3.394 \(\int \frac{(a+b x)^{3/2} (A+B x)}{x^4} \, dx\)

Optimal. Leaf size=112 \[ \frac{b^2 (A b-6 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{3/2}}+\frac{(a+b x)^{3/2} (A b-6 a B)}{12 a x^2}+\frac{b \sqrt{a+b x} (A b-6 a B)}{8 a x}-\frac{A (a+b x)^{5/2}}{3 a x^3} \]

[Out]

(b*(A*b - 6*a*B)*Sqrt[a + b*x])/(8*a*x) + ((A*b - 6*a*B)*(a + b*x)^(3/2))/(12*a*
x^2) - (A*(a + b*x)^(5/2))/(3*a*x^3) + (b^2*(A*b - 6*a*B)*ArcTanh[Sqrt[a + b*x]/
Sqrt[a]])/(8*a^(3/2))

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Rubi [A]  time = 0.150309, antiderivative size = 112, 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 \[ \frac{b^2 (A b-6 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{3/2}}+\frac{(a+b x)^{3/2} (A b-6 a B)}{12 a x^2}+\frac{b \sqrt{a+b x} (A b-6 a B)}{8 a x}-\frac{A (a+b x)^{5/2}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(3/2)*(A + B*x))/x^4,x]

[Out]

(b*(A*b - 6*a*B)*Sqrt[a + b*x])/(8*a*x) + ((A*b - 6*a*B)*(a + b*x)^(3/2))/(12*a*
x^2) - (A*(a + b*x)^(5/2))/(3*a*x^3) + (b^2*(A*b - 6*a*B)*ArcTanh[Sqrt[a + b*x]/
Sqrt[a]])/(8*a^(3/2))

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Rubi in Sympy [A]  time = 12.7519, size = 97, normalized size = 0.87 \[ - \frac{A \left (a + b x\right )^{\frac{5}{2}}}{3 a x^{3}} + \frac{b \sqrt{a + b x} \left (A b - 6 B a\right )}{8 a x} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (A b - 6 B a\right )}{12 a x^{2}} + \frac{b^{2} \left (A b - 6 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{8 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/2)*(B*x+A)/x**4,x)

[Out]

-A*(a + b*x)**(5/2)/(3*a*x**3) + b*sqrt(a + b*x)*(A*b - 6*B*a)/(8*a*x) + (a + b*
x)**(3/2)*(A*b - 6*B*a)/(12*a*x**2) + b**2*(A*b - 6*B*a)*atanh(sqrt(a + b*x)/sqr
t(a))/(8*a**(3/2))

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Mathematica [A]  time = 0.134278, size = 93, normalized size = 0.83 \[ \sqrt{a+b x} \left (\frac{-6 a B-7 A b}{12 x^2}-\frac{b (10 a B+A b)}{8 a x}-\frac{a A}{3 x^3}\right )-\frac{b^2 (6 a B-A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(3/2)*(A + B*x))/x^4,x]

[Out]

(-(a*A)/(3*x^3) + (-7*A*b - 6*a*B)/(12*x^2) - (b*(A*b + 10*a*B))/(8*a*x))*Sqrt[a
+ b*x] - (b^2*(-(A*b) + 6*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(8*a^(3/2))

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Maple [A]  time = 0.019, size = 96, normalized size = 0.9 \[ 2\,{b}^{2} \left ({\frac{1}{{x}^{3}{b}^{3}} \left ( -1/16\,{\frac{ \left ( Ab+10\,Ba \right ) \left ( bx+a \right ) ^{5/2}}{a}}+ \left ( -1/6\,Ab+Ba \right ) \left ( bx+a \right ) ^{3/2}+ \left ( -3/8\,B{a}^{2}+1/16\,Aab \right ) \sqrt{bx+a} \right ) }+1/16\,{\frac{Ab-6\,Ba}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)*(B*x+A)/x^4,x)

[Out]

2*b^2*((-1/16*(A*b+10*B*a)/a*(b*x+a)^(5/2)+(-1/6*A*b+B*a)*(b*x+a)^(3/2)+(-3/8*B*
a^2+1/16*A*a*b)*(b*x+a)^(1/2))/x^3/b^3+1/16*(A*b-6*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)*(b*x + a)^(3/2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221777, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (6 \, B a b^{2} - A b^{3}\right )} x^{3} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (8 \, A a^{2} + 3 \,{\left (10 \, B a b + A b^{2}\right )} x^{2} + 2 \,{\left (6 \, B a^{2} + 7 \, A a b\right )} x\right )} \sqrt{b x + a} \sqrt{a}}{48 \, a^{\frac{3}{2}} x^{3}}, \frac{3 \,{\left (6 \, B a b^{2} - A b^{3}\right )} x^{3} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) -{\left (8 \, A a^{2} + 3 \,{\left (10 \, B a b + A b^{2}\right )} x^{2} + 2 \,{\left (6 \, B a^{2} + 7 \, A a b\right )} x\right )} \sqrt{b x + a} \sqrt{-a}}{24 \, \sqrt{-a} a x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(3/2)/x^4,x, algorithm="fricas")

[Out]

