3.2177 \(\int \frac {(a+b \sqrt {x})^{15}}{x^3} \, dx\)

Optimal. Leaf size=190 \[ -\frac {a^{15}}{2 x^2}-\frac {10 a^{14} b}{x^{3/2}}-\frac {105 a^{13} b^2}{x}-\frac {910 a^{12} b^3}{\sqrt {x}}+1365 a^{11} b^4 \log (x)+6006 a^{10} b^5 \sqrt {x}+5005 a^9 b^6 x+4290 a^8 b^7 x^{3/2}+\frac {6435}{2} a^7 b^8 x^2+2002 a^6 b^9 x^{5/2}+1001 a^5 b^{10} x^3+390 a^4 b^{11} x^{7/2}+\frac {455}{4} a^3 b^{12} x^4+\frac {70}{3} a^2 b^{13} x^{9/2}+3 a b^{14} x^5+\frac {2}{11} b^{15} x^{11/2} \]

[Out]

-1/2*a^15/x^2-10*a^14*b/x^(3/2)-105*a^13*b^2/x+5005*a^9*b^6*x+4290*a^8*b^7*x^(3/2)+6435/2*a^7*b^8*x^2+2002*a^6
*b^9*x^(5/2)+1001*a^5*b^10*x^3+390*a^4*b^11*x^(7/2)+455/4*a^3*b^12*x^4+70/3*a^2*b^13*x^(9/2)+3*a*b^14*x^5+2/11
*b^15*x^(11/2)+1365*a^11*b^4*ln(x)-910*a^12*b^3/x^(1/2)+6006*a^10*b^5*x^(1/2)

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Rubi [A]  time = 0.12, antiderivative size = 190, 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} \[ 4290 a^8 b^7 x^{3/2}+\frac {6435}{2} a^7 b^8 x^2+2002 a^6 b^9 x^{5/2}+1001 a^5 b^{10} x^3+390 a^4 b^{11} x^{7/2}+\frac {455}{4} a^3 b^{12} x^4+\frac {70}{3} a^2 b^{13} x^{9/2}-\frac {105 a^{13} b^2}{x}-\frac {910 a^{12} b^3}{\sqrt {x}}+6006 a^{10} b^5 \sqrt {x}+5005 a^9 b^6 x+1365 a^{11} b^4 \log (x)-\frac {10 a^{14} b}{x^{3/2}}-\frac {a^{15}}{2 x^2}+3 a b^{14} x^5+\frac {2}{11} b^{15} x^{11/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^15/x^3,x]

[Out]

-a^15/(2*x^2) - (10*a^14*b)/x^(3/2) - (105*a^13*b^2)/x - (910*a^12*b^3)/Sqrt[x] + 6006*a^10*b^5*Sqrt[x] + 5005
*a^9*b^6*x + 4290*a^8*b^7*x^(3/2) + (6435*a^7*b^8*x^2)/2 + 2002*a^6*b^9*x^(5/2) + 1001*a^5*b^10*x^3 + 390*a^4*
b^11*x^(7/2) + (455*a^3*b^12*x^4)/4 + (70*a^2*b^13*x^(9/2))/3 + 3*a*b^14*x^5 + (2*b^15*x^(11/2))/11 + 1365*a^1
1*b^4*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^3} \, dx &=2 \operatorname {Subst}\left (\int \frac {(a+b x)^{15}}{x^5} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (3003 a^{10} b^5+\frac {a^{15}}{x^5}+\frac {15 a^{14} b}{x^4}+\frac {105 a^{13} b^2}{x^3}+\frac {455 a^{12} b^3}{x^2}+\frac {1365 a^{11} b^4}{x}+5005 a^9 b^6 x+6435 a^8 b^7 x^2+6435 a^7 b^8 x^3+5005 a^6 b^9 x^4+3003 a^5 b^{10} x^5+1365 a^4 b^{11} x^6+455 a^3 b^{12} x^7+105 a^2 b^{13} x^8+15 a b^{14} x^9+b^{15} x^{10}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {a^{15}}{2 x^2}-\frac {10 a^{14} b}{x^{3/2}}-\frac {105 a^{13} b^2}{x}-\frac {910 a^{12} b^3}{\sqrt {x}}+6006 a^{10} b^5 \sqrt {x}+5005 a^9 b^6 x+4290 a^8 b^7 x^{3/2}+\frac {6435}{2} a^7 b^8 x^2+2002 a^6 b^9 x^{5/2}+1001 a^5 b^{10} x^3+390 a^4 b^{11} x^{7/2}+\frac {455}{4} a^3 b^{12} x^4+\frac {70}{3} a^2 b^{13} x^{9/2}+3 a b^{14} x^5+\frac {2}{11} b^{15} x^{11/2}+1365 a^{11} b^4 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 190, normalized size = 1.00 \[ -\frac {a^{15}}{2 x^2}-\frac {10 a^{14} b}{x^{3/2}}-\frac {105 a^{13} b^2}{x}-\frac {910 a^{12} b^3}{\sqrt {x}}+1365 a^{11} b^4 \log (x)+6006 a^{10} b^5 \sqrt {x}+5005 a^9 b^6 x+4290 a^8 b^7 x^{3/2}+\frac {6435}{2} a^7 b^8 x^2+2002 a^6 b^9 x^{5/2}+1001 a^5 b^{10} x^3+390 a^4 b^{11} x^{7/2}+\frac {455}{4} a^3 b^{12} x^4+\frac {70}{3} a^2 b^{13} x^{9/2}+3 a b^{14} x^5+\frac {2}{11} b^{15} x^{11/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^15/x^3,x]

