e x)^{9/2}}{9 e^4} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)*Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 44.1776, size = 92, normalized size = 0.92 \[ \frac{2 b^{3} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{4}} + \frac{6 b^{2} \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )}{7 e^{4}} + \frac{6 b \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{2}}{5 e^{4}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{3}}{3 e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)*(e*x+d)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.118867, size = 102, normalized size = 1.02 \[ \frac{2 (d+e x)^{3/2} \left (105 a^3 e^3+63 a^2 b e^2 (3 e x-2 d)+9 a b^2 e \left (8 d^2-12 d e x+15 e^2 x^2\right )+b^3 \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )\right )}{315 e^4} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)*Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.012, size = 116, normalized size = 1.2 \[{\frac{70\,{x}^{3}{b}^{3}{e}^{3}+270\,{x}^{2}a{b}^{2}{e}^{3}-60\,{x}^{2}{b}^{3}d{e}^{2}+378\,x{a}^{2}b{e}^{3}-216\,xa{b}^{2}d{e}^{2}+48\,x{b}^{3}{d}^{2}e+210\,{a}^{3}{e}^{3}-252\,{a}^{2}bd{e}^{2}+144\,a{b}^{2}{d}^{2}e-32\,{b}^{3}{d}^{3}}{315\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)*(e*x+d)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.722319, size = 159, normalized size = 1.59 \[ \frac{2 \,{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} b^{3} - 135 \,{\left (b^{3} d - a b^{2} e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 189 \,{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 105 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{315 \, e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)*(b*x + a)*sqrt(e*x + d),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.280715, size = 221, normalized size = 2.21 \[ \frac{2 \,{\left (35 \, b^{3} e^{4} x^{4} - 16 \, b^{3} d^{4} + 72 \, a b^{2} d^{3} e - 126 \, a^{2} b d^{2} e^{2} + 105 \, a^{3} d e^{3} + 5 \,{\left (b^{3} d e^{3} + 27 \, a b^{2} e^{4}\right )} x^{3} - 3 \,{\left (2 \, b^{3} d^{2} e^{2} - 9 \, a b^{2} d e^{3} - 63 \, a^{2} b e^{4}\right )} x^{2} +{\left (8 \, b^{3} d^{3} e - 36 \, a b^{2} d^{2} e^{2} + 63 \, a^{2} b d e^{3} + 105 \, a^{3} e^{4}\right )} x\right )} \sqrt{e x + d}}{315 \, e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)*(b*x + a)*sqrt(e*x + d),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 6.42327, size = 146, normalized size = 1.46 \[ \frac{2 \left (\frac{b^{3} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{3}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (3 a b^{2} e - 3 b^{3} d\right )}{7 e^{3}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (3 a^{2} b e^{2} - 6 a b^{2} d e + 3 b^{3} d^{2}\right )}{5 e^{3}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (a^{3} e^{3} - 3 a^{2} b d e^{2} + 3 a b^{2} d^{2} e - b^{3} d^{3}\right )}{3 e^{3}}\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)*(e*x+d)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.281948, size = 215, normalized size = 2.15 \[ \frac{2}{315} \,{\left (63 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a^{2} b e^{\left (-1\right )} + 9 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} a b^{2} e^{\left (-14\right )} +{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} b^{3} e^{\left (-27\right )} + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)*(b*x + a)*sqrt(e*x + d),x, algorithm="giac")
[Out]