3.383 \(\int \frac{\sqrt{a+b x} (A+B x)}{x^4} \, dx\)

Optimal. Leaf size=112 \[ -\frac{b^2 (A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{5/2}}+\frac{b \sqrt{a+b x} (A b-2 a B)}{8 a^2 x}+\frac{\sqrt{a+b x} (A b-2 a B)}{4 a x^2}-\frac{A (a+b x)^{3/2}}{3 a x^3} \]

[Out]

((A*b - 2*a*B)*Sqrt[a + b*x])/(4*a*x^2) + (b*(A*b - 2*a*B)*Sqrt[a + b*x])/(8*a^2
*x) - (A*(a + b*x)^(3/2))/(3*a*x^3) - (b^2*(A*b - 2*a*B)*ArcTanh[Sqrt[a + b*x]/S
qrt[a]])/(8*a^(5/2))

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Rubi [A]  time = 0.155907, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ -\frac{b^2 (A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{5/2}}+\frac{b \sqrt{a+b x} (A b-2 a B)}{8 a^2 x}+\frac{\sqrt{a+b x} (A b-2 a B)}{4 a x^2}-\frac{A (a+b x)^{3/2}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[a + b*x]*(A + B*x))/x^4,x]

[Out]

((A*b - 2*a*B)*Sqrt[a + b*x])/(4*a*x^2) + (b*(A*b - 2*a*B)*Sqrt[a + b*x])/(8*a^2
*x) - (A*(a + b*x)^(3/2))/(3*a*x^3) - (b^2*(A*b - 2*a*B)*ArcTanh[Sqrt[a + b*x]/S
qrt[a]])/(8*a^(5/2))

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Rubi in Sympy [A]  time = 13.505, size = 99, normalized size = 0.88 \[ - \frac{A \left (a + b x\right )^{\frac{3}{2}}}{3 a x^{3}} + \frac{\sqrt{a + b x} \left (\frac{A b}{2} - B a\right )}{2 a x^{2}} + \frac{b \sqrt{a + b x} \left (A b - 2 B a\right )}{8 a^{2} x} - \frac{b^{2} \left (\frac{A b}{2} - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{4 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b*x+a)**(1/2)/x**4,x)

[Out]

-A*(a + b*x)**(3/2)/(3*a*x**3) + sqrt(a + b*x)*(A*b/2 - B*a)/(2*a*x**2) + b*sqrt
(a + b*x)*(A*b - 2*B*a)/(8*a**2*x) - b**2*(A*b/2 - B*a)*atanh(sqrt(a + b*x)/sqrt
(a))/(4*a**(5/2))

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Mathematica [A]  time = 0.165876, size = 91, normalized size = 0.81 \[ \frac{b^2 (2 a B-A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{5/2}}+\frac{\sqrt{a+b x} \left (-4 a^2 (2 A+3 B x)-2 a b x (A+3 B x)+3 A b^2 x^2\right )}{24 a^2 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[a + b*x]*(A + B*x))/x^4,x]

[Out]

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

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Maple [A]  time = 0.019, size = 91, normalized size = 0.8 \[ 2\,{b}^{2} \left ({\frac{1}{{x}^{3}{b}^{3}} \left ( 1/16\,{\frac{ \left ( Ab-2\,Ba \right ) \left ( bx+a \right ) ^{5/2}}{{a}^{2}}}-1/6\,{\frac{Ab \left ( bx+a \right ) ^{3/2}}{a}}+ \left ( -1/16\,Ab+1/8\,Ba \right ) \sqrt{bx+a} \right ) }-1/16\,{\frac{Ab-2\,Ba}{{a}^{5/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b*x+a)^(1/2)/x^4,x)

[Out]

