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

Optimal. Leaf size=112 \[ -\frac{3 b (5 A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{7/2}}+\frac{3 \sqrt{a+b x} (5 A b-4 a B)}{4 a^3 x}-\frac{5 A b-4 a B}{2 a^2 x \sqrt{a+b x}}-\frac{A}{2 a x^2 \sqrt{a+b x}} \]

[Out]

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

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Rubi [A]  time = 0.148271, 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 \[ -\frac{3 b (5 A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{7/2}}+\frac{3 \sqrt{a+b x} (5 A b-4 a B)}{4 a^3 x}-\frac{5 A b-4 a B}{2 a^2 x \sqrt{a+b x}}-\frac{A}{2 a x^2 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.8986, size = 104, normalized size = 0.93 \[ - \frac{A}{2 a x^{2} \sqrt{a + b x}} - \frac{5 A b - 4 B a}{2 a^{2} x \sqrt{a + b x}} + \frac{3 \sqrt{a + b x} \left (5 A b - 4 B a\right )}{4 a^{3} x} - \frac{3 b \left (5 A b - 4 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{4 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.159402, size = 88, normalized size = 0.79 \[ \frac{3 b (4 a B-5 A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{7/2}}+\frac{-2 a^2 (A+2 B x)+a b x (5 A-12 B x)+15 A b^2 x^2}{4 a^3 x^2 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.02, size = 101, normalized size = 0.9 \[ 2\,b \left ( -{\frac{-Ab+Ba}{{a}^{3}\sqrt{bx+a}}}+{\frac{1}{{a}^{3}} \left ({\frac{1}{{b}^{2}{x}^{2}} \left ( \left ({\frac{7\,Ab}{8}}-1/2\,Ba \right ) \left ( bx+a \right ) ^{3/2}+ \left ( -{\frac{9\,Aab}{8}}+1/2\,B{a}^{2} \right ) \sqrt{bx+a} \right ) }-3/8\,{\frac{5\,Ab-4\,Ba}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232918, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (4 \, B a b - 5 \, A b^{2}\right )} \sqrt{b x + a} x^{2} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (2 \, A a^{2} + 3 \,{\left (4 \, B a b - 5 \, A b^{2}\right )} x^{2} +{\left (4 \, B a^{2} - 5 \, A a b\right )} x\right )} \sqrt{a}}{8 \, \sqrt{b x + a} a^{\frac{7}{2}} x^{2}}, -\frac{3 \,{\left (4 \, B a b - 5 \, A b^{2}\right )} \sqrt{b x + a} x^{2} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) +{\left (2 \, A a^{2} + 3 \,{\left (4 \, B a b - 5 \, A b^{2}\right )} x^{2} +{\left (4 \, B a^{2} - 5 \, A a b\right )} x\right )} \sqrt{-a}}{4 \, \sqrt{b x + a} \sqrt{-a} a^{3} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 38.2708, size = 185, normalized size = 1.65 \[ A \left (- \frac{1}{2 a \sqrt{b} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{5 \sqrt{b}}{4 a^{2} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{15 b^{\frac{3}{2}}}{4 a^{3} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} - \frac{15 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4 a^{\frac{7}{2}}}\right ) + B \left (- \frac{1}{a \sqrt{b} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{3 \sqrt{b}}{a^{2} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} + \frac{3 b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{a^{\frac{5}{2}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(-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))) + B*(-1/(a*sqrt(b)*x**(3/2)*sqrt(a/(b*x
) + 1)) - 3*sqrt(b)/(a**2*sqrt(x)*sqrt(a/(b*x) + 1)) + 3*b*asinh(sqrt(a)/(sqrt(b
)*sqrt(x)))/a**(5/2))

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GIAC/XCAS [A]  time = 0.218802, size = 169, normalized size = 1.51 \[ -\frac{3 \,{\left (4 \, B a b - 5 \, A b^{2}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{4 \, \sqrt{-a} a^{3}} - \frac{2 \,{\left (B a b - A b^{2}\right )}}{\sqrt{b x + a} a^{3}} - \frac{4 \,{\left (b x + a\right )}^{\frac{3}{2}} B a b - 4 \, \sqrt{b x + a} B a^{2} b - 7 \,{\left (b x + a\right )}^{\frac{3}{2}} A b^{2} + 9 \, \sqrt{b x + a} A a b^{2}}{4 \, a^{3} b^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/4*(4*B*a*b - 5*A*b^2)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^3) - 2*(B*a*
b - A*b^2)/(sqrt(b*x + a)*a^3) - 1/4*(4*(b*x + a)^(3/2)*B*a*b - 4*sqrt(b*x + a)*
B*a^2*b - 7*(b*x + a)^(3/2)*A*b^2 + 9*sqrt(b*x + a)*A*a*b^2)/(a^3*b^2*x^2)