3.384 \(\int \frac{\sqrt{a+b x} (A+B x)}{x^5} \, dx\)

Optimal. Leaf size=146 \[ \frac{b^3 (5 A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{64 a^{7/2}}-\frac{b^2 \sqrt{a+b x} (5 A b-8 a B)}{64 a^3 x}+\frac{b \sqrt{a+b x} (5 A b-8 a B)}{96 a^2 x^2}+\frac{\sqrt{a+b x} (5 A b-8 a B)}{24 a x^3}-\frac{A (a+b x)^{3/2}}{4 a x^4} \]

[Out]

((5*A*b - 8*a*B)*Sqrt[a + b*x])/(24*a*x^3) + (b*(5*A*b - 8*a*B)*Sqrt[a + b*x])/(
96*a^2*x^2) - (b^2*(5*A*b - 8*a*B)*Sqrt[a + b*x])/(64*a^3*x) - (A*(a + b*x)^(3/2
))/(4*a*x^4) + (b^3*(5*A*b - 8*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(64*a^(7/2))

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Rubi [A]  time = 0.198887, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ \frac{b^3 (5 A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{64 a^{7/2}}-\frac{b^2 \sqrt{a+b x} (5 A b-8 a B)}{64 a^3 x}+\frac{b \sqrt{a+b x} (5 A b-8 a B)}{96 a^2 x^2}+\frac{\sqrt{a+b x} (5 A b-8 a B)}{24 a x^3}-\frac{A (a+b x)^{3/2}}{4 a x^4} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[a + b*x]*(A + B*x))/x^5,x]

[Out]

((5*A*b - 8*a*B)*Sqrt[a + b*x])/(24*a*x^3) + (b*(5*A*b - 8*a*B)*Sqrt[a + b*x])/(
96*a^2*x^2) - (b^2*(5*A*b - 8*a*B)*Sqrt[a + b*x])/(64*a^3*x) - (A*(a + b*x)^(3/2
))/(4*a*x^4) + (b^3*(5*A*b - 8*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(64*a^(7/2))

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Rubi in Sympy [A]  time = 17.6599, size = 134, normalized size = 0.92 \[ - \frac{A \left (a + b x\right )^{\frac{3}{2}}}{4 a x^{4}} + \frac{\sqrt{a + b x} \left (5 A b - 8 B a\right )}{24 a x^{3}} + \frac{b \sqrt{a + b x} \left (5 A b - 8 B a\right )}{96 a^{2} x^{2}} - \frac{b^{2} \sqrt{a + b x} \left (5 A b - 8 B a\right )}{64 a^{3} x} + \frac{b^{3} \left (5 A b - 8 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{64 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b*x+a)**(1/2)/x**5,x)

[Out]

-A*(a + b*x)**(3/2)/(4*a*x**4) + sqrt(a + b*x)*(5*A*b - 8*B*a)/(24*a*x**3) + b*s
qrt(a + b*x)*(5*A*b - 8*B*a)/(96*a**2*x**2) - b**2*sqrt(a + b*x)*(5*A*b - 8*B*a)
/(64*a**3*x) + b**3*(5*A*b - 8*B*a)*atanh(sqrt(a + b*x)/sqrt(a))/(64*a**(7/2))

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Mathematica [A]  time = 0.17541, size = 110, normalized size = 0.75 \[ \frac{b^3 (5 A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{64 a^{7/2}}-\frac{\sqrt{a+b x} \left (16 a^3 (3 A+4 B x)+8 a^2 b x (A+2 B x)-2 a b^2 x^2 (5 A+12 B x)+15 A b^3 x^3\right )}{192 a^3 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[a + b*x]*(A + B*x))/x^5,x]

[Out]

-(Sqrt[a + b*x]*(15*A*b^3*x^3 + 8*a^2*b*x*(A + 2*B*x) + 16*a^3*(3*A + 4*B*x) - 2
*a*b^2*x^2*(5*A + 12*B*x)))/(192*a^3*x^4) + (b^3*(5*A*b - 8*a*B)*ArcTanh[Sqrt[a
+ b*x]/Sqrt[a]])/(64*a^(7/2))

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Maple [A]  time = 0.018, size = 121, normalized size = 0.8 \[ 2\,{b}^{3} \left ({\frac{1}{{x}^{4}{b}^{4}} \left ( -{\frac{ \left ( 5\,Ab-8\,Ba \right ) \left ( bx+a \right ) ^{7/2}}{128\,{a}^{3}}}+{\frac{ \left ( 55\,Ab-88\,Ba \right ) \left ( bx+a \right ) ^{5/2}}{384\,{a}^{2}}}-{\frac{ \left ( 73\,Ab-40\,Ba \right ) \left ( bx+a \right ) ^{3/2}}{384\,a}}+ \left ( -{\frac{5\,Ab}{128}}+1/16\,Ba \right ) \sqrt{bx+a} \right ) }+{\frac{5\,Ab-8\,Ba}{128\,{a}^{7/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b*x+a)^(1/2)/x^5,x)

[Out]

