3.421 \(\int \frac{A+B x}{x^6 \sqrt{a+b x}} \, dx\)

Optimal. Leaf size=177 \[ \frac{7 b^4 (9 A b-10 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{128 a^{11/2}}-\frac{7 b^3 \sqrt{a+b x} (9 A b-10 a B)}{128 a^5 x}+\frac{7 b^2 \sqrt{a+b x} (9 A b-10 a B)}{192 a^4 x^2}-\frac{7 b \sqrt{a+b x} (9 A b-10 a B)}{240 a^3 x^3}+\frac{\sqrt{a+b x} (9 A b-10 a B)}{40 a^2 x^4}-\frac{A \sqrt{a+b x}}{5 a x^5} \]

[Out]

-(A*Sqrt[a + b*x])/(5*a*x^5) + ((9*A*b - 10*a*B)*Sqrt[a + b*x])/(40*a^2*x^4) - (
7*b*(9*A*b - 10*a*B)*Sqrt[a + b*x])/(240*a^3*x^3) + (7*b^2*(9*A*b - 10*a*B)*Sqrt
[a + b*x])/(192*a^4*x^2) - (7*b^3*(9*A*b - 10*a*B)*Sqrt[a + b*x])/(128*a^5*x) +
(7*b^4*(9*A*b - 10*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(128*a^(11/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.239845, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{7 b^4 (9 A b-10 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{128 a^{11/2}}-\frac{7 b^3 \sqrt{a+b x} (9 A b-10 a B)}{128 a^5 x}+\frac{7 b^2 \sqrt{a+b x} (9 A b-10 a B)}{192 a^4 x^2}-\frac{7 b \sqrt{a+b x} (9 A b-10 a B)}{240 a^3 x^3}+\frac{\sqrt{a+b x} (9 A b-10 a B)}{40 a^2 x^4}-\frac{A \sqrt{a+b x}}{5 a x^5} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^6*Sqrt[a + b*x]),x]

[Out]

-(A*Sqrt[a + b*x])/(5*a*x^5) + ((9*A*b - 10*a*B)*Sqrt[a + b*x])/(40*a^2*x^4) - (
7*b*(9*A*b - 10*a*B)*Sqrt[a + b*x])/(240*a^3*x^3) + (7*b^2*(9*A*b - 10*a*B)*Sqrt
[a + b*x])/(192*a^4*x^2) - (7*b^3*(9*A*b - 10*a*B)*Sqrt[a + b*x])/(128*a^5*x) +
(7*b^4*(9*A*b - 10*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(128*a^(11/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 22.2395, size = 173, normalized size = 0.98 \[ - \frac{A \sqrt{a + b x}}{5 a x^{5}} + \frac{\sqrt{a + b x} \left (9 A b - 10 B a\right )}{40 a^{2} x^{4}} - \frac{7 b \sqrt{a + b x} \left (9 A b - 10 B a\right )}{240 a^{3} x^{3}} + \frac{7 b^{2} \sqrt{a + b x} \left (9 A b - 10 B a\right )}{192 a^{4} x^{2}} - \frac{7 b^{3} \sqrt{a + b x} \left (9 A b - 10 B a\right )}{128 a^{5} x} + \frac{7 b^{4} \left (9 A b - 10 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{128 a^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**6/(b*x+a)**(1/2),x)

[Out]

-A*sqrt(a + b*x)/(5*a*x**5) + sqrt(a + b*x)*(9*A*b - 10*B*a)/(40*a**2*x**4) - 7*
b*sqrt(a + b*x)*(9*A*b - 10*B*a)/(240*a**3*x**3) + 7*b**2*sqrt(a + b*x)*(9*A*b -
10*B*a)/(192*a**4*x**2) - 7*b**3*sqrt(a + b*x)*(9*A*b - 10*B*a)/(128*a**5*x) +
7*b**4*(9*A*b - 10*B*a)*atanh(sqrt(a + b*x)/sqrt(a))/(128*a**(11/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.243972, size = 131, normalized size = 0.74 \[ \frac{7 b^4 (9 A b-10 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{128 a^{11/2}}+\frac{\sqrt{a+b x} \left (-96 a^4 (4 A+5 B x)+16 a^3 b x (27 A+35 B x)-28 a^2 b^2 x^2 (18 A+25 B x)+210 a b^3 x^3 (3 A+5 B x)-945 A b^4 x^4\right )}{1920 a^5 x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^6*Sqrt[a + b*x]),x]

[Out]

(Sqrt[a + b*x]*(-945*A*b^4*x^4 + 210*a*b^3*x^3*(3*A + 5*B*x) - 96*a^4*(4*A + 5*B
*x) - 28*a^2*b^2*x^2*(18*A + 25*B*x) + 16*a^3*b*x*(27*A + 35*B*x)))/(1920*a^5*x^
5) + (7*b^4*(9*A*b - 10*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(128*a^(11/2))

_______________________________________________________________________________________

Maple [A]  time = 0.02, size = 146, normalized size = 0.8 \[ 2\,{b}^{4} \left ({\frac{1}{{b}^{5}{x}^{5}} \left ( -{\frac{ \left ( 63\,Ab-70\,Ba \right ) \left ( bx+a \right ) ^{9/2}}{256\,{a}^{5}}}+{\frac{ \left ( 441\,Ab-490\,Ba \right ) \left ( bx+a \right ) ^{7/2}}{384\,{a}^{4}}}-{\frac{ \left ( 63\,Ab-70\,Ba \right ) \left ( bx+a \right ) ^{5/2}}{30\,{a}^{3}}}+{\frac{ \left ( 711\,Ab-790\,Ba \right ) \left ( bx+a \right ) ^{3/2}}{384\,{a}^{2}}}-{\frac{ \left ( 193\,Ab-186\,Ba \right ) \sqrt{bx+a}}{256\,a}} \right ) }+{\frac{63\,Ab-70\,Ba}{256\,{a}^{11/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^6/(b*x+a)^(1/2),x)

