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

Optimal. Leaf size=95 \[ \frac{(a+b x)^{3/2} (2 a B+3 A b)}{3 a}+\sqrt{a+b x} (2 a B+3 A b)-\sqrt{a} (2 a B+3 A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )-\frac{A (a+b x)^{5/2}}{a x} \]

[Out]

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

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Rubi [A]  time = 0.135588, antiderivative size = 95, 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 \[ \frac{(a+b x)^{3/2} (2 a B+3 A b)}{3 a}+\sqrt{a+b x} (2 a B+3 A b)-\sqrt{a} (2 a B+3 A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )-\frac{A (a+b x)^{5/2}}{a x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.5949, size = 87, normalized size = 0.92 \[ - \frac{A \left (a + b x\right )^{\frac{5}{2}}}{a x} - 2 \sqrt{a} \left (\frac{3 A b}{2} + B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )} + \sqrt{a + b x} \left (3 A b + 2 B a\right ) + \frac{2 \left (a + b x\right )^{\frac{3}{2}} \left (\frac{3 A b}{2} + B a\right )}{3 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0996302, size = 71, normalized size = 0.75 \[ \sqrt{a+b x} \left (\frac{2}{3} (4 a B+3 A b)-\frac{a A}{x}+\frac{2 b B x}{3}\right )-\sqrt{a} (2 a B+3 A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.017, size = 77, normalized size = 0.8 \[{\frac{2\,B}{3} \left ( bx+a \right ) ^{{\frac{3}{2}}}}+2\,Ab\sqrt{bx+a}+2\,Ba\sqrt{bx+a}+2\,a \left ( -1/2\,{\frac{A\sqrt{bx+a}}{x}}-1/2\,{\frac{3\,Ab+2\,Ba}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222003, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (2 \, B a + 3 \, A b\right )} \sqrt{a} x \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (2 \, B b x^{2} - 3 \, A a + 2 \,{\left (4 \, B a + 3 \, A b\right )} x\right )} \sqrt{b x + a}}{6 \, x}, -\frac{3 \,{\left (2 \, B a + 3 \, A b\right )} \sqrt{-a} x \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) -{\left (2 \, B b x^{2} - 3 \, A a + 2 \,{\left (4 \, B a + 3 \, A b\right )} x\right )} \sqrt{b x + a}}{3 \, x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(3*(2*B*a + 3*A*b)*sqrt(a)*x*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) +
2*(2*B*b*x^2 - 3*A*a + 2*(4*B*a + 3*A*b)*x)*sqrt(b*x + a))/x, -1/3*(3*(2*B*a +
3*A*b)*sqrt(-a)*x*arctan(sqrt(b*x + a)/sqrt(-a)) - (2*B*b*x^2 - 3*A*a + 2*(4*B*a
+ 3*A*b)*x)*sqrt(b*x + a))/x]

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Sympy [A]  time = 23.3673, size = 314, normalized size = 3.31 \[ - \frac{A a^{2} b \sqrt{\frac{1}{a^{3}}} \log{\left (- a^{2} \sqrt{\frac{1}{a^{3}}} + \sqrt{a + b x} \right )}}{2} + \frac{A a^{2} b \sqrt{\frac{1}{a^{3}}} \log{\left (a^{2} \sqrt{\frac{1}{a^{3}}} + \sqrt{a + b x} \right )}}{2} - 4 A a b \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{a + b x}}{\sqrt{- a}} \right )}}{\sqrt{- a}} & \text{for}\: - a > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: - a < 0 \wedge a < a + b x \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: a > a + b x \wedge - a < 0 \end{cases}\right ) - \frac{A a \sqrt{a + b x}}{x} + 2 A b \sqrt{a + b x} - 2 B a^{2} \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{a + b x}}{\sqrt{- a}} \right )}}{\sqrt{- a}} & \text{for}\: - a > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: - a < 0 \wedge a < a + b x \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: a > a + b x \wedge - a < 0 \end{cases}\right ) + 2 B a \sqrt{a + b x} + B b \left (\begin{cases} \sqrt{a} x & \text{for}\: b = 0 \\\frac{2 \left (a + b x\right )^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a**2*b*sqrt(a**(-3))*log(-a**2*sqrt(a**(-3)) + sqrt(a + b*x))/2 + A*a**2*b*sq
rt(a**(-3))*log(a**2*sqrt(a**(-3)) + sqrt(a + b*x))/2 - 4*A*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))/sqrt(a), (-a < 0) &
(a > a + b*x))) - A*a*sqrt(a + b*x)/x + 2*A*b*sqrt(a + b*x) - 2*B*a**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*B*a*sqrt(a + b*x) + B*b*Piecewise((sqrt(a)*x, Eq(b, 0
)), (2*(a + b*x)**(3/2)/(3*b), True))

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GIAC/XCAS [A]  time = 0.213141, size = 126, normalized size = 1.33 \[ \frac{2 \,{\left (b x + a\right )}^{\frac{3}{2}} B b + 6 \, \sqrt{b x + a} B a b + 6 \, \sqrt{b x + a} A b^{2} - \frac{3 \, \sqrt{b x + a} A a b}{x} + \frac{3 \,{\left (2 \, B a^{2} b + 3 \, A a b^{2}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(2*(b*x + a)^(3/2)*B*b + 6*sqrt(b*x + a)*B*a*b + 6*sqrt(b*x + a)*A*b^2 - 3*s
qrt(b*x + a)*A*a*b/x + 3*(2*B*a^2*b + 3*A*a*b^2)*arctan(sqrt(b*x + a)/sqrt(-a))/
sqrt(-a))/b