3.2225 \(\int \frac{x^3}{(a+b \sqrt{x})^8} \, dx\)

Optimal. Leaf size=157 \[ \frac{2 a^7}{7 b^8 \left (a+b \sqrt{x}\right )^7}-\frac{7 a^6}{3 b^8 \left (a+b \sqrt{x}\right )^6}+\frac{42 a^5}{5 b^8 \left (a+b \sqrt{x}\right )^5}-\frac{35 a^4}{2 b^8 \left (a+b \sqrt{x}\right )^4}+\frac{70 a^3}{3 b^8 \left (a+b \sqrt{x}\right )^3}-\frac{21 a^2}{b^8 \left (a+b \sqrt{x}\right )^2}+\frac{14 a}{b^8 \left (a+b \sqrt{x}\right )}+\frac{2 \log \left (a+b \sqrt{x}\right )}{b^8} \]

[Out]

(2*a^7)/(7*b^8*(a + b*Sqrt[x])^7) - (7*a^6)/(3*b^8*(a + b*Sqrt[x])^6) + (42*a^5)/(5*b^8*(a + b*Sqrt[x])^5) - (
35*a^4)/(2*b^8*(a + b*Sqrt[x])^4) + (70*a^3)/(3*b^8*(a + b*Sqrt[x])^3) - (21*a^2)/(b^8*(a + b*Sqrt[x])^2) + (1
4*a)/(b^8*(a + b*Sqrt[x])) + (2*Log[a + b*Sqrt[x]])/b^8

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Rubi [A]  time = 0.106499, antiderivative size = 157, 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} \[ \frac{2 a^7}{7 b^8 \left (a+b \sqrt{x}\right )^7}-\frac{7 a^6}{3 b^8 \left (a+b \sqrt{x}\right )^6}+\frac{42 a^5}{5 b^8 \left (a+b \sqrt{x}\right )^5}-\frac{35 a^4}{2 b^8 \left (a+b \sqrt{x}\right )^4}+\frac{70 a^3}{3 b^8 \left (a+b \sqrt{x}\right )^3}-\frac{21 a^2}{b^8 \left (a+b \sqrt{x}\right )^2}+\frac{14 a}{b^8 \left (a+b \sqrt{x}\right )}+\frac{2 \log \left (a+b \sqrt{x}\right )}{b^8} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*Sqrt[x])^8,x]

[Out]

(2*a^7)/(7*b^8*(a + b*Sqrt[x])^7) - (7*a^6)/(3*b^8*(a + b*Sqrt[x])^6) + (42*a^5)/(5*b^8*(a + b*Sqrt[x])^5) - (
35*a^4)/(2*b^8*(a + b*Sqrt[x])^4) + (70*a^3)/(3*b^8*(a + b*Sqrt[x])^3) - (21*a^2)/(b^8*(a + b*Sqrt[x])^2) + (1
4*a)/(b^8*(a + b*Sqrt[x])) + (2*Log[a + b*Sqrt[x]])/b^8

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{x^3}{\left (a+b \sqrt{x}\right )^8} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^7}{(a+b x)^8} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{a^7}{b^7 (a+b x)^8}+\frac{7 a^6}{b^7 (a+b x)^7}-\frac{21 a^5}{b^7 (a+b x)^6}+\frac{35 a^4}{b^7 (a+b x)^5}-\frac{35 a^3}{b^7 (a+b x)^4}+\frac{21 a^2}{b^7 (a+b x)^3}-\frac{7 a}{b^7 (a+b x)^2}+\frac{1}{b^7 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 a^7}{7 b^8 \left (a+b \sqrt{x}\right )^7}-\frac{7 a^6}{3 b^8 \left (a+b \sqrt{x}\right )^6}+\frac{42 a^5}{5 b^8 \left (a+b \sqrt{x}\right )^5}-\frac{35 a^4}{2 b^8 \left (a+b \sqrt{x}\right )^4}+\frac{70 a^3}{3 b^8 \left (a+b \sqrt{x}\right )^3}-\frac{21 a^2}{b^8 \left (a+b \sqrt{x}\right )^2}+\frac{14 a}{b^8 \left (a+b \sqrt{x}\right )}+\frac{2 \log \left (a+b \sqrt{x}\right )}{b^8}\\ \end{align*}

Mathematica [A]  time = 0.100937, size = 102, normalized size = 0.65 \[ \frac{\frac{a \left (30625 a^3 b^3 x^{3/2}+26950 a^2 b^4 x^2+20139 a^4 b^2 x+7203 a^5 b \sqrt{x}+1089 a^6+13230 a b^5 x^{5/2}+2940 b^6 x^3\right )}{\left (a+b \sqrt{x}\right )^7}+420 \log \left (a+b \sqrt{x}\right )}{210 b^8} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*Sqrt[x])^8,x]

