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

Optimal. Leaf size=143 \[ \frac{5 b^2 (7 A b-6 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{9/2}}-\frac{5 b \sqrt{a+b x} (7 A b-6 a B)}{8 a^4 x}+\frac{5 \sqrt{a+b x} (7 A b-6 a B)}{12 a^3 x^2}-\frac{7 A b-6 a B}{3 a^2 x^2 \sqrt{a+b x}}-\frac{A}{3 a x^3 \sqrt{a+b x}} \]

[Out]

-A/(3*a*x^3*Sqrt[a + b*x]) - (7*A*b - 6*a*B)/(3*a^2*x^2*Sqrt[a + b*x]) + (5*(7*A
*b - 6*a*B)*Sqrt[a + b*x])/(12*a^3*x^2) - (5*b*(7*A*b - 6*a*B)*Sqrt[a + b*x])/(8
*a^4*x) + (5*b^2*(7*A*b - 6*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(8*a^(9/2))

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Rubi [A]  time = 0.192136, antiderivative size = 143, 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 \[ \frac{5 b^2 (7 A b-6 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{9/2}}-\frac{5 b \sqrt{a+b x} (7 A b-6 a B)}{8 a^4 x}+\frac{5 \sqrt{a+b x} (7 A b-6 a B)}{12 a^3 x^2}-\frac{7 A b-6 a B}{3 a^2 x^2 \sqrt{a+b x}}-\frac{A}{3 a x^3 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

-A/(3*a*x^3*Sqrt[a + b*x]) - (7*A*b - 6*a*B)/(3*a^2*x^2*Sqrt[a + b*x]) + (5*(7*A
*b - 6*a*B)*Sqrt[a + b*x])/(12*a^3*x^2) - (5*b*(7*A*b - 6*a*B)*Sqrt[a + b*x])/(8
*a^4*x) + (5*b^2*(7*A*b - 6*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(8*a^(9/2))

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Rubi in Sympy [A]  time = 17.2234, size = 138, normalized size = 0.97 \[ - \frac{A}{3 a x^{3} \sqrt{a + b x}} - \frac{7 A b - 6 B a}{3 a^{2} x^{2} \sqrt{a + b x}} + \frac{5 \sqrt{a + b x} \left (7 A b - 6 B a\right )}{12 a^{3} x^{2}} - \frac{5 b \sqrt{a + b x} \left (7 A b - 6 B a\right )}{8 a^{4} x} + \frac{5 b^{2} \left (7 A b - 6 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{8 a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/(3*a*x**3*sqrt(a + b*x)) - (7*A*b - 6*B*a)/(3*a**2*x**2*sqrt(a + b*x)) + 5*sq
rt(a + b*x)*(7*A*b - 6*B*a)/(12*a**3*x**2) - 5*b*sqrt(a + b*x)*(7*A*b - 6*B*a)/(
8*a**4*x) + 5*b**2*(7*A*b - 6*B*a)*atanh(sqrt(a + b*x)/sqrt(a))/(8*a**(9/2))

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Mathematica [A]  time = 0.212838, size = 112, normalized size = 0.78 \[ \frac{5 b^2 (7 A b-6 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{9/2}}+\frac{-4 a^3 (2 A+3 B x)+2 a^2 b x (7 A+15 B x)+5 a b^2 x^2 (18 B x-7 A)-105 A b^3 x^3}{24 a^4 x^3 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(-105*A*b^3*x^3 - 4*a^3*(2*A + 3*B*x) + 2*a^2*b*x*(7*A + 15*B*x) + 5*a*b^2*x^2*(
-7*A + 18*B*x))/(24*a^4*x^3*Sqrt[a + b*x]) + (5*b^2*(7*A*b - 6*a*B)*ArcTanh[Sqrt
[a + b*x]/Sqrt[a]])/(8*a^(9/2))

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Maple [A]  time = 0.024, size = 126, normalized size = 0.9 \[ 2\,{b}^{2} \left ( -{\frac{Ab-Ba}{{a}^{4}\sqrt{bx+a}}}-{\frac{1}{{a}^{4}} \left ({\frac{1}{{x}^{3}{b}^{3}} \left ( \left ({\frac{19\,Ab}{16}}-{\frac{7\,Ba}{8}} \right ) \left ( bx+a \right ) ^{5/2}+ \left ( -{\frac{17\,Aab}{6}}+2\,B{a}^{2} \right ) \left ( bx+a \right ) ^{3/2}+ \left ({\frac{29\,A{a}^{2}b}{16}}-{\frac{9\,B{a}^{3}}{8}} \right ) \sqrt{bx+a} \right ) }-{\frac{35\,Ab-30\,Ba}{16\,\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b^2*(-(A*b-B*a)/a^4/(b*x+a)^(1/2)-1/a^4*(((19/16*A*b-7/8*B*a)*(b*x+a)^(5/2)+(-
17/6*A*a*b+2*B*a^2)*(b*x+a)^(3/2)+(29/16*A*a^2*b-9/8*B*a^3)*(b*x+a)^(1/2))/x^3/b
^3-5/16*(7*A*b-6*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^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227935, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (6 \, B a b^{2} - 7 \, A b^{3}\right )} \sqrt{b x + a} x^{3} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (8 \, A a^{3} - 15 \,{\left (6 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} - 5 \,{\left (6 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 2 \,{\left (6 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt{a}}{48 \, \sqrt{b x + a} a^{\frac{9}{2}} x^{3}}, \frac{15 \,{\left (6 \, B a b^{2} - 7 \, A b^{3}\right )} \sqrt{b x + a} x^{3} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) -{\left (8 \, A a^{3} - 15 \,{\left (6 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} - 5 \,{\left (6 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 2 \,{\left (6 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt{-a}}{24 \, \sqrt{b x + a} \sqrt{-a} a^{4} x^{3}}\right ] \]

