3.2179 \(\int \frac{(a+b \sqrt{x})^{15}}{x^6} \, dx\)

Optimal. Leaf size=194 \[ -\frac{105 a^{13} b^2}{4 x^4}-\frac{130 a^{12} b^3}{x^{7/2}}-\frac{455 a^{11} b^4}{x^3}-\frac{6006 a^{10} b^5}{5 x^{5/2}}-\frac{5005 a^9 b^6}{2 x^2}-\frac{4290 a^8 b^7}{x^{3/2}}+70 a^2 b^{13} x^{3/2}-\frac{6435 a^7 b^8}{x}-\frac{10010 a^6 b^9}{\sqrt{x}}+2730 a^4 b^{11} \sqrt{x}+455 a^3 b^{12} x+3003 a^5 b^{10} \log (x)-\frac{10 a^{14} b}{3 x^{9/2}}-\frac{a^{15}}{5 x^5}+\frac{15}{2} a b^{14} x^2+\frac{2}{5} b^{15} x^{5/2} \]

[Out]

-a^15/(5*x^5) - (10*a^14*b)/(3*x^(9/2)) - (105*a^13*b^2)/(4*x^4) - (130*a^12*b^3)/x^(7/2) - (455*a^11*b^4)/x^3
- (6006*a^10*b^5)/(5*x^(5/2)) - (5005*a^9*b^6)/(2*x^2) - (4290*a^8*b^7)/x^(3/2) - (6435*a^7*b^8)/x - (10010*a
^6*b^9)/Sqrt[x] + 2730*a^4*b^11*Sqrt[x] + 455*a^3*b^12*x + 70*a^2*b^13*x^(3/2) + (15*a*b^14*x^2)/2 + (2*b^15*x
^(5/2))/5 + 3003*a^5*b^10*Log[x]

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Rubi [A]  time = 0.1167, antiderivative size = 194, 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} \[ -\frac{105 a^{13} b^2}{4 x^4}-\frac{130 a^{12} b^3}{x^{7/2}}-\frac{455 a^{11} b^4}{x^3}-\frac{6006 a^{10} b^5}{5 x^{5/2}}-\frac{5005 a^9 b^6}{2 x^2}-\frac{4290 a^8 b^7}{x^{3/2}}+70 a^2 b^{13} x^{3/2}-\frac{6435 a^7 b^8}{x}-\frac{10010 a^6 b^9}{\sqrt{x}}+2730 a^4 b^{11} \sqrt{x}+455 a^3 b^{12} x+3003 a^5 b^{10} \log (x)-\frac{10 a^{14} b}{3 x^{9/2}}-\frac{a^{15}}{5 x^5}+\frac{15}{2} a b^{14} x^2+\frac{2}{5} b^{15} x^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^15/x^6,x]

[Out]

-a^15/(5*x^5) - (10*a^14*b)/(3*x^(9/2)) - (105*a^13*b^2)/(4*x^4) - (130*a^12*b^3)/x^(7/2) - (455*a^11*b^4)/x^3
- (6006*a^10*b^5)/(5*x^(5/2)) - (5005*a^9*b^6)/(2*x^2) - (4290*a^8*b^7)/x^(3/2) - (6435*a^7*b^8)/x - (10010*a
^6*b^9)/Sqrt[x] + 2730*a^4*b^11*Sqrt[x] + 455*a^3*b^12*x + 70*a^2*b^13*x^(3/2) + (15*a*b^14*x^2)/2 + (2*b^15*x
^(5/2))/5 + 3003*a^5*b^10*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^6} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{11}} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (1365 a^4 b^{11}+\frac{a^{15}}{x^{11}}+\frac{15 a^{14} b}{x^{10}}+\frac{105 a^{13} b^2}{x^9}+\frac{455 a^{12} b^3}{x^8}+\frac{1365 a^{11} b^4}{x^7}+\frac{3003 a^{10} b^5}{x^6}+\frac{5005 a^9 b^6}{x^5}+\frac{6435 a^8 b^7}{x^4}+\frac{6435 a^7 b^8}{x^3}+\frac{5005 a^6 b^9}{x^2}+\frac{3003 a^5 b^{10}}{x}+455 a^3 b^{12} x+105 a^2 b^{13} x^2+15 a b^{14} x^3+b^{15} x^4\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a^{15}}{5 x^5}-\frac{10 a^{14} b}{3 x^{9/2}}-\frac{105 a^{13} b^2}{4 x^4}-\frac{130 a^{12} b^3}{x^{7/2}}-\frac{455 a^{11} b^4}{x^3}-\frac{6006 a^{10} b^5}{5 x^{5/2}}-\frac{5005 a^9 b^6}{2 x^2}-\frac{4290 a^8 b^7}{x^{3/2}}-\frac{6435 a^7 b^8}{x}-\frac{10010 a^6 b^9}{\sqrt{x}}+2730 a^4 b^{11} \sqrt{x}+455 a^3 b^{12} x+70 a^2 b^{13} x^{3/2}+\frac{15}{2} a b^{14} x^2+\frac{2}{5} b^{15} x^{5/2}+3003 a^5 b^{10} \log (x)\\ \end{align*}

