Optimal. Leaf size=438 \[ \frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{9 e^7 (a+b x) (d+e x)^9}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{2 e^7 (a+b x) (d+e x)^{10}}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{11 e^7 (a+b x) (d+e x)^{11}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{12 e^7 (a+b x) (d+e x)^{12}}+\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (-5 a B e-A b e+6 b B d)}{7 e^7 (a+b x) (d+e x)^7}-\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-2 a B e-A b e+3 b B d)}{8 e^7 (a+b x) (d+e x)^8}-\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{6 e^7 (a+b x) (d+e x)^6} \]
[Out]
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Rubi [A] time = 1.04875, antiderivative size = 438, 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 \[ \frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{9 e^7 (a+b x) (d+e x)^9}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{2 e^7 (a+b x) (d+e x)^{10}}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{11 e^7 (a+b x) (d+e x)^{11}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{12 e^7 (a+b x) (d+e x)^{12}}+\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (-5 a B e-A b e+6 b B d)}{7 e^7 (a+b x) (d+e x)^7}-\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-2 a B e-A b e+3 b B d)}{8 e^7 (a+b x) (d+e x)^8}-\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{6 e^7 (a+b x) (d+e x)^6} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^13,x]
[Out]
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Rubi in Sympy [A] time = 79.3936, size = 418, normalized size = 0.95 \[ \frac{b^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (A b e - 2 B a e + B b d\right )}{792 e^{6} \left (d + e x\right )^{7} \left (a e - b d\right )} - \frac{b^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (A b e - 2 B a e + B b d\right )}{5544 e^{7} \left (a + b x\right ) \left (d + e x\right )^{7}} + \frac{b^{3} \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (A b e - 2 B a e + B b d\right )}{792 e^{5} \left (d + e x\right )^{8} \left (a e - b d\right )} + \frac{b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (A b e - 2 B a e + B b d\right )}{99 e^{4} \left (d + e x\right )^{9} \left (a e - b d\right )} + \frac{b \left (5 a + 5 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (A b e - 2 B a e + B b d\right )}{220 e^{3} \left (d + e x\right )^{10} \left (a e - b d\right )} - \frac{\left (2 a + 2 b x\right ) \left (A e - B d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{24 e \left (d + e x\right )^{12} \left (a e - b d\right )} + \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}} \left (A b e - 2 B a e + B b d\right )}{22 e^{2} \left (d + e x\right )^{11} \left (a e - b d\right )} \]
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)**(5/2)/(e*x+d)**13,x)
[Out]
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Mathematica [A] time = 0.426037, size = 465, normalized size = 1.06 \[ -\frac{\sqrt{(a+b x)^2} \left (42 a^5 e^5 (11 A e+B (d+12 e x))+42 a^4 b e^4 \left (5 A e (d+12 e x)+B \left (d^2+12 d e x+66 e^2 x^2\right )\right )+28 a^3 b^2 e^3 \left (3 A e \left (d^2+12 d e x+66 e^2 x^2\right )+B \left (d^3+12 d^2 e x+66 d e^2 x^2+220 e^3 x^3\right )\right )+14 a^2 b^3 e^2 \left (2 A e \left (d^3+12 d^2 e x+66 d e^2 x^2+220 e^3 x^3\right )+B \left (d^4+12 d^3 e x+66 d^2 e^2 x^2+220 d e^3 x^3+495 e^4 x^4\right )\right )+a b^4 e \left (7 A e \left (d^4+12 d^3 e x+66 d^2 e^2 x^2+220 d e^3 x^3+495 e^4 x^4\right )+5 B \left (d^5+12 d^4 e x+66 d^3 e^2 x^2+220 d^2 e^3 x^3+495 d e^4 x^4+792 e^5 x^5\right )\right )+b^5 \left (A e \left (d^5+12 d^4 e x+66 d^3 e^2 x^2+220 d^2 e^3 x^3+495 d e^4 x^4+792 e^5 x^5\right )+B \left (d^6+12 d^5 e x+66 d^4 e^2 x^2+220 d^3 e^3 x^3+495 d^2 e^4 x^4+792 d e^5 x^5+924 e^6 x^6\right )\right )\right )}{5544 e^7 (a+b x) (d+e x)^{12}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^13,x]
[Out]
