3.419 \(\int \frac{A+B x}{x^4 \sqrt{a+b x}} \, dx\)

Optimal. Leaf size=115 \[ \frac{b^2 (5 A b-6 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{7/2}}-\frac{b \sqrt{a+b x} (5 A b-6 a B)}{8 a^3 x}+\frac{\sqrt{a+b x} (5 A b-6 a B)}{12 a^2 x^2}-\frac{A \sqrt{a+b x}}{3 a x^3} \]

[Out]

-(A*Sqrt[a + b*x])/(3*a*x^3) + ((5*A*b - 6*a*B)*Sqrt[a + b*x])/(12*a^2*x^2) - (b
*(5*A*b - 6*a*B)*Sqrt[a + b*x])/(8*a^3*x) + (b^2*(5*A*b - 6*a*B)*ArcTanh[Sqrt[a
+ b*x]/Sqrt[a]])/(8*a^(7/2))

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Rubi [A]  time = 0.151893, antiderivative size = 115, 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 \[ \frac{b^2 (5 A b-6 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{7/2}}-\frac{b \sqrt{a+b x} (5 A b-6 a B)}{8 a^3 x}+\frac{\sqrt{a+b x} (5 A b-6 a B)}{12 a^2 x^2}-\frac{A \sqrt{a+b x}}{3 a x^3} \]

Antiderivative was successfully verified.

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

[Out]

-(A*Sqrt[a + b*x])/(3*a*x^3) + ((5*A*b - 6*a*B)*Sqrt[a + b*x])/(12*a^2*x^2) - (b
*(5*A*b - 6*a*B)*Sqrt[a + b*x])/(8*a^3*x) + (b^2*(5*A*b - 6*a*B)*ArcTanh[Sqrt[a
+ b*x]/Sqrt[a]])/(8*a^(7/2))

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Rubi in Sympy [A]  time = 12.7658, size = 105, normalized size = 0.91 \[ - \frac{A \sqrt{a + b x}}{3 a x^{3}} + \frac{\sqrt{a + b x} \left (5 A b - 6 B a\right )}{12 a^{2} x^{2}} - \frac{b \sqrt{a + b x} \left (5 A b - 6 B a\right )}{8 a^{3} x} + \frac{b^{2} \left (5 A b - 6 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{8 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**4/(b*x+a)**(1/2),x)

[Out]

-A*sqrt(a + b*x)/(3*a*x**3) + sqrt(a + b*x)*(5*A*b - 6*B*a)/(12*a**2*x**2) - b*s
qrt(a + b*x)*(5*A*b - 6*B*a)/(8*a**3*x) + b**2*(5*A*b - 6*B*a)*atanh(sqrt(a + b*
x)/sqrt(a))/(8*a**(7/2))

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Mathematica [A]  time = 0.155461, size = 93, normalized size = 0.81 \[ \frac{b^2 (5 A b-6 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{7/2}}+\frac{\sqrt{a+b x} \left (-4 a^2 (2 A+3 B x)+2 a b x (5 A+9 B x)-15 A b^2 x^2\right )}{24 a^3 x^3} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*(-15*A*b^2*x^2 - 4*a^2*(2*A + 3*B*x) + 2*a*b*x*(5*A + 9*B*x)))/(2
4*a^3*x^3) + (b^2*(5*A*b - 6*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(8*a^(7/2))

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Maple [A]  time = 0.017, size = 104, normalized size = 0.9 \[ 2\,{b}^{2} \left ({\frac{1}{{x}^{3}{b}^{3}} \left ( -1/16\,{\frac{ \left ( 5\,Ab-6\,Ba \right ) \left ( bx+a \right ) ^{5/2}}{{a}^{3}}}+1/6\,{\frac{ \left ( 5\,Ab-6\,Ba \right ) \left ( bx+a \right ) ^{3/2}}{{a}^{2}}}-1/16\,{\frac{ \left ( 11\,Ab-10\,Ba \right ) \sqrt{bx+a}}{a}} \right ) }+1/16\,{\frac{5\,Ab-6\,Ba}{{a}^{7/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^4/(b*x+a)^(1/2),x)

[Out]

