3.2231 \(\int \frac{1}{(a+b \sqrt{x})^8 x^3} \, dx\)

Optimal. Leaf size=217 \[ \frac{420 b^4}{a^{11} \left (a+b \sqrt{x}\right )}+\frac{126 b^4}{a^{10} \left (a+b \sqrt{x}\right )^2}+\frac{140 b^4}{3 a^9 \left (a+b \sqrt{x}\right )^3}+\frac{35 b^4}{2 a^8 \left (a+b \sqrt{x}\right )^4}+\frac{6 b^4}{a^7 \left (a+b \sqrt{x}\right )^5}+\frac{5 b^4}{3 a^6 \left (a+b \sqrt{x}\right )^6}+\frac{2 b^4}{7 a^5 \left (a+b \sqrt{x}\right )^7}+\frac{240 b^3}{a^{11} \sqrt{x}}-\frac{36 b^2}{a^{10} x}-\frac{660 b^4 \log \left (a+b \sqrt{x}\right )}{a^{12}}+\frac{330 b^4 \log (x)}{a^{12}}+\frac{16 b}{3 a^9 x^{3/2}}-\frac{1}{2 a^8 x^2} \]

[Out]

(2*b^4)/(7*a^5*(a + b*Sqrt[x])^7) + (5*b^4)/(3*a^6*(a + b*Sqrt[x])^6) + (6*b^4)/(a^7*(a + b*Sqrt[x])^5) + (35*
b^4)/(2*a^8*(a + b*Sqrt[x])^4) + (140*b^4)/(3*a^9*(a + b*Sqrt[x])^3) + (126*b^4)/(a^10*(a + b*Sqrt[x])^2) + (4
20*b^4)/(a^11*(a + b*Sqrt[x])) - 1/(2*a^8*x^2) + (16*b)/(3*a^9*x^(3/2)) - (36*b^2)/(a^10*x) + (240*b^3)/(a^11*
Sqrt[x]) - (660*b^4*Log[a + b*Sqrt[x]])/a^12 + (330*b^4*Log[x])/a^12

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Rubi [A]  time = 0.19141, antiderivative size = 217, 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, 44} \[ \frac{420 b^4}{a^{11} \left (a+b \sqrt{x}\right )}+\frac{126 b^4}{a^{10} \left (a+b \sqrt{x}\right )^2}+\frac{140 b^4}{3 a^9 \left (a+b \sqrt{x}\right )^3}+\frac{35 b^4}{2 a^8 \left (a+b \sqrt{x}\right )^4}+\frac{6 b^4}{a^7 \left (a+b \sqrt{x}\right )^5}+\frac{5 b^4}{3 a^6 \left (a+b \sqrt{x}\right )^6}+\frac{2 b^4}{7 a^5 \left (a+b \sqrt{x}\right )^7}+\frac{240 b^3}{a^{11} \sqrt{x}}-\frac{36 b^2}{a^{10} x}-\frac{660 b^4 \log \left (a+b \sqrt{x}\right )}{a^{12}}+\frac{330 b^4 \log (x)}{a^{12}}+\frac{16 b}{3 a^9 x^{3/2}}-\frac{1}{2 a^8 x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*Sqrt[x])^8*x^3),x]

[Out]

(2*b^4)/(7*a^5*(a + b*Sqrt[x])^7) + (5*b^4)/(3*a^6*(a + b*Sqrt[x])^6) + (6*b^4)/(a^7*(a + b*Sqrt[x])^5) + (35*
b^4)/(2*a^8*(a + b*Sqrt[x])^4) + (140*b^4)/(3*a^9*(a + b*Sqrt[x])^3) + (126*b^4)/(a^10*(a + b*Sqrt[x])^2) + (4
20*b^4)/(a^11*(a + b*Sqrt[x])) - 1/(2*a^8*x^2) + (16*b)/(3*a^9*x^(3/2)) - (36*b^2)/(a^10*x) + (240*b^3)/(a^11*
Sqrt[x]) - (660*b^4*Log[a + b*Sqrt[x]])/a^12 + (330*b^4*Log[x])/a^12