[Out]

2*b^4*((-7/256*(9*A*b-10*B*a)/a^5*(b*x+a)^(9/2)+49/384/a^4*(9*A*b-10*B*a)*(b*x+a
)^(7/2)-7/30/a^3*(9*A*b-10*B*a)*(b*x+a)^(5/2)+79/384/a^2*(9*A*b-10*B*a)*(b*x+a)^
(3/2)-1/256*(193*A*b-186*B*a)/a*(b*x+a)^(1/2))/x^5/b^5+7/256*(9*A*b-10*B*a)/a^(1
1/2)*arctanh((b*x+a)^(1/2)/a^(1/2)))

_______________________________________________________________________________________

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)/(sqrt(b*x + a)*x^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.225951, size = 1, normalized size = 0.01 \[ \left [-\frac{105 \,{\left (10 \, B a b^{4} - 9 \, A b^{5}\right )} x^{5} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (384 \, A a^{4} - 105 \,{\left (10 \, B a b^{3} - 9 \, A b^{4}\right )} x^{4} + 70 \,{\left (10 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{3} - 56 \,{\left (10 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{2} + 48 \,{\left (10 \, B a^{4} - 9 \, A a^{3} b\right )} x\right )} \sqrt{b x + a} \sqrt{a}}{3840 \, a^{\frac{11}{2}} x^{5}}, \frac{105 \,{\left (10 \, B a b^{4} - 9 \, A b^{5}\right )} x^{5} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) -{\left (384 \, A a^{4} - 105 \,{\left (10 \, B a b^{3} - 9 \, A b^{4}\right )} x^{4} + 70 \,{\left (10 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{3} - 56 \,{\left (10 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{2} + 48 \,{\left (10 \, B a^{4} - 9 \, A a^{3} b\right )} x\right )} \sqrt{b x + a} \sqrt{-a}}{1920 \, \sqrt{-a} a^{5} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(b*x + a)*x^6),x, algorithm="fricas")

[Out]

[-1/3840*(105*(10*B*a*b^4 - 9*A*b^5)*x^5*log(((b*x + 2*a)*sqrt(a) + 2*sqrt(b*x +
a)*a)/x) + 2*(384*A*a^4 - 105*(10*B*a*b^3 - 9*A*b^4)*x^4 + 70*(10*B*a^2*b^2 - 9
*A*a*b^3)*x^3 - 56*(10*B*a^3*b - 9*A*a^2*b^2)*x^2 + 48*(10*B*a^4 - 9*A*a^3*b)*x)
*sqrt(b*x + a)*sqrt(a))/(a^(11/2)*x^5), 1/1920*(105*(10*B*a*b^4 - 9*A*b^5)*x^5*a
rctan(a/(sqrt(b*x + a)*sqrt(-a))) - (384*A*a^4 - 105*(10*B*a*b^3 - 9*A*b^4)*x^4
+ 70*(10*B*a^2*b^2 - 9*A*a*b^3)*x^3 - 56*(10*B*a^3*b - 9*A*a^2*b^2)*x^2 + 48*(10
*B*a^4 - 9*A*a^3*b)*x)*sqrt(b*x + a)*sqrt(-a))/(sqrt(-a)*a^5*x^5)]

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/x**6/(b*x+a)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.212279, size = 281, normalized size = 1.59 \[ \frac{\frac{105 \,{\left (10 \, B a b^{5} - 9 \, A b^{6}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{5}} + \frac{1050 \,{\left (b x + a\right )}^{\frac{9}{2}} B a b^{5} - 4900 \,{\left (b x + a\right )}^{\frac{7}{2}} B a^{2} b^{5} + 8960 \,{\left (b x + a\right )}^{\frac{5}{2}} B a^{3} b^{5} - 7900 \,{\left (b x + a\right )}^{\frac{3}{2}} B a^{4} b^{5} + 2790 \, \sqrt{b x + a} B a^{5} b^{5} - 945 \,{\left (b x + a\right )}^{\frac{9}{2}} A b^{6} + 4410 \,{\left (b x + a\right )}^{\frac{7}{2}} A a b^{6} - 8064 \,{\left (b x + a\right )}^{\frac{5}{2}} A a^{2} b^{6} + 7110 \,{\left (b x + a\right )}^{\frac{3}{2}} A a^{3} b^{6} - 2895 \, \sqrt{b x + a} A a^{4} b^{6}}{a^{5} b^{5} x^{5}}}{1920 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(b*x + a)*x^6),x, algorithm="giac")

[Out]

1/1920*(105*(10*B*a*b^5 - 9*A*b^6)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^5)
+ (1050*(b*x + a)^(9/2)*B*a*b^5 - 4900*(b*x + a)^(7/2)*B*a^2*b^5 + 8960*(b*x +
a)^(5/2)*B*a^3*b^5 - 7900*(b*x + a)^(3/2)*B*a^4*b^5 + 2790*sqrt(b*x + a)*B*a^5*b
^5 - 945*(b*x + a)^(9/2)*A*b^6 + 4410*(b*x + a)^(7/2)*A*a*b^6 - 8064*(b*x + a)^(
5/2)*A*a^2*b^6 + 7110*(b*x + a)^(3/2)*A*a^3*b^6 - 2895*sqrt(b*x + a)*A*a^4*b^6)/
(a^5*b^5*x^5))/b