Optimal. Leaf size=128 \[ -\frac{2 b (d+e x)^{7/2} (-2 a B e-A b e+3 b B d)}{7 e^4}+\frac{2 (d+e x)^{5/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{5 e^4}-\frac{2 (d+e x)^{3/2} (b d-a e)^2 (B d-A e)}{3 e^4}+\frac{2 b^2 B (d+e x)^{9/2}}{9 e^4} \]
[Out]
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Rubi [A] time = 0.155805, antiderivative size = 128, 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 \[ -\frac{2 b (d+e x)^{7/2} (-2 a B e-A b e+3 b B d)}{7 e^4}+\frac{2 (d+e x)^{5/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{5 e^4}-\frac{2 (d+e x)^{3/2} (b d-a e)^2 (B d-A e)}{3 e^4}+\frac{2 b^2 B (d+e x)^{9/2}}{9 e^4} \]
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]
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Rubi in Sympy [A] time = 54.0808, size = 126, normalized size = 0.98 \[ \frac{2 B b^{2} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{4}} + \frac{2 b \left (d + e x\right )^{\frac{7}{2}} \left (A b e + 2 B a e - 3 B b d\right )}{7 e^{4}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right ) \left (2 A b e + B a e - 3 B b d\right )}{5 e^{4}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (A e - B d\right ) \left (a e - b d\right )^{2}}{3 e^{4}} \]
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]
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Mathematica [A] time = 0.187455, size = 138, normalized size = 1.08 \[ \frac{2 (d+e x)^{3/2} \left (21 a^2 e^2 (5 A e-2 B d+3 B e x)+6 a b e \left (7 A e (3 e x-2 d)+B \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )+b^2 \left (3 A e \left (8 d^2-12 d e x+15 e^2 x^2\right )+B \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )\right )\right )}{315 e^4} \]
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]
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Maple [A] time = 0.015, size = 169, normalized size = 1.3 \[{\frac{70\,B{x}^{3}{b}^{2}{e}^{3}+90\,A{b}^{2}{e}^{3}{x}^{2}+180\,Bab{e}^{3}{x}^{2}-60\,B{b}^{2}d{e}^{2}{x}^{2}+252\,Axab{e}^{3}-72\,Ax{b}^{2}d{e}^{2}+126\,Bx{a}^{2}{e}^{3}-144\,Bxabd{e}^{2}+48\,B{b}^{2}{d}^{2}ex+210\,A{a}^{2}{e}^{3}-168\,Aabd{e}^{2}+48\,A{b}^{2}{d}^{2}e-84\,Bd{e}^{2}{a}^{2}+96\,B{d}^{2}abe-32\,B{b}^{2}{d}^{3}}{315\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \]
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]
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Maxima [A] time = 0.735892, size = 215, normalized size = 1.68 \[ \frac{2 \,{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} B b^{2} - 45 \,{\left (3 \, B b^{2} d -{\left (2 \, B a b + A b^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 63 \,{\left (3 \, B b^{2} d^{2} - 2 \,{\left (2 \, B a b + A b^{2}\right )} d e +{\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 105 \,{\left (B b^{2} d^{3} - A a^{2} e^{3} -{\left (2 \, B a b + A b^{2}\right )} d^{2} e +{\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{315 \, e^{4}} \]
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]
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Fricas [A] time = 0.304369, size = 297, normalized size = 2.32 \[ \frac{2 \,{\left (35 \, B b^{2} e^{4} x^{4} - 16 \, B b^{2} d^{4} + 105 \, A a^{2} d e^{3} + 24 \,{\left (2 \, B a b + A b^{2}\right )} d^{3} e - 42 \,{\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{2} + 5 \,{\left (B b^{2} d e^{3} + 9 \,{\left (2 \, B a b + A b^{2}\right )} e^{4}\right )} x^{3} - 3 \,{\left (2 \, B b^{2} d^{2} e^{2} - 3 \,{\left (2 \, B a b + A b^{2}\right )} d e^{3} - 21 \,{\left (B a^{2} + 2 \, A a b\right )} e^{4}\right )} x^{2} +{\left (8 \, B b^{2} d^{3} e + 105 \, A a^{2} e^{4} - 12 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e^{2} + 21 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{3}\right )} x\right )} \sqrt{e x + d}}{315 \, e^{4}} \]
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]
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Sympy [A] time = 8.42333, size = 201, normalized size = 1.57 \[ \frac{2 \left (\frac{B b^{2} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{3}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (A b^{2} e + 2 B a b e - 3 B b^{2} d\right )}{7 e^{3}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (2 A a b e^{2} - 2 A b^{2} d e + B a^{2} e^{2} - 4 B a b d e + 3 B b^{2} d^{2}\right )}{5 e^{3}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (A a^{2} e^{3} - 2 A a b d e^{2} + A b^{2} d^{2} e - B a^{2} d e^{2} + 2 B a b d^{2} e - B b^{2} d^{3}\right )}{3 e^{3}}\right )}{e} \]
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]
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GIAC/XCAS [A] time = 0.289927, size = 321, normalized size = 2.51 \[ \frac{2}{315} \,{\left (21 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} B a^{2} e^{\left (-1\right )} + 42 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} A a b e^{\left (-1\right )} + 6 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} B a b e^{\left (-14\right )} + 3 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} A b^{2} e^{\left (-14\right )} +{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} B b^{2} e^{\left (-27\right )} + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} A a^{2}\right )} e^{\left (-1\right )} \]
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]