3.2180 \(\int \frac {(a+b \sqrt {x})^{15}}{x^7} \, dx\)

Optimal. Leaf size=196 \[ -\frac {a^{15}}{6 x^6}-\frac {30 a^{14} b}{11 x^{11/2}}-\frac {21 a^{13} b^2}{x^5}-\frac {910 a^{12} b^3}{9 x^{9/2}}-\frac {1365 a^{11} b^4}{4 x^4}-\frac {858 a^{10} b^5}{x^{7/2}}-\frac {5005 a^9 b^6}{3 x^3}-\frac {2574 a^8 b^7}{x^{5/2}}-\frac {6435 a^7 b^8}{2 x^2}-\frac {10010 a^6 b^9}{3 x^{3/2}}-\frac {3003 a^5 b^{10}}{x}-\frac {2730 a^4 b^{11}}{\sqrt {x}}+455 a^3 b^{12} \log (x)+210 a^2 b^{13} \sqrt {x}+15 a b^{14} x+\frac {2}{3} b^{15} x^{3/2} \]

[Out]

-1/6*a^15/x^6-30/11*a^14*b/x^(11/2)-21*a^13*b^2/x^5-910/9*a^12*b^3/x^(9/2)-1365/4*a^11*b^4/x^4-858*a^10*b^5/x^
(7/2)-5005/3*a^9*b^6/x^3-2574*a^8*b^7/x^(5/2)-6435/2*a^7*b^8/x^2-10010/3*a^6*b^9/x^(3/2)-3003*a^5*b^10/x+15*a*
b^14*x+2/3*b^15*x^(3/2)+455*a^3*b^12*ln(x)-2730*a^4*b^11/x^(1/2)+210*a^2*b^13*x^(1/2)

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Rubi [A]  time = 0.11, antiderivative size = 196, 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} \[ -\frac {21 a^{13} b^2}{x^5}-\frac {910 a^{12} b^3}{9 x^{9/2}}-\frac {1365 a^{11} b^4}{4 x^4}-\frac {858 a^{10} b^5}{x^{7/2}}-\frac {5005 a^9 b^6}{3 x^3}-\frac {2574 a^8 b^7}{x^{5/2}}-\frac {6435 a^7 b^8}{2 x^2}-\frac {10010 a^6 b^9}{3 x^{3/2}}-\frac {3003 a^5 b^{10}}{x}-\frac {2730 a^4 b^{11}}{\sqrt {x}}+210 a^2 b^{13} \sqrt {x}+455 a^3 b^{12} \log (x)-\frac {30 a^{14} b}{11 x^{11/2}}-\frac {a^{15}}{6 x^6}+15 a b^{14} x+\frac {2}{3} b^{15} x^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^15/x^7,x]

[Out]

-a^15/(6*x^6) - (30*a^14*b)/(11*x^(11/2)) - (21*a^13*b^2)/x^5 - (910*a^12*b^3)/(9*x^(9/2)) - (1365*a^11*b^4)/(
4*x^4) - (858*a^10*b^5)/x^(7/2) - (5005*a^9*b^6)/(3*x^3) - (2574*a^8*b^7)/x^(5/2) - (6435*a^7*b^8)/(2*x^2) - (
10010*a^6*b^9)/(3*x^(3/2)) - (3003*a^5*b^10)/x - (2730*a^4*b^11)/Sqrt[x] + 210*a^2*b^13*Sqrt[x] + 15*a*b^14*x
+ (2*b^15*x^(3/2))/3 + 455*a^3*b^12*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^7} \, dx &=2 \operatorname {Subst}\left (\int \frac {(a+b x)^{15}}{x^{13}} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (105 a^2 b^{13}+\frac {a^{15}}{x^{13}}+\frac {15 a^{14} b}{x^{12}}+\frac {105 a^{13} b^2}{x^{11}}+\frac {455 a^{12} b^3}{x^{10}}+\frac {1365 a^{11} b^4}{x^9}+\frac {3003 a^{10} b^5}{x^8}+\frac {5005 a^9 b^6}{x^7}+\frac {6435 a^8 b^7}{x^6}+\frac {6435 a^7 b^8}{x^5}+\frac {5005 a^6 b^9}{x^4}+\frac {3003 a^5 b^{10}}{x^3}+\frac {1365 a^4 b^{11}}{x^2}+\frac {455 a^3 b^{12}}{x}+15 a b^{14} x+b^{15} x^2\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {a^{15}}{6 x^6}-\frac {30 a^{14} b}{11 x^{11/2}}-\frac {21 a^{13} b^2}{x^5}-\frac {910 a^{12} b^3}{9 x^{9/2}}-\frac {1365 a^{11} b^4}{4 x^4}-\frac {858 a^{10} b^5}{x^{7/2}}-\frac {5005 a^9 b^6}{3 x^3}-\frac {2574 a^8 b^7}{x^{5/2}}-\frac {6435 a^7 b^8}{2 x^2}-\frac {10010 a^6 b^9}{3 x^{3/2}}-\frac {3003 a^5 b^{10}}{x}-\frac {2730 a^4 b^{11}}{\sqrt {x}}+210 a^2 b^{13} \sqrt {x}+15 a b^{14} x+\frac {2}{3} b^{15} x^{3/2}+455 a^3 b^{12} \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 196, normalized size = 1.00 \[ -\frac {a^{15}}{6 x^6}-\frac {30 a^{14} b}{11 x^{11/2}}-\frac {21 a^{13} b^2}{x^5}-\frac {910 a^{12} b^3}{9 x^{9/2}}-\frac {1365 a^{11} b^4}{4 x^4}-\frac {858 a^{10} b^5}{x^{7/2}}-\frac {5005 a^9 b^6}{3 x^3}-\frac {2574 a^8 b^7}{x^{5/2}}-\frac {6435 a^7 b^8}{2 x^2}-\frac {10010 a^6 b^9}{3 x^{3/2}}-\frac {3003 a^5 b^{10}}{x}-\frac {2730 a^4 b^{11}}{\sqrt {x}}+455 a^3 b^{12} \log (x)+210 a^2 b^{13} \sqrt {x}+15 a b^{14} x+\frac {2}{3} b^{15} x^{3/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^15/x^7,x]

