3.2170 \(\int \left (a+b \sqrt{x}\right )^{15} x^2 \, dx\)

Optimal. Leaf size=122 \[ -\frac{a^5 \left (a+b \sqrt{x}\right )^{16}}{8 b^6}+\frac{10 a^4 \left (a+b \sqrt{x}\right )^{17}}{17 b^6}-\frac{10 a^3 \left (a+b \sqrt{x}\right )^{18}}{9 b^6}+\frac{20 a^2 \left (a+b \sqrt{x}\right )^{19}}{19 b^6}+\frac{2 \left (a+b \sqrt{x}\right )^{21}}{21 b^6}-\frac{a \left (a+b \sqrt{x}\right )^{20}}{2 b^6} \]

[Out]

-(a^5*(a + b*Sqrt[x])^16)/(8*b^6) + (10*a^4*(a + b*Sqrt[x])^17)/(17*b^6) - (10*a
^3*(a + b*Sqrt[x])^18)/(9*b^6) + (20*a^2*(a + b*Sqrt[x])^19)/(19*b^6) - (a*(a +
b*Sqrt[x])^20)/(2*b^6) + (2*(a + b*Sqrt[x])^21)/(21*b^6)

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Rubi [A]  time = 0.226807, antiderivative size = 122, 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 \[ -\frac{a^5 \left (a+b \sqrt{x}\right )^{16}}{8 b^6}+\frac{10 a^4 \left (a+b \sqrt{x}\right )^{17}}{17 b^6}-\frac{10 a^3 \left (a+b \sqrt{x}\right )^{18}}{9 b^6}+\frac{20 a^2 \left (a+b \sqrt{x}\right )^{19}}{19 b^6}+\frac{2 \left (a+b \sqrt{x}\right )^{21}}{21 b^6}-\frac{a \left (a+b \sqrt{x}\right )^{20}}{2 b^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^15*x^2,x]

[Out]

-(a^5*(a + b*Sqrt[x])^16)/(8*b^6) + (10*a^4*(a + b*Sqrt[x])^17)/(17*b^6) - (10*a
^3*(a + b*Sqrt[x])^18)/(9*b^6) + (20*a^2*(a + b*Sqrt[x])^19)/(19*b^6) - (a*(a +
b*Sqrt[x])^20)/(2*b^6) + (2*(a + b*Sqrt[x])^21)/(21*b^6)

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Rubi in Sympy [A]  time = 41.822, size = 112, normalized size = 0.92 \[ - \frac{a^{5} \left (a + b \sqrt{x}\right )^{16}}{8 b^{6}} + \frac{10 a^{4} \left (a + b \sqrt{x}\right )^{17}}{17 b^{6}} - \frac{10 a^{3} \left (a + b \sqrt{x}\right )^{18}}{9 b^{6}} + \frac{20 a^{2} \left (a + b \sqrt{x}\right )^{19}}{19 b^{6}} - \frac{a \left (a + b \sqrt{x}\right )^{20}}{2 b^{6}} + \frac{2 \left (a + b \sqrt{x}\right )^{21}}{21 b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**5*(a + b*sqrt(x))**16/(8*b**6) + 10*a**4*(a + b*sqrt(x))**17/(17*b**6) - 10*
a**3*(a + b*sqrt(x))**18/(9*b**6) + 20*a**2*(a + b*sqrt(x))**19/(19*b**6) - a*(a
+ b*sqrt(x))**20/(2*b**6) + 2*(a + b*sqrt(x))**21/(21*b**6)

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Mathematica [A]  time = 0.0307612, size = 209, normalized size = 1.71 \[ \frac{a^{15} x^3}{3}+\frac{30}{7} a^{14} b x^{7/2}+\frac{105}{4} a^{13} b^2 x^4+\frac{910}{9} a^{12} b^3 x^{9/2}+273 a^{11} b^4 x^5+546 a^{10} b^5 x^{11/2}+\frac{5005}{6} a^9 b^6 x^6+990 a^8 b^7 x^{13/2}+\frac{6435}{7} a^7 b^8 x^7+\frac{2002}{3} a^6 b^9 x^{15/2}+\frac{3003}{8} a^5 b^{10} x^8+\frac{2730}{17} a^4 b^{11} x^{17/2}+\frac{455}{9} a^3 b^{12} x^9+\frac{210}{19} a^2 b^{13} x^{19/2}+\frac{3}{2} a b^{14} x^{10}+\frac{2}{21} b^{15} x^{21/2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^15*x^2,x]

[Out]

(a^15*x^3)/3 + (30*a^14*b*x^(7/2))/7 + (105*a^13*b^2*x^4)/4 + (910*a^12*b^3*x^(9
/2))/9 + 273*a^11*b^4*x^5 + 546*a^10*b^5*x^(11/2) + (5005*a^9*b^6*x^6)/6 + 990*a
^8*b^7*x^(13/2) + (6435*a^7*b^8*x^7)/7 + (2002*a^6*b^9*x^(15/2))/3 + (3003*a^5*b
^10*x^8)/8 + (2730*a^4*b^11*x^(17/2))/17 + (455*a^3*b^12*x^9)/9 + (210*a^2*b^13*
x^(19/2))/19 + (3*a*b^14*x^10)/2 + (2*b^15*x^(21/2))/21

