3.2176 \(\int \frac{\left (a+b \sqrt{x}\right )^{15}}{x^4} \, dx\)

Optimal. Leaf size=196 \[ -\frac{a^{15}}{3 x^3}-\frac{6 a^{14} b}{x^{5/2}}-\frac{105 a^{13} b^2}{2 x^2}-\frac{910 a^{12} b^3}{3 x^{3/2}}-\frac{1365 a^{11} b^4}{x}-\frac{6006 a^{10} b^5}{\sqrt{x}}+5005 a^9 b^6 \log (x)+12870 a^8 b^7 \sqrt{x}+6435 a^7 b^8 x+\frac{10010}{3} a^6 b^9 x^{3/2}+\frac{3003}{2} a^5 b^{10} x^2+546 a^4 b^{11} x^{5/2}+\frac{455}{3} a^3 b^{12} x^3+30 a^2 b^{13} x^{7/2}+\frac{15}{4} a b^{14} x^4+\frac{2}{9} b^{15} x^{9/2} \]

[Out]

-a^15/(3*x^3) - (6*a^14*b)/x^(5/2) - (105*a^13*b^2)/(2*x^2) - (910*a^12*b^3)/(3*
x^(3/2)) - (1365*a^11*b^4)/x - (6006*a^10*b^5)/Sqrt[x] + 12870*a^8*b^7*Sqrt[x] +
6435*a^7*b^8*x + (10010*a^6*b^9*x^(3/2))/3 + (3003*a^5*b^10*x^2)/2 + 546*a^4*b^
11*x^(5/2) + (455*a^3*b^12*x^3)/3 + 30*a^2*b^13*x^(7/2) + (15*a*b^14*x^4)/4 + (2
*b^15*x^(9/2))/9 + 5005*a^9*b^6*Log[x]

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Rubi [A]  time = 0.310543, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{a^{15}}{3 x^3}-\frac{6 a^{14} b}{x^{5/2}}-\frac{105 a^{13} b^2}{2 x^2}-\frac{910 a^{12} b^3}{3 x^{3/2}}-\frac{1365 a^{11} b^4}{x}-\frac{6006 a^{10} b^5}{\sqrt{x}}+5005 a^9 b^6 \log (x)+12870 a^8 b^7 \sqrt{x}+6435 a^7 b^8 x+\frac{10010}{3} a^6 b^9 x^{3/2}+\frac{3003}{2} a^5 b^{10} x^2+546 a^4 b^{11} x^{5/2}+\frac{455}{3} a^3 b^{12} x^3+30 a^2 b^{13} x^{7/2}+\frac{15}{4} a b^{14} x^4+\frac{2}{9} b^{15} x^{9/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^15/x^4,x]

[Out]

-a^15/(3*x^3) - (6*a^14*b)/x^(5/2) - (105*a^13*b^2)/(2*x^2) - (910*a^12*b^3)/(3*
x^(3/2)) - (1365*a^11*b^4)/x - (6006*a^10*b^5)/Sqrt[x] + 12870*a^8*b^7*Sqrt[x] +
6435*a^7*b^8*x + (10010*a^6*b^9*x^(3/2))/3 + (3003*a^5*b^10*x^2)/2 + 546*a^4*b^
11*x^(5/2) + (455*a^3*b^12*x^3)/3 + 30*a^2*b^13*x^(7/2) + (15*a*b^14*x^4)/4 + (2
*b^15*x^(9/2))/9 + 5005*a^9*b^6*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{15}}{3 x^{3}} - \frac{6 a^{14} b}{x^{\frac{5}{2}}} - \frac{105 a^{13} b^{2}}{2 x^{2}} - \frac{910 a^{12} b^{3}}{3 x^{\frac{3}{2}}} - \frac{1365 a^{11} b^{4}}{x} - \frac{6006 a^{10} b^{5}}{\sqrt{x}} + 10010 a^{9} b^{6} \log{\left (\sqrt{x} \right )} + 12870 a^{8} b^{7} \sqrt{x} + 12870 a^{7} b^{8} \int ^{\sqrt{x}} x\, dx + \frac{10010 a^{6} b^{9} x^{\frac{3}{2}}}{3} + \frac{3003 a^{5} b^{10} x^{2}}{2} + 546 a^{4} b^{11} x^{\frac{5}{2}} + \frac{455 a^{3} b^{12} x^{3}}{3} + 30 a^{2} b^{13} x^{\frac{7}{2}} + \frac{15 a b^{14} x^{4}}{4} + \frac{2 b^{15} x^{\frac{9}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/2))**15/x**4,x)

