3.2223 \(\int \frac{x^3}{\left (a+b \sqrt{x}\right )^8} \, dx\)

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) + (14*a)/(b^8*(a + b*Sqr
t[x])) + (2*Log[a + b*Sqrt[x]])/b^8

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Rubi [A]  time = 0.244912, 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 \[ \frac{2 a^7}{7 b^8 \left (a+b \sqrt{x}\right )^7}-\frac{7 a^6}{3 b^8 \left (a+b \sqrt{x}\right )^6}+\frac{42 a^5}{5 b^8 \left (a+b \sqrt{x}\right )^5}-\frac{35 a^4}{2 b^8 \left (a+b \sqrt{x}\right )^4}+\frac{70 a^3}{3 b^8 \left (a+b \sqrt{x}\right )^3}-\frac{21 a^2}{b^8 \left (a+b \sqrt{x}\right )^2}+\frac{14 a}{b^8 \left (a+b \sqrt{x}\right )}+\frac{2 \log \left (a+b \sqrt{x}\right )}{b^8} \]

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) + (14*a)/(b^8*(a + b*Sqr
t[x])) + (2*Log[a + b*Sqrt[x]])/b^8

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Rubi in Sympy [A]  time = 37.6382, size = 150, normalized size = 0.96 \[ \frac{2 a^{7}}{7 b^{8} \left (a + b \sqrt{x}\right )^{7}} - \frac{7 a^{6}}{3 b^{8} \left (a + b \sqrt{x}\right )^{6}} + \frac{42 a^{5}}{5 b^{8} \left (a + b \sqrt{x}\right )^{5}} - \frac{35 a^{4}}{2 b^{8} \left (a + b \sqrt{x}\right )^{4}} + \frac{70 a^{3}}{3 b^{8} \left (a + b \sqrt{x}\right )^{3}} - \frac{21 a^{2}}{b^{8} \left (a + b \sqrt{x}\right )^{2}} + \frac{14 a}{b^{8} \left (a + b \sqrt{x}\right )} + \frac{2 \log{\left (a + b \sqrt{x} \right )}}{b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(a+b*x**(1/2))**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) + 14*a/(b**8*(a +
b*sqrt(x))) + 2*log(a + b*sqrt(x))/b**8

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Mathematica [A]  time = 0.0662979, size = 102, normalized size = 0.65 \[ \frac{\frac{a \left (1089 a^6+7203 a^5 b \sqrt{x}+20139 a^4 b^2 x+30625 a^3 b^3 x^{3/2}+26950 a^2 b^4 x^2+13230 a b^5 x^{5/2}+2940 b^6 x^3\right )}{\left (a+b \sqrt{x}\right )^7}+420 \log \left (a+b \sqrt{x}\right )}{210 b^8} \]

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) + 2
6950*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 = 0.013, size = 132, normalized size = 0.8 \[ 2\,{\frac{\ln \left ( a+b\sqrt{x} \right ) }{{b}^{8}}}+{\frac{2\,{a}^{7}}{7\,{b}^{8}} \left ( a+b\sqrt{x} \right ) ^{-7}}-{\frac{7\,{a}^{6}}{3\,{b}^{8}} \left ( a+b\sqrt{x} \right ) ^{-6}}+{\frac{42\,{a}^{5}}{5\,{b}^{8}} \left ( a+b\sqrt{x} \right ) ^{-5}}-{\frac{35\,{a}^{4}}{2\,{b}^{8}} \left ( a+b\sqrt{x} \right ) ^{-4}}+{\frac{70\,{a}^{3}}{3\,{b}^{8}} \left ( a+b\sqrt{x} \right ) ^{-3}}-21\,{\frac{{a}^{2}}{{b}^{8} \left ( a+b\sqrt{x} \right ) ^{2}}}+14\,{\frac{a}{{b}^{8} \left ( a+b\sqrt{x} \right ) }} \]

