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

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

[Out]

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

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Rubi [A]  time = 0.242388, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ -\frac{b^4 (7 A b-10 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{128 a^{9/2}}+\frac{b^3 \sqrt{a+b x} (7 A b-10 a B)}{128 a^4 x}-\frac{b^2 \sqrt{a+b x} (7 A b-10 a B)}{192 a^3 x^2}+\frac{b \sqrt{a+b x} (7 A b-10 a B)}{240 a^2 x^3}+\frac{\sqrt{a+b x} (7 A b-10 a B)}{40 a x^4}-\frac{A (a+b x)^{3/2}}{5 a x^5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.7244, size = 165, normalized size = 0.93 \[ - \frac{A \left (a + b x\right )^{\frac{3}{2}}}{5 a x^{5}} + \frac{\sqrt{a + b x} \left (7 A b - 10 B a\right )}{40 a x^{4}} + \frac{b \sqrt{a + b x} \left (7 A b - 10 B a\right )}{240 a^{2} x^{3}} - \frac{b^{2} \sqrt{a + b x} \left (7 A b - 10 B a\right )}{192 a^{3} x^{2}} + \frac{b^{3} \sqrt{a + b x} \left (7 A b - 10 B a\right )}{128 a^{4} x} - \frac{b^{4} \left (7 A b - 10 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{128 a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.26069, size = 132, normalized size = 0.75 \[ \frac{\frac{\sqrt{a} \sqrt{a+b x} \left (-96 a^4 (4 A+5 B x)-16 a^3 b x (3 A+5 B x)+4 a^2 b^2 x^2 (14 A+25 B x)-10 a b^3 x^3 (7 A+15 B x)+105 A b^4 x^4\right )}{x^5}-15 b^4 (7 A b-10 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{1920 a^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[a]*Sqrt[a + b*x]*(105*A*b^4*x^4 - 16*a^3*b*x*(3*A + 5*B*x) - 96*a^4*(4*A
+ 5*B*x) - 10*a*b^3*x^3*(7*A + 15*B*x) + 4*a^2*b^2*x^2*(14*A + 25*B*x)))/x^5 - 1
5*b^4*(7*A*b - 10*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(1920*a^(9/2))

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Maple [A]  time = 0.022, size = 142, normalized size = 0.8 \[ 2\,{b}^{4} \left ({\frac{1}{{b}^{5}{x}^{5}} \left ({\frac{ \left ( 7\,Ab-10\,Ba \right ) \left ( bx+a \right ) ^{9/2}}{256\,{a}^{4}}}-{\frac{ \left ( 49\,Ab-70\,Ba \right ) \left ( bx+a \right ) ^{7/2}}{384\,{a}^{3}}}+1/30\,{\frac{ \left ( 7\,Ab-10\,Ba \right ) \left ( bx+a \right ) ^{5/2}}{{a}^{2}}}-{\frac{ \left ( 79\,Ab-58\,Ba \right ) \left ( bx+a \right ) ^{3/2}}{384\,a}}+ \left ( -{\frac{7\,Ab}{256}}+{\frac{5\,Ba}{128}} \right ) \sqrt{bx+a} \right ) }-{\frac{7\,Ab-10\,Ba}{256\,{a}^{9/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b^4*((1/256*(7*A*b-10*B*a)/a^4*(b*x+a)^(9/2)-7/384/a^3*(7*A*b-10*B*a)*(b*x+a)^
(7/2)+1/30/a^2*(7*A*b-10*B*a)*(b*x+a)^(5/2)-1/384*(79*A*b-58*B*a)/a*(b*x+a)^(3/2
)+(-7/256*A*b+5/128*B*a)*(b*x+a)^(1/2))/x^5/b^5-1/256*(7*A*b-10*B*a)/a^(9/2)*arc
tanh((b*x+a)^(1/2)/a^(1/2)))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222426, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (10 \, B a b^{4} - 7 \, A b^{5}\right )} x^{5} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (384 \, A a^{4} + 15 \,{\left (10 \, B a b^{3} - 7 \, A b^{4}\right )} x^{4} - 10 \,{\left (10 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{3} + 8 \,{\left (10 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{2} + 48 \,{\left (10 \, B a^{4} + A a^{3} b\right )} x\right )} \sqrt{b x + a} \sqrt{a}}{3840 \, a^{\frac{9}{2}} x^{5}}, -\frac{15 \,{\left (10 \, B a b^{4} - 7 \, A b^{5}\right )} x^{5} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) +{\left (384 \, A a^{4} + 15 \,{\left (10 \, B a b^{3} - 7 \, A b^{4}\right )} x^{4} - 10 \,{\left (10 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{3} + 8 \,{\left (10 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{2} + 48 \,{\left (10 \, B a^{4} + A a^{3} b\right )} x\right )} \sqrt{b x + a} \sqrt{-a}}{1920 \, \sqrt{-a} a^{4} x^{5}}\right ] \]