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Maple [A] time = 0.019, size = 687, normalized size = 1.6 \[ -{\frac{924\,B{x}^{6}{b}^{5}{e}^{6}+792\,A{x}^{5}{b}^{5}{e}^{6}+3960\,B{x}^{5}a{b}^{4}{e}^{6}+792\,B{x}^{5}{b}^{5}d{e}^{5}+3465\,A{x}^{4}a{b}^{4}{e}^{6}+495\,A{x}^{4}{b}^{5}d{e}^{5}+6930\,B{x}^{4}{a}^{2}{b}^{3}{e}^{6}+2475\,B{x}^{4}a{b}^{4}d{e}^{5}+495\,B{x}^{4}{b}^{5}{d}^{2}{e}^{4}+6160\,A{x}^{3}{a}^{2}{b}^{3}{e}^{6}+1540\,A{x}^{3}a{b}^{4}d{e}^{5}+220\,A{x}^{3}{b}^{5}{d}^{2}{e}^{4}+6160\,B{x}^{3}{a}^{3}{b}^{2}{e}^{6}+3080\,B{x}^{3}{a}^{2}{b}^{3}d{e}^{5}+1100\,B{x}^{3}a{b}^{4}{d}^{2}{e}^{4}+220\,B{x}^{3}{b}^{5}{d}^{3}{e}^{3}+5544\,A{x}^{2}{a}^{3}{b}^{2}{e}^{6}+1848\,A{x}^{2}{a}^{2}{b}^{3}d{e}^{5}+462\,A{x}^{2}a{b}^{4}{d}^{2}{e}^{4}+66\,A{x}^{2}{b}^{5}{d}^{3}{e}^{3}+2772\,B{x}^{2}{a}^{4}b{e}^{6}+1848\,B{x}^{2}{a}^{3}{b}^{2}d{e}^{5}+924\,B{x}^{2}{a}^{2}{b}^{3}{d}^{2}{e}^{4}+330\,B{x}^{2}a{b}^{4}{d}^{3}{e}^{3}+66\,B{x}^{2}{b}^{5}{d}^{4}{e}^{2}+2520\,Ax{a}^{4}b{e}^{6}+1008\,Ax{a}^{3}{b}^{2}d{e}^{5}+336\,Ax{a}^{2}{b}^{3}{d}^{2}{e}^{4}+84\,Axa{b}^{4}{d}^{3}{e}^{3}+12\,Ax{b}^{5}{d}^{4}{e}^{2}+504\,Bx{a}^{5}{e}^{6}+504\,Bx{a}^{4}bd{e}^{5}+336\,Bx{a}^{3}{b}^{2}{d}^{2}{e}^{4}+168\,Bx{a}^{2}{b}^{3}{d}^{3}{e}^{3}+60\,Bxa{b}^{4}{d}^{4}{e}^{2}+12\,Bx{b}^{5}{d}^{5}e+462\,A{a}^{5}{e}^{6}+210\,Ad{e}^{5}{a}^{4}b+84\,A{a}^{3}{b}^{2}{d}^{2}{e}^{4}+28\,A{a}^{2}{b}^{3}{d}^{3}{e}^{3}+7\,Aa{b}^{4}{d}^{4}{e}^{2}+A{b}^{5}{d}^{5}e+42\,Bd{e}^{5}{a}^{5}+42\,B{a}^{4}b{d}^{2}{e}^{4}+28\,B{a}^{3}{b}^{2}{d}^{3}{e}^{3}+14\,B{a}^{2}{b}^{3}{d}^{4}{e}^{2}+5\,Ba{b}^{4}{d}^{5}e+B{b}^{5}{d}^{6}}{5544\,{e}^{7} \left ( ex+d \right ) ^{12} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]
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)^(5/2)/(e*x+d)^13,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)^(5/2)*(B*x + A)/(e*x + d)^13,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.281534, size = 907, normalized size = 2.07 \[ -\frac{924 \, B b^{5} e^{6} x^{6} + B b^{5} d^{6} + 462 \, A a^{5} e^{6} +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 7 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 28 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 42 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} + 42 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} + 792 \,{\left (B b^{5} d e^{5} +{\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 495 \,{\left (B b^{5} d^{2} e^{4} +{\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 7 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} + 220 \,{\left (B b^{5} d^{3} e^{3} +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 7 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} + 28 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 66 \,{\left (B b^{5} d^{4} e^{2} +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 7 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} + 28 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 42 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} + 12 \,{\left (B b^{5} d^{5} e +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 7 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} + 28 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 42 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} + 42 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x}{5544 \,{\left (e^{19} x^{12} + 12 \, d e^{18} x^{11} + 66 \, d^{2} e^{17} x^{10} + 220 \, d^{3} e^{16} x^{9} + 495 \, d^{4} e^{15} x^{8} + 792 \, d^{5} e^{14} x^{7} + 924 \, d^{6} e^{13} x^{6} + 792 \, d^{7} e^{12} x^{5} + 495 \, d^{8} e^{11} x^{4} + 220 \, d^{9} e^{10} x^{3} + 66 \, d^{10} e^{9} x^{2} + 12 \, d^{11} e^{8} x + d^{12} e^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)/(e*x + d)^13,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)**(5/2)/(e*x+d)**13,x)
[Out]
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GIAC/XCAS [A] time = 0.306588, 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)^(5/2)*(B*x + A)/(e*x + d)^13,x, algorithm="giac")
[Out]