3.382 \(\int \frac{\sqrt{a+b x} (A+B x)}{x^3} \, dx\)

Optimal. Leaf size=82 \[ \frac{b (A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{3/2}}+\frac{\sqrt{a+b x} (A b-4 a B)}{4 a x}-\frac{A (a+b x)^{3/2}}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.114176, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{b (A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{3/2}}+\frac{\sqrt{a+b x} (A b-4 a B)}{4 a x}-\frac{A (a+b x)^{3/2}}{2 a x^2} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[a + b*x]*(A + B*x))/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.82648, size = 70, normalized size = 0.85 \[ - \frac{A \left (a + b x\right )^{\frac{3}{2}}}{2 a x^{2}} + \frac{\sqrt{a + b x} \left (A b - 4 B a\right )}{4 a x} + \frac{b \left (A b - 4 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b*x+a)**(1/2)/x**3,x)

[Out]

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

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Mathematica [A]  time = 0.0949322, size = 68, normalized size = 0.83 \[ \frac{b (A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{\sqrt{a+b x} (2 a (A+2 B x)+A b x)}{4 a x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[a + b*x]*(A + B*x))/x^3,x]

[Out]

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

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Maple [A]  time = 0.02, size = 75, normalized size = 0.9 \[ 2\,b \left ({\frac{1}{{b}^{2}{x}^{2}} \left ( -1/8\,{\frac{ \left ( Ab+4\,Ba \right ) \left ( bx+a \right ) ^{3/2}}{a}}+ \left ( 1/2\,Ba-1/8\,Ab \right ) \sqrt{bx+a} \right ) }+1/8\,{\frac{Ab-4\,Ba}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b*x+a)^(1/2)/x^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220675, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (4 \, B a b - A b^{2}\right )} x^{2} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (2 \, A a +{\left (4 \, B a + A b\right )} x\right )} \sqrt{b x + a} \sqrt{a}}{8 \, a^{\frac{3}{2}} x^{2}}, \frac{{\left (4 \, B a b - A b^{2}\right )} x^{2} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) -{\left (2 \, A a +{\left (4 \, B a + A b\right )} x\right )} \sqrt{b x + a} \sqrt{-a}}{4 \, \sqrt{-a} a x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(b*x + a)/x^3,x, algorithm="fricas")

[Out]

[-1/8*((4*B*a*b - A*b^2)*x^2*log(((b*x + 2*a)*sqrt(a) + 2*sqrt(b*x + a)*a)/x) +
2*(2*A*a + (4*B*a + A*b)*x)*sqrt(b*x + a)*sqrt(a))/(a^(3/2)*x^2), 1/4*((4*B*a*b
- A*b^2)*x^2*arctan(a/(sqrt(b*x + a)*sqrt(-a))) - (2*A*a + (4*B*a + A*b)*x)*sqrt
(b*x + a)*sqrt(-a))/(sqrt(-a)*a*x^2)]

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Sympy [A]  time = 26.8639, size = 428, normalized size = 5.22 \[ - \frac{10 A a^{2} b^{2} \sqrt{a + b x}}{- 8 a^{4} - 16 a^{3} b x + 8 a^{2} \left (a + b x\right )^{2}} + \frac{6 A a b^{2} \left (a + b x\right )^{\frac{3}{2}}}{- 8 a^{4} - 16 a^{3} b x + 8 a^{2} \left (a + b x\right )^{2}} + \frac{3 A a b^{2} \sqrt{\frac{1}{a^{5}}} \log{\left (- a^{3} \sqrt{\frac{1}{a^{5}}} + \sqrt{a + b x} \right )}}{8} - \frac{3 A a b^{2} \sqrt{\frac{1}{a^{5}}} \log{\left (a^{3} \sqrt{\frac{1}{a^{5}}} + \sqrt{a + b x} \right )}}{8} - \frac{A b^{2} \sqrt{\frac{1}{a^{3}}} \log{\left (- a^{2} \sqrt{\frac{1}{a^{3}}} + \sqrt{a + b x} \right )}}{2} + \frac{A b^{2} \sqrt{\frac{1}{a^{3}}} \log{\left (a^{2} \sqrt{\frac{1}{a^{3}}} + \sqrt{a + b x} \right )}}{2} - \frac{A b \sqrt{a + b x}}{a x} - \frac{B a b \sqrt{\frac{1}{a^{3}}} \log{\left (- a^{2} \sqrt{\frac{1}{a^{3}}} + \sqrt{a + b x} \right )}}{2} + \frac{B a b \sqrt{\frac{1}{a^{3}}} \log{\left (a^{2} \sqrt{\frac{1}{a^{3}}} + \sqrt{a + b x} \right )}}{2} - 2 B b \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{a + b x}}{\sqrt{- a}} \right )}}{\sqrt{- a}} & \text{for}\: - a > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: - a < 0 \wedge a < a + b x \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: a > a + b x \wedge - a < 0 \end{cases}\right ) - \frac{B \sqrt{a + b x}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(b*x+a)**(1/2)/x**3,x)

[Out]

-10*A*a**2*b**2*sqrt(a + b*x)/(-8*a**4 - 16*a**3*b*x + 8*a**2*(a + b*x)**2) + 6*
A*a*b**2*(a + b*x)**(3/2)/(-8*a**4 - 16*a**3*b*x + 8*a**2*(a + b*x)**2) + 3*A*a*
b**2*sqrt(a**(-5))*log(-a**3*sqrt(a**(-5)) + sqrt(a + b*x))/8 - 3*A*a*b**2*sqrt(
a**(-5))*log(a**3*sqrt(a**(-5)) + sqrt(a + b*x))/8 - A*b**2*sqrt(a**(-3))*log(-a
**2*sqrt(a**(-3)) + sqrt(a + b*x))/2 + A*b**2*sqrt(a**(-3))*log(a**2*sqrt(a**(-3
)) + sqrt(a + b*x))/2 - A*b*sqrt(a + b*x)/(a*x) - B*a*b*sqrt(a**(-3))*log(-a**2*
sqrt(a**(-3)) + sqrt(a + b*x))/2 + B*a*b*sqrt(a**(-3))*log(a**2*sqrt(a**(-3)) +
sqrt(a + b*x))/2 - 2*B*b*Piecewise((-atan(sqrt(a + b*x)/sqrt(-a))/sqrt(-a), -a >
0), (acoth(sqrt(a + b*x)/sqrt(a))/sqrt(a), (-a < 0) & (a < a + b*x)), (atanh(sq
rt(a + b*x)/sqrt(a))/sqrt(a), (-a < 0) & (a > a + b*x))) - B*sqrt(a + b*x)/x

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GIAC/XCAS [A]  time = 0.270893, size = 149, normalized size = 1.82 \[ \frac{\frac{{\left (4 \, B a b^{2} - A b^{3}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} - \frac{4 \,{\left (b x + a\right )}^{\frac{3}{2}} B a b^{2} - 4 \, \sqrt{b x + a} B a^{2} b^{2} +{\left (b x + a\right )}^{\frac{3}{2}} A b^{3} + \sqrt{b x + a} A a b^{3}}{a b^{2} x^{2}}}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(b*x + a)/x^3,x, algorithm="giac")

[Out]

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