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

Optimal. Leaf size=126 \[ -\frac{3 (5 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{7/2} \sqrt{b}}-\frac{3 (5 A b-a B)}{4 a^3 b \sqrt{x}}+\frac{5 A b-a B}{4 a^2 b \sqrt{x} (a+b x)}+\frac{A b-a B}{2 a b \sqrt{x} (a+b x)^2} \]

[Out]

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

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Rubi [A]  time = 0.142911, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ -\frac{3 (5 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{7/2} \sqrt{b}}-\frac{3 (5 A b-a B)}{4 a^3 b \sqrt{x}}+\frac{5 A b-a B}{4 a^2 b \sqrt{x} (a+b x)}+\frac{A b-a B}{2 a b \sqrt{x} (a+b x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.6048, size = 109, normalized size = 0.87 \[ \frac{A b - B a}{2 a b \sqrt{x} \left (a + b x\right )^{2}} + \frac{5 A b - B a}{4 a^{2} b \sqrt{x} \left (a + b x\right )} - \frac{3 \left (5 A b - B a\right )}{4 a^{3} b \sqrt{x}} - \frac{3 \left (5 A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 a^{\frac{7}{2}} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.115873, size = 93, normalized size = 0.74 \[ \frac{3 (a B-5 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{7/2} \sqrt{b}}+\frac{a^2 (5 B x-8 A)+a b x (3 B x-25 A)-15 A b^2 x^2}{4 a^3 \sqrt{x} (a+b x)^2} \]

Antiderivative was successfully verified.

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

[Out]

(-15*A*b^2*x^2 + a*b*x*(-25*A + 3*B*x) + a^2*(-8*A + 5*B*x))/(4*a^3*Sqrt[x]*(a +
b*x)^2) + (3*(-5*A*b + a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^(7/2)*Sqrt[
b])

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Maple [A]  time = 0.024, size = 125, normalized size = 1. \[ -2\,{\frac{A}{{a}^{3}\sqrt{x}}}-{\frac{7\,{b}^{2}A}{4\,{a}^{3} \left ( bx+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{3\,Bb}{4\,{a}^{2} \left ( bx+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{9\,Ab}{4\,{a}^{2} \left ( bx+a \right ) ^{2}}\sqrt{x}}+{\frac{5\,B}{4\,a \left ( bx+a \right ) ^{2}}\sqrt{x}}-{\frac{15\,Ab}{4\,{a}^{3}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,B}{4\,{a}^{2}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A/a^3/x^(1/2)-7/4/a^3/(b*x+a)^2*x^(3/2)*b^2*A+3/4/a^2/(b*x+a)^2*x^(3/2)*B*b-9
/4/a^2/(b*x+a)^2*A*x^(1/2)*b+5/4/a/(b*x+a)^2*B*x^(1/2)-15/4/a^3/(a*b)^(1/2)*arct
an(x^(1/2)*b/(a*b)^(1/2))*A*b+3/4/a^2/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*
B

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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*x^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227963, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (B a^{3} - 5 \, A a^{2} b +{\left (B a b^{2} - 5 \, A b^{3}\right )} x^{2} + 2 \,{\left (B a^{2} b - 5 \, A a b^{2}\right )} x\right )} \sqrt{x} \log \left (-\frac{2 \, a b \sqrt{x} - \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right ) + 2 \,{\left (8 \, A a^{2} - 3 \,{\left (B a b - 5 \, A b^{2}\right )} x^{2} - 5 \,{\left (B a^{2} - 5 \, A a b\right )} x\right )} \sqrt{-a b}}{8 \,{\left (a^{3} b^{2} x^{2} + 2 \, a^{4} b x + a^{5}\right )} \sqrt{-a b} \sqrt{x}}, -\frac{3 \,{\left (B a^{3} - 5 \, A a^{2} b +{\left (B a b^{2} - 5 \, A b^{3}\right )} x^{2} + 2 \,{\left (B a^{2} b - 5 \, A a b^{2}\right )} x\right )} \sqrt{x} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right ) +{\left (8 \, A a^{2} - 3 \,{\left (B a b - 5 \, A b^{2}\right )} x^{2} - 5 \,{\left (B a^{2} - 5 \, A a b\right )} x\right )} \sqrt{a b}}{4 \,{\left (a^{3} b^{2} x^{2} + 2 \, a^{4} b x + a^{5}\right )} \sqrt{a b} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.213831, size = 116, normalized size = 0.92 \[ \frac{3 \,{\left (B a - 5 \, A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} a^{3}} - \frac{2 \, A}{a^{3} \sqrt{x}} + \frac{3 \, B a b x^{\frac{3}{2}} - 7 \, A b^{2} x^{\frac{3}{2}} + 5 \, B a^{2} \sqrt{x} - 9 \, A a b \sqrt{x}}{4 \,{\left (b x + a\right )}^{2} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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