Optimal. Leaf size=298 \[ -\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^9 (-3 a B e-A b e+4 b B d)}{9 e^5 (a+b x)}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e) (-a B e-A b e+2 b B d)}{8 e^5 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^3 (B d-A e)}{6 e^5 (a+b x)}+\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{10}}{10 e^5 (a+b x)} \]
[Out]
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Rubi [A] time = 1.26452, 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 \[ -\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^9 (-3 a B e-A b e+4 b B d)}{9 e^5 (a+b x)}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e) (-a B e-A b e+2 b B d)}{8 e^5 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^3 (B d-A e)}{6 e^5 (a+b x)}+\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{10}}{10 e^5 (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 49.1708, size = 296, normalized size = 0.99 \[ \frac{B \left (2 a + 2 b x\right ) \left (d + e x\right )^{6} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{20 b e} + \frac{\left (d + e x\right )^{6} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (5 A b e - 3 B a e - 2 B b d\right )}{45 b e^{2}} + \frac{\left (3 a + 3 b x\right ) \left (d + e x\right )^{6} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (5 A b e - 3 B a e - 2 B b d\right )}{360 b e^{3}} + \frac{\left (d + e x\right )^{6} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (5 A b e - 3 B a e - 2 B b d\right )}{420 b e^{4}} + \frac{\left (d + e x\right )^{6} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (5 A b e - 3 B a e - 2 B b d\right )}{2520 b e^{5} \left (a + b x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**5*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
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Mathematica [A] time = 0.359388, size = 496, normalized size = 1.66 \[ \frac{x \sqrt{(a+b x)^2} \left (60 a^3 \left (7 A \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )+B x \left (21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right )\right )+45 a^2 b x \left (4 A \left (21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right )+B x \left (56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right )\right )+15 a b^2 x^2 \left (3 A \left (56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right )+B x \left (126 d^5+504 d^4 e x+840 d^3 e^2 x^2+720 d^2 e^3 x^3+315 d e^4 x^4+56 e^5 x^5\right )\right )+b^3 x^3 \left (5 A \left (126 d^5+504 d^4 e x+840 d^3 e^2 x^2+720 d^2 e^3 x^3+315 d e^4 x^4+56 e^5 x^5\right )+2 B x \left (252 d^5+1050 d^4 e x+1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+700 d e^4 x^4+126 e^5 x^5\right )\right )\right )}{2520 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
[Out]
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Maple [B] time = 0.014, size = 676, normalized size = 2.3 \[{\frac{x \left ( 252\,B{b}^{3}{e}^{5}{x}^{9}+280\,{x}^{8}A{b}^{3}{e}^{5}+840\,{x}^{8}Ba{b}^{2}{e}^{5}+1400\,{x}^{8}B{b}^{3}d{e}^{4}+945\,{x}^{7}Aa{b}^{2}{e}^{5}+1575\,{x}^{7}A{b}^{3}d{e}^{4}+945\,{x}^{7}B{a}^{2}b{e}^{5}+4725\,{x}^{7}Ba{b}^{2}d{e}^{4}+3150\,{x}^{7}B{b}^{3}{d}^{2}{e}^{3}+1080\,{x}^{6}A{a}^{2}b{e}^{5}+5400\,{x}^{6}Aa{b}^{2}d{e}^{4}+3600\,{x}^{6}A{b}^{3}{d}^{2}{e}^{3}+360\,{x}^{6}B{a}^{3}{e}^{5}+5400\,{x}^{6}B{a}^{2}bd{e}^{4}+10800\,{x}^{6}Ba{b}^{2}{d}^{2}{e}^{3}+3600\,{x}^{6}B{b}^{3}{d}^{3}{e}^{2}+420\,{x}^{5}A{a}^{3}{e}^{5}+6300\,{x}^{5}A{a}^{2}bd{e}^{4}+12600\,{x}^{5}Aa{b}^{2}{d}^{2}{e}^{3}+4200\,{x}^{5}A{b}^{3}{d}^{3}{e}^{2}+2100\,{x}^{5}B{a}^{3}d{e}^{4}+12600\,{x}^{5}B{a}^{2}b{d}^{2}{e}^{3}+12600\,{x}^{5}Ba{b}^{2}{d}^{3}{e}^{2}+2100\,{x}^{5}B{b}^{3}{d}^{4}e+2520\,{x}^{4}A{a}^{3}d{e}^{4}+15120\,{x}^{4}A{a}^{2}b{d}^{2}{e}^{3}+15120\,{x}^{4}Aa{b}^{2}{d}^{3}{e}^{2}+2520\,{x}^{4}A{b}^{3}{d}^{4}e+5040\,{x}^{4}B{a}^{3}{d}^{2}{e}^{3}+15120\,{x}^{4}B{a}^{2}b{d}^{3}{e}^{2}+7560\,{x}^{4}Ba{b}^{2}{d}^{4}e+504\,{x}^{4}B{b}^{3}{d}^{5}+6300\,{x}^{3}A{a}^{3}{d}^{2}{e}^{3}+18900\,{x}^{3}A{a}^{2}b{d}^{3}{e}^{2}+9450\,{x}^{3}Aa{b}^{2}{d}^{4}e+630\,{x}^{3}A{b}^{3}{d}^{5}+6300\,{x}^{3}B{a}^{3}{d}^{3}{e}^{2}+9450\,{x}^{3}B{a}^{2}b{d}^{4}e+1890\,{x}^{3}Ba{b}^{2}{d}^{5}+8400\,{x}^{2}A{a}^{3}{d}^{3}{e}^{2}+12600\,{x}^{2}A{a}^{2}b{d}^{4}e+2520\,{x}^{2}Aa{b}^{2}{d}^{5}+4200\,{x}^{2}B{a}^{3}{d}^{4}e+2520\,{x}^{2}B{a}^{2}b{d}^{5}+6300\,xA{a}^{3}{d}^{4}e+3780\,xA{a}^{2}b{d}^{5}+1260\,xB{a}^{3}{d}^{5}+2520\,A{a}^{3}{d}^{5} \right ) }{2520\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^(3/2),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)^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.270934, size = 699, normalized size = 2.35 \[ \frac{1}{10} \, B b^{3} e^{5} x^{10} + A a^{3} d^{5} x + \frac{1}{9} \,{\left (5 \, B b^{3} d e^{4} +{\left (3 \, B a b^{2} + A b^{3}\right )} e^{5}\right )} x^{9} + \frac{1}{8} \,{\left (10 \, B b^{3} d^{2} e^{3} + 5 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{4} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} e^{5}\right )} x^{8} + \frac{1}{7} \,{\left (10 \, B b^{3} d^{3} e^{2} + 10 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{3} + 15 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{4} +{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{5}\right )} x^{7} + \frac{1}{6} \,{\left (5 \, B b^{3} d^{4} e + A a^{3} e^{5} + 10 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{2} + 30 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{3} + 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (B b^{3} d^{5} + 5 \, A a^{3} d e^{4} + 5 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e + 30 \,{\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{2} + 10 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (10 \, A a^{3} d^{2} e^{3} +{\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} + 15 \,{\left (B a^{2} b + A a b^{2}\right )} d^{4} e + 10 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (10 \, A a^{3} d^{3} e^{2} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d^{5} + 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4} e\right )} x^{3} + \frac{1}{2} \,{\left (5 \, A a^{3} d^{4} e +{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{5}\right )} x^{2} \]
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)^5,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (A + B x\right ) \left (d + e x\right )^{5} \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)**5*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.289234, size = 1, normalized size = 0. \[ \mathit{Done} \]
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)^5,x, algorithm="giac")
[Out]