3.2176 \(\int \frac {(a+b \sqrt {x})^{15}}{x^2} \, dx\)

Optimal. Leaf size=192 \[ -\frac {a^{15}}{x}-\frac {30 a^{14} b}{\sqrt {x}}+105 a^{13} b^2 \log (x)+910 a^{12} b^3 \sqrt {x}+1365 a^{11} b^4 x+2002 a^{10} b^5 x^{3/2}+\frac {5005}{2} a^9 b^6 x^2+2574 a^8 b^7 x^{5/2}+2145 a^7 b^8 x^3+1430 a^6 b^9 x^{7/2}+\frac {3003}{4} a^5 b^{10} x^4+\frac {910}{3} a^4 b^{11} x^{9/2}+91 a^3 b^{12} x^5+\frac {210}{11} a^2 b^{13} x^{11/2}+\frac {5}{2} a b^{14} x^6+\frac {2}{13} b^{15} x^{13/2} \]

[Out]

-a^15/x+1365*a^11*b^4*x+2002*a^10*b^5*x^(3/2)+5005/2*a^9*b^6*x^2+2574*a^8*b^7*x^(5/2)+2145*a^7*b^8*x^3+1430*a^
6*b^9*x^(7/2)+3003/4*a^5*b^10*x^4+910/3*a^4*b^11*x^(9/2)+91*a^3*b^12*x^5+210/11*a^2*b^13*x^(11/2)+5/2*a*b^14*x
^6+2/13*b^15*x^(13/2)+105*a^13*b^2*ln(x)-30*a^14*b/x^(1/2)+910*a^12*b^3*x^(1/2)

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Rubi [A]  time = 0.12, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ 2002 a^{10} b^5 x^{3/2}+\frac {5005}{2} a^9 b^6 x^2+2574 a^8 b^7 x^{5/2}+2145 a^7 b^8 x^3+1430 a^6 b^9 x^{7/2}+\frac {3003}{4} a^5 b^{10} x^4+\frac {910}{3} a^4 b^{11} x^{9/2}+91 a^3 b^{12} x^5+\frac {210}{11} a^2 b^{13} x^{11/2}+910 a^{12} b^3 \sqrt {x}+1365 a^{11} b^4 x+105 a^{13} b^2 \log (x)-\frac {30 a^{14} b}{\sqrt {x}}-\frac {a^{15}}{x}+\frac {5}{2} a b^{14} x^6+\frac {2}{13} b^{15} x^{13/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^15/x^2,x]

[Out]

-(a^15/x) - (30*a^14*b)/Sqrt[x] + 910*a^12*b^3*Sqrt[x] + 1365*a^11*b^4*x + 2002*a^10*b^5*x^(3/2) + (5005*a^9*b
^6*x^2)/2 + 2574*a^8*b^7*x^(5/2) + 2145*a^7*b^8*x^3 + 1430*a^6*b^9*x^(7/2) + (3003*a^5*b^10*x^4)/4 + (910*a^4*
b^11*x^(9/2))/3 + 91*a^3*b^12*x^5 + (210*a^2*b^13*x^(11/2))/11 + (5*a*b^14*x^6)/2 + (2*b^15*x^(13/2))/13 + 105
*a^13*b^2*Log[x]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {(a+b x)^{15}}{x^3} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (455 a^{12} b^3+\frac {a^{15}}{x^3}+\frac {15 a^{14} b}{x^2}+\frac {105 a^{13} b^2}{x}+1365 a^{11} b^4 x+3003 a^{10} b^5 x^2+5005 a^9 b^6 x^3+6435 a^8 b^7 x^4+6435 a^7 b^8 x^5+5005 a^6 b^9 x^6+3003 a^5 b^{10} x^7+1365 a^4 b^{11} x^8+455 a^3 b^{12} x^9+105 a^2 b^{13} x^{10}+15 a b^{14} x^{11}+b^{15} x^{12}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {a^{15}}{x}-\frac {30 a^{14} b}{\sqrt {x}}+910 a^{12} b^3 \sqrt {x}+1365 a^{11} b^4 x+2002 a^{10} b^5 x^{3/2}+\frac {5005}{2} a^9 b^6 x^2+2574 a^8 b^7 x^{5/2}+2145 a^7 b^8 x^3+1430 a^6 b^9 x^{7/2}+\frac {3003}{4} a^5 b^{10} x^4+\frac {910}{3} a^4 b^{11} x^{9/2}+91 a^3 b^{12} x^5+\frac {210}{11} a^2 b^{13} x^{11/2}+\frac {5}{2} a b^{14} x^6+\frac {2}{13} b^{15} x^{13/2}+105 a^{13} b^2 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 192, normalized size = 1.00 \[ -\frac {a^{15}}{x}-\frac {30 a^{14} b}{\sqrt {x}}+105 a^{13} b^2 \log (x)+910 a^{12} b^3 \sqrt {x}+1365 a^{11} b^4 x+2002 a^{10} b^5 x^{3/2}+\frac {5005}{2} a^9 b^6 x^2+2574 a^8 b^7 x^{5/2}+2145 a^7 b^8 x^3+1430 a^6 b^9 x^{7/2}+\frac {3003}{4} a^5 b^{10} x^4+\frac {910}{3} a^4 b^{11} x^{9/2}+91 a^3 b^{12} x^5+\frac {210}{11} a^2 b^{13} x^{11/2}+\frac {5}{2} a b^{14} x^6+\frac {2}{13} b^{15} x^{13/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^15/x^2,x]