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*(15*(6*B*a*b^2 - 7*A*b^3)*sqrt(b*x + a)*x^3*log(((b*x + 2*a)*sqrt(a) + 2*
sqrt(b*x + a)*a)/x) + 2*(8*A*a^3 - 15*(6*B*a*b^2 - 7*A*b^3)*x^3 - 5*(6*B*a^2*b -
7*A*a*b^2)*x^2 + 2*(6*B*a^3 - 7*A*a^2*b)*x)*sqrt(a))/(sqrt(b*x + a)*a^(9/2)*x^3
), 1/24*(15*(6*B*a*b^2 - 7*A*b^3)*sqrt(b*x + a)*x^3*arctan(a/(sqrt(b*x + a)*sqrt
(-a))) - (8*A*a^3 - 15*(6*B*a*b^2 - 7*A*b^3)*x^3 - 5*(6*B*a^2*b - 7*A*a*b^2)*x^2
+ 2*(6*B*a^3 - 7*A*a^2*b)*x)*sqrt(-a))/(sqrt(b*x + a)*sqrt(-a)*a^4*x^3)]

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Sympy [A]  time = 55.8453, size = 246, normalized size = 1.72 \[ A \left (- \frac{1}{3 a \sqrt{b} x^{\frac{7}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{7 \sqrt{b}}{12 a^{2} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{35 b^{\frac{3}{2}}}{24 a^{3} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{35 b^{\frac{5}{2}}}{8 a^{4} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} + \frac{35 b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{8 a^{\frac{9}{2}}}\right ) + B \left (- \frac{1}{2 a \sqrt{b} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{5 \sqrt{b}}{4 a^{2} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{15 b^{\frac{3}{2}}}{4 a^{3} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} - \frac{15 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4 a^{\frac{7}{2}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(-1/(3*a*sqrt(b)*x**(7/2)*sqrt(a/(b*x) + 1)) + 7*sqrt(b)/(12*a**2*x**(5/2)*sqr
t(a/(b*x) + 1)) - 35*b**(3/2)/(24*a**3*x**(3/2)*sqrt(a/(b*x) + 1)) - 35*b**(5/2)
/(8*a**4*sqrt(x)*sqrt(a/(b*x) + 1)) + 35*b**3*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/(
8*a**(9/2))) + B*(-1/(2*a*sqrt(b)*x**(5/2)*sqrt(a/(b*x) + 1)) + 5*sqrt(b)/(4*a**
2*x**(3/2)*sqrt(a/(b*x) + 1)) + 15*b**(3/2)/(4*a**3*sqrt(x)*sqrt(a/(b*x) + 1)) -
15*b**2*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/(4*a**(7/2)))

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GIAC/XCAS [A]  time = 0.217093, size = 223, normalized size = 1.56 \[ \frac{5 \,{\left (6 \, B a b^{2} - 7 \, A b^{3}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{8 \, \sqrt{-a} a^{4}} + \frac{2 \,{\left (B a b^{2} - A b^{3}\right )}}{\sqrt{b x + a} a^{4}} + \frac{42 \,{\left (b x + a\right )}^{\frac{5}{2}} B a b^{2} - 96 \,{\left (b x + a\right )}^{\frac{3}{2}} B a^{2} b^{2} + 54 \, \sqrt{b x + a} B a^{3} b^{2} - 57 \,{\left (b x + a\right )}^{\frac{5}{2}} A b^{3} + 136 \,{\left (b x + a\right )}^{\frac{3}{2}} A a b^{3} - 87 \, \sqrt{b x + a} A a^{2} b^{3}}{24 \, a^{4} b^{3} x^{3}} \]

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]

5/8*(6*B*a*b^2 - 7*A*b^3)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^4) + 2*(B*a
*b^2 - A*b^3)/(sqrt(b*x + a)*a^4) + 1/24*(42*(b*x + a)^(5/2)*B*a*b^2 - 96*(b*x +
a)^(3/2)*B*a^2*b^2 + 54*sqrt(b*x + a)*B*a^3*b^2 - 57*(b*x + a)^(5/2)*A*b^3 + 13
6*(b*x + a)^(3/2)*A*a*b^3 - 87*sqrt(b*x + a)*A*a^2*b^3)/(a^4*b^3*x^3)