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*(15*(10*B*a*b^4 - 7*A*b^5)*x^5*log(((b*x + 2*a)*sqrt(a) - 2*sqrt(b*x +
a)*a)/x) + 2*(384*A*a^4 + 15*(10*B*a*b^3 - 7*A*b^4)*x^4 - 10*(10*B*a^2*b^2 - 7*A
*a*b^3)*x^3 + 8*(10*B*a^3*b - 7*A*a^2*b^2)*x^2 + 48*(10*B*a^4 + A*a^3*b)*x)*sqrt
(b*x + a)*sqrt(a))/(a^(9/2)*x^5), -1/1920*(15*(10*B*a*b^4 - 7*A*b^5)*x^5*arctan(
a/(sqrt(b*x + a)*sqrt(-a))) + (384*A*a^4 + 15*(10*B*a*b^3 - 7*A*b^4)*x^4 - 10*(1
0*B*a^2*b^2 - 7*A*a*b^3)*x^3 + 8*(10*B*a^3*b - 7*A*a^2*b^2)*x^2 + 48*(10*B*a^4 +
A*a^3*b)*x)*sqrt(b*x + a)*sqrt(-a))/(sqrt(-a)*a^4*x^5)]

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Sympy [A]  time = 76.3841, size = 1416, normalized size = 8. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1930*A*a**5*b**5*sqrt(a + b*x)/(5120*a**10 + 6400*a**9*b*x - 12800*a**8*(a + b*
x)**2 + 12800*a**7*(a + b*x)**3 - 6400*a**6*(a + b*x)**4 + 1280*a**5*(a + b*x)**
5) + 4740*A*a**4*b**5*(a + b*x)**(3/2)/(5120*a**10 + 6400*a**9*b*x - 12800*a**8*
(a + b*x)**2 + 12800*a**7*(a + b*x)**3 - 6400*a**6*(a + b*x)**4 + 1280*a**5*(a +
b*x)**5) - 5376*A*a**3*b**5*(a + b*x)**(5/2)/(5120*a**10 + 6400*a**9*b*x - 1280
0*a**8*(a + b*x)**2 + 12800*a**7*(a + b*x)**3 - 6400*a**6*(a + b*x)**4 + 1280*a*
*5*(a + b*x)**5) - 558*A*a**3*b**5*sqrt(a + b*x)/(-1152*a**8 - 1536*a**7*b*x + 2
304*a**6*(a + b*x)**2 - 1536*a**5*(a + b*x)**3 + 384*a**4*(a + b*x)**4) + 2940*A
*a**2*b**5*(a + b*x)**(7/2)/(5120*a**10 + 6400*a**9*b*x - 12800*a**8*(a + b*x)**
2 + 12800*a**7*(a + b*x)**3 - 6400*a**6*(a + b*x)**4 + 1280*a**5*(a + b*x)**5) +
1022*A*a**2*b**5*(a + b*x)**(3/2)/(-1152*a**8 - 1536*a**7*b*x + 2304*a**6*(a +
b*x)**2 - 1536*a**5*(a + b*x)**3 + 384*a**4*(a + b*x)**4) - 630*A*a*b**5*(a + b*
x)**(9/2)/(5120*a**10 + 6400*a**9*b*x - 12800*a**8*(a + b*x)**2 + 12800*a**7*(a
+ b*x)**3 - 6400*a**6*(a + b*x)**4 + 1280*a**5*(a + b*x)**5) - 770*A*a*b**5*(a +
b*x)**(5/2)/(-1152*a**8 - 1536*a**7*b*x + 2304*a**6*(a + b*x)**2 - 1536*a**5*(a
+ b*x)**3 + 384*a**4*(a + b*x)**4) - 63*A*a*b**5*sqrt(a**(-11))*log(-a**6*sqrt(
a**(-11)) + sqrt(a + b*x))/256 + 63*A*a*b**5*sqrt(a**(-11))*log(a**6*sqrt(a**(-1
1)) + sqrt(a + b*x))/256 + 210*A*b**5*(a + b*x)**(7/2)/(-1152*a**8 - 1536*a**7*b
*x + 2304*a**6*(a + b*x)**2 - 1536*a**5*(a + b*x)**3 + 384*a**4*(a + b*x)**4) +
35*A*b**5*sqrt(a**(-9))*log(-a**5*sqrt(a**(-9)) + sqrt(a + b*x))/128 - 35*A*b**5
*sqrt(a**(-9))*log(a**5*sqrt(a**(-9)) + sqrt(a + b*x))/128 - 558*B*a**4*b**4*sqr
t(a + b*x)/(-1152*a**8 - 1536*a**7*b*x + 2304*a**6*(a + b*x)**2 - 1536*a**5*(a +
b*x)**3 + 384*a**4*(a + b*x)**4) + 1022*B*a**3*b**4*(a + b*x)**(3/2)/(-1152*a**
8 - 1536*a**7*b*x + 2304*a**6*(a + b*x)**2 - 1536*a**5*(a + b*x)**3 + 384*a**4*(
a + b*x)**4) - 770*B*a**2*b**4*(a + b*x)**(5/2)/(-1152*a**8 - 1536*a**7*b*x + 23
04*a**6*(a + b*x)**2 - 1536*a**5*(a + b*x)**3 + 384*a**4*(a + b*x)**4) - 66*B*a*
*2*b**4*sqrt(a + b*x)/(96*a**6 + 144*a**5*b*x - 144*a**4*(a + b*x)**2 + 48*a**3*
(a + b*x)**3) + 210*B*a*b**4*(a + b*x)**(7/2)/(-1152*a**8 - 1536*a**7*b*x + 2304
*a**6*(a + b*x)**2 - 1536*a**5*(a + b*x)**3 + 384*a**4*(a + b*x)**4) + 80*B*a*b*
*4*(a + b*x)**(3/2)/(96*a**6 + 144*a**5*b*x - 144*a**4*(a + b*x)**2 + 48*a**3*(a
+ b*x)**3) + 35*B*a*b**4*sqrt(a**(-9))*log(-a**5*sqrt(a**(-9)) + sqrt(a + b*x))
/128 - 35*B*a*b**4*sqrt(a**(-9))*log(a**5*sqrt(a**(-9)) + sqrt(a + b*x))/128 - 3
0*B*b**4*(a + b*x)**(5/2)/(96*a**6 + 144*a**5*b*x - 144*a**4*(a + b*x)**2 + 48*a
**3*(a + b*x)**3) - 5*B*b**4*sqrt(a**(-7))*log(-a**4*sqrt(a**(-7)) + sqrt(a + b*
x))/16 + 5*B*b**4*sqrt(a**(-7))*log(a**4*sqrt(a**(-7)) + sqrt(a + b*x))/16