[Out]

-1/6*a^15/x^6 - (30*a^14*b)/(11*x^(11/2)) - (21*a^13*b^2)/x^5 - (910*a^12*b^3)/(9*x^(9/2)) - (1365*a^11*b^4)/(
4*x^4) - (858*a^10*b^5)/x^(7/2) - (5005*a^9*b^6)/(3*x^3) - (2574*a^8*b^7)/x^(5/2) - (6435*a^7*b^8)/(2*x^2) - (
10010*a^6*b^9)/(3*x^(3/2)) - (3003*a^5*b^10)/x - (2730*a^4*b^11)/Sqrt[x] + 210*a^2*b^13*Sqrt[x] + 15*a*b^14*x
+ (2*b^15*x^(3/2))/3 + 455*a^3*b^12*Log[x]

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fricas [A]  time = 0.82, size = 172, normalized size = 0.88 \[ \frac {5940 \, a b^{14} x^{7} + 360360 \, a^{3} b^{12} x^{6} \log \left (\sqrt {x}\right ) - 1189188 \, a^{5} b^{10} x^{5} - 1274130 \, a^{7} b^{8} x^{4} - 660660 \, a^{9} b^{6} x^{3} - 135135 \, a^{11} b^{4} x^{2} - 8316 \, a^{13} b^{2} x - 66 \, a^{15} + 8 \, {\left (33 \, b^{15} x^{7} + 10395 \, a^{2} b^{13} x^{6} - 135135 \, a^{4} b^{11} x^{5} - 165165 \, a^{6} b^{9} x^{4} - 127413 \, a^{8} b^{7} x^{3} - 42471 \, a^{10} b^{5} x^{2} - 5005 \, a^{12} b^{3} x - 135 \, a^{14} b\right )} \sqrt {x}}{396 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^7,x, algorithm="fricas")

[Out]

1/396*(5940*a*b^14*x^7 + 360360*a^3*b^12*x^6*log(sqrt(x)) - 1189188*a^5*b^10*x^5 - 1274130*a^7*b^8*x^4 - 66066
0*a^9*b^6*x^3 - 135135*a^11*b^4*x^2 - 8316*a^13*b^2*x - 66*a^15 + 8*(33*b^15*x^7 + 10395*a^2*b^13*x^6 - 135135
*a^4*b^11*x^5 - 165165*a^6*b^9*x^4 - 127413*a^8*b^7*x^3 - 42471*a^10*b^5*x^2 - 5005*a^12*b^3*x - 135*a^14*b)*s
qrt(x))/x^6

