3.324 \(\int \frac{(a+b x)^{9/2}}{x^8} \, dx\)

Optimal. Leaf size=163 \[ -\frac{9 b^7 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{1024 a^{5/2}}+\frac{9 b^6 \sqrt{a+b x}}{1024 a^2 x}-\frac{3 b^5 \sqrt{a+b x}}{512 a x^2}-\frac{3 b^4 \sqrt{a+b x}}{128 x^3}-\frac{3 b^3 (a+b x)^{3/2}}{64 x^4}-\frac{3 b^2 (a+b x)^{5/2}}{40 x^5}-\frac{(a+b x)^{9/2}}{7 x^7}-\frac{3 b (a+b x)^{7/2}}{28 x^6} \]

[Out]

(-3*b^4*Sqrt[a + b*x])/(128*x^3) - (3*b^5*Sqrt[a + b*x])/(512*a*x^2) + (9*b^6*Sq
rt[a + b*x])/(1024*a^2*x) - (3*b^3*(a + b*x)^(3/2))/(64*x^4) - (3*b^2*(a + b*x)^
(5/2))/(40*x^5) - (3*b*(a + b*x)^(7/2))/(28*x^6) - (a + b*x)^(9/2)/(7*x^7) - (9*
b^7*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(1024*a^(5/2))

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Rubi [A]  time = 0.181111, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{9 b^7 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{1024 a^{5/2}}+\frac{9 b^6 \sqrt{a+b x}}{1024 a^2 x}-\frac{3 b^5 \sqrt{a+b x}}{512 a x^2}-\frac{3 b^4 \sqrt{a+b x}}{128 x^3}-\frac{3 b^3 (a+b x)^{3/2}}{64 x^4}-\frac{3 b^2 (a+b x)^{5/2}}{40 x^5}-\frac{(a+b x)^{9/2}}{7 x^7}-\frac{3 b (a+b x)^{7/2}}{28 x^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(9/2)/x^8,x]

[Out]

(-3*b^4*Sqrt[a + b*x])/(128*x^3) - (3*b^5*Sqrt[a + b*x])/(512*a*x^2) + (9*b^6*Sq
rt[a + b*x])/(1024*a^2*x) - (3*b^3*(a + b*x)^(3/2))/(64*x^4) - (3*b^2*(a + b*x)^
(5/2))/(40*x^5) - (3*b*(a + b*x)^(7/2))/(28*x^6) - (a + b*x)^(9/2)/(7*x^7) - (9*
b^7*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(1024*a^(5/2))

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Rubi in Sympy [A]  time = 25.4724, size = 153, normalized size = 0.94 \[ - \frac{3 b^{4} \sqrt{a + b x}}{128 x^{3}} - \frac{3 b^{3} \left (a + b x\right )^{\frac{3}{2}}}{64 x^{4}} - \frac{3 b^{2} \left (a + b x\right )^{\frac{5}{2}}}{40 x^{5}} - \frac{3 b \left (a + b x\right )^{\frac{7}{2}}}{28 x^{6}} - \frac{\left (a + b x\right )^{\frac{9}{2}}}{7 x^{7}} - \frac{3 b^{5} \sqrt{a + b x}}{512 a x^{2}} + \frac{9 b^{6} \sqrt{a + b x}}{1024 a^{2} x} - \frac{9 b^{7} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{1024 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(9/2)/x**8,x)

[Out]

-3*b**4*sqrt(a + b*x)/(128*x**3) - 3*b**3*(a + b*x)**(3/2)/(64*x**4) - 3*b**2*(a
+ b*x)**(5/2)/(40*x**5) - 3*b*(a + b*x)**(7/2)/(28*x**6) - (a + b*x)**(9/2)/(7*
x**7) - 3*b**5*sqrt(a + b*x)/(512*a*x**2) + 9*b**6*sqrt(a + b*x)/(1024*a**2*x) -
9*b**7*atanh(sqrt(a + b*x)/sqrt(a))/(1024*a**(5/2))

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Mathematica [A]  time = 0.118012, size = 111, normalized size = 0.68 \[ -\frac{9 b^7 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{1024 a^{5/2}}-\frac{\sqrt{a+b x} \left (5120 a^6+24320 a^5 b x+44928 a^4 b^2 x^2+39056 a^3 b^3 x^3+14168 a^2 b^4 x^4+210 a b^5 x^5-315 b^6 x^6\right )}{35840 a^2 x^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(9/2)/x^8,x]

[Out]

-(Sqrt[a + b*x]*(5120*a^6 + 24320*a^5*b*x + 44928*a^4*b^2*x^2 + 39056*a^3*b^3*x^
3 + 14168*a^2*b^4*x^4 + 210*a*b^5*x^5 - 315*b^6*x^6))/(35840*a^2*x^7) - (9*b^7*A
rcTanh[Sqrt[a + b*x]/Sqrt[a]])/(1024*a^(5/2))

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Maple [A]  time = 0.022, size = 111, normalized size = 0.7 \[ 2\,{b}^{7} \left ({\frac{1}{{b}^{7}{x}^{7}} \left ({\frac{9\, \left ( bx+a \right ) ^{13/2}}{2048\,{a}^{2}}}-{\frac{15\, \left ( bx+a \right ) ^{11/2}}{512\,a}}-{\frac{1199\, \left ( bx+a \right ) ^{9/2}}{10240}}+{\frac{9\,a \left ( bx+a \right ) ^{7/2}}{70}}-{\frac{849\,{a}^{2} \left ( bx+a \right ) ^{5/2}}{10240}}+{\frac{15\,{a}^{3} \left ( bx+a \right ) ^{3/2}}{512}}-{\frac{9\,{a}^{4}\sqrt{bx+a}}{2048}} \right ) }-{\frac{9}{2048\,{a}^{5/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(9/2)/x^8,x)