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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b \sqrt{x}\right )^8 x^3} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x^5 (a+b x)^8} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{1}{a^8 x^5}-\frac{8 b}{a^9 x^4}+\frac{36 b^2}{a^{10} x^3}-\frac{120 b^3}{a^{11} x^2}+\frac{330 b^4}{a^{12} x}-\frac{b^5}{a^5 (a+b x)^8}-\frac{5 b^5}{a^6 (a+b x)^7}-\frac{15 b^5}{a^7 (a+b x)^6}-\frac{35 b^5}{a^8 (a+b x)^5}-\frac{70 b^5}{a^9 (a+b x)^4}-\frac{126 b^5}{a^{10} (a+b x)^3}-\frac{210 b^5}{a^{11} (a+b x)^2}-\frac{330 b^5}{a^{12} (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 b^4}{7 a^5 \left (a+b \sqrt{x}\right )^7}+\frac{5 b^4}{3 a^6 \left (a+b \sqrt{x}\right )^6}+\frac{6 b^4}{a^7 \left (a+b \sqrt{x}\right )^5}+\frac{35 b^4}{2 a^8 \left (a+b \sqrt{x}\right )^4}+\frac{140 b^4}{3 a^9 \left (a+b \sqrt{x}\right )^3}+\frac{126 b^4}{a^{10} \left (a+b \sqrt{x}\right )^2}+\frac{420 b^4}{a^{11} \left (a+b \sqrt{x}\right )}-\frac{1}{2 a^8 x^2}+\frac{16 b}{3 a^9 x^{3/2}}-\frac{36 b^2}{a^{10} x}+\frac{240 b^3}{a^{11} \sqrt{x}}-\frac{660 b^4 \log \left (a+b \sqrt{x}\right )}{a^{12}}+\frac{330 b^4 \log (x)}{a^{12}}\\ \end{align*}

Mathematica [A]  time = 0.24693, size = 163, normalized size = 0.75 \[ \frac{\frac{a \left (3465 a^7 b^3 x^{3/2}+71874 a^6 b^4 x^2+309078 a^5 b^5 x^{5/2}+636174 a^4 b^6 x^3+736890 a^3 b^7 x^{7/2}+494340 a^2 b^8 x^4-385 a^8 b^2 x+77 a^9 b \sqrt{x}-21 a^{10}+180180 a b^9 x^{9/2}+27720 b^{10} x^5\right )}{x^2 \left (a+b \sqrt{x}\right )^7}-27720 b^4 \log \left (a+b \sqrt{x}\right )+13860 b^4 \log (x)}{42 a^{12}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*Sqrt[x])^8*x^3),x]

[Out]

((a*(-21*a^10 + 77*a^9*b*Sqrt[x] - 385*a^8*b^2*x + 3465*a^7*b^3*x^(3/2) + 71874*a^6*b^4*x^2 + 309078*a^5*b^5*x
^(5/2) + 636174*a^4*b^6*x^3 + 736890*a^3*b^7*x^(7/2) + 494340*a^2*b^8*x^4 + 180180*a*b^9*x^(9/2) + 27720*b^10*
x^5))/((a + b*Sqrt[x])^7*x^2) - 27720*b^4*Log[a + b*Sqrt[x]] + 13860*b^4*Log[x])/(42*a^12)

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Maple [A]  time = 0.014, size = 186, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{a}^{8}{x}^{2}}}+{\frac{16\,b}{3\,{a}^{9}}{x}^{-{\frac{3}{2}}}}-36\,{\frac{{b}^{2}}{x{a}^{10}}}+330\,{\frac{{b}^{4}\ln \left ( x \right ) }{{a}^{12}}}-660\,{\frac{{b}^{4}\ln \left ( a+b\sqrt{x} \right ) }{{a}^{12}}}+240\,{\frac{{b}^{3}}{{a}^{11}\sqrt{x}}}+{\frac{2\,{b}^{4}}{7\,{a}^{5}} \left ( a+b\sqrt{x} \right ) ^{-7}}+{\frac{5\,{b}^{4}}{3\,{a}^{6}} \left ( a+b\sqrt{x} \right ) ^{-6}}+6\,{\frac{{b}^{4}}{{a}^{7} \left ( a+b\sqrt{x} \right ) ^{5}}}+{\frac{35\,{b}^{4}}{2\,{a}^{8}} \left ( a+b\sqrt{x} \right ) ^{-4}}+{\frac{140\,{b}^{4}}{3\,{a}^{9}} \left ( a+b\sqrt{x} \right ) ^{-3}}+126\,{\frac{{b}^{4}}{{a}^{10} \left ( a+b\sqrt{x} \right ) ^{2}}}+420\,{\frac{{b}^{4}}{{a}^{11} \left ( a+b\sqrt{x} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(a+b*x^(1/2))^8,x)

[Out]