[Out]

-1/2*a^15/x^2 - (10*a^14*b)/x^(3/2) - (105*a^13*b^2)/x - (910*a^12*b^3)/Sqrt[x] + 6006*a^10*b^5*Sqrt[x] + 5005
*a^9*b^6*x + 4290*a^8*b^7*x^(3/2) + (6435*a^7*b^8*x^2)/2 + 2002*a^6*b^9*x^(5/2) + 1001*a^5*b^10*x^3 + 390*a^4*
b^11*x^(7/2) + (455*a^3*b^12*x^4)/4 + (70*a^2*b^13*x^(9/2))/3 + 3*a*b^14*x^5 + (2*b^15*x^(11/2))/11 + 1365*a^1
1*b^4*Log[x]

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fricas [A]  time = 0.88, size = 172, normalized size = 0.91 \[ \frac {396 \, a b^{14} x^{7} + 15015 \, a^{3} b^{12} x^{6} + 132132 \, a^{5} b^{10} x^{5} + 424710 \, a^{7} b^{8} x^{4} + 660660 \, a^{9} b^{6} x^{3} + 360360 \, a^{11} b^{4} x^{2} \log \left (\sqrt {x}\right ) - 13860 \, a^{13} b^{2} x - 66 \, a^{15} + 8 \, {\left (3 \, b^{15} x^{7} + 385 \, a^{2} b^{13} x^{6} + 6435 \, a^{4} b^{11} x^{5} + 33033 \, a^{6} b^{9} x^{4} + 70785 \, a^{8} b^{7} x^{3} + 99099 \, a^{10} b^{5} x^{2} - 15015 \, a^{12} b^{3} x - 165 \, a^{14} b\right )} \sqrt {x}}{132 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^3,x, algorithm="fricas")

[Out]

1/132*(396*a*b^14*x^7 + 15015*a^3*b^12*x^6 + 132132*a^5*b^10*x^5 + 424710*a^7*b^8*x^4 + 660660*a^9*b^6*x^3 + 3
60360*a^11*b^4*x^2*log(sqrt(x)) - 13860*a^13*b^2*x - 66*a^15 + 8*(3*b^15*x^7 + 385*a^2*b^13*x^6 + 6435*a^4*b^1
1*x^5 + 33033*a^6*b^9*x^4 + 70785*a^8*b^7*x^3 + 99099*a^10*b^5*x^2 - 15015*a^12*b^3*x - 165*a^14*b)*sqrt(x))/x
^2