Mathematica [A]  time = 0.108614, size = 194, normalized size = 1. \[ -\frac{105 a^{13} b^2}{4 x^4}-\frac{130 a^{12} b^3}{x^{7/2}}-\frac{455 a^{11} b^4}{x^3}-\frac{6006 a^{10} b^5}{5 x^{5/2}}-\frac{5005 a^9 b^6}{2 x^2}-\frac{4290 a^8 b^7}{x^{3/2}}+70 a^2 b^{13} x^{3/2}-\frac{6435 a^7 b^8}{x}-\frac{10010 a^6 b^9}{\sqrt{x}}+2730 a^4 b^{11} \sqrt{x}+455 a^3 b^{12} x+3003 a^5 b^{10} \log (x)-\frac{10 a^{14} b}{3 x^{9/2}}-\frac{a^{15}}{5 x^5}+\frac{15}{2} a b^{14} x^2+\frac{2}{5} b^{15} x^{5/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^15/x^6,x]

[Out]

-a^15/(5*x^5) - (10*a^14*b)/(3*x^(9/2)) - (105*a^13*b^2)/(4*x^4) - (130*a^12*b^3)/x^(7/2) - (455*a^11*b^4)/x^3
- (6006*a^10*b^5)/(5*x^(5/2)) - (5005*a^9*b^6)/(2*x^2) - (4290*a^8*b^7)/x^(3/2) - (6435*a^7*b^8)/x - (10010*a
^6*b^9)/Sqrt[x] + 2730*a^4*b^11*Sqrt[x] + 455*a^3*b^12*x + 70*a^2*b^13*x^(3/2) + (15*a*b^14*x^2)/2 + (2*b^15*x
^(5/2))/5 + 3003*a^5*b^10*Log[x]

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Maple [A]  time = 0.004, size = 165, normalized size = 0.9 \begin{align*} -{\frac{{a}^{15}}{5\,{x}^{5}}}-{\frac{10\,{a}^{14}b}{3}{x}^{-{\frac{9}{2}}}}-{\frac{105\,{a}^{13}{b}^{2}}{4\,{x}^{4}}}-130\,{\frac{{a}^{12}{b}^{3}}{{x}^{7/2}}}-455\,{\frac{{a}^{11}{b}^{4}}{{x}^{3}}}-{\frac{6006\,{a}^{10}{b}^{5}}{5}{x}^{-{\frac{5}{2}}}}-{\frac{5005\,{a}^{9}{b}^{6}}{2\,{x}^{2}}}-4290\,{\frac{{a}^{8}{b}^{7}}{{x}^{3/2}}}-6435\,{\frac{{a}^{7}{b}^{8}}{x}}+455\,{a}^{3}{b}^{12}x+70\,{a}^{2}{b}^{13}{x}^{3/2}+{\frac{15\,a{b}^{14}{x}^{2}}{2}}+{\frac{2\,{b}^{15}}{5}{x}^{{\frac{5}{2}}}}+3003\,{a}^{5}{b}^{10}\ln \left ( x \right ) -10010\,{\frac{{a}^{6}{b}^{9}}{\sqrt{x}}}+2730\,{a}^{4}{b}^{11}\sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/2))^15/x^6,x)

