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

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

[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]

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Rubi [A]  time = 0.113786, 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, 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 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]]

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])

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*}

Mathematica [A]  time = 0.116963, size = 196, normalized size = 1. \[ -\frac{21 a^{13} b^2}{x^5}-\frac{910 a^{12} b^3}{9 x^{9/2}}-\frac{1365 a^{11} b^4}{4 x^4}-\frac{858 a^{10} b^5}{x^{7/2}}-\frac{5005 a^9 b^6}{3 x^3}-\frac{2574 a^8 b^7}{x^{5/2}}-\frac{6435 a^7 b^8}{2 x^2}-\frac{10010 a^6 b^9}{3 x^{3/2}}-\frac{3003 a^5 b^{10}}{x}-\frac{2730 a^4 b^{11}}{\sqrt{x}}+210 a^2 b^{13} \sqrt{x}+455 a^3 b^{12} \log (x)-\frac{30 a^{14} b}{11 x^{11/2}}-\frac{a^{15}}{6 x^6}+15 a b^{14} x+\frac{2}{3} b^{15} x^{3/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(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]

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Maple [A]  time = 0.004, size = 165, normalized size = 0.8 \begin{align*} -{\frac{{a}^{15}}{6\,{x}^{6}}}-{\frac{30\,{a}^{14}b}{11}{x}^{-{\frac{11}{2}}}}-21\,{\frac{{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}}}-858\,{\frac{{a}^{10}{b}^{5}}{{x}^{7/2}}}-{\frac{5005\,{a}^{9}{b}^{6}}{3\,{x}^{3}}}-2574\,{\frac{{a}^{8}{b}^{7}}{{x}^{5/2}}}-{\frac{6435\,{a}^{7}{b}^{8}}{2\,{x}^{2}}}-{\frac{10010\,{a}^{6}{b}^{9}}{3}{x}^{-{\frac{3}{2}}}}-3003\,{\frac{{a}^{5}{b}^{10}}{x}}+15\,a{b}^{14}x+{\frac{2\,{b}^{15}}{3}{x}^{{\frac{3}{2}}}}+455\,{a}^{3}{b}^{12}\ln \left ( x \right ) -2730\,{\frac{{a}^{4}{b}^{11}}{\sqrt{x}}}+210\,{a}^{2}{b}^{13}\sqrt{x} \end{align*}

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 = 0.981972, size = 223, normalized size = 1.14 \begin{align*} \frac{2}{3} \, b^{15} x^{\frac{3}{2}} + 15 \, a b^{14} x + 455 \, a^{3} b^{12} \log \left (x\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}} \end{align*}

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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Fricas [A]  time = 1.26888, size = 463, normalized size = 2.36 \begin{align*} \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}} \end{align*}

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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Sympy [A]  time = 5.51175, size = 201, normalized size = 1.03 \begin{align*} - \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{\left (x \right )} + 210 a^{2} b^{13} \sqrt{x} + 15 a b^{14} x + \frac{2 b^{15} x^{\frac{3}{2}}}{3} \end{align*}

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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Giac [A]  time = 1.1158, size = 224, normalized size = 1.14 \begin{align*} \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}} \end{align*}

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