3.393 \(\int \frac{(a+b x)^{3/2} (A+B x)}{x^3} \, dx\)

Optimal. Leaf size=107 \[ -\frac{(a+b x)^{3/2} (4 a B+A b)}{4 a x}+\frac{3 b \sqrt{a+b x} (4 a B+A b)}{4 a}-\frac{3 b (4 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 \sqrt{a}}-\frac{A (a+b x)^{5/2}}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.138254, antiderivative size = 107, 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 \[ -\frac{(a+b x)^{3/2} (4 a B+A b)}{4 a x}+\frac{3 b \sqrt{a+b x} (4 a B+A b)}{4 a}-\frac{3 b (4 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 \sqrt{a}}-\frac{A (a+b x)^{5/2}}{2 a x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.7369, size = 95, normalized size = 0.89 \[ - \frac{A \left (a + b x\right )^{\frac{5}{2}}}{2 a x^{2}} + \frac{3 b \sqrt{a + b x} \left (A b + 4 B a\right )}{4 a} - \frac{\left (a + b x\right )^{\frac{3}{2}} \left (A b + 4 B a\right )}{4 a x} - \frac{3 b \left (A b + 4 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{4 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.145586, size = 72, normalized size = 0.67 \[ -\frac{\sqrt{a+b x} (2 a (A+2 B x)+b x (5 A-8 B x))}{4 x^2}-\frac{3 b (4 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 \sqrt{a}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.019, size = 84, normalized size = 0.8 \[ 2\,b \left ( B\sqrt{bx+a}+{\frac{ \left ( -5/8\,Ab-1/2\,Ba \right ) \left ( bx+a \right ) ^{3/2}+ \left ( 1/2\,B{a}^{2}+3/8\,Aab \right ) \sqrt{bx+a}}{{b}^{2}{x}^{2}}}-3/8\,{\frac{Ab+4\,Ba}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232996, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (4 \, B a b + A b^{2}\right )} x^{2} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (8 \, B b x^{2} - 2 \, A a -{\left (4 \, B a + 5 \, A b\right )} x\right )} \sqrt{b x + a} \sqrt{a}}{8 \, \sqrt{a} x^{2}}, \frac{3 \,{\left (4 \, B a b + A b^{2}\right )} x^{2} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) +{\left (8 \, B b x^{2} - 2 \, A a -{\left (4 \, B a + 5 \, A b\right )} x\right )} \sqrt{b x + a} \sqrt{-a}}{4 \, \sqrt{-a} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(3/2)/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 49.6004, size = 541, normalized size = 5.06 \[ - \frac{10 A a^{3} b^{2} \sqrt{a + b x}}{- 8 a^{4} - 16 a^{3} b x + 8 a^{2} \left (a + b x\right )^{2}} + \frac{6 A a^{2} b^{2} \left (a + b x\right )^{\frac{3}{2}}}{- 8 a^{4} - 16 a^{3} b x + 8 a^{2} \left (a + b x\right )^{2}} + \frac{3 A a^{2} b^{2} \sqrt{\frac{1}{a^{5}}} \log{\left (- a^{3} \sqrt{\frac{1}{a^{5}}} + \sqrt{a + b x} \right )}}{8} - \frac{3 A a^{2} b^{2} \sqrt{\frac{1}{a^{5}}} \log{\left (a^{3} \sqrt{\frac{1}{a^{5}}} + \sqrt{a + b x} \right )}}{8} - A a b^{2} \sqrt{\frac{1}{a^{3}}} \log{\left (- a^{2} \sqrt{\frac{1}{a^{3}}} + \sqrt{a + b x} \right )} + A a b^{2} \sqrt{\frac{1}{a^{3}}} \log{\left (a^{2} \sqrt{\frac{1}{a^{3}}} + \sqrt{a + b x} \right )} - 2 A b^{2} \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{a + b x}}{\sqrt{- a}} \right )}}{\sqrt{- a}} & \text{for}\: - a > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: - a < 0 \wedge a < a + b x \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: a > a + b x \wedge - a < 0 \end{cases}\right ) - \frac{2 A b \sqrt{a + b x}}{x} - \frac{B a^{2} b \sqrt{\frac{1}{a^{3}}} \log{\left (- a^{2} \sqrt{\frac{1}{a^{3}}} + \sqrt{a + b x} \right )}}{2} + \frac{B a^{2} b \sqrt{\frac{1}{a^{3}}} \log{\left (a^{2} \sqrt{\frac{1}{a^{3}}} + \sqrt{a + b x} \right )}}{2} - 4 B a b \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{a + b x}}{\sqrt{- a}} \right )}}{\sqrt{- a}} & \text{for}\: - a > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: - a < 0 \wedge a < a + b x \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: a > a + b x \wedge - a < 0 \end{cases}\right ) - \frac{B a \sqrt{a + b x}}{x} + 2 B b \sqrt{a + b x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-10*A*a**3*b**2*sqrt(a + b*x)/(-8*a**4 - 16*a**3*b*x + 8*a**2*(a + b*x)**2) + 6*
A*a**2*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**2*b**2*sqrt(a**(-5))*log(-a**3*sqrt(a**(-5)) + sqrt(a + b*x))/8 - 3*A*a**2*b
**2*sqrt(a**(-5))*log(a**3*sqrt(a**(-5)) + sqrt(a + b*x))/8 - A*a*b**2*sqrt(a**(
-3))*log(-a**2*sqrt(a**(-3)) + sqrt(a + b*x)) + A*a*b**2*sqrt(a**(-3))*log(a**2*
sqrt(a**(-3)) + sqrt(a + b*x)) - 2*A*b**2*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(sqrt(a + b*x)/sqrt(a))/sqrt(a), (-a < 0) & (a > a + b*x))) - 2*A*
b*sqrt(a + b*x)/x - B*a**2*b*sqrt(a**(-3))*log(-a**2*sqrt(a**(-3)) + sqrt(a + b*
x))/2 + B*a**2*b*sqrt(a**(-3))*log(a**2*sqrt(a**(-3)) + sqrt(a + b*x))/2 - 4*B*a
*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(sqrt(a + b*x)/sqrt(a))/s
qrt(a), (-a < 0) & (a > a + b*x))) - B*a*sqrt(a + b*x)/x + 2*B*b*sqrt(a + b*x)

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GIAC/XCAS [A]  time = 0.228823, size = 161, normalized size = 1.5 \[ \frac{8 \, \sqrt{b x + a} B b^{2} + \frac{3 \,{\left (4 \, B a b^{2} + A b^{3}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{4 \,{\left (b x + a\right )}^{\frac{3}{2}} B a b^{2} - 4 \, \sqrt{b x + a} B a^{2} b^{2} + 5 \,{\left (b x + a\right )}^{\frac{3}{2}} A b^{3} - 3 \, \sqrt{b x + a} A a b^{3}}{b^{2} x^{2}}}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(3/2)/x^3,x, algorithm="giac")

[Out]

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