[Out]

((a*(1089*a^6 + 7203*a^5*b*Sqrt[x] + 20139*a^4*b^2*x + 30625*a^3*b^3*x^(3/2) + 26950*a^2*b^4*x^2 + 13230*a*b^5
*x^(5/2) + 2940*b^6*x^3))/(a + b*Sqrt[x])^7 + 420*Log[a + b*Sqrt[x]])/(210*b^8)

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Maple [A]  time = 0.007, size = 132, normalized size = 0.8 \begin{align*} 2\,{\frac{\ln \left ( a+b\sqrt{x} \right ) }{{b}^{8}}}+{\frac{2\,{a}^{7}}{7\,{b}^{8}} \left ( a+b\sqrt{x} \right ) ^{-7}}-{\frac{7\,{a}^{6}}{3\,{b}^{8}} \left ( a+b\sqrt{x} \right ) ^{-6}}+{\frac{42\,{a}^{5}}{5\,{b}^{8}} \left ( a+b\sqrt{x} \right ) ^{-5}}-{\frac{35\,{a}^{4}}{2\,{b}^{8}} \left ( a+b\sqrt{x} \right ) ^{-4}}+{\frac{70\,{a}^{3}}{3\,{b}^{8}} \left ( a+b\sqrt{x} \right ) ^{-3}}-21\,{\frac{{a}^{2}}{{b}^{8} \left ( a+b\sqrt{x} \right ) ^{2}}}+14\,{\frac{a}{{b}^{8} \left ( a+b\sqrt{x} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*ln(a+b*x^(1/2))/b^8+2/7*a^7/b^8/(a+b*x^(1/2))^7-7/3*a^6/b^8/(a+b*x^(1/2))^6+42/5*a^5/b^8/(a+b*x^(1/2))^5-35/
2*a^4/b^8/(a+b*x^(1/2))^4+70/3*a^3/b^8/(a+b*x^(1/2))^3-21*a^2/b^8/(a+b*x^(1/2))^2+14*a/b^8/(a+b*x^(1/2))

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Maxima [A]  time = 0.968184, size = 177, normalized size = 1.13 \begin{align*} \frac{2 \, \log \left (b \sqrt{x} + a\right )}{b^{8}} + \frac{14 \, a}{{\left (b \sqrt{x} + a\right )} b^{8}} - \frac{21 \, a^{2}}{{\left (b \sqrt{x} + a\right )}^{2} b^{8}} + \frac{70 \, a^{3}}{3 \,{\left (b \sqrt{x} + a\right )}^{3} b^{8}} - \frac{35 \, a^{4}}{2 \,{\left (b \sqrt{x} + a\right )}^{4} b^{8}} + \frac{42 \, a^{5}}{5 \,{\left (b \sqrt{x} + a\right )}^{5} b^{8}} - \frac{7 \, a^{6}}{3 \,{\left (b \sqrt{x} + a\right )}^{6} b^{8}} + \frac{2 \, a^{7}}{7 \,{\left (b \sqrt{x} + a\right )}^{7} b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(b*sqrt(x) + a)/b^8 + 14*a/((b*sqrt(x) + a)*b^8) - 21*a^2/((b*sqrt(x) + a)^2*b^8) + 70/3*a^3/((b*sqrt(x)
+ a)^3*b^8) - 35/2*a^4/((b*sqrt(x) + a)^4*b^8) + 42/5*a^5/((b*sqrt(x) + a)^5*b^8) - 7/3*a^6/((b*sqrt(x) + a)^6
*b^8) + 2/7*a^7/((b*sqrt(x) + a)^7*b^8)

