3.404 \(\int \frac{(a+b x)^{5/2} (A+B x)}{x^2} \, dx\)

Optimal. Leaf size=118 \[ -a^{3/2} (2 a B+5 A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+\frac{(a+b x)^{5/2} (2 a B+5 A b)}{5 a}+\frac{1}{3} (a+b x)^{3/2} (2 a B+5 A b)+a \sqrt{a+b x} (2 a B+5 A b)-\frac{A (a+b x)^{7/2}}{a x} \]

[Out]

a*(5*A*b + 2*a*B)*Sqrt[a + b*x] + ((5*A*b + 2*a*B)*(a + b*x)^(3/2))/3 + ((5*A*b
+ 2*a*B)*(a + b*x)^(5/2))/(5*a) - (A*(a + b*x)^(7/2))/(a*x) - a^(3/2)*(5*A*b + 2
*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]]

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Rubi [A]  time = 0.160128, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -a^{3/2} (2 a B+5 A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+\frac{(a+b x)^{5/2} (2 a B+5 A b)}{5 a}+\frac{1}{3} (a+b x)^{3/2} (2 a B+5 A b)+a \sqrt{a+b x} (2 a B+5 A b)-\frac{A (a+b x)^{7/2}}{a x} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(5/2)*(A + B*x))/x^2,x]

[Out]

a*(5*A*b + 2*a*B)*Sqrt[a + b*x] + ((5*A*b + 2*a*B)*(a + b*x)^(3/2))/3 + ((5*A*b
+ 2*a*B)*(a + b*x)^(5/2))/(5*a) - (A*(a + b*x)^(7/2))/(a*x) - a^(3/2)*(5*A*b + 2
*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]]

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Rubi in Sympy [A]  time = 16.9449, size = 110, normalized size = 0.93 \[ - \frac{A \left (a + b x\right )^{\frac{7}{2}}}{a x} - 2 a^{\frac{3}{2}} \left (\frac{5 A b}{2} + B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )} + a \sqrt{a + b x} \left (5 A b + 2 B a\right ) + \left (a + b x\right )^{\frac{3}{2}} \left (\frac{5 A b}{3} + \frac{2 B a}{3}\right ) + \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (\frac{5 A b}{2} + B a\right )}{5 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)*(B*x+A)/x**2,x)

[Out]

-A*(a + b*x)**(7/2)/(a*x) - 2*a**(3/2)*(5*A*b/2 + B*a)*atanh(sqrt(a + b*x)/sqrt(
a)) + a*sqrt(a + b*x)*(5*A*b + 2*B*a) + (a + b*x)**(3/2)*(5*A*b/3 + 2*B*a/3) + 2
*(a + b*x)**(5/2)*(5*A*b/2 + B*a)/(5*a)

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Mathematica [A]  time = 0.145423, size = 93, normalized size = 0.79 \[ \sqrt{a+b x} \left (-\frac{a^2 A}{x}+\frac{2}{15} b x (11 a B+5 A b)+\frac{2}{15} a (23 a B+35 A b)+\frac{2}{5} b^2 B x^2\right )-a^{3/2} (2 a B+5 A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(5/2)*(A + B*x))/x^2,x]

[Out]

Sqrt[a + b*x]*((2*a*(35*A*b + 23*a*B))/15 - (a^2*A)/x + (2*b*(5*A*b + 11*a*B)*x)
/15 + (2*b^2*B*x^2)/5) - a^(3/2)*(5*A*b + 2*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]]

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Maple [A]  time = 0.018, size = 104, normalized size = 0.9 \[{\frac{2\,B}{5} \left ( bx+a \right ) ^{{\frac{5}{2}}}}+{\frac{2\,Ab}{3} \left ( bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{2\,Ba}{3} \left ( bx+a \right ) ^{{\frac{3}{2}}}}+4\,abA\sqrt{bx+a}+2\,{a}^{2}B\sqrt{bx+a}+2\,{a}^{2} \left ( -1/2\,{\frac{A\sqrt{bx+a}}{x}}-1/2\,{\frac{5\,Ab+2\,Ba}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(B*x+A)/x^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226917, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (2 \, B a^{2} + 5 \, A a b\right )} \sqrt{a} x \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (6 \, B b^{2} x^{3} - 15 \, A a^{2} + 2 \,{\left (11 \, B a b + 5 \, A b^{2}\right )} x^{2} + 2 \,{\left (23 \, B a^{2} + 35 \, A a b\right )} x\right )} \sqrt{b x + a}}{30 \, x}, -\frac{15 \,{\left (2 \, B a^{2} + 5 \, A a b\right )} \sqrt{-a} x \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) -{\left (6 \, B b^{2} x^{3} - 15 \, A a^{2} + 2 \,{\left (11 \, B a b + 5 \, A b^{2}\right )} x^{2} + 2 \,{\left (23 \, B a^{2} + 35 \, A a b\right )} x\right )} \sqrt{b x + a}}{15 \, x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/x^2,x, algorithm="fricas")

[Out]