[Out]

-a**15/(3*x**3) - 6*a**14*b/x**(5/2) - 105*a**13*b**2/(2*x**2) - 910*a**12*b**3/
(3*x**(3/2)) - 1365*a**11*b**4/x - 6006*a**10*b**5/sqrt(x) + 10010*a**9*b**6*log
(sqrt(x)) + 12870*a**8*b**7*sqrt(x) + 12870*a**7*b**8*Integral(x, (x, sqrt(x)))
+ 10010*a**6*b**9*x**(3/2)/3 + 3003*a**5*b**10*x**2/2 + 546*a**4*b**11*x**(5/2)
+ 455*a**3*b**12*x**3/3 + 30*a**2*b**13*x**(7/2) + 15*a*b**14*x**4/4 + 2*b**15*x
**(9/2)/9

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Mathematica [A]  time = 0.104421, size = 196, normalized size = 1. \[ -\frac{a^{15}}{3 x^3}-\frac{6 a^{14} b}{x^{5/2}}-\frac{105 a^{13} b^2}{2 x^2}-\frac{910 a^{12} b^3}{3 x^{3/2}}-\frac{1365 a^{11} b^4}{x}-\frac{6006 a^{10} b^5}{\sqrt{x}}+5005 a^9 b^6 \log (x)+12870 a^8 b^7 \sqrt{x}+6435 a^7 b^8 x+\frac{10010}{3} a^6 b^9 x^{3/2}+\frac{3003}{2} a^5 b^{10} x^2+546 a^4 b^{11} x^{5/2}+\frac{455}{3} a^3 b^{12} x^3+30 a^2 b^{13} x^{7/2}+\frac{15}{4} a b^{14} x^4+\frac{2}{9} b^{15} x^{9/2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^15/x^4,x]

[Out]

-a^15/(3*x^3) - (6*a^14*b)/x^(5/2) - (105*a^13*b^2)/(2*x^2) - (910*a^12*b^3)/(3*
x^(3/2)) - (1365*a^11*b^4)/x - (6006*a^10*b^5)/Sqrt[x] + 12870*a^8*b^7*Sqrt[x] +
6435*a^7*b^8*x + (10010*a^6*b^9*x^(3/2))/3 + (3003*a^5*b^10*x^2)/2 + 546*a^4*b^
11*x^(5/2) + (455*a^3*b^12*x^3)/3 + 30*a^2*b^13*x^(7/2) + (15*a*b^14*x^4)/4 + (2
*b^15*x^(9/2))/9 + 5005*a^9*b^6*Log[x]

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Maple [A]  time = 0.006, size = 165, normalized size = 0.8 \[ -{\frac{{a}^{15}}{3\,{x}^{3}}}-6\,{\frac{{a}^{14}b}{{x}^{5/2}}}-{\frac{105\,{a}^{13}{b}^{2}}{2\,{x}^{2}}}-{\frac{910\,{a}^{12}{b}^{3}}{3}{x}^{-{\frac{3}{2}}}}-1365\,{\frac{{a}^{11}{b}^{4}}{x}}+6435\,{a}^{7}{b}^{8}x+{\frac{10010\,{a}^{6}{b}^{9}}{3}{x}^{{\frac{3}{2}}}}+{\frac{3003\,{a}^{5}{b}^{10}{x}^{2}}{2}}+546\,{a}^{4}{b}^{11}{x}^{5/2}+{\frac{455\,{a}^{3}{b}^{12}{x}^{3}}{3}}+30\,{a}^{2}{b}^{13}{x}^{7/2}+{\frac{15\,a{b}^{14}{x}^{4}}{4}}+{\frac{2\,{b}^{15}}{9}{x}^{{\frac{9}{2}}}}+5005\,{a}^{9}{b}^{6}\ln \left ( x \right ) -6006\,{\frac{{a}^{10}{b}^{5}}{\sqrt{x}}}+12870\,{a}^{8}{b}^{7}\sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^15/x^4,x)