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giac [A]  time = 0.18, size = 166, normalized size = 0.85 \[ \frac {2}{3} \, b^{15} x^{\frac {3}{2}} + 15 \, a b^{14} x + 455 \, a^{3} b^{12} \log \left ({\left | x \right |}\right ) + 210 \, a^{2} b^{13} \sqrt {x} - \frac {1081080 \, a^{4} b^{11} x^{\frac {11}{2}} + 1189188 \, a^{5} b^{10} x^{5} + 1321320 \, a^{6} b^{9} x^{\frac {9}{2}} + 1274130 \, a^{7} b^{8} x^{4} + 1019304 \, a^{8} b^{7} x^{\frac {7}{2}} + 660660 \, a^{9} b^{6} x^{3} + 339768 \, a^{10} b^{5} x^{\frac {5}{2}} + 135135 \, a^{11} b^{4} x^{2} + 40040 \, a^{12} b^{3} x^{\frac {3}{2}} + 8316 \, a^{13} b^{2} x + 1080 \, a^{14} b \sqrt {x} + 66 \, a^{15}}{396 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^7,x, algorithm="giac")

[Out]

2/3*b^15*x^(3/2) + 15*a*b^14*x + 455*a^3*b^12*log(abs(x)) + 210*a^2*b^13*sqrt(x) - 1/396*(1081080*a^4*b^11*x^(
11/2) + 1189188*a^5*b^10*x^5 + 1321320*a^6*b^9*x^(9/2) + 1274130*a^7*b^8*x^4 + 1019304*a^8*b^7*x^(7/2) + 66066
0*a^9*b^6*x^3 + 339768*a^10*b^5*x^(5/2) + 135135*a^11*b^4*x^2 + 40040*a^12*b^3*x^(3/2) + 8316*a^13*b^2*x + 108
0*a^14*b*sqrt(x) + 66*a^15)/x^6

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maple [A]  time = 0.00, size = 165, normalized size = 0.84 \[ \frac {2 b^{15} x^{\frac {3}{2}}}{3}+455 a^{3} b^{12} \ln \relax (x )+15 a \,b^{14} x +210 a^{2} b^{13} \sqrt {x}-\frac {2730 a^{4} b^{11}}{\sqrt {x}}-\frac {3003 a^{5} b^{10}}{x}-\frac {10010 a^{6} b^{9}}{3 x^{\frac {3}{2}}}-\frac {6435 a^{7} b^{8}}{2 x^{2}}-\frac {2574 a^{8} b^{7}}{x^{\frac {5}{2}}}-\frac {5005 a^{9} b^{6}}{3 x^{3}}-\frac {858 a^{10} b^{5}}{x^{\frac {7}{2}}}-\frac {1365 a^{11} b^{4}}{4 x^{4}}-\frac {910 a^{12} b^{3}}{9 x^{\frac {9}{2}}}-\frac {21 a^{13} b^{2}}{x^{5}}-\frac {30 a^{14} b}{11 x^{\frac {11}{2}}}-\frac {a^{15}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/2))^15/x^7,x)

[Out]

-1/6*a^15/x^6-30/11*a^14*b/x^(11/2)-21*a^13*b^2/x^5-910/9*a^12*b^3/x^(9/2)-1365/4*a^11*b^4/x^4-858*a^10*b^5/x^
(7/2)-5005/3*a^9*b^6/x^3-2574*a^8*b^7/x^(5/2)-6435/2*a^7*b^8/x^2-10010/3*a^6*b^9/x^(3/2)-3003*a^5*b^10/x+15*a*
b^14*x+2/3*b^15*x^(3/2)+455*a^3*b^12*ln(x)-2730*a^4*b^11/x^(1/2)+210*a^2*b^13*x^(1/2)

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maxima [A]  time = 0.90, size = 165, normalized size = 0.84 \[ \frac {2}{3} \, b^{15} x^{\frac {3}{2}} + 15 \, a b^{14} x + 455 \, a^{3} b^{12} \log \relax (x) + 210 \, a^{2} b^{13} \sqrt {x} - \frac {1081080 \, a^{4} b^{11} x^{\frac {11}{2}} + 1189188 \, a^{5} b^{10} x^{5} + 1321320 \, a^{6} b^{9} x^{\frac {9}{2}} + 1274130 \, a^{7} b^{8} x^{4} + 1019304 \, a^{8} b^{7} x^{\frac {7}{2}} + 660660 \, a^{9} b^{6} x^{3} + 339768 \, a^{10} b^{5} x^{\frac {5}{2}} + 135135 \, a^{11} b^{4} x^{2} + 40040 \, a^{12} b^{3} x^{\frac {3}{2}} + 8316 \, a^{13} b^{2} x + 1080 \, a^{14} b \sqrt {x} + 66 \, a^{15}}{396 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^7,x, algorithm="maxima")

[Out]