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giac [A]  time = 0.18, size = 164, normalized size = 0.86 \[ \frac {2}{11} \, b^{15} x^{\frac {11}{2}} + 3 \, a b^{14} x^{5} + \frac {70}{3} \, a^{2} b^{13} x^{\frac {9}{2}} + \frac {455}{4} \, a^{3} b^{12} x^{4} + 390 \, a^{4} b^{11} x^{\frac {7}{2}} + 1001 \, a^{5} b^{10} x^{3} + 2002 \, a^{6} b^{9} x^{\frac {5}{2}} + \frac {6435}{2} \, a^{7} b^{8} x^{2} + 4290 \, a^{8} b^{7} x^{\frac {3}{2}} + 5005 \, a^{9} b^{6} x + 1365 \, a^{11} b^{4} \log \left ({\left | x \right |}\right ) + 6006 \, a^{10} b^{5} \sqrt {x} - \frac {1820 \, a^{12} b^{3} x^{\frac {3}{2}} + 210 \, a^{13} b^{2} x + 20 \, a^{14} b \sqrt {x} + a^{15}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^3,x, algorithm="giac")

[Out]

2/11*b^15*x^(11/2) + 3*a*b^14*x^5 + 70/3*a^2*b^13*x^(9/2) + 455/4*a^3*b^12*x^4 + 390*a^4*b^11*x^(7/2) + 1001*a
^5*b^10*x^3 + 2002*a^6*b^9*x^(5/2) + 6435/2*a^7*b^8*x^2 + 4290*a^8*b^7*x^(3/2) + 5005*a^9*b^6*x + 1365*a^11*b^
4*log(abs(x)) + 6006*a^10*b^5*sqrt(x) - 1/2*(1820*a^12*b^3*x^(3/2) + 210*a^13*b^2*x + 20*a^14*b*sqrt(x) + a^15
)/x^2

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maple [A]  time = 0.00, size = 165, normalized size = 0.87 \[ \frac {2 b^{15} x^{\frac {11}{2}}}{11}+3 a \,b^{14} x^{5}+\frac {70 a^{2} b^{13} x^{\frac {9}{2}}}{3}+\frac {455 a^{3} b^{12} x^{4}}{4}+390 a^{4} b^{11} x^{\frac {7}{2}}+1001 a^{5} b^{10} x^{3}+2002 a^{6} b^{9} x^{\frac {5}{2}}+\frac {6435 a^{7} b^{8} x^{2}}{2}+4290 a^{8} b^{7} x^{\frac {3}{2}}+1365 a^{11} b^{4} \ln \relax (x )+5005 a^{9} b^{6} x +6006 a^{10} b^{5} \sqrt {x}-\frac {910 a^{12} b^{3}}{\sqrt {x}}-\frac {105 a^{13} b^{2}}{x}-\frac {10 a^{14} b}{x^{\frac {3}{2}}}-\frac {a^{15}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/2))^15/x^3,x)

[Out]

-1/2*a^15/x^2-10*a^14*b/x^(3/2)-105*a^13*b^2/x+5005*a^9*b^6*x+4290*a^8*b^7*x^(3/2)+6435/2*a^7*b^8*x^2+2002*a^6
*b^9*x^(5/2)+1001*a^5*b^10*x^3+390*a^4*b^11*x^(7/2)+455/4*a^3*b^12*x^4+70/3*a^2*b^13*x^(9/2)+3*a*b^14*x^5+2/11
*b^15*x^(11/2)+1365*a^11*b^4*ln(x)-910*a^12*b^3/x^(1/2)+6006*a^10*b^5*x^(1/2)

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maxima [A]  time = 0.90, size = 163, normalized size = 0.86 \[ \frac {2}{11} \, b^{15} x^{\frac {11}{2}} + 3 \, a b^{14} x^{5} + \frac {70}{3} \, a^{2} b^{13} x^{\frac {9}{2}} + \frac {455}{4} \, a^{3} b^{12} x^{4} + 390 \, a^{4} b^{11} x^{\frac {7}{2}} + 1001 \, a^{5} b^{10} x^{3} + 2002 \, a^{6} b^{9} x^{\frac {5}{2}} + \frac {6435}{2} \, a^{7} b^{8} x^{2} + 4290 \, a^{8} b^{7} x^{\frac {3}{2}} + 5005 \, a^{9} b^{6} x + 1365 \, a^{11} b^{4} \log \relax (x) + 6006 \, a^{10} b^{5} \sqrt {x} - \frac {1820 \, a^{12} b^{3} x^{\frac {3}{2}} + 210 \, a^{13} b^{2} x + 20 \, a^{14} b \sqrt {x} + a^{15}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^3,x, algorithm="maxima")

[Out]

