3.323 \(\int \frac{(a+b x)^{9/2}}{x^7} \, dx\)

Optimal. Leaf size=141 \[ \frac{21 b^6 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{512 a^{3/2}}-\frac{21 b^5 \sqrt{a+b x}}{512 a x}-\frac{21 b^4 \sqrt{a+b x}}{256 x^2}-\frac{7 b^3 (a+b x)^{3/2}}{64 x^3}-\frac{21 b^2 (a+b x)^{5/2}}{160 x^4}-\frac{(a+b x)^{9/2}}{6 x^6}-\frac{3 b (a+b x)^{7/2}}{20 x^5} \]

[Out]

(-21*b^4*Sqrt[a + b*x])/(256*x^2) - (21*b^5*Sqrt[a + b*x])/(512*a*x) - (7*b^3*(a
+ b*x)^(3/2))/(64*x^3) - (21*b^2*(a + b*x)^(5/2))/(160*x^4) - (3*b*(a + b*x)^(7
/2))/(20*x^5) - (a + b*x)^(9/2)/(6*x^6) + (21*b^6*ArcTanh[Sqrt[a + b*x]/Sqrt[a]]
)/(512*a^(3/2))

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Rubi [A]  time = 0.145196, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{21 b^6 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{512 a^{3/2}}-\frac{21 b^5 \sqrt{a+b x}}{512 a x}-\frac{21 b^4 \sqrt{a+b x}}{256 x^2}-\frac{7 b^3 (a+b x)^{3/2}}{64 x^3}-\frac{21 b^2 (a+b x)^{5/2}}{160 x^4}-\frac{(a+b x)^{9/2}}{6 x^6}-\frac{3 b (a+b x)^{7/2}}{20 x^5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(9/2)/x^7,x]

[Out]

(-21*b^4*Sqrt[a + b*x])/(256*x^2) - (21*b^5*Sqrt[a + b*x])/(512*a*x) - (7*b^3*(a
+ b*x)^(3/2))/(64*x^3) - (21*b^2*(a + b*x)^(5/2))/(160*x^4) - (3*b*(a + b*x)^(7
/2))/(20*x^5) - (a + b*x)^(9/2)/(6*x^6) + (21*b^6*ArcTanh[Sqrt[a + b*x]/Sqrt[a]]
)/(512*a^(3/2))

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Rubi in Sympy [A]  time = 20.0939, size = 131, normalized size = 0.93 \[ - \frac{21 b^{4} \sqrt{a + b x}}{256 x^{2}} - \frac{7 b^{3} \left (a + b x\right )^{\frac{3}{2}}}{64 x^{3}} - \frac{21 b^{2} \left (a + b x\right )^{\frac{5}{2}}}{160 x^{4}} - \frac{3 b \left (a + b x\right )^{\frac{7}{2}}}{20 x^{5}} - \frac{\left (a + b x\right )^{\frac{9}{2}}}{6 x^{6}} - \frac{21 b^{5} \sqrt{a + b x}}{512 a x} + \frac{21 b^{6} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{512 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(9/2)/x**7,x)

[Out]

-21*b**4*sqrt(a + b*x)/(256*x**2) - 7*b**3*(a + b*x)**(3/2)/(64*x**3) - 21*b**2*
(a + b*x)**(5/2)/(160*x**4) - 3*b*(a + b*x)**(7/2)/(20*x**5) - (a + b*x)**(9/2)/
(6*x**6) - 21*b**5*sqrt(a + b*x)/(512*a*x) + 21*b**6*atanh(sqrt(a + b*x)/sqrt(a)
)/(512*a**(3/2))

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Mathematica [A]  time = 0.0970179, size = 100, normalized size = 0.71 \[ \frac{21 b^6 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{512 a^{3/2}}-\frac{\sqrt{a+b x} \left (1280 a^5+6272 a^4 b x+12144 a^3 b^2 x^2+11432 a^2 b^3 x^3+4910 a b^4 x^4+315 b^5 x^5\right )}{7680 a x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(9/2)/x^7,x]

[Out]

-(Sqrt[a + b*x]*(1280*a^5 + 6272*a^4*b*x + 12144*a^3*b^2*x^2 + 11432*a^2*b^3*x^3
+ 4910*a*b^4*x^4 + 315*b^5*x^5))/(7680*a*x^6) + (21*b^6*ArcTanh[Sqrt[a + b*x]/S
qrt[a]])/(512*a^(3/2))

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Maple [A]  time = 0.02, size = 99, normalized size = 0.7 \[ 2\,{b}^{6} \left ({\frac{1}{{x}^{6}{b}^{6}} \left ( -{\frac{21\, \left ( bx+a \right ) ^{11/2}}{1024\,a}}-{\frac{667\, \left ( bx+a \right ) ^{9/2}}{3072}}+{\frac{843\,a \left ( bx+a \right ) ^{7/2}}{2560}}-{\frac{693\,{a}^{2} \left ( bx+a \right ) ^{5/2}}{2560}}+{\frac{119\,{a}^{3} \left ( bx+a \right ) ^{3/2}}{1024}}-{\frac{21\,{a}^{4}\sqrt{bx+a}}{1024}} \right ) }+{\frac{21}{1024\,{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(9/2)/x^7,x)

