Optimal. Leaf size=298 \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (-3 a B e-A b e+4 b B d)}{6 e^5 (a+b x) (d+e x)^6}-\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-a B e-A b e+2 b B d)}{7 e^5 (a+b x) (d+e x)^7}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{8 e^5 (a+b x) (d+e x)^8}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{9 e^5 (a+b x) (d+e x)^9}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x) (d+e x)^5} \]
[Out]
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Rubi [A] time = 0.598178, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (-3 a B e-A b e+4 b B d)}{6 e^5 (a+b x) (d+e x)^6}-\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-a B e-A b e+2 b B d)}{7 e^5 (a+b x) (d+e x)^7}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{8 e^5 (a+b x) (d+e x)^8}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{9 e^5 (a+b x) (d+e x)^9}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x) (d+e x)^5} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^10,x]
[Out]
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Rubi in Sympy [A] time = 54.3923, size = 304, normalized size = 1.02 \[ \frac{b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (5 A b e - 9 B a e + 4 B b d\right )}{420 e^{4} \left (d + e x\right )^{6} \left (a e - b d\right )} - \frac{b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (5 A b e - 9 B a e + 4 B b d\right )}{2520 e^{5} \left (a + b x\right ) \left (d + e x\right )^{6}} + \frac{b \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (5 A b e - 9 B a e + 4 B b d\right )}{504 e^{3} \left (d + e x\right )^{7} \left (a e - b d\right )} - \frac{\left (2 a + 2 b x\right ) \left (A e - B d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{18 e \left (d + e x\right )^{9} \left (a e - b d\right )} + \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (5 A b e - 9 B a e + 4 B b d\right )}{72 e^{2} \left (d + e x\right )^{8} \left (a e - b d\right )} \]
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)**(3/2)/(e*x+d)**10,x)
[Out]
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Mathematica [A] time = 0.2023, size = 232, normalized size = 0.78 \[ -\frac{\sqrt{(a+b x)^2} \left (35 a^3 e^3 (8 A e+B (d+9 e x))+15 a^2 b e^2 \left (7 A e (d+9 e x)+2 B \left (d^2+9 d e x+36 e^2 x^2\right )\right )+15 a b^2 e \left (2 A e \left (d^2+9 d e x+36 e^2 x^2\right )+B \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )\right )+b^3 \left (5 A e \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+4 B \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )\right )\right )}{2520 e^5 (a+b x) (d+e x)^9} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^10,x]
[Out]
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Maple [A] time = 0.016, size = 317, normalized size = 1.1 \[ -{\frac{504\,B{x}^{4}{b}^{3}{e}^{4}+420\,A{x}^{3}{b}^{3}{e}^{4}+1260\,B{x}^{3}a{b}^{2}{e}^{4}+336\,B{x}^{3}{b}^{3}d{e}^{3}+1080\,A{x}^{2}a{b}^{2}{e}^{4}+180\,A{x}^{2}{b}^{3}d{e}^{3}+1080\,B{x}^{2}{a}^{2}b{e}^{4}+540\,B{x}^{2}a{b}^{2}d{e}^{3}+144\,B{x}^{2}{b}^{3}{d}^{2}{e}^{2}+945\,Ax{a}^{2}b{e}^{4}+270\,Axa{b}^{2}d{e}^{3}+45\,Ax{b}^{3}{d}^{2}{e}^{2}+315\,Bx{a}^{3}{e}^{4}+270\,Bx{a}^{2}bd{e}^{3}+135\,Bxa{b}^{2}{d}^{2}{e}^{2}+36\,Bx{b}^{3}{d}^{3}e+280\,A{a}^{3}{e}^{4}+105\,Ad{e}^{3}{a}^{2}b+30\,Aa{b}^{2}{d}^{2}{e}^{2}+5\,A{b}^{3}{d}^{3}e+35\,Bd{e}^{3}{a}^{3}+30\,B{a}^{2}b{d}^{2}{e}^{2}+15\,Ba{b}^{2}{d}^{3}e+4\,B{b}^{3}{d}^{4}}{2520\,{e}^{5} \left ( ex+d \right ) ^{9} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]
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)^(3/2)/(e*x+d)^10,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)/(e*x + d)^10,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.301739, size = 478, normalized size = 1.6 \[ -\frac{504 \, B b^{3} e^{4} x^{4} + 4 \, B b^{3} d^{4} + 280 \, A a^{3} e^{4} + 5 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 30 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 35 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 84 \,{\left (4 \, B b^{3} d e^{3} + 5 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 36 \,{\left (4 \, B b^{3} d^{2} e^{2} + 5 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 30 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 9 \,{\left (4 \, B b^{3} d^{3} e + 5 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 30 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 35 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{2520 \,{\left (e^{14} x^{9} + 9 \, d e^{13} x^{8} + 36 \, d^{2} e^{12} x^{7} + 84 \, d^{3} e^{11} x^{6} + 126 \, d^{4} e^{10} x^{5} + 126 \, d^{5} e^{9} x^{4} + 84 \, d^{6} e^{8} x^{3} + 36 \, d^{7} e^{7} x^{2} + 9 \, d^{8} e^{6} x + d^{9} e^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)/(e*x + d)^10,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
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)**(3/2)/(e*x+d)**10,x)
[Out]
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GIAC/XCAS [A] time = 0.290789, size = 576, normalized size = 1.93 \[ -\frac{{\left (504 \, B b^{3} x^{4} e^{4}{\rm sign}\left (b x + a\right ) + 336 \, B b^{3} d x^{3} e^{3}{\rm sign}\left (b x + a\right ) + 144 \, B b^{3} d^{2} x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 36 \, B b^{3} d^{3} x e{\rm sign}\left (b x + a\right ) + 4 \, B b^{3} d^{4}{\rm sign}\left (b x + a\right ) + 1260 \, B a b^{2} x^{3} e^{4}{\rm sign}\left (b x + a\right ) + 420 \, A b^{3} x^{3} e^{4}{\rm sign}\left (b x + a\right ) + 540 \, B a b^{2} d x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 180 \, A b^{3} d x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 135 \, B a b^{2} d^{2} x e^{2}{\rm sign}\left (b x + a\right ) + 45 \, A b^{3} d^{2} x e^{2}{\rm sign}\left (b x + a\right ) + 15 \, B a b^{2} d^{3} e{\rm sign}\left (b x + a\right ) + 5 \, A b^{3} d^{3} e{\rm sign}\left (b x + a\right ) + 1080 \, B a^{2} b x^{2} e^{4}{\rm sign}\left (b x + a\right ) + 1080 \, A a b^{2} x^{2} e^{4}{\rm sign}\left (b x + a\right ) + 270 \, B a^{2} b d x e^{3}{\rm sign}\left (b x + a\right ) + 270 \, A a b^{2} d x e^{3}{\rm sign}\left (b x + a\right ) + 30 \, B a^{2} b d^{2} e^{2}{\rm sign}\left (b x + a\right ) + 30 \, A a b^{2} d^{2} e^{2}{\rm sign}\left (b x + a\right ) + 315 \, B a^{3} x e^{4}{\rm sign}\left (b x + a\right ) + 945 \, A a^{2} b x e^{4}{\rm sign}\left (b x + a\right ) + 35 \, B a^{3} d e^{3}{\rm sign}\left (b x + a\right ) + 105 \, A a^{2} b d e^{3}{\rm sign}\left (b x + a\right ) + 280 \, A a^{3} e^{4}{\rm sign}\left (b x + a\right )\right )} e^{\left (-5\right )}}{2520 \,{\left (x e + d\right )}^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)/(e*x + d)^10,x, algorithm="giac")
[Out]