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Fricas [B]  time = 1.26963, size = 740, normalized size = 4.71 \begin{align*} -\frac{7350 \, a^{2} b^{12} x^{6} - 16905 \, a^{4} b^{10} x^{5} + 32585 \, a^{6} b^{8} x^{4} - 34370 \, a^{8} b^{6} x^{3} + 21504 \, a^{10} b^{4} x^{2} - 7413 \, a^{12} b^{2} x + 1089 \, a^{14} - 420 \,{\left (b^{14} x^{7} - 7 \, a^{2} b^{12} x^{6} + 21 \, a^{4} b^{10} x^{5} - 35 \, a^{6} b^{8} x^{4} + 35 \, a^{8} b^{6} x^{3} - 21 \, a^{10} b^{4} x^{2} + 7 \, a^{12} b^{2} x - a^{14}\right )} \log \left (b \sqrt{x} + a\right ) - 4 \,{\left (735 \, a b^{13} x^{6} - 980 \, a^{3} b^{11} x^{5} + 2891 \, a^{5} b^{9} x^{4} - 3072 \, a^{7} b^{7} x^{3} + 1981 \, a^{9} b^{5} x^{2} - 700 \, a^{11} b^{3} x + 105 \, a^{13} b\right )} \sqrt{x}}{210 \,{\left (b^{22} x^{7} - 7 \, a^{2} b^{20} x^{6} + 21 \, a^{4} b^{18} x^{5} - 35 \, a^{6} b^{16} x^{4} + 35 \, a^{8} b^{14} x^{3} - 21 \, a^{10} b^{12} x^{2} + 7 \, a^{12} b^{10} x - a^{14} b^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/210*(7350*a^2*b^12*x^6 - 16905*a^4*b^10*x^5 + 32585*a^6*b^8*x^4 - 34370*a^8*b^6*x^3 + 21504*a^10*b^4*x^2 -
7413*a^12*b^2*x + 1089*a^14 - 420*(b^14*x^7 - 7*a^2*b^12*x^6 + 21*a^4*b^10*x^5 - 35*a^6*b^8*x^4 + 35*a^8*b^6*x
^3 - 21*a^10*b^4*x^2 + 7*a^12*b^2*x - a^14)*log(b*sqrt(x) + a) - 4*(735*a*b^13*x^6 - 980*a^3*b^11*x^5 + 2891*a
^5*b^9*x^4 - 3072*a^7*b^7*x^3 + 1981*a^9*b^5*x^2 - 700*a^11*b^3*x + 105*a^13*b)*sqrt(x))/(b^22*x^7 - 7*a^2*b^2
0*x^6 + 21*a^4*b^18*x^5 - 35*a^6*b^16*x^4 + 35*a^8*b^14*x^3 - 21*a^10*b^12*x^2 + 7*a^12*b^10*x - a^14*b^8)

