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

Optimal. Leaf size=69 \[ -2 a^{3/2} A \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+\frac{2}{3} A (a+b x)^{3/2}+2 a A \sqrt{a+b x}+\frac{2 B (a+b x)^{5/2}}{5 b} \]

[Out]

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

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Rubi [A]  time = 0.0874744, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -2 a^{3/2} A \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+\frac{2}{3} A (a+b x)^{3/2}+2 a A \sqrt{a+b x}+\frac{2 B (a+b x)^{5/2}}{5 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.35232, size = 65, normalized size = 0.94 \[ - 2 A a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )} + 2 A a \sqrt{a + b x} + \frac{2 A \left (a + b x\right )^{\frac{3}{2}}}{3} + \frac{2 B \left (a + b x\right )^{\frac{5}{2}}}{5 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.135795, size = 73, normalized size = 1.06 \[ \frac{2 \sqrt{a+b x} \left (3 a^2 B+a (20 A b+6 b B x)+b^2 x (5 A+3 B x)\right )}{15 b}-2 a^{3/2} A \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[a + b*x]*(3*a^2*B + b^2*x*(5*A + 3*B*x) + a*(20*A*b + 6*b*B*x)))/(15*b)
- 2*a^(3/2)*A*ArcTanh[Sqrt[a + b*x]/Sqrt[a]]

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Maple [A]  time = 0.01, size = 58, normalized size = 0.8 \[ 2\,{\frac{1}{b} \left ( 1/5\,B \left ( bx+a \right ) ^{5/2}+1/3\,Ab \left ( bx+a \right ) ^{3/2}+abA\sqrt{bx+a}-A{a}^{3/2}b{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)*(B*x+A)/x,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227443, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, A a^{\frac{3}{2}} b \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (3 \, B b^{2} x^{2} + 3 \, B a^{2} + 20 \, A a b +{\left (6 \, B a b + 5 \, A b^{2}\right )} x\right )} \sqrt{b x + a}}{15 \, b}, -\frac{2 \,{\left (15 \, A \sqrt{-a} a b \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) -{\left (3 \, B b^{2} x^{2} + 3 \, B a^{2} + 20 \, A a b +{\left (6 \, B a b + 5 \, A b^{2}\right )} x\right )} \sqrt{b x + a}\right )}}{15 \, b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/15*(15*A*a^(3/2)*b*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*(3*B*b^2*
x^2 + 3*B*a^2 + 20*A*a*b + (6*B*a*b + 5*A*b^2)*x)*sqrt(b*x + a))/b, -2/15*(15*A*
sqrt(-a)*a*b*arctan(sqrt(b*x + a)/sqrt(-a)) - (3*B*b^2*x^2 + 3*B*a^2 + 20*A*a*b
+ (6*B*a*b + 5*A*b^2)*x)*sqrt(b*x + a))/b]

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Sympy [A]  time = 11.2265, size = 128, normalized size = 1.86 \[ - 2 A a^{2} \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{a + b x}}{\sqrt{- a}} \right )}}{\sqrt{- a}} & \text{for}\: - a > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: - a < 0 \wedge a < a + b x \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: a > a + b x \wedge - a < 0 \end{cases}\right ) + 2 A a \sqrt{a + b x} + \frac{2 A \left (a + b x\right )^{\frac{3}{2}}}{3} + \frac{2 B \left (a + b x\right )^{\frac{5}{2}}}{5 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*a**2*Piecewise((-atan(sqrt(a + b*x)/sqrt(-a))/sqrt(-a), -a > 0), (acoth(sqr
t(a + b*x)/sqrt(a))/sqrt(a), (-a < 0) & (a < a + b*x)), (atanh(sqrt(a + b*x)/sqr
t(a))/sqrt(a), (-a < 0) & (a > a + b*x))) + 2*A*a*sqrt(a + b*x) + 2*A*(a + b*x)*
*(3/2)/3 + 2*B*(a + b*x)**(5/2)/(5*b)

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GIAC/XCAS [A]  time = 0.210209, size = 97, normalized size = 1.41 \[ \frac{2 \, A a^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \frac{2 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} B b^{4} + 5 \,{\left (b x + a\right )}^{\frac{3}{2}} A b^{5} + 15 \, \sqrt{b x + a} A a b^{5}\right )}}{15 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a^2*arctan(sqrt(b*x + a)/sqrt(-a))/sqrt(-a) + 2/15*(3*(b*x + a)^(5/2)*B*b^4
+ 5*(b*x + a)^(3/2)*A*b^5 + 15*sqrt(b*x + a)*A*a*b^5)/b^5