[Out]

2*b^6*((-21/1024/a*(b*x+a)^(11/2)-667/3072*(b*x+a)^(9/2)+843/2560*a*(b*x+a)^(7/2
)-693/2560*a^2*(b*x+a)^(5/2)+119/1024*a^3*(b*x+a)^(3/2)-21/1024*a^4*(b*x+a)^(1/2
))/x^6/b^6+21/1024*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(3/2))

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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^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231354, size = 1, normalized size = 0.01 \[ \left [\frac{315 \, b^{6} x^{6} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x + a} a}{x}\right ) - 2 \,{\left (315 \, b^{5} x^{5} + 4910 \, a b^{4} x^{4} + 11432 \, a^{2} b^{3} x^{3} + 12144 \, a^{3} b^{2} x^{2} + 6272 \, a^{4} b x + 1280 \, a^{5}\right )} \sqrt{b x + a} \sqrt{a}}{15360 \, a^{\frac{3}{2}} x^{6}}, -\frac{315 \, b^{6} x^{6} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) +{\left (315 \, b^{5} x^{5} + 4910 \, a b^{4} x^{4} + 11432 \, a^{2} b^{3} x^{3} + 12144 \, a^{3} b^{2} x^{2} + 6272 \, a^{4} b x + 1280 \, a^{5}\right )} \sqrt{b x + a} \sqrt{-a}}{7680 \, \sqrt{-a} a x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(9/2)/x^7,x, algorithm="fricas")

[Out]

[1/15360*(315*b^6*x^6*log(((b*x + 2*a)*sqrt(a) + 2*sqrt(b*x + a)*a)/x) - 2*(315*
b^5*x^5 + 4910*a*b^4*x^4 + 11432*a^2*b^3*x^3 + 12144*a^3*b^2*x^2 + 6272*a^4*b*x
+ 1280*a^5)*sqrt(b*x + a)*sqrt(a))/(a^(3/2)*x^6), -1/7680*(315*b^6*x^6*arctan(a/
(sqrt(b*x + a)*sqrt(-a))) + (315*b^5*x^5 + 4910*a*b^4*x^4 + 11432*a^2*b^3*x^3 +
12144*a^3*b^2*x^2 + 6272*a^4*b*x + 1280*a^5)*sqrt(b*x + a)*sqrt(-a))/(sqrt(-a)*a
*x^6)]

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Sympy [A]  time = 52.0609, size = 209, normalized size = 1.48 \[ - \frac{a^{5}}{6 \sqrt{b} x^{\frac{13}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{59 a^{4} \sqrt{b}}{60 x^{\frac{11}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{1151 a^{3} b^{\frac{3}{2}}}{480 x^{\frac{9}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{2947 a^{2} b^{\frac{5}{2}}}{960 x^{\frac{7}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{8171 a b^{\frac{7}{2}}}{3840 x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{1045 b^{\frac{9}{2}}}{1536 x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{21 b^{\frac{11}{2}}}{512 a \sqrt{x} \sqrt{\frac{a}{b x} + 1}} + \frac{21 b^{6} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{512 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(9/2)/x**7,x)

[Out]

-a**5/(6*sqrt(b)*x**(13/2)*sqrt(a/(b*x) + 1)) - 59*a**4*sqrt(b)/(60*x**(11/2)*sq
rt(a/(b*x) + 1)) - 1151*a**3*b**(3/2)/(480*x**(9/2)*sqrt(a/(b*x) + 1)) - 2947*a*
*2*b**(5/2)/(960*x**(7/2)*sqrt(a/(b*x) + 1)) - 8171*a*b**(7/2)/(3840*x**(5/2)*sq
rt(a/(b*x) + 1)) - 1045*b**(9/2)/(1536*x**(3/2)*sqrt(a/(b*x) + 1)) - 21*b**(11/2
)/(512*a*sqrt(x)*sqrt(a/(b*x) + 1)) + 21*b**6*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/(
512*a**(3/2))

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GIAC/XCAS [A]  time = 0.214301, size = 174, normalized size = 1.23 \[ -\frac{\frac{315 \, b^{7} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{315 \,{\left (b x + a\right )}^{\frac{11}{2}} b^{7} + 3335 \,{\left (b x + a\right )}^{\frac{9}{2}} a b^{7} - 5058 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{2} b^{7} + 4158 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{3} b^{7} - 1785 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{4} b^{7} + 315 \, \sqrt{b x + a} a^{5} b^{7}}{a b^{6} x^{6}}}{7680 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(9/2)/x^7,x, algorithm="giac")

[Out]

-1/7680*(315*b^7*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a) + (315*(b*x + a)^(1
1/2)*b^7 + 3335*(b*x + a)^(9/2)*a*b^7 - 5058*(b*x + a)^(7/2)*a^2*b^7 + 4158*(b*x
+ a)^(5/2)*a^3*b^7 - 1785*(b*x + a)^(3/2)*a^4*b^7 + 315*sqrt(b*x + a)*a^5*b^7)/
(a*b^6*x^6))/b