[Out]

-1/3*a^15/x^3-6*a^14*b/x^(5/2)-105/2*a^13*b^2/x^2-910/3*a^12*b^3/x^(3/2)-1365*a^
11*b^4/x+6435*a^7*b^8*x+10010/3*a^6*b^9*x^(3/2)+3003/2*a^5*b^10*x^2+546*a^4*b^11
*x^(5/2)+455/3*a^3*b^12*x^3+30*a^2*b^13*x^(7/2)+15/4*a*b^14*x^4+2/9*b^15*x^(9/2)
+5005*a^9*b^6*ln(x)-6006*a^10*b^5/x^(1/2)+12870*a^8*b^7*x^(1/2)

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Maxima [A]  time = 1.43043, size = 223, normalized size = 1.14 \[ \frac{2}{9} \, b^{15} x^{\frac{9}{2}} + \frac{15}{4} \, a b^{14} x^{4} + 30 \, a^{2} b^{13} x^{\frac{7}{2}} + \frac{455}{3} \, a^{3} b^{12} x^{3} + 546 \, a^{4} b^{11} x^{\frac{5}{2}} + \frac{3003}{2} \, a^{5} b^{10} x^{2} + \frac{10010}{3} \, a^{6} b^{9} x^{\frac{3}{2}} + 6435 \, a^{7} b^{8} x + 5005 \, a^{9} b^{6} \log \left (x\right ) + 12870 \, a^{8} b^{7} \sqrt{x} - \frac{36036 \, a^{10} b^{5} x^{\frac{5}{2}} + 8190 \, a^{11} b^{4} x^{2} + 1820 \, a^{12} b^{3} x^{\frac{3}{2}} + 315 \, a^{13} b^{2} x + 36 \, a^{14} b \sqrt{x} + 2 \, a^{15}}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^15/x^4,x, algorithm="maxima")

[Out]

2/9*b^15*x^(9/2) + 15/4*a*b^14*x^4 + 30*a^2*b^13*x^(7/2) + 455/3*a^3*b^12*x^3 +
546*a^4*b^11*x^(5/2) + 3003/2*a^5*b^10*x^2 + 10010/3*a^6*b^9*x^(3/2) + 6435*a^7*
b^8*x + 5005*a^9*b^6*log(x) + 12870*a^8*b^7*sqrt(x) - 1/6*(36036*a^10*b^5*x^(5/2
) + 8190*a^11*b^4*x^2 + 1820*a^12*b^3*x^(3/2) + 315*a^13*b^2*x + 36*a^14*b*sqrt(
x) + 2*a^15)/x^3

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Fricas [A]  time = 0.238551, size = 231, normalized size = 1.18 \[ \frac{135 \, a b^{14} x^{7} + 5460 \, a^{3} b^{12} x^{6} + 54054 \, a^{5} b^{10} x^{5} + 231660 \, a^{7} b^{8} x^{4} + 360360 \, a^{9} b^{6} x^{3} \log \left (\sqrt{x}\right ) - 49140 \, a^{11} b^{4} x^{2} - 1890 \, a^{13} b^{2} x - 12 \, a^{15} + 8 \,{\left (b^{15} x^{7} + 135 \, a^{2} b^{13} x^{6} + 2457 \, a^{4} b^{11} x^{5} + 15015 \, a^{6} b^{9} x^{4} + 57915 \, a^{8} b^{7} x^{3} - 27027 \, a^{10} b^{5} x^{2} - 1365 \, a^{12} b^{3} x - 27 \, a^{14} b\right )} \sqrt{x}}{36 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^15/x^4,x, algorithm="fricas")

[Out]

1/36*(135*a*b^14*x^7 + 5460*a^3*b^12*x^6 + 54054*a^5*b^10*x^5 + 231660*a^7*b^8*x
^4 + 360360*a^9*b^6*x^3*log(sqrt(x)) - 49140*a^11*b^4*x^2 - 1890*a^13*b^2*x - 12
*a^15 + 8*(b^15*x^7 + 135*a^2*b^13*x^6 + 2457*a^4*b^11*x^5 + 15015*a^6*b^9*x^4 +
57915*a^8*b^7*x^3 - 27027*a^10*b^5*x^2 - 1365*a^12*b^3*x - 27*a^14*b)*sqrt(x))/
x^3