[1/30*(15*(2*B*a^2 + 5*A*a*b)*sqrt(a)*x*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a
)/x) + 2*(6*B*b^2*x^3 - 15*A*a^2 + 2*(11*B*a*b + 5*A*b^2)*x^2 + 2*(23*B*a^2 + 35
*A*a*b)*x)*sqrt(b*x + a))/x, -1/15*(15*(2*B*a^2 + 5*A*a*b)*sqrt(-a)*x*arctan(sqr
t(b*x + a)/sqrt(-a)) - (6*B*b^2*x^3 - 15*A*a^2 + 2*(11*B*a*b + 5*A*b^2)*x^2 + 2*
(23*B*a^2 + 35*A*a*b)*x)*sqrt(b*x + a))/x]

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Sympy [A]  time = 30.8081, size = 379, normalized size = 3.21 \[ - \frac{A a^{3} b \sqrt{\frac{1}{a^{3}}} \log{\left (- a^{2} \sqrt{\frac{1}{a^{3}}} + \sqrt{a + b x} \right )}}{2} + \frac{A a^{3} b \sqrt{\frac{1}{a^{3}}} \log{\left (a^{2} \sqrt{\frac{1}{a^{3}}} + \sqrt{a + b x} \right )}}{2} - 6 A a^{2} b \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{a + b x}}{\sqrt{- a}} \right )}}{\sqrt{- a}} & \text{for}\: - a > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: - a < 0 \wedge a < a + b x \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: a > a + b x \wedge - a < 0 \end{cases}\right ) - \frac{A a^{2} \sqrt{a + b x}}{x} + 4 A a b \sqrt{a + b x} + A b^{2} \left (\begin{cases} \sqrt{a} x & \text{for}\: b = 0 \\\frac{2 \left (a + b x\right )^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases}\right ) - 2 B a^{3} \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{a + b x}}{\sqrt{- a}} \right )}}{\sqrt{- a}} & \text{for}\: - a > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: - a < 0 \wedge a < a + b x \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: a > a + b x \wedge - a < 0 \end{cases}\right ) + 2 B a^{2} \sqrt{a + b x} + 2 B a b \left (\begin{cases} \sqrt{a} x & \text{for}\: b = 0 \\\frac{2 \left (a + b x\right )^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases}\right ) - \frac{2 B a \left (a + b x\right )^{\frac{3}{2}}}{3} + \frac{2 B \left (a + b x\right )^{\frac{5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(5/2)*(B*x+A)/x**2,x)

[Out]

-A*a**3*b*sqrt(a**(-3))*log(-a**2*sqrt(a**(-3)) + sqrt(a + b*x))/2 + A*a**3*b*sq
rt(a**(-3))*log(a**2*sqrt(a**(-3)) + sqrt(a + b*x))/2 - 6*A*a**2*b*Piecewise((-a
tan(sqrt(a + b*x)/sqrt(-a))/sqrt(-a), -a > 0), (acoth(sqrt(a + b*x)/sqrt(a))/sqr
t(a), (-a < 0) & (a < a + b*x)), (atanh(sqrt(a + b*x)/sqrt(a))/sqrt(a), (-a < 0)
& (a > a + b*x))) - A*a**2*sqrt(a + b*x)/x + 4*A*a*b*sqrt(a + b*x) + A*b**2*Pie
cewise((sqrt(a)*x, Eq(b, 0)), (2*(a + b*x)**(3/2)/(3*b), True)) - 2*B*a**3*Piece
wise((-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**2*sqrt(a + b*x) + 2*B*a*b*Piecewise((sqrt(a)
*x, Eq(b, 0)), (2*(a + b*x)**(3/2)/(3*b), True)) - 2*B*a*(a + b*x)**(3/2)/3 + 2*
B*(a + b*x)**(5/2)/5

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GIAC/XCAS [A]  time = 0.2358, size = 169, normalized size = 1.43 \[ \frac{6 \,{\left (b x + a\right )}^{\frac{5}{2}} B b + 10 \,{\left (b x + a\right )}^{\frac{3}{2}} B a b + 30 \, \sqrt{b x + a} B a^{2} b + 10 \,{\left (b x + a\right )}^{\frac{3}{2}} A b^{2} + 60 \, \sqrt{b x + a} A a b^{2} - \frac{15 \, \sqrt{b x + a} A a^{2} b}{x} + \frac{15 \,{\left (2 \, B a^{3} b + 5 \, A a^{2} b^{2}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}}}{15 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/x^2,x, algorithm="giac")

[Out]

1/15*(6*(b*x + a)^(5/2)*B*b + 10*(b*x + a)^(3/2)*B*a*b + 30*sqrt(b*x + a)*B*a^2*
b + 10*(b*x + a)^(3/2)*A*b^2 + 60*sqrt(b*x + a)*A*a*b^2 - 15*sqrt(b*x + a)*A*a^2
*b/x + 15*(2*B*a^3*b + 5*A*a^2*b^2)*arctan(sqrt(b*x + a)/sqrt(-a))/sqrt(-a))/b