2/11*b^15*x^(11/2) + 3*a*b^14*x^5 + 70/3*a^2*b^13*x^(9/2) + 455/4*a^3*b^12*x^4 + 390*a^4*b^11*x^(7/2) + 1001*a
^5*b^10*x^3 + 2002*a^6*b^9*x^(5/2) + 6435/2*a^7*b^8*x^2 + 4290*a^8*b^7*x^(3/2) + 5005*a^9*b^6*x + 1365*a^11*b^
4*log(x) + 6006*a^10*b^5*sqrt(x) - 1/2*(1820*a^12*b^3*x^(3/2) + 210*a^13*b^2*x + 20*a^14*b*sqrt(x) + a^15)/x^2

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mupad [B]  time = 0.10, size = 167, normalized size = 0.88 \[ \frac {2\,b^{15}\,x^{11/2}}{11}-\frac {\frac {a^{15}}{2}+105\,a^{13}\,b^2\,x+10\,a^{14}\,b\,\sqrt {x}+910\,a^{12}\,b^3\,x^{3/2}}{x^2}+2730\,a^{11}\,b^4\,\ln \left (\sqrt {x}\right )+5005\,a^9\,b^6\,x+3\,a\,b^{14}\,x^5+\frac {6435\,a^7\,b^8\,x^2}{2}+1001\,a^5\,b^{10}\,x^3+6006\,a^{10}\,b^5\,\sqrt {x}+\frac {455\,a^3\,b^{12}\,x^4}{4}+4290\,a^8\,b^7\,x^{3/2}+2002\,a^6\,b^9\,x^{5/2}+390\,a^4\,b^{11}\,x^{7/2}+\frac {70\,a^2\,b^{13}\,x^{9/2}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(1/2))^15/x^3,x)

[Out]

(2*b^15*x^(11/2))/11 - (a^15/2 + 105*a^13*b^2*x + 10*a^14*b*x^(1/2) + 910*a^12*b^3*x^(3/2))/x^2 + 2730*a^11*b^
4*log(x^(1/2)) + 5005*a^9*b^6*x + 3*a*b^14*x^5 + (6435*a^7*b^8*x^2)/2 + 1001*a^5*b^10*x^3 + 6006*a^10*b^5*x^(1
/2) + (455*a^3*b^12*x^4)/4 + 4290*a^8*b^7*x^(3/2) + 2002*a^6*b^9*x^(5/2) + 390*a^4*b^11*x^(7/2) + (70*a^2*b^13
*x^(9/2))/3

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sympy [A]  time = 5.63, size = 196, normalized size = 1.03 \[ - \frac {a^{15}}{2 x^{2}} - \frac {10 a^{14} b}{x^{\frac {3}{2}}} - \frac {105 a^{13} b^{2}}{x} - \frac {910 a^{12} b^{3}}{\sqrt {x}} + 1365 a^{11} b^{4} \log {\relax (x )} + 6006 a^{10} b^{5} \sqrt {x} + 5005 a^{9} b^{6} x + 4290 a^{8} b^{7} x^{\frac {3}{2}} + \frac {6435 a^{7} b^{8} x^{2}}{2} + 2002 a^{6} b^{9} x^{\frac {5}{2}} + 1001 a^{5} b^{10} x^{3} + 390 a^{4} b^{11} x^{\frac {7}{2}} + \frac {455 a^{3} b^{12} x^{4}}{4} + \frac {70 a^{2} b^{13} x^{\frac {9}{2}}}{3} + 3 a b^{14} x^{5} + \frac {2 b^{15} x^{\frac {11}{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/2))**15/x**3,x)

[Out]

-a**15/(2*x**2) - 10*a**14*b/x**(3/2) - 105*a**13*b**2/x - 910*a**12*b**3/sqrt(x) + 1365*a**11*b**4*log(x) + 6
006*a**10*b**5*sqrt(x) + 5005*a**9*b**6*x + 4290*a**8*b**7*x**(3/2) + 6435*a**7*b**8*x**2/2 + 2002*a**6*b**9*x
**(5/2) + 1001*a**5*b**10*x**3 + 390*a**4*b**11*x**(7/2) + 455*a**3*b**12*x**4/4 + 70*a**2*b**13*x**(9/2)/3 +
3*a*b**14*x**5 + 2*b**15*x**(11/2)/11

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