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Sympy [A]  time = 17.5361, size = 201, normalized size = 1.03 \[ - \frac{a^{15}}{3 x^{3}} - \frac{6 a^{14} b}{x^{\frac{5}{2}}} - \frac{105 a^{13} b^{2}}{2 x^{2}} - \frac{910 a^{12} b^{3}}{3 x^{\frac{3}{2}}} - \frac{1365 a^{11} b^{4}}{x} - \frac{6006 a^{10} b^{5}}{\sqrt{x}} + 5005 a^{9} b^{6} \log{\left (x \right )} + 12870 a^{8} b^{7} \sqrt{x} + 6435 a^{7} b^{8} x + \frac{10010 a^{6} b^{9} x^{\frac{3}{2}}}{3} + \frac{3003 a^{5} b^{10} x^{2}}{2} + 546 a^{4} b^{11} x^{\frac{5}{2}} + \frac{455 a^{3} b^{12} x^{3}}{3} + 30 a^{2} b^{13} x^{\frac{7}{2}} + \frac{15 a b^{14} x^{4}}{4} + \frac{2 b^{15} x^{\frac{9}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/2))**15/x**4,x)

[Out]

-a**15/(3*x**3) - 6*a**14*b/x**(5/2) - 105*a**13*b**2/(2*x**2) - 910*a**12*b**3/
(3*x**(3/2)) - 1365*a**11*b**4/x - 6006*a**10*b**5/sqrt(x) + 5005*a**9*b**6*log(
x) + 12870*a**8*b**7*sqrt(x) + 6435*a**7*b**8*x + 10010*a**6*b**9*x**(3/2)/3 + 3
003*a**5*b**10*x**2/2 + 546*a**4*b**11*x**(5/2) + 455*a**3*b**12*x**3/3 + 30*a**
2*b**13*x**(7/2) + 15*a*b**14*x**4/4 + 2*b**15*x**(9/2)/9

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GIAC/XCAS [A]  time = 0.219209, size = 224, normalized size = 1.14 \[ \frac{2}{9} \, b^{15} x^{\frac{9}{2}} + \frac{15}{4} \, a b^{14} x^{4} + 30 \, a^{2} b^{13} x^{\frac{7}{2}} + \frac{455}{3} \, a^{3} b^{12} x^{3} + 546 \, a^{4} b^{11} x^{\frac{5}{2}} + \frac{3003}{2} \, a^{5} b^{10} x^{2} + \frac{10010}{3} \, a^{6} b^{9} x^{\frac{3}{2}} + 6435 \, a^{7} b^{8} x + 5005 \, a^{9} b^{6}{\rm ln}\left ({\left | x \right |}\right ) + 12870 \, a^{8} b^{7} \sqrt{x} - \frac{36036 \, a^{10} b^{5} x^{\frac{5}{2}} + 8190 \, a^{11} b^{4} x^{2} + 1820 \, a^{12} b^{3} x^{\frac{3}{2}} + 315 \, a^{13} b^{2} x + 36 \, a^{14} b \sqrt{x} + 2 \, a^{15}}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^15/x^4,x, algorithm="giac")

[Out]

2/9*b^15*x^(9/2) + 15/4*a*b^14*x^4 + 30*a^2*b^13*x^(7/2) + 455/3*a^3*b^12*x^3 +
546*a^4*b^11*x^(5/2) + 3003/2*a^5*b^10*x^2 + 10010/3*a^6*b^9*x^(3/2) + 6435*a^7*
b^8*x + 5005*a^9*b^6*ln(abs(x)) + 12870*a^8*b^7*sqrt(x) - 1/6*(36036*a^10*b^5*x^
(5/2) + 8190*a^11*b^4*x^2 + 1820*a^12*b^3*x^(3/2) + 315*a^13*b^2*x + 36*a^14*b*s
qrt(x) + 2*a^15)/x^3