[-1/48*(3*(6*B*a*b^2 - A*b^3)*x^3*log(((b*x + 2*a)*sqrt(a) + 2*sqrt(b*x + a)*a)/
x) + 2*(8*A*a^2 + 3*(10*B*a*b + A*b^2)*x^2 + 2*(6*B*a^2 + 7*A*a*b)*x)*sqrt(b*x +
a)*sqrt(a))/(a^(3/2)*x^3), 1/24*(3*(6*B*a*b^2 - A*b^3)*x^3*arctan(a/(sqrt(b*x +
a)*sqrt(-a))) - (8*A*a^2 + 3*(10*B*a*b + A*b^2)*x^2 + 2*(6*B*a^2 + 7*A*a*b)*x)*
sqrt(b*x + a)*sqrt(-a))/(sqrt(-a)*a*x^3)]

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Sympy [A]  time = 78.317, size = 862, normalized size = 7.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(3/2)*(B*x+A)/x**4,x)

[Out]

-66*A*a**4*b**3*sqrt(a + b*x)/(96*a**6 + 144*a**5*b*x - 144*a**4*(a + b*x)**2 +
48*a**3*(a + b*x)**3) + 80*A*a**3*b**3*(a + b*x)**(3/2)/(96*a**6 + 144*a**5*b*x
- 144*a**4*(a + b*x)**2 + 48*a**3*(a + b*x)**3) - 30*A*a**2*b**3*(a + b*x)**(5/2
)/(96*a**6 + 144*a**5*b*x - 144*a**4*(a + b*x)**2 + 48*a**3*(a + b*x)**3) - 20*A
*a**2*b**3*sqrt(a + b*x)/(-8*a**4 - 16*a**3*b*x + 8*a**2*(a + b*x)**2) - 5*A*a**
2*b**3*sqrt(a**(-7))*log(-a**4*sqrt(a**(-7)) + sqrt(a + b*x))/16 + 5*A*a**2*b**3
*sqrt(a**(-7))*log(a**4*sqrt(a**(-7)) + sqrt(a + b*x))/16 + 12*A*a*b**3*(a + b*x
)**(3/2)/(-8*a**4 - 16*a**3*b*x + 8*a**2*(a + b*x)**2) + 3*A*a*b**3*sqrt(a**(-5)
)*log(-a**3*sqrt(a**(-5)) + sqrt(a + b*x))/4 - 3*A*a*b**3*sqrt(a**(-5))*log(a**3
*sqrt(a**(-5)) + sqrt(a + b*x))/4 - A*b**3*sqrt(a**(-3))*log(-a**2*sqrt(a**(-3))
+ sqrt(a + b*x))/2 + A*b**3*sqrt(a**(-3))*log(a**2*sqrt(a**(-3)) + sqrt(a + b*x
))/2 - A*b**2*sqrt(a + b*x)/(a*x) - 10*B*a**3*b**2*sqrt(a + b*x)/(-8*a**4 - 16*a
**3*b*x + 8*a**2*(a + b*x)**2) + 6*B*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*B*a**2*b**2*sqrt(a**(-5))*log(-a**3*sqrt(a**(-
5)) + sqrt(a + b*x))/8 - 3*B*a**2*b**2*sqrt(a**(-5))*log(a**3*sqrt(a**(-5)) + sq
rt(a + b*x))/8 - B*a*b**2*sqrt(a**(-3))*log(-a**2*sqrt(a**(-3)) + sqrt(a + b*x))
+ B*a*b**2*sqrt(a**(-3))*log(a**2*sqrt(a**(-3)) + sqrt(a + b*x)) - 2*B*b**2*Pie
cewise((-atan(sqrt(a + b*x)/sqrt(-a))/sqrt(-a), -a > 0), (acoth(sqrt(a + b*x)/sq
rt(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*b*sqrt(a + b*x)/x

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GIAC/XCAS [A]  time = 0.234752, size = 196, normalized size = 1.75 \[ \frac{\frac{3 \,{\left (6 \, B a b^{3} - A b^{4}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} - \frac{30 \,{\left (b x + a\right )}^{\frac{5}{2}} B a b^{3} - 48 \,{\left (b x + a\right )}^{\frac{3}{2}} B a^{2} b^{3} + 18 \, \sqrt{b x + a} B a^{3} b^{3} + 3 \,{\left (b x + a\right )}^{\frac{5}{2}} A b^{4} + 8 \,{\left (b x + a\right )}^{\frac{3}{2}} A a b^{4} - 3 \, \sqrt{b x + a} A a^{2} b^{4}}{a b^{3} x^{3}}}{24 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(3/2)/x^4,x, algorithm="giac")

[Out]

1/24*(3*(6*B*a*b^3 - A*b^4)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a) - (30*(b
*x + a)^(5/2)*B*a*b^3 - 48*(b*x + a)^(3/2)*B*a^2*b^3 + 18*sqrt(b*x + a)*B*a^3*b^
3 + 3*(b*x + a)^(5/2)*A*b^4 + 8*(b*x + a)^(3/2)*A*a*b^4 - 3*sqrt(b*x + a)*A*a^2*
b^4)/(a*b^3*x^3))/b