2*b^3*((-1/128*(5*A*b-8*B*a)/a^3*(b*x+a)^(7/2)+11/384/a^2*(5*A*b-8*B*a)*(b*x+a)^
(5/2)-1/384*(73*A*b-40*B*a)/a*(b*x+a)^(3/2)+(-5/128*A*b+1/16*B*a)*(b*x+a)^(1/2))
/x^4/b^4+1/128*(5*A*b-8*B*a)/a^(7/2)*arctanh((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^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224252, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (8 \, B a b^{3} - 5 \, A b^{4}\right )} x^{4} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (48 \, A a^{3} - 3 \,{\left (8 \, B a b^{2} - 5 \, A b^{3}\right )} x^{3} + 2 \,{\left (8 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{2} + 8 \,{\left (8 \, B a^{3} + A a^{2} b\right )} x\right )} \sqrt{b x + a} \sqrt{a}}{384 \, a^{\frac{7}{2}} x^{4}}, \frac{3 \,{\left (8 \, B a b^{3} - 5 \, A b^{4}\right )} x^{4} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) -{\left (48 \, A a^{3} - 3 \,{\left (8 \, B a b^{2} - 5 \, A b^{3}\right )} x^{3} + 2 \,{\left (8 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{2} + 8 \,{\left (8 \, B a^{3} + A a^{2} b\right )} x\right )} \sqrt{b x + a} \sqrt{-a}}{192 \, \sqrt{-a} a^{3} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(b*x + a)/x^5,x, algorithm="fricas")

[Out]

[-1/384*(3*(8*B*a*b^3 - 5*A*b^4)*x^4*log(((b*x + 2*a)*sqrt(a) + 2*sqrt(b*x + a)*
a)/x) + 2*(48*A*a^3 - 3*(8*B*a*b^2 - 5*A*b^3)*x^3 + 2*(8*B*a^2*b - 5*A*a*b^2)*x^
2 + 8*(8*B*a^3 + A*a^2*b)*x)*sqrt(b*x + a)*sqrt(a))/(a^(7/2)*x^4), 1/192*(3*(8*B
*a*b^3 - 5*A*b^4)*x^4*arctan(a/(sqrt(b*x + a)*sqrt(-a))) - (48*A*a^3 - 3*(8*B*a*
b^2 - 5*A*b^3)*x^3 + 2*(8*B*a^2*b - 5*A*a*b^2)*x^2 + 8*(8*B*a^3 + A*a^2*b)*x)*sq
rt(b*x + a)*sqrt(-a))/(sqrt(-a)*a^3*x^4)]

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Sympy [A]  time = 58.3618, size = 1001, normalized size = 6.86 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(b*x+a)**(1/2)/x**5,x)

[Out]

-558*A*a**4*b**4*sqrt(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*A*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*A*a**2*b**4*(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) - 66*A*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*A*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*A*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*A*a*b**4*sqrt(a**(-9))*log(-a**5*sqrt(a**(-
9)) + sqrt(a + b*x))/128 - 35*A*a*b**4*sqrt(a**(-9))*log(a**5*sqrt(a**(-9)) + sq
rt(a + b*x))/128 - 30*A*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*A*b**4*sqrt(a**(-7))*log(-a**4*sqrt(a*
*(-7)) + sqrt(a + b*x))/16 + 5*A*b**4*sqrt(a**(-7))*log(a**4*sqrt(a**(-7)) + sqr
t(a + b*x))/16 - 66*B*a**3*b**3*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) + 80*B*a**2*b**3*(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) - 30*B*a*b**3*(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) - 10*B*a*b**3*sqrt(a + b*x)/(-8*a**4 - 16*a**3*b*x + 8*a**2*(a + b*x)**2)
- 5*B*a*b**3*sqrt(a**(-7))*log(-a**4*sqrt(a**(-7)) + sqrt(a + b*x))/16 + 5*B*a*
b**3*sqrt(a**(-7))*log(a**4*sqrt(a**(-7)) + sqrt(a + b*x))/16 + 6*B*b**3*(a + b*
x)**(3/2)/(-8*a**4 - 16*a**3*b*x + 8*a**2*(a + b*x)**2) + 3*B*b**3*sqrt(a**(-5))
*log(-a**3*sqrt(a**(-5)) + sqrt(a + b*x))/8 - 3*B*b**3*sqrt(a**(-5))*log(a**3*sq
rt(a**(-5)) + sqrt(a + b*x))/8

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GIAC/XCAS [A]  time = 0.237769, size = 238, normalized size = 1.63 \[ \frac{\frac{3 \,{\left (8 \, B a b^{4} - 5 \, A b^{5}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} + \frac{24 \,{\left (b x + a\right )}^{\frac{7}{2}} B a b^{4} - 88 \,{\left (b x + a\right )}^{\frac{5}{2}} B a^{2} b^{4} + 40 \,{\left (b x + a\right )}^{\frac{3}{2}} B a^{3} b^{4} + 24 \, \sqrt{b x + a} B a^{4} b^{4} - 15 \,{\left (b x + a\right )}^{\frac{7}{2}} A b^{5} + 55 \,{\left (b x + a\right )}^{\frac{5}{2}} A a b^{5} - 73 \,{\left (b x + a\right )}^{\frac{3}{2}} A a^{2} b^{5} - 15 \, \sqrt{b x + a} A a^{3} b^{5}}{a^{3} b^{4} x^{4}}}{192 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(b*x + a)/x^5,x, algorithm="giac")

[Out]

1/192*(3*(8*B*a*b^4 - 5*A*b^5)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^3) + (
24*(b*x + a)^(7/2)*B*a*b^4 - 88*(b*x + a)^(5/2)*B*a^2*b^4 + 40*(b*x + a)^(3/2)*B
*a^3*b^4 + 24*sqrt(b*x + a)*B*a^4*b^4 - 15*(b*x + a)^(7/2)*A*b^5 + 55*(b*x + a)^
(5/2)*A*a*b^5 - 73*(b*x + a)^(3/2)*A*a^2*b^5 - 15*sqrt(b*x + a)*A*a^3*b^5)/(a^3*
b^4*x^4))/b