[Out]

-1/5*a^15/x^5-10/3*a^14*b/x^(9/2)-105/4*a^13*b^2/x^4-130*a^12*b^3/x^(7/2)-455*a^11*b^4/x^3-6006/5*a^10*b^5/x^(
5/2)-5005/2*a^9*b^6/x^2-4290*a^8*b^7/x^(3/2)-6435*a^7*b^8/x+455*a^3*b^12*x+70*a^2*b^13*x^(3/2)+15/2*a*b^14*x^2
+2/5*b^15*x^(5/2)+3003*a^5*b^10*ln(x)-10010*a^6*b^9/x^(1/2)+2730*a^4*b^11*x^(1/2)

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Maxima [A]  time = 0.962745, size = 223, normalized size = 1.15 \begin{align*} \frac{2}{5} \, b^{15} x^{\frac{5}{2}} + \frac{15}{2} \, a b^{14} x^{2} + 70 \, a^{2} b^{13} x^{\frac{3}{2}} + 455 \, a^{3} b^{12} x + 3003 \, a^{5} b^{10} \log \left (x\right ) + 2730 \, a^{4} b^{11} \sqrt{x} - \frac{600600 \, a^{6} b^{9} x^{\frac{9}{2}} + 386100 \, a^{7} b^{8} x^{4} + 257400 \, a^{8} b^{7} x^{\frac{7}{2}} + 150150 \, a^{9} b^{6} x^{3} + 72072 \, a^{10} b^{5} x^{\frac{5}{2}} + 27300 \, a^{11} b^{4} x^{2} + 7800 \, a^{12} b^{3} x^{\frac{3}{2}} + 1575 \, a^{13} b^{2} x + 200 \, a^{14} b \sqrt{x} + 12 \, a^{15}}{60 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^6,x, algorithm="maxima")

[Out]

2/5*b^15*x^(5/2) + 15/2*a*b^14*x^2 + 70*a^2*b^13*x^(3/2) + 455*a^3*b^12*x + 3003*a^5*b^10*log(x) + 2730*a^4*b^
11*sqrt(x) - 1/60*(600600*a^6*b^9*x^(9/2) + 386100*a^7*b^8*x^4 + 257400*a^8*b^7*x^(7/2) + 150150*a^9*b^6*x^3 +
72072*a^10*b^5*x^(5/2) + 27300*a^11*b^4*x^2 + 7800*a^12*b^3*x^(3/2) + 1575*a^13*b^2*x + 200*a^14*b*sqrt(x) +
12*a^15)/x^5

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Fricas [A]  time = 1.32668, size = 443, normalized size = 2.28 \begin{align*} \frac{450 \, a b^{14} x^{7} + 27300 \, a^{3} b^{12} x^{6} + 360360 \, a^{5} b^{10} x^{5} \log \left (\sqrt{x}\right ) - 386100 \, a^{7} b^{8} x^{4} - 150150 \, a^{9} b^{6} x^{3} - 27300 \, a^{11} b^{4} x^{2} - 1575 \, a^{13} b^{2} x - 12 \, a^{15} + 8 \,{\left (3 \, b^{15} x^{7} + 525 \, a^{2} b^{13} x^{6} + 20475 \, a^{4} b^{11} x^{5} - 75075 \, a^{6} b^{9} x^{4} - 32175 \, a^{8} b^{7} x^{3} - 9009 \, a^{10} b^{5} x^{2} - 975 \, a^{12} b^{3} x - 25 \, a^{14} b\right )} \sqrt{x}}{60 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^6,x, algorithm="fricas")

[Out]

1/60*(450*a*b^14*x^7 + 27300*a^3*b^12*x^6 + 360360*a^5*b^10*x^5*log(sqrt(x)) - 386100*a^7*b^8*x^4 - 150150*a^9
*b^6*x^3 - 27300*a^11*b^4*x^2 - 1575*a^13*b^2*x - 12*a^15 + 8*(3*b^15*x^7 + 525*a^2*b^13*x^6 + 20475*a^4*b^11*
x^5 - 75075*a^6*b^9*x^4 - 32175*a^8*b^7*x^3 - 9009*a^10*b^5*x^2 - 975*a^12*b^3*x - 25*a^14*b)*sqrt(x))/x^5