2*b^2*((-1/16*(5*A*b-6*B*a)/a^3*(b*x+a)^(5/2)+1/6/a^2*(5*A*b-6*B*a)*(b*x+a)^(3/2
)-1/16*(11*A*b-10*B*a)/a*(b*x+a)^(1/2))/x^3/b^3+1/16*(5*A*b-6*B*a)/a^(7/2)*arcta
nh((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)/(sqrt(b*x + a)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229695, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (6 \, B a b^{2} - 5 \, A b^{3}\right )} x^{3} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (8 \, A a^{2} - 3 \,{\left (6 \, B a b - 5 \, A b^{2}\right )} x^{2} + 2 \,{\left (6 \, B a^{2} - 5 \, A a b\right )} x\right )} \sqrt{b x + a} \sqrt{a}}{48 \, a^{\frac{7}{2}} x^{3}}, \frac{3 \,{\left (6 \, B a b^{2} - 5 \, A b^{3}\right )} x^{3} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) -{\left (8 \, A a^{2} - 3 \,{\left (6 \, B a b - 5 \, A b^{2}\right )} x^{2} + 2 \,{\left (6 \, B a^{2} - 5 \, A a b\right )} x\right )} \sqrt{b x + a} \sqrt{-a}}{24 \, \sqrt{-a} a^{3} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/48*(3*(6*B*a*b^2 - 5*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*(6*B*a*b - 5*A*b^2)*x^2 + 2*(6*B*a^2 - 5*A*a*b)*x)*sqrt(b*
x + a)*sqrt(a))/(a^(7/2)*x^3), 1/24*(3*(6*B*a*b^2 - 5*A*b^3)*x^3*arctan(a/(sqrt(
b*x + a)*sqrt(-a))) - (8*A*a^2 - 3*(6*B*a*b - 5*A*b^2)*x^2 + 2*(6*B*a^2 - 5*A*a*
b)*x)*sqrt(b*x + a)*sqrt(-a))/(sqrt(-a)*a^3*x^3)]

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Sympy [A]  time = 90.4866, size = 245, normalized size = 2.13 \[ - \frac{A}{3 \sqrt{b} x^{\frac{7}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{A \sqrt{b}}{12 a x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{5 A b^{\frac{3}{2}}}{24 a^{2} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{5 A b^{\frac{5}{2}}}{8 a^{3} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} + \frac{5 A b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{8 a^{\frac{7}{2}}} - \frac{B}{2 \sqrt{b} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{B \sqrt{b}}{4 a x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{3 B b^{\frac{3}{2}}}{4 a^{2} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} - \frac{3 B b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/x**4/(b*x+a)**(1/2),x)

[Out]

-A/(3*sqrt(b)*x**(7/2)*sqrt(a/(b*x) + 1)) + A*sqrt(b)/(12*a*x**(5/2)*sqrt(a/(b*x
) + 1)) - 5*A*b**(3/2)/(24*a**2*x**(3/2)*sqrt(a/(b*x) + 1)) - 5*A*b**(5/2)/(8*a*
*3*sqrt(x)*sqrt(a/(b*x) + 1)) + 5*A*b**3*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/(8*a**
(7/2)) - B/(2*sqrt(b)*x**(5/2)*sqrt(a/(b*x) + 1)) + B*sqrt(b)/(4*a*x**(3/2)*sqrt
(a/(b*x) + 1)) + 3*B*b**(3/2)/(4*a**2*sqrt(x)*sqrt(a/(b*x) + 1)) - 3*B*b**2*asin
h(sqrt(a)/(sqrt(b)*sqrt(x)))/(4*a**(5/2))

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GIAC/XCAS [A]  time = 0.214179, size = 194, normalized size = 1.69 \[ \frac{\frac{3 \,{\left (6 \, B a b^{3} - 5 \, A b^{4}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} + \frac{18 \,{\left (b x + a\right )}^{\frac{5}{2}} B a b^{3} - 48 \,{\left (b x + a\right )}^{\frac{3}{2}} B a^{2} b^{3} + 30 \, \sqrt{b x + a} B a^{3} b^{3} - 15 \,{\left (b x + a\right )}^{\frac{5}{2}} A b^{4} + 40 \,{\left (b x + a\right )}^{\frac{3}{2}} A a b^{4} - 33 \, \sqrt{b x + a} A a^{2} b^{4}}{a^{3} b^{3} x^{3}}}{24 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(3*(6*B*a*b^3 - 5*A*b^4)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^3) + (1
8*(b*x + a)^(5/2)*B*a*b^3 - 48*(b*x + a)^(3/2)*B*a^2*b^3 + 30*sqrt(b*x + a)*B*a^
3*b^3 - 15*(b*x + a)^(5/2)*A*b^4 + 40*(b*x + a)^(3/2)*A*a*b^4 - 33*sqrt(b*x + a)
*A*a^2*b^4)/(a^3*b^3*x^3))/b