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Sympy [A]  time = 8.9634, size = 1627, normalized size = 10.36 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((420*a**7*log(a/b + sqrt(x))/(210*a**7*b**8 + 1470*a**6*b**9*sqrt(x) + 4410*a**5*b**10*x + 7350*a**4
*b**11*x**(3/2) + 7350*a**3*b**12*x**2 + 4410*a**2*b**13*x**(5/2) + 1470*a*b**14*x**3 + 210*b**15*x**(7/2)) +
60*a**7/(210*a**7*b**8 + 1470*a**6*b**9*sqrt(x) + 4410*a**5*b**10*x + 7350*a**4*b**11*x**(3/2) + 7350*a**3*b**
12*x**2 + 4410*a**2*b**13*x**(5/2) + 1470*a*b**14*x**3 + 210*b**15*x**(7/2)) + 2940*a**6*b*sqrt(x)*log(a/b + s
qrt(x))/(210*a**7*b**8 + 1470*a**6*b**9*sqrt(x) + 4410*a**5*b**10*x + 7350*a**4*b**11*x**(3/2) + 7350*a**3*b**
12*x**2 + 4410*a**2*b**13*x**(5/2) + 1470*a*b**14*x**3 + 210*b**15*x**(7/2)) + 8820*a**5*b**2*x*log(a/b + sqrt
(x))/(210*a**7*b**8 + 1470*a**6*b**9*sqrt(x) + 4410*a**5*b**10*x + 7350*a**4*b**11*x**(3/2) + 7350*a**3*b**12*
x**2 + 4410*a**2*b**13*x**(5/2) + 1470*a*b**14*x**3 + 210*b**15*x**(7/2)) - 1470*a**5*b**2*x/(210*a**7*b**8 +
1470*a**6*b**9*sqrt(x) + 4410*a**5*b**10*x + 7350*a**4*b**11*x**(3/2) + 7350*a**3*b**12*x**2 + 4410*a**2*b**13
*x**(5/2) + 1470*a*b**14*x**3 + 210*b**15*x**(7/2)) + 14700*a**4*b**3*x**(3/2)*log(a/b + sqrt(x))/(210*a**7*b*
*8 + 1470*a**6*b**9*sqrt(x) + 4410*a**5*b**10*x + 7350*a**4*b**11*x**(3/2) + 7350*a**3*b**12*x**2 + 4410*a**2*
b**13*x**(5/2) + 1470*a*b**14*x**3 + 210*b**15*x**(7/2)) - 5390*a**4*b**3*x**(3/2)/(210*a**7*b**8 + 1470*a**6*
b**9*sqrt(x) + 4410*a**5*b**10*x + 7350*a**4*b**11*x**(3/2) + 7350*a**3*b**12*x**2 + 4410*a**2*b**13*x**(5/2)
+ 1470*a*b**14*x**3 + 210*b**15*x**(7/2)) + 14700*a**3*b**4*x**2*log(a/b + sqrt(x))/(210*a**7*b**8 + 1470*a**6
*b**9*sqrt(x) + 4410*a**5*b**10*x + 7350*a**4*b**11*x**(3/2) + 7350*a**3*b**12*x**2 + 4410*a**2*b**13*x**(5/2)
+ 1470*a*b**14*x**3 + 210*b**15*x**(7/2)) - 9065*a**3*b**4*x**2/(210*a**7*b**8 + 1470*a**6*b**9*sqrt(x) + 441
0*a**5*b**10*x + 7350*a**4*b**11*x**(3/2) + 7350*a**3*b**12*x**2 + 4410*a**2*b**13*x**(5/2) + 1470*a*b**14*x**
3 + 210*b**15*x**(7/2)) + 8820*a**2*b**5*x**(5/2)*log(a/b + sqrt(x))/(210*a**7*b**8 + 1470*a**6*b**9*sqrt(x) +
4410*a**5*b**10*x + 7350*a**4*b**11*x**(3/2) + 7350*a**3*b**12*x**2 + 4410*a**2*b**13*x**(5/2) + 1470*a*b**14
*x**3 + 210*b**15*x**(7/2)) - 8379*a**2*b**5*x**(5/2)/(210*a**7*b**8 + 1470*a**6*b**9*sqrt(x) + 4410*a**5*b**1
0*x + 7350*a**4*b**11*x**(3/2) + 7350*a**3*b**12*x**2 + 4410*a**2*b**13*x**(5/2) + 1470*a*b**14*x**3 + 210*b**
15*x**(7/2)) + 2940*a*b**6*x**3*log(a/b + sqrt(x))/(210*a**7*b**8 + 1470*a**6*b**9*sqrt(x) + 4410*a**5*b**10*x
+ 7350*a**4*b**11*x**(3/2) + 7350*a**3*b**12*x**2 + 4410*a**2*b**13*x**(5/2) + 1470*a*b**14*x**3 + 210*b**15*
x**(7/2)) - 4263*a*b**6*x**3/(210*a**7*b**8 + 1470*a**6*b**9*sqrt(x) + 4410*a**5*b**10*x + 7350*a**4*b**11*x**
(3/2) + 7350*a**3*b**12*x**2 + 4410*a**2*b**13*x**(5/2) + 1470*a*b**14*x**3 + 210*b**15*x**(7/2)) + 420*b**7*x
**(7/2)*log(a/b + sqrt(x))/(210*a**7*b**8 + 1470*a**6*b**9*sqrt(x) + 4410*a**5*b**10*x + 7350*a**4*b**11*x**(3
/2) + 7350*a**3*b**12*x**2 + 4410*a**2*b**13*x**(5/2) + 1470*a*b**14*x**3 + 210*b**15*x**(7/2)) - 1029*b**7*x*
*(7/2)/(210*a**7*b**8 + 1470*a**6*b**9*sqrt(x) + 4410*a**5*b**10*x + 7350*a**4*b**11*x**(3/2) + 7350*a**3*b**1
2*x**2 + 4410*a**2*b**13*x**(5/2) + 1470*a*b**14*x**3 + 210*b**15*x**(7/2)), Ne(b, 0)), (x**4/(4*a**8), True))

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Giac [A]  time = 1.10431, size = 128, normalized size = 0.82 \begin{align*} \frac{2 \, \log \left ({\left | b \sqrt{x} + a \right |}\right )}{b^{8}} + \frac{2940 \, a b^{5} x^{3} + 13230 \, a^{2} b^{4} x^{\frac{5}{2}} + 26950 \, a^{3} b^{3} x^{2} + 30625 \, a^{4} b^{2} x^{\frac{3}{2}} + 20139 \, a^{5} b x + 7203 \, a^{6} \sqrt{x} + \frac{1089 \, a^{7}}{b}}{210 \,{\left (b \sqrt{x} + a\right )}^{7} b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(abs(b*sqrt(x) + a))/b^8 + 1/210*(2940*a*b^5*x^3 + 13230*a^2*b^4*x^(5/2) + 26950*a^3*b^3*x^2 + 30625*a^4*
b^2*x^(3/2) + 20139*a^5*b*x + 7203*a^6*sqrt(x) + 1089*a^7/b)/((b*sqrt(x) + a)^7*b^7)