2*b^2*((1/16*(A*b-2*B*a)/a^2*(b*x+a)^(5/2)-1/6*A*b/a*(b*x+a)^(3/2)+(-1/16*A*b+1/
8*B*a)*(b*x+a)^(1/2))/x^3/b^3-1/16*(A*b-2*B*a)/a^(5/2)*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^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221291, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (2 \, B a b^{2} - A b^{3}\right )} x^{3} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (8 \, A a^{2} + 3 \,{\left (2 \, B a b - A b^{2}\right )} x^{2} + 2 \,{\left (6 \, B a^{2} + A a b\right )} x\right )} \sqrt{b x + a} \sqrt{a}}{48 \, a^{\frac{5}{2}} x^{3}}, -\frac{3 \,{\left (2 \, B a b^{2} - A b^{3}\right )} x^{3} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) +{\left (8 \, A a^{2} + 3 \,{\left (2 \, B a b - A b^{2}\right )} x^{2} + 2 \,{\left (6 \, B a^{2} + A a b\right )} x\right )} \sqrt{b x + a} \sqrt{-a}}{24 \, \sqrt{-a} a^{2} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(b*x + a)/x^4,x, algorithm="fricas")

[Out]

[-1/48*(3*(2*B*a*b^2 - A*b^3)*x^3*log(((b*x + 2*a)*sqrt(a) - 2*sqrt(b*x + a)*a)/
x) + 2*(8*A*a^2 + 3*(2*B*a*b - A*b^2)*x^2 + 2*(6*B*a^2 + A*a*b)*x)*sqrt(b*x + a)
*sqrt(a))/(a^(5/2)*x^3), -1/24*(3*(2*B*a*b^2 - A*b^3)*x^3*arctan(a/(sqrt(b*x + a
)*sqrt(-a))) + (8*A*a^2 + 3*(2*B*a*b - A*b^2)*x^2 + 2*(6*B*a^2 + A*a*b)*x)*sqrt(
b*x + a)*sqrt(-a))/(sqrt(-a)*a^2*x^3)]

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Sympy [A]  time = 31.1645, size = 666, normalized size = 5.95 \[ - \frac{66 A a^{3} b^{3} \sqrt{a + b x}}{96 a^{6} + 144 a^{5} b x - 144 a^{4} \left (a + b x\right )^{2} + 48 a^{3} \left (a + b x\right )^{3}} + \frac{80 A a^{2} b^{3} \left (a + b x\right )^{\frac{3}{2}}}{96 a^{6} + 144 a^{5} b x - 144 a^{4} \left (a + b x\right )^{2} + 48 a^{3} \left (a + b x\right )^{3}} - \frac{30 A a b^{3} \left (a + b x\right )^{\frac{5}{2}}}{96 a^{6} + 144 a^{5} b x - 144 a^{4} \left (a + b x\right )^{2} + 48 a^{3} \left (a + b x\right )^{3}} - \frac{10 A a b^{3} \sqrt{a + b x}}{- 8 a^{4} - 16 a^{3} b x + 8 a^{2} \left (a + b x\right )^{2}} - \frac{5 A a b^{3} \sqrt{\frac{1}{a^{7}}} \log{\left (- a^{4} \sqrt{\frac{1}{a^{7}}} + \sqrt{a + b x} \right )}}{16} + \frac{5 A a b^{3} \sqrt{\frac{1}{a^{7}}} \log{\left (a^{4} \sqrt{\frac{1}{a^{7}}} + \sqrt{a + b x} \right )}}{16} + \frac{6 A b^{3} \left (a + b x\right )^{\frac{3}{2}}}{- 8 a^{4} - 16 a^{3} b x + 8 a^{2} \left (a + b x\right )^{2}} + \frac{3 A b^{3} \sqrt{\frac{1}{a^{5}}} \log{\left (- a^{3} \sqrt{\frac{1}{a^{5}}} + \sqrt{a + b x} \right )}}{8} - \frac{3 A b^{3} \sqrt{\frac{1}{a^{5}}} \log{\left (a^{3} \sqrt{\frac{1}{a^{5}}} + \sqrt{a + b x} \right )}}{8} - \frac{10 B a^{2} b^{2} \sqrt{a + b x}}{- 8 a^{4} - 16 a^{3} b x + 8 a^{2} \left (a + b x\right )^{2}} + \frac{6 B a b^{2} \left (a + b x\right )^{\frac{3}{2}}}{- 8 a^{4} - 16 a^{3} b x + 8 a^{2} \left (a + b x\right )^{2}} + \frac{3 B a b^{2} \sqrt{\frac{1}{a^{5}}} \log{\left (- a^{3} \sqrt{\frac{1}{a^{5}}} + \sqrt{a + b x} \right )}}{8} - \frac{3 B a b^{2} \sqrt{\frac{1}{a^{5}}} \log{\left (a^{3} \sqrt{\frac{1}{a^{5}}} + \sqrt{a + b x} \right )}}{8} - \frac{B b^{2} \sqrt{\frac{1}{a^{3}}} \log{\left (- a^{2} \sqrt{\frac{1}{a^{3}}} + \sqrt{a + b x} \right )}}{2} + \frac{B b^{2} \sqrt{\frac{1}{a^{3}}} \log{\left (a^{2} \sqrt{\frac{1}{a^{3}}} + \sqrt{a + b x} \right )}}{2} - \frac{B b \sqrt{a + b x}}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(b*x+a)**(1/2)/x**4,x)