[Out]

2*b^7*((9/2048/a^2*(b*x+a)^(13/2)-15/512/a*(b*x+a)^(11/2)-1199/10240*(b*x+a)^(9/
2)+9/70*a*(b*x+a)^(7/2)-849/10240*a^2*(b*x+a)^(5/2)+15/512*a^3*(b*x+a)^(3/2)-9/2
048*a^4*(b*x+a)^(1/2))/x^7/b^7-9/2048*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(5/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(9/2)/x^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221069, size = 1, normalized size = 0.01 \[ \left [\frac{315 \, b^{7} x^{7} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (315 \, b^{6} x^{6} - 210 \, a b^{5} x^{5} - 14168 \, a^{2} b^{4} x^{4} - 39056 \, a^{3} b^{3} x^{3} - 44928 \, a^{4} b^{2} x^{2} - 24320 \, a^{5} b x - 5120 \, a^{6}\right )} \sqrt{b x + a} \sqrt{a}}{71680 \, a^{\frac{5}{2}} x^{7}}, \frac{315 \, b^{7} x^{7} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) +{\left (315 \, b^{6} x^{6} - 210 \, a b^{5} x^{5} - 14168 \, a^{2} b^{4} x^{4} - 39056 \, a^{3} b^{3} x^{3} - 44928 \, a^{4} b^{2} x^{2} - 24320 \, a^{5} b x - 5120 \, a^{6}\right )} \sqrt{b x + a} \sqrt{-a}}{35840 \, \sqrt{-a} a^{2} x^{7}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(9/2)/x^8,x, algorithm="fricas")

[Out]

[1/71680*(315*b^7*x^7*log(((b*x + 2*a)*sqrt(a) - 2*sqrt(b*x + a)*a)/x) + 2*(315*
b^6*x^6 - 210*a*b^5*x^5 - 14168*a^2*b^4*x^4 - 39056*a^3*b^3*x^3 - 44928*a^4*b^2*
x^2 - 24320*a^5*b*x - 5120*a^6)*sqrt(b*x + a)*sqrt(a))/(a^(5/2)*x^7), 1/35840*(3
15*b^7*x^7*arctan(a/(sqrt(b*x + a)*sqrt(-a))) + (315*b^6*x^6 - 210*a*b^5*x^5 - 1
4168*a^2*b^4*x^4 - 39056*a^3*b^3*x^3 - 44928*a^4*b^2*x^2 - 24320*a^5*b*x - 5120*
a^6)*sqrt(b*x + a)*sqrt(-a))/(sqrt(-a)*a^2*x^7)]

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Sympy [A]  time = 69.2434, size = 236, normalized size = 1.45 \[ - \frac{a^{5}}{7 \sqrt{b} x^{\frac{15}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{23 a^{4} \sqrt{b}}{28 x^{\frac{13}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{541 a^{3} b^{\frac{3}{2}}}{280 x^{\frac{11}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{5249 a^{2} b^{\frac{5}{2}}}{2240 x^{\frac{9}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{6653 a b^{\frac{7}{2}}}{4480 x^{\frac{7}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{1027 b^{\frac{9}{2}}}{2560 x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{3 b^{\frac{11}{2}}}{1024 a x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{9 b^{\frac{13}{2}}}{1024 a^{2} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} - \frac{9 b^{7} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{1024 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(9/2)/x**8,x)

[Out]

-a**5/(7*sqrt(b)*x**(15/2)*sqrt(a/(b*x) + 1)) - 23*a**4*sqrt(b)/(28*x**(13/2)*sq
rt(a/(b*x) + 1)) - 541*a**3*b**(3/2)/(280*x**(11/2)*sqrt(a/(b*x) + 1)) - 5249*a*
*2*b**(5/2)/(2240*x**(9/2)*sqrt(a/(b*x) + 1)) - 6653*a*b**(7/2)/(4480*x**(7/2)*s
qrt(a/(b*x) + 1)) - 1027*b**(9/2)/(2560*x**(5/2)*sqrt(a/(b*x) + 1)) + 3*b**(11/2
)/(1024*a*x**(3/2)*sqrt(a/(b*x) + 1)) + 9*b**(13/2)/(1024*a**2*sqrt(x)*sqrt(a/(b
*x) + 1)) - 9*b**7*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/(1024*a**(5/2))

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GIAC/XCAS [A]  time = 0.213173, size = 194, normalized size = 1.19 \[ \frac{\frac{315 \, b^{8} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{315 \,{\left (b x + a\right )}^{\frac{13}{2}} b^{8} - 2100 \,{\left (b x + a\right )}^{\frac{11}{2}} a b^{8} - 8393 \,{\left (b x + a\right )}^{\frac{9}{2}} a^{2} b^{8} + 9216 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{3} b^{8} - 5943 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{4} b^{8} + 2100 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{5} b^{8} - 315 \, \sqrt{b x + a} a^{6} b^{8}}{a^{2} b^{7} x^{7}}}{35840 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(9/2)/x^8,x, algorithm="giac")

[Out]

1/35840*(315*b^8*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^2) + (315*(b*x + a)^
(13/2)*b^8 - 2100*(b*x + a)^(11/2)*a*b^8 - 8393*(b*x + a)^(9/2)*a^2*b^8 + 9216*(
b*x + a)^(7/2)*a^3*b^8 - 5943*(b*x + a)^(5/2)*a^4*b^8 + 2100*(b*x + a)^(3/2)*a^5
*b^8 - 315*sqrt(b*x + a)*a^6*b^8)/(a^2*b^7*x^7))/b