[Out]

-(a^15/x) - (30*a^14*b)/Sqrt[x] + 910*a^12*b^3*Sqrt[x] + 1365*a^11*b^4*x + 2002*a^10*b^5*x^(3/2) + (5005*a^9*b
^6*x^2)/2 + 2574*a^8*b^7*x^(5/2) + 2145*a^7*b^8*x^3 + 1430*a^6*b^9*x^(7/2) + (3003*a^5*b^10*x^4)/4 + (910*a^4*
b^11*x^(9/2))/3 + 91*a^3*b^12*x^5 + (210*a^2*b^13*x^(11/2))/11 + (5*a*b^14*x^6)/2 + (2*b^15*x^(13/2))/13 + 105
*a^13*b^2*Log[x]

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fricas [A]  time = 1.18, size = 172, normalized size = 0.90 \[ \frac {4290 \, a b^{14} x^{7} + 156156 \, a^{3} b^{12} x^{6} + 1288287 \, a^{5} b^{10} x^{5} + 3680820 \, a^{7} b^{8} x^{4} + 4294290 \, a^{9} b^{6} x^{3} + 2342340 \, a^{11} b^{4} x^{2} + 360360 \, a^{13} b^{2} x \log \left (\sqrt {x}\right ) - 1716 \, a^{15} + 8 \, {\left (33 \, b^{15} x^{7} + 4095 \, a^{2} b^{13} x^{6} + 65065 \, a^{4} b^{11} x^{5} + 306735 \, a^{6} b^{9} x^{4} + 552123 \, a^{8} b^{7} x^{3} + 429429 \, a^{10} b^{5} x^{2} + 195195 \, a^{12} b^{3} x - 6435 \, a^{14} b\right )} \sqrt {x}}{1716 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^2,x, algorithm="fricas")

[Out]

1/1716*(4290*a*b^14*x^7 + 156156*a^3*b^12*x^6 + 1288287*a^5*b^10*x^5 + 3680820*a^7*b^8*x^4 + 4294290*a^9*b^6*x
^3 + 2342340*a^11*b^4*x^2 + 360360*a^13*b^2*x*log(sqrt(x)) - 1716*a^15 + 8*(33*b^15*x^7 + 4095*a^2*b^13*x^6 +
65065*a^4*b^11*x^5 + 306735*a^6*b^9*x^4 + 552123*a^8*b^7*x^3 + 429429*a^10*b^5*x^2 + 195195*a^12*b^3*x - 6435*
a^14*b)*sqrt(x))/x