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Maple [A]  time = 0.004, size = 168, normalized size = 1.4 \[{\frac{2\,{b}^{15}}{21}{x}^{{\frac{21}{2}}}}+{\frac{3\,{x}^{10}a{b}^{14}}{2}}+{\frac{210\,{a}^{2}{b}^{13}}{19}{x}^{{\frac{19}{2}}}}+{\frac{455\,{a}^{3}{b}^{12}{x}^{9}}{9}}+{\frac{2730\,{a}^{4}{b}^{11}}{17}{x}^{{\frac{17}{2}}}}+{\frac{3003\,{x}^{8}{a}^{5}{b}^{10}}{8}}+{\frac{2002\,{a}^{6}{b}^{9}}{3}{x}^{{\frac{15}{2}}}}+{\frac{6435\,{x}^{7}{a}^{7}{b}^{8}}{7}}+990\,{x}^{13/2}{a}^{8}{b}^{7}+{\frac{5005\,{x}^{6}{a}^{9}{b}^{6}}{6}}+546\,{x}^{11/2}{a}^{10}{b}^{5}+273\,{x}^{5}{a}^{11}{b}^{4}+{\frac{910\,{a}^{12}{b}^{3}}{9}{x}^{{\frac{9}{2}}}}+{\frac{105\,{x}^{4}{a}^{13}{b}^{2}}{4}}+{\frac{30\,{a}^{14}b}{7}{x}^{{\frac{7}{2}}}}+{\frac{{x}^{3}{a}^{15}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(a+b*x^(1/2))^15,x)

[Out]

2/21*x^(21/2)*b^15+3/2*x^10*a*b^14+210/19*x^(19/2)*a^2*b^13+455/9*a^3*b^12*x^9+2
730/17*x^(17/2)*a^4*b^11+3003/8*x^8*a^5*b^10+2002/3*x^(15/2)*a^6*b^9+6435/7*x^7*
a^7*b^8+990*x^(13/2)*a^8*b^7+5005/6*x^6*a^9*b^6+546*x^(11/2)*a^10*b^5+273*x^5*a^
11*b^4+910/9*x^(9/2)*a^12*b^3+105/4*x^4*a^13*b^2+30/7*x^(7/2)*a^14*b+1/3*x^3*a^1
5

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Maxima [A]  time = 1.42034, size = 132, normalized size = 1.08 \[ \frac{2 \,{\left (b \sqrt{x} + a\right )}^{21}}{21 \, b^{6}} - \frac{{\left (b \sqrt{x} + a\right )}^{20} a}{2 \, b^{6}} + \frac{20 \,{\left (b \sqrt{x} + a\right )}^{19} a^{2}}{19 \, b^{6}} - \frac{10 \,{\left (b \sqrt{x} + a\right )}^{18} a^{3}}{9 \, b^{6}} + \frac{10 \,{\left (b \sqrt{x} + a\right )}^{17} a^{4}}{17 \, b^{6}} - \frac{{\left (b \sqrt{x} + a\right )}^{16} a^{5}}{8 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^15*x^2,x, algorithm="maxima")

[Out]

2/21*(b*sqrt(x) + a)^21/b^6 - 1/2*(b*sqrt(x) + a)^20*a/b^6 + 20/19*(b*sqrt(x) +
a)^19*a^2/b^6 - 10/9*(b*sqrt(x) + a)^18*a^3/b^6 + 10/17*(b*sqrt(x) + a)^17*a^4/b
^6 - 1/8*(b*sqrt(x) + a)^16*a^5/b^6

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Fricas [A]  time = 0.232144, size = 234, normalized size = 1.92 \[ \frac{3}{2} \, a b^{14} x^{10} + \frac{455}{9} \, a^{3} b^{12} x^{9} + \frac{3003}{8} \, a^{5} b^{10} x^{8} + \frac{6435}{7} \, a^{7} b^{8} x^{7} + \frac{5005}{6} \, a^{9} b^{6} x^{6} + 273 \, a^{11} b^{4} x^{5} + \frac{105}{4} \, a^{13} b^{2} x^{4} + \frac{1}{3} \, a^{15} x^{3} + \frac{2}{20349} \,{\left (969 \, b^{15} x^{10} + 112455 \, a^{2} b^{13} x^{9} + 1633905 \, a^{4} b^{11} x^{8} + 6789783 \, a^{6} b^{9} x^{7} + 10072755 \, a^{8} b^{7} x^{6} + 5555277 \, a^{10} b^{5} x^{5} + 1028755 \, a^{12} b^{3} x^{4} + 43605 \, a^{14} b x^{3}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^15*x^2,x, algorithm="fricas")