-1/2/a^8/x^2+16/3*b/a^9/x^(3/2)-36*b^2/a^10/x+330*b^4*ln(x)/a^12-660*b^4*ln(a+b*x^(1/2))/a^12+240*b^3/a^11/x^(
1/2)+2/7*b^4/a^5/(a+b*x^(1/2))^7+5/3*b^4/a^6/(a+b*x^(1/2))^6+6*b^4/a^7/(a+b*x^(1/2))^5+35/2*b^4/a^8/(a+b*x^(1/
2))^4+140/3*b^4/a^9/(a+b*x^(1/2))^3+126*b^4/a^10/(a+b*x^(1/2))^2+420*b^4/a^11/(a+b*x^(1/2))

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Maxima [A]  time = 1.03358, size = 297, normalized size = 1.37 \begin{align*} \frac{27720 \, b^{10} x^{5} + 180180 \, a b^{9} x^{\frac{9}{2}} + 494340 \, a^{2} b^{8} x^{4} + 736890 \, a^{3} b^{7} x^{\frac{7}{2}} + 636174 \, a^{4} b^{6} x^{3} + 309078 \, a^{5} b^{5} x^{\frac{5}{2}} + 71874 \, a^{6} b^{4} x^{2} + 3465 \, a^{7} b^{3} x^{\frac{3}{2}} - 385 \, a^{8} b^{2} x + 77 \, a^{9} b \sqrt{x} - 21 \, a^{10}}{42 \,{\left (a^{11} b^{7} x^{\frac{11}{2}} + 7 \, a^{12} b^{6} x^{5} + 21 \, a^{13} b^{5} x^{\frac{9}{2}} + 35 \, a^{14} b^{4} x^{4} + 35 \, a^{15} b^{3} x^{\frac{7}{2}} + 21 \, a^{16} b^{2} x^{3} + 7 \, a^{17} b x^{\frac{5}{2}} + a^{18} x^{2}\right )}} - \frac{660 \, b^{4} \log \left (b \sqrt{x} + a\right )}{a^{12}} + \frac{330 \, b^{4} \log \left (x\right )}{a^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(a+b*x^(1/2))^8,x, algorithm="maxima")

[Out]

1/42*(27720*b^10*x^5 + 180180*a*b^9*x^(9/2) + 494340*a^2*b^8*x^4 + 736890*a^3*b^7*x^(7/2) + 636174*a^4*b^6*x^3
+ 309078*a^5*b^5*x^(5/2) + 71874*a^6*b^4*x^2 + 3465*a^7*b^3*x^(3/2) - 385*a^8*b^2*x + 77*a^9*b*sqrt(x) - 21*a
^10)/(a^11*b^7*x^(11/2) + 7*a^12*b^6*x^5 + 21*a^13*b^5*x^(9/2) + 35*a^14*b^4*x^4 + 35*a^15*b^3*x^(7/2) + 21*a^
16*b^2*x^3 + 7*a^17*b*x^(5/2) + a^18*x^2) - 660*b^4*log(b*sqrt(x) + a)/a^12 + 330*b^4*log(x)/a^12

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Fricas [B]  time = 1.45051, size = 1106, normalized size = 5.1 \begin{align*} -\frac{13860 \, a^{2} b^{16} x^{8} - 90090 \, a^{4} b^{14} x^{7} + 247170 \, a^{6} b^{12} x^{6} - 368445 \, a^{8} b^{10} x^{5} + 318087 \, a^{10} b^{8} x^{4} - 154532 \, a^{12} b^{6} x^{3} + 36104 \, a^{14} b^{4} x^{2} - 1365 \, a^{16} b^{2} x - 21 \, a^{18} + 27720 \,{\left (b^{18} x^{9} - 7 \, a^{2} b^{16} x^{8} + 21 \, a^{4} b^{14} x^{7} - 35 \, a^{6} b^{12} x^{6} + 35 \, a^{8} b^{10} x^{5} - 21 \, a^{10} b^{8} x^{4} + 7 \, a^{12} b^{6} x^{3} - a^{14} b^{4} x^{2}\right )} \log \left (b \sqrt{x} + a\right ) - 27720 \,{\left (b^{18} x^{9} - 7 \, a^{2} b^{16} x^{8} + 21 \, a^{4} b^{14} x^{7} - 35 \, a^{6} b^{12} x^{6} + 35 \, a^{8} b^{10} x^{5} - 21 \, a^{10} b^{8} x^{4} + 7 \, a^{12} b^{6} x^{3} - a^{14} b^{4} x^{2}\right )} \log \left (\sqrt{x}\right ) - 8 \,{\left (3465 \, a b^{17} x^{8} - 23100 \, a^{3} b^{15} x^{7} + 65373 \, a^{5} b^{13} x^{6} - 101376 \, a^{7} b^{11} x^{5} + 92323 \, a^{9} b^{9} x^{4} - 48580 \, a^{11} b^{7} x^{3} + 13083 \, a^{13} b^{5} x^{2} - 1064 \, a^{15} b^{3} x - 28 \, a^{17} b\right )} \sqrt{x}}{42 \,{\left (a^{12} b^{14} x^{9} - 7 \, a^{14} b^{12} x^{8} + 21 \, a^{16} b^{10} x^{7} - 35 \, a^{18} b^{8} x^{6} + 35 \, a^{20} b^{6} x^{5} - 21 \, a^{22} b^{4} x^{4} + 7 \, a^{24} b^{2} x^{3} - a^{26} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(a+b*x^(1/2))^8,x, algorithm="fricas")