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GIAC/XCAS [A]  time = 0.214152, size = 281, normalized size = 1.59 \[ -\frac{\frac{15 \,{\left (10 \, B a b^{5} - 7 \, A b^{6}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{4}} + \frac{150 \,{\left (b x + a\right )}^{\frac{9}{2}} B a b^{5} - 700 \,{\left (b x + a\right )}^{\frac{7}{2}} B a^{2} b^{5} + 1280 \,{\left (b x + a\right )}^{\frac{5}{2}} B a^{3} b^{5} - 580 \,{\left (b x + a\right )}^{\frac{3}{2}} B a^{4} b^{5} - 150 \, \sqrt{b x + a} B a^{5} b^{5} - 105 \,{\left (b x + a\right )}^{\frac{9}{2}} A b^{6} + 490 \,{\left (b x + a\right )}^{\frac{7}{2}} A a b^{6} - 896 \,{\left (b x + a\right )}^{\frac{5}{2}} A a^{2} b^{6} + 790 \,{\left (b x + a\right )}^{\frac{3}{2}} A a^{3} b^{6} + 105 \, \sqrt{b x + a} A a^{4} b^{6}}{a^{4} b^{5} x^{5}}}{1920 \, b} \]

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*(15*(10*B*a*b^5 - 7*A*b^6)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^4)
+ (150*(b*x + a)^(9/2)*B*a*b^5 - 700*(b*x + a)^(7/2)*B*a^2*b^5 + 1280*(b*x + a)
^(5/2)*B*a^3*b^5 - 580*(b*x + a)^(3/2)*B*a^4*b^5 - 150*sqrt(b*x + a)*B*a^5*b^5 -
105*(b*x + a)^(9/2)*A*b^6 + 490*(b*x + a)^(7/2)*A*a*b^6 - 896*(b*x + a)^(5/2)*A
*a^2*b^6 + 790*(b*x + a)^(3/2)*A*a^3*b^6 + 105*sqrt(b*x + a)*A*a^4*b^6)/(a^4*b^5
*x^5))/b