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giac [A]  time = 0.19, size = 166, normalized size = 0.86 \[ \frac {2}{13} \, b^{15} x^{\frac {13}{2}} + \frac {5}{2} \, a b^{14} x^{6} + \frac {210}{11} \, a^{2} b^{13} x^{\frac {11}{2}} + 91 \, a^{3} b^{12} x^{5} + \frac {910}{3} \, a^{4} b^{11} x^{\frac {9}{2}} + \frac {3003}{4} \, a^{5} b^{10} x^{4} + 1430 \, a^{6} b^{9} x^{\frac {7}{2}} + 2145 \, a^{7} b^{8} x^{3} + 2574 \, a^{8} b^{7} x^{\frac {5}{2}} + \frac {5005}{2} \, a^{9} b^{6} x^{2} + 2002 \, a^{10} b^{5} x^{\frac {3}{2}} + 1365 \, a^{11} b^{4} x + 105 \, a^{13} b^{2} \log \left ({\left | x \right |}\right ) + 910 \, a^{12} b^{3} \sqrt {x} - \frac {30 \, a^{14} b \sqrt {x} + a^{15}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^2,x, algorithm="giac")

[Out]

2/13*b^15*x^(13/2) + 5/2*a*b^14*x^6 + 210/11*a^2*b^13*x^(11/2) + 91*a^3*b^12*x^5 + 910/3*a^4*b^11*x^(9/2) + 30
03/4*a^5*b^10*x^4 + 1430*a^6*b^9*x^(7/2) + 2145*a^7*b^8*x^3 + 2574*a^8*b^7*x^(5/2) + 5005/2*a^9*b^6*x^2 + 2002
*a^10*b^5*x^(3/2) + 1365*a^11*b^4*x + 105*a^13*b^2*log(abs(x)) + 910*a^12*b^3*sqrt(x) - (30*a^14*b*sqrt(x) + a
^15)/x

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maple [A]  time = 0.00, size = 165, normalized size = 0.86 \[ \frac {2 b^{15} x^{\frac {13}{2}}}{13}+\frac {5 a \,b^{14} x^{6}}{2}+\frac {210 a^{2} b^{13} x^{\frac {11}{2}}}{11}+91 a^{3} b^{12} x^{5}+\frac {910 a^{4} b^{11} x^{\frac {9}{2}}}{3}+\frac {3003 a^{5} b^{10} x^{4}}{4}+1430 a^{6} b^{9} x^{\frac {7}{2}}+2145 a^{7} b^{8} x^{3}+2574 a^{8} b^{7} x^{\frac {5}{2}}+\frac {5005 a^{9} b^{6} x^{2}}{2}+2002 a^{10} b^{5} x^{\frac {3}{2}}+105 a^{13} b^{2} \ln \relax (x )+1365 a^{11} b^{4} x +910 a^{12} b^{3} \sqrt {x}-\frac {30 a^{14} b}{\sqrt {x}}-\frac {a^{15}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/2))^15/x^2,x)

[Out]

-a^15/x+1365*a^11*b^4*x+2002*a^10*b^5*x^(3/2)+5005/2*a^9*b^6*x^2+2574*a^8*b^7*x^(5/2)+2145*a^7*b^8*x^3+1430*a^
6*b^9*x^(7/2)+3003/4*a^5*b^10*x^4+910/3*a^4*b^11*x^(9/2)+91*a^3*b^12*x^5+210/11*a^2*b^13*x^(11/2)+5/2*a*b^14*x
^6+2/13*b^15*x^(13/2)+105*a^13*b^2*ln(x)-30*a^14*b/x^(1/2)+910*a^12*b^3*x^(1/2)

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maxima [A]  time = 0.87, size = 165, normalized size = 0.86 \[ \frac {2}{13} \, b^{15} x^{\frac {13}{2}} + \frac {5}{2} \, a b^{14} x^{6} + \frac {210}{11} \, a^{2} b^{13} x^{\frac {11}{2}} + 91 \, a^{3} b^{12} x^{5} + \frac {910}{3} \, a^{4} b^{11} x^{\frac {9}{2}} + \frac {3003}{4} \, a^{5} b^{10} x^{4} + 1430 \, a^{6} b^{9} x^{\frac {7}{2}} + 2145 \, a^{7} b^{8} x^{3} + 2574 \, a^{8} b^{7} x^{\frac {5}{2}} + \frac {5005}{2} \, a^{9} b^{6} x^{2} + 2002 \, a^{10} b^{5} x^{\frac {3}{2}} + 1365 \, a^{11} b^{4} x + 105 \, a^{13} b^{2} \log \relax (x) + 910 \, a^{12} b^{3} \sqrt {x} - \frac {30 \, a^{14} b \sqrt {x} + a^{15}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^2,x, algorithm="maxima")

[Out]