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 = 1.45616, size = 177, normalized size = 1.13 \[ \frac{2 \, \log \left (b \sqrt{x} + a\right )}{b^{8}} + \frac{14 \, a}{{\left (b \sqrt{x} + a\right )} b^{8}} - \frac{21 \, a^{2}}{{\left (b \sqrt{x} + a\right )}^{2} b^{8}} + \frac{70 \, a^{3}}{3 \,{\left (b \sqrt{x} + a\right )}^{3} b^{8}} - \frac{35 \, a^{4}}{2 \,{\left (b \sqrt{x} + a\right )}^{4} b^{8}} + \frac{42 \, a^{5}}{5 \,{\left (b \sqrt{x} + a\right )}^{5} b^{8}} - \frac{7 \, a^{6}}{3 \,{\left (b \sqrt{x} + a\right )}^{6} b^{8}} + \frac{2 \, a^{7}}{7 \,{\left (b \sqrt{x} + a\right )}^{7} b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*sqrt(x) + a)^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) + 4
2/5*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 [A]  time = 0.237058, size = 309, normalized size = 1.97 \[ \frac{2940 \, a b^{6} x^{3} + 26950 \, a^{3} b^{4} x^{2} + 20139 \, a^{5} b^{2} x + 1089 \, a^{7} + 420 \,{\left (7 \, a b^{6} x^{3} + 35 \, a^{3} b^{4} x^{2} + 21 \, a^{5} b^{2} x + a^{7} +{\left (b^{7} x^{3} + 21 \, a^{2} b^{5} x^{2} + 35 \, a^{4} b^{3} x + 7 \, a^{6} b\right )} \sqrt{x}\right )} \log \left (b \sqrt{x} + a\right ) + 49 \,{\left (270 \, a^{2} b^{5} x^{2} + 625 \, a^{4} b^{3} x + 147 \, a^{6} b\right )} \sqrt{x}}{210 \,{\left (7 \, a b^{14} x^{3} + 35 \, a^{3} b^{12} x^{2} + 21 \, a^{5} b^{10} x + a^{7} b^{8} +{\left (b^{15} x^{3} + 21 \, a^{2} b^{13} x^{2} + 35 \, a^{4} b^{11} x + 7 \, a^{6} b^{9}\right )} \sqrt{x}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*sqrt(x) + a)^8,x, algorithm="fricas")

[Out]

1/210*(2940*a*b^6*x^3 + 26950*a^3*b^4*x^2 + 20139*a^5*b^2*x + 1089*a^7 + 420*(7*
a*b^6*x^3 + 35*a^3*b^4*x^2 + 21*a^5*b^2*x + a^7 + (b^7*x^3 + 21*a^2*b^5*x^2 + 35
*a^4*b^3*x + 7*a^6*b)*sqrt(x))*log(b*sqrt(x) + a) + 49*(270*a^2*b^5*x^2 + 625*a^
4*b^3*x + 147*a^6*b)*sqrt(x))/(7*a*b^14*x^3 + 35*a^3*b^12*x^2 + 21*a^5*b^10*x +
a^7*b^8 + (b^15*x^3 + 21*a^2*b^13*x^2 + 35*a^4*b^11*x + 7*a^6*b^9)*sqrt(x))

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Sympy [A]  time = 32.6686, size = 1629, normalized size = 10.38 \[ \text{result too large to display} \]

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)) + 1089*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) + 7
350*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 + 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)) + 7203*a**6
*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)) + 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) + 735
0*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)) + 20139*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)) + 30625*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)) + 1
4700*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)) + 26950*a**3*b**4*x**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)) + 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)) + 13230*a**2*b**5*x**(5/2)/(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)) + 2940*a*b**6*x**3*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)) + 2940*a*b**6*x**3/(210*a**7*b**8 + 1470*a**6*b**9*sq
rt(x) + 4410*a**5*b**10*x + 7350*a**4*b**11*x**(3/2) + 7350*a**3*b**12*x**2 + 44
10*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**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)), Ne(b, 0)), (x**4/(4*a**8), True))

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GIAC/XCAS [A]  time = 0.242489, size = 128, normalized size = 0.82 \[ \frac{2 \,{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{b^{8}} + \frac{2940 \, a b^{5} x^{3} + 13230 \, a^{2} b^{4} x^{\frac{5}{2}} + 26950 \, a^{3} b^{3} x^{2} + 30625 \, a^{4} b^{2} x^{\frac{3}{2}} + 20139 \, a^{5} b x + 7203 \, a^{6} \sqrt{x} + \frac{1089 \, a^{7}}{b}}{210 \,{\left (b \sqrt{x} + a\right )}^{7} b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*sqrt(x) + a)^8,x, algorithm="giac")

[Out]

2*ln(abs(b*sqrt(x) + a))/b^8 + 1/210*(2940*a*b^5*x^3 + 13230*a^2*b^4*x^(5/2) + 2
6950*a^3*b^3*x^2 + 30625*a^4*b^2*x^(3/2) + 20139*a^5*b*x + 7203*a^6*sqrt(x) + 10
89*a^7/b)/((b*sqrt(x) + a)^7*b^7)