[Out]

-1/42*(13860*a^2*b^16*x^8 - 90090*a^4*b^14*x^7 + 247170*a^6*b^12*x^6 - 368445*a^8*b^10*x^5 + 318087*a^10*b^8*x
^4 - 154532*a^12*b^6*x^3 + 36104*a^14*b^4*x^2 - 1365*a^16*b^2*x - 21*a^18 + 27720*(b^18*x^9 - 7*a^2*b^16*x^8 +
21*a^4*b^14*x^7 - 35*a^6*b^12*x^6 + 35*a^8*b^10*x^5 - 21*a^10*b^8*x^4 + 7*a^12*b^6*x^3 - a^14*b^4*x^2)*log(b*
sqrt(x) + a) - 27720*(b^18*x^9 - 7*a^2*b^16*x^8 + 21*a^4*b^14*x^7 - 35*a^6*b^12*x^6 + 35*a^8*b^10*x^5 - 21*a^1
0*b^8*x^4 + 7*a^12*b^6*x^3 - a^14*b^4*x^2)*log(sqrt(x)) - 8*(3465*a*b^17*x^8 - 23100*a^3*b^15*x^7 + 65373*a^5*
b^13*x^6 - 101376*a^7*b^11*x^5 + 92323*a^9*b^9*x^4 - 48580*a^11*b^7*x^3 + 13083*a^13*b^5*x^2 - 1064*a^15*b^3*x
- 28*a^17*b)*sqrt(x))/(a^12*b^14*x^9 - 7*a^14*b^12*x^8 + 21*a^16*b^10*x^7 - 35*a^18*b^8*x^6 + 35*a^20*b^6*x^5
- 21*a^22*b^4*x^4 + 7*a^24*b^2*x^3 - a^26*x^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(a+b*x**(1/2))**8,x)

[Out]

Timed out

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Giac [A]  time = 1.10313, size = 211, normalized size = 0.97 \begin{align*} -\frac{660 \, b^{4} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{a^{12}} + \frac{330 \, b^{4} \log \left ({\left | x \right |}\right )}{a^{12}} + \frac{27720 \, a b^{10} x^{5} + 180180 \, a^{2} b^{9} x^{\frac{9}{2}} + 494340 \, a^{3} b^{8} x^{4} + 736890 \, a^{4} b^{7} x^{\frac{7}{2}} + 636174 \, a^{5} b^{6} x^{3} + 309078 \, a^{6} b^{5} x^{\frac{5}{2}} + 71874 \, a^{7} b^{4} x^{2} + 3465 \, a^{8} b^{3} x^{\frac{3}{2}} - 385 \, a^{9} b^{2} x + 77 \, a^{10} b \sqrt{x} - 21 \, a^{11}}{42 \,{\left (b \sqrt{x} + a\right )}^{7} a^{12} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(a+b*x^(1/2))^8,x, algorithm="giac")

[Out]

-660*b^4*log(abs(b*sqrt(x) + a))/a^12 + 330*b^4*log(abs(x))/a^12 + 1/42*(27720*a*b^10*x^5 + 180180*a^2*b^9*x^(
9/2) + 494340*a^3*b^8*x^4 + 736890*a^4*b^7*x^(7/2) + 636174*a^5*b^6*x^3 + 309078*a^6*b^5*x^(5/2) + 71874*a^7*b
^4*x^2 + 3465*a^8*b^3*x^(3/2) - 385*a^9*b^2*x + 77*a^10*b*sqrt(x) - 21*a^11)/((b*sqrt(x) + a)^7*a^12*x^2)