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Sympy [A]  time = 4.97942, size = 199, normalized size = 1.03 \begin{align*} - \frac{a^{15}}{5 x^{5}} - \frac{10 a^{14} b}{3 x^{\frac{9}{2}}} - \frac{105 a^{13} b^{2}}{4 x^{4}} - \frac{130 a^{12} b^{3}}{x^{\frac{7}{2}}} - \frac{455 a^{11} b^{4}}{x^{3}} - \frac{6006 a^{10} b^{5}}{5 x^{\frac{5}{2}}} - \frac{5005 a^{9} b^{6}}{2 x^{2}} - \frac{4290 a^{8} b^{7}}{x^{\frac{3}{2}}} - \frac{6435 a^{7} b^{8}}{x} - \frac{10010 a^{6} b^{9}}{\sqrt{x}} + 3003 a^{5} b^{10} \log{\left (x \right )} + 2730 a^{4} b^{11} \sqrt{x} + 455 a^{3} b^{12} x + 70 a^{2} b^{13} x^{\frac{3}{2}} + \frac{15 a b^{14} x^{2}}{2} + \frac{2 b^{15} x^{\frac{5}{2}}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/2))**15/x**6,x)

[Out]

-a**15/(5*x**5) - 10*a**14*b/(3*x**(9/2)) - 105*a**13*b**2/(4*x**4) - 130*a**12*b**3/x**(7/2) - 455*a**11*b**4
/x**3 - 6006*a**10*b**5/(5*x**(5/2)) - 5005*a**9*b**6/(2*x**2) - 4290*a**8*b**7/x**(3/2) - 6435*a**7*b**8/x -
10010*a**6*b**9/sqrt(x) + 3003*a**5*b**10*log(x) + 2730*a**4*b**11*sqrt(x) + 455*a**3*b**12*x + 70*a**2*b**13*
x**(3/2) + 15*a*b**14*x**2/2 + 2*b**15*x**(5/2)/5

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Giac [A]  time = 1.11773, size = 224, normalized size = 1.15 \begin{align*} \frac{2}{5} \, b^{15} x^{\frac{5}{2}} + \frac{15}{2} \, a b^{14} x^{2} + 70 \, a^{2} b^{13} x^{\frac{3}{2}} + 455 \, a^{3} b^{12} x + 3003 \, a^{5} b^{10} \log \left ({\left | x \right |}\right ) + 2730 \, a^{4} b^{11} \sqrt{x} - \frac{600600 \, a^{6} b^{9} x^{\frac{9}{2}} + 386100 \, a^{7} b^{8} x^{4} + 257400 \, a^{8} b^{7} x^{\frac{7}{2}} + 150150 \, a^{9} b^{6} x^{3} + 72072 \, a^{10} b^{5} x^{\frac{5}{2}} + 27300 \, a^{11} b^{4} x^{2} + 7800 \, a^{12} b^{3} x^{\frac{3}{2}} + 1575 \, a^{13} b^{2} x + 200 \, a^{14} b \sqrt{x} + 12 \, a^{15}}{60 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^15/x^6,x, algorithm="giac")

[Out]

2/5*b^15*x^(5/2) + 15/2*a*b^14*x^2 + 70*a^2*b^13*x^(3/2) + 455*a^3*b^12*x + 3003*a^5*b^10*log(abs(x)) + 2730*a
^4*b^11*sqrt(x) - 1/60*(600600*a^6*b^9*x^(9/2) + 386100*a^7*b^8*x^4 + 257400*a^8*b^7*x^(7/2) + 150150*a^9*b^6*
x^3 + 72072*a^10*b^5*x^(5/2) + 27300*a^11*b^4*x^2 + 7800*a^12*b^3*x^(3/2) + 1575*a^13*b^2*x + 200*a^14*b*sqrt(
x) + 12*a^15)/x^5