2/13*b^15*x^(13/2) + 5/2*a*b^14*x^6 + 210/11*a^2*b^13*x^(11/2) + 91*a^3*b^12*x^5 + 910/3*a^4*b^11*x^(9/2) + 30
03/4*a^5*b^10*x^4 + 1430*a^6*b^9*x^(7/2) + 2145*a^7*b^8*x^3 + 2574*a^8*b^7*x^(5/2) + 5005/2*a^9*b^6*x^2 + 2002
*a^10*b^5*x^(3/2) + 1365*a^11*b^4*x + 105*a^13*b^2*log(x) + 910*a^12*b^3*sqrt(x) - (30*a^14*b*sqrt(x) + a^15)/
x

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mupad [B]  time = 0.14, size = 167, normalized size = 0.87 \[ \frac {2\,b^{15}\,x^{13/2}}{13}-\frac {a^{15}+30\,a^{14}\,b\,\sqrt {x}}{x}+210\,a^{13}\,b^2\,\ln \left (\sqrt {x}\right )+1365\,a^{11}\,b^4\,x+\frac {5\,a\,b^{14}\,x^6}{2}+\frac {5005\,a^9\,b^6\,x^2}{2}+2145\,a^7\,b^8\,x^3+910\,a^{12}\,b^3\,\sqrt {x}+\frac {3003\,a^5\,b^{10}\,x^4}{4}+91\,a^3\,b^{12}\,x^5+2002\,a^{10}\,b^5\,x^{3/2}+2574\,a^8\,b^7\,x^{5/2}+1430\,a^6\,b^9\,x^{7/2}+\frac {910\,a^4\,b^{11}\,x^{9/2}}{3}+\frac {210\,a^2\,b^{13}\,x^{11/2}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(1/2))^15/x^2,x)

[Out]

(2*b^15*x^(13/2))/13 - (a^15 + 30*a^14*b*x^(1/2))/x + 210*a^13*b^2*log(x^(1/2)) + 1365*a^11*b^4*x + (5*a*b^14*
x^6)/2 + (5005*a^9*b^6*x^2)/2 + 2145*a^7*b^8*x^3 + 910*a^12*b^3*x^(1/2) + (3003*a^5*b^10*x^4)/4 + 91*a^3*b^12*
x^5 + 2002*a^10*b^5*x^(3/2) + 2574*a^8*b^7*x^(5/2) + 1430*a^6*b^9*x^(7/2) + (910*a^4*b^11*x^(9/2))/3 + (210*a^
2*b^13*x^(11/2))/11

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sympy [A]  time = 5.81, size = 197, normalized size = 1.03 \[ - \frac {a^{15}}{x} - \frac {30 a^{14} b}{\sqrt {x}} + 105 a^{13} b^{2} \log {\relax (x )} + 910 a^{12} b^{3} \sqrt {x} + 1365 a^{11} b^{4} x + 2002 a^{10} b^{5} x^{\frac {3}{2}} + \frac {5005 a^{9} b^{6} x^{2}}{2} + 2574 a^{8} b^{7} x^{\frac {5}{2}} + 2145 a^{7} b^{8} x^{3} + 1430 a^{6} b^{9} x^{\frac {7}{2}} + \frac {3003 a^{5} b^{10} x^{4}}{4} + \frac {910 a^{4} b^{11} x^{\frac {9}{2}}}{3} + 91 a^{3} b^{12} x^{5} + \frac {210 a^{2} b^{13} x^{\frac {11}{2}}}{11} + \frac {5 a b^{14} x^{6}}{2} + \frac {2 b^{15} x^{\frac {13}{2}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**15/x - 30*a**14*b/sqrt(x) + 105*a**13*b**2*log(x) + 910*a**12*b**3*sqrt(x) + 1365*a**11*b**4*x + 2002*a**1
0*b**5*x**(3/2) + 5005*a**9*b**6*x**2/2 + 2574*a**8*b**7*x**(5/2) + 2145*a**7*b**8*x**3 + 1430*a**6*b**9*x**(7
/2) + 3003*a**5*b**10*x**4/4 + 910*a**4*b**11*x**(9/2)/3 + 91*a**3*b**12*x**5 + 210*a**2*b**13*x**(11/2)/11 +
5*a*b**14*x**6/2 + 2*b**15*x**(13/2)/13

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