[Out]

3/2*a*b^14*x^10 + 455/9*a^3*b^12*x^9 + 3003/8*a^5*b^10*x^8 + 6435/7*a^7*b^8*x^7
+ 5005/6*a^9*b^6*x^6 + 273*a^11*b^4*x^5 + 105/4*a^13*b^2*x^4 + 1/3*a^15*x^3 + 2/
20349*(969*b^15*x^10 + 112455*a^2*b^13*x^9 + 1633905*a^4*b^11*x^8 + 6789783*a^6*
b^9*x^7 + 10072755*a^8*b^7*x^6 + 5555277*a^10*b^5*x^5 + 1028755*a^12*b^3*x^4 + 4
3605*a^14*b*x^3)*sqrt(x)

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Sympy [A]  time = 14.1527, size = 212, normalized size = 1.74 \[ \frac{a^{15} x^{3}}{3} + \frac{30 a^{14} b x^{\frac{7}{2}}}{7} + \frac{105 a^{13} b^{2} x^{4}}{4} + \frac{910 a^{12} b^{3} x^{\frac{9}{2}}}{9} + 273 a^{11} b^{4} x^{5} + 546 a^{10} b^{5} x^{\frac{11}{2}} + \frac{5005 a^{9} b^{6} x^{6}}{6} + 990 a^{8} b^{7} x^{\frac{13}{2}} + \frac{6435 a^{7} b^{8} x^{7}}{7} + \frac{2002 a^{6} b^{9} x^{\frac{15}{2}}}{3} + \frac{3003 a^{5} b^{10} x^{8}}{8} + \frac{2730 a^{4} b^{11} x^{\frac{17}{2}}}{17} + \frac{455 a^{3} b^{12} x^{9}}{9} + \frac{210 a^{2} b^{13} x^{\frac{19}{2}}}{19} + \frac{3 a b^{14} x^{10}}{2} + \frac{2 b^{15} x^{\frac{21}{2}}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**15*x**3/3 + 30*a**14*b*x**(7/2)/7 + 105*a**13*b**2*x**4/4 + 910*a**12*b**3*x*
*(9/2)/9 + 273*a**11*b**4*x**5 + 546*a**10*b**5*x**(11/2) + 5005*a**9*b**6*x**6/
6 + 990*a**8*b**7*x**(13/2) + 6435*a**7*b**8*x**7/7 + 2002*a**6*b**9*x**(15/2)/3
+ 3003*a**5*b**10*x**8/8 + 2730*a**4*b**11*x**(17/2)/17 + 455*a**3*b**12*x**9/9
+ 210*a**2*b**13*x**(19/2)/19 + 3*a*b**14*x**10/2 + 2*b**15*x**(21/2)/21

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GIAC/XCAS [A]  time = 0.217238, size = 225, normalized size = 1.84 \[ \frac{2}{21} \, b^{15} x^{\frac{21}{2}} + \frac{3}{2} \, a b^{14} x^{10} + \frac{210}{19} \, a^{2} b^{13} x^{\frac{19}{2}} + \frac{455}{9} \, a^{3} b^{12} x^{9} + \frac{2730}{17} \, a^{4} b^{11} x^{\frac{17}{2}} + \frac{3003}{8} \, a^{5} b^{10} x^{8} + \frac{2002}{3} \, a^{6} b^{9} x^{\frac{15}{2}} + \frac{6435}{7} \, a^{7} b^{8} x^{7} + 990 \, a^{8} b^{7} x^{\frac{13}{2}} + \frac{5005}{6} \, a^{9} b^{6} x^{6} + 546 \, a^{10} b^{5} x^{\frac{11}{2}} + 273 \, a^{11} b^{4} x^{5} + \frac{910}{9} \, a^{12} b^{3} x^{\frac{9}{2}} + \frac{105}{4} \, a^{13} b^{2} x^{4} + \frac{30}{7} \, a^{14} b x^{\frac{7}{2}} + \frac{1}{3} \, a^{15} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^15*x^2,x, algorithm="giac")

[Out]

2/21*b^15*x^(21/2) + 3/2*a*b^14*x^10 + 210/19*a^2*b^13*x^(19/2) + 455/9*a^3*b^12
*x^9 + 2730/17*a^4*b^11*x^(17/2) + 3003/8*a^5*b^10*x^8 + 2002/3*a^6*b^9*x^(15/2)
+ 6435/7*a^7*b^8*x^7 + 990*a^8*b^7*x^(13/2) + 5005/6*a^9*b^6*x^6 + 546*a^10*b^5
*x^(11/2) + 273*a^11*b^4*x^5 + 910/9*a^12*b^3*x^(9/2) + 105/4*a^13*b^2*x^4 + 30/
7*a^14*b*x^(7/2) + 1/3*a^15*x^3