2/3*b^15*x^(3/2) + 15*a*b^14*x + 455*a^3*b^12*log(x) + 210*a^2*b^13*sqrt(x) - 1/396*(1081080*a^4*b^11*x^(11/2)
+ 1189188*a^5*b^10*x^5 + 1321320*a^6*b^9*x^(9/2) + 1274130*a^7*b^8*x^4 + 1019304*a^8*b^7*x^(7/2) + 660660*a^9
*b^6*x^3 + 339768*a^10*b^5*x^(5/2) + 135135*a^11*b^4*x^2 + 40040*a^12*b^3*x^(3/2) + 8316*a^13*b^2*x + 1080*a^1
4*b*sqrt(x) + 66*a^15)/x^6

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mupad [B]  time = 1.12, size = 167, normalized size = 0.85 \[ \frac {2\,b^{15}\,x^{3/2}}{3}-\frac {\frac {a^{15}}{6}+21\,a^{13}\,b^2\,x+\frac {30\,a^{14}\,b\,\sqrt {x}}{11}+\frac {1365\,a^{11}\,b^4\,x^2}{4}+\frac {5005\,a^9\,b^6\,x^3}{3}+\frac {6435\,a^7\,b^8\,x^4}{2}+3003\,a^5\,b^{10}\,x^5+\frac {910\,a^{12}\,b^3\,x^{3/2}}{9}+858\,a^{10}\,b^5\,x^{5/2}+2574\,a^8\,b^7\,x^{7/2}+\frac {10010\,a^6\,b^9\,x^{9/2}}{3}+2730\,a^4\,b^{11}\,x^{11/2}}{x^6}+910\,a^3\,b^{12}\,\ln \left (\sqrt {x}\right )+210\,a^2\,b^{13}\,\sqrt {x}+15\,a\,b^{14}\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(1/2))^15/x^7,x)

[Out]

(2*b^15*x^(3/2))/3 - (a^15/6 + 21*a^13*b^2*x + (30*a^14*b*x^(1/2))/11 + (1365*a^11*b^4*x^2)/4 + (5005*a^9*b^6*
x^3)/3 + (6435*a^7*b^8*x^4)/2 + 3003*a^5*b^10*x^5 + (910*a^12*b^3*x^(3/2))/9 + 858*a^10*b^5*x^(5/2) + 2574*a^8
*b^7*x^(7/2) + (10010*a^6*b^9*x^(9/2))/3 + 2730*a^4*b^11*x^(11/2))/x^6 + 910*a^3*b^12*log(x^(1/2)) + 210*a^2*b
^13*x^(1/2) + 15*a*b^14*x

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sympy [A]  time = 5.04, size = 201, normalized size = 1.03 \[ - \frac {a^{15}}{6 x^{6}} - \frac {30 a^{14} b}{11 x^{\frac {11}{2}}} - \frac {21 a^{13} b^{2}}{x^{5}} - \frac {910 a^{12} b^{3}}{9 x^{\frac {9}{2}}} - \frac {1365 a^{11} b^{4}}{4 x^{4}} - \frac {858 a^{10} b^{5}}{x^{\frac {7}{2}}} - \frac {5005 a^{9} b^{6}}{3 x^{3}} - \frac {2574 a^{8} b^{7}}{x^{\frac {5}{2}}} - \frac {6435 a^{7} b^{8}}{2 x^{2}} - \frac {10010 a^{6} b^{9}}{3 x^{\frac {3}{2}}} - \frac {3003 a^{5} b^{10}}{x} - \frac {2730 a^{4} b^{11}}{\sqrt {x}} + 455 a^{3} b^{12} \log {\relax (x )} + 210 a^{2} b^{13} \sqrt {x} + 15 a b^{14} x + \frac {2 b^{15} x^{\frac {3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/2))**15/x**7,x)

[Out]

-a**15/(6*x**6) - 30*a**14*b/(11*x**(11/2)) - 21*a**13*b**2/x**5 - 910*a**12*b**3/(9*x**(9/2)) - 1365*a**11*b*
*4/(4*x**4) - 858*a**10*b**5/x**(7/2) - 5005*a**9*b**6/(3*x**3) - 2574*a**8*b**7/x**(5/2) - 6435*a**7*b**8/(2*
x**2) - 10010*a**6*b**9/(3*x**(3/2)) - 3003*a**5*b**10/x - 2730*a**4*b**11/sqrt(x) + 455*a**3*b**12*log(x) + 2
10*a**2*b**13*sqrt(x) + 15*a*b**14*x + 2*b**15*x**(3/2)/3

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