[Out]

-66*A*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*A*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*A*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*A*a*
b**3*sqrt(a + b*x)/(-8*a**4 - 16*a**3*b*x + 8*a**2*(a + b*x)**2) - 5*A*a*b**3*sq
rt(a**(-7))*log(-a**4*sqrt(a**(-7)) + sqrt(a + b*x))/16 + 5*A*a*b**3*sqrt(a**(-7
))*log(a**4*sqrt(a**(-7)) + sqrt(a + b*x))/16 + 6*A*b**3*(a + b*x)**(3/2)/(-8*a*
*4 - 16*a**3*b*x + 8*a**2*(a + b*x)**2) + 3*A*b**3*sqrt(a**(-5))*log(-a**3*sqrt(
a**(-5)) + sqrt(a + b*x))/8 - 3*A*b**3*sqrt(a**(-5))*log(a**3*sqrt(a**(-5)) + sq
rt(a + b*x))/8 - 10*B*a**2*b**2*sqrt(a + b*x)/(-8*a**4 - 16*a**3*b*x + 8*a**2*(a
+ b*x)**2) + 6*B*a*b**2*(a + b*x)**(3/2)/(-8*a**4 - 16*a**3*b*x + 8*a**2*(a + b
*x)**2) + 3*B*a*b**2*sqrt(a**(-5))*log(-a**3*sqrt(a**(-5)) + sqrt(a + b*x))/8 -
3*B*a*b**2*sqrt(a**(-5))*log(a**3*sqrt(a**(-5)) + sqrt(a + b*x))/8 - B*b**2*sqrt
(a**(-3))*log(-a**2*sqrt(a**(-3)) + sqrt(a + b*x))/2 + B*b**2*sqrt(a**(-3))*log(
a**2*sqrt(a**(-3)) + sqrt(a + b*x))/2 - B*b*sqrt(a + b*x)/(a*x)

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GIAC/XCAS [A]  time = 0.227746, size = 173, normalized size = 1.54 \[ -\frac{\frac{3 \,{\left (2 \, B a b^{3} - A b^{4}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{6 \,{\left (b x + a\right )}^{\frac{5}{2}} B a b^{3} - 6 \, \sqrt{b x + a} B a^{3} b^{3} - 3 \,{\left (b x + a\right )}^{\frac{5}{2}} A b^{4} + 8 \,{\left (b x + a\right )}^{\frac{3}{2}} A a b^{4} + 3 \, \sqrt{b x + a} A a^{2} b^{4}}{a^{2} b^{3} x^{3}}}{24 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(b*x + a)/x^4,x, algorithm="giac")

[Out]

-1/24*(3*(2*B*a*b^3 - A*b^4)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^2) + (6*
(b*x + a)^(5/2)*B*a*b^3 - 6*sqrt(b*x + a)*B*a^3*b^3 - 3*(b*x + a)^(5/2)*A*b^4 +
8*(b*x + a)^(3/2)*A*a*b^4 + 3*sqrt(b*x + a)*A*a^2*b^4)/(a^2*b^3*x^3))/b