3.520 \(\int \frac{A+B x}{x^{5/2} (a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=83 \[ \frac{4 \sqrt{a+b x} (4 A b-3 a B)}{3 a^3 \sqrt{x}}-\frac{2 (4 A b-3 a B)}{3 a^2 \sqrt{x} \sqrt{a+b x}}-\frac{2 A}{3 a x^{3/2} \sqrt{a+b x}} \]

[Out]

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

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Rubi [A]  time = 0.0971705, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{4 \sqrt{a+b x} (4 A b-3 a B)}{3 a^3 \sqrt{x}}-\frac{2 (4 A b-3 a B)}{3 a^2 \sqrt{x} \sqrt{a+b x}}-\frac{2 A}{3 a x^{3/2} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^(5/2)*(a + b*x)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 8.09775, size = 78, normalized size = 0.94 \[ - \frac{2 A}{3 a x^{\frac{3}{2}} \sqrt{a + b x}} - \frac{2 \left (4 A b - 3 B a\right )}{3 a^{2} \sqrt{x} \sqrt{a + b x}} + \frac{4 \sqrt{a + b x} \left (4 A b - 3 B a\right )}{3 a^{3} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**(5/2)/(b*x+a)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0655463, size = 54, normalized size = 0.65 \[ -\frac{2 \left (a^2 (A+3 B x)+2 a b x (3 B x-2 A)-8 A b^2 x^2\right )}{3 a^3 x^{3/2} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^(5/2)*(a + b*x)^(3/2)),x]

[Out]

(-2*(-8*A*b^2*x^2 + 2*a*b*x*(-2*A + 3*B*x) + a^2*(A + 3*B*x)))/(3*a^3*x^(3/2)*Sq
rt[a + b*x])

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Maple [A]  time = 0.007, size = 52, normalized size = 0.6 \[ -{\frac{-16\,A{b}^{2}{x}^{2}+12\,B{x}^{2}ab-8\,aAbx+6\,{a}^{2}Bx+2\,A{a}^{2}}{3\,{a}^{3}}{x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^(5/2)/(b*x+a)^(3/2),x)

[Out]

-2/3*(-8*A*b^2*x^2+6*B*a*b*x^2-4*A*a*b*x+3*B*a^2*x+A*a^2)/x^(3/2)/(b*x+a)^(1/2)/
a^3

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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^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.217769, size = 70, normalized size = 0.84 \[ -\frac{2 \,{\left (A a^{2} + 2 \,{\left (3 \, B a b - 4 \, A b^{2}\right )} x^{2} +{\left (3 \, B a^{2} - 4 \, A a b\right )} x\right )}}{3 \, \sqrt{b x + a} a^{3} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^(3/2)*x^(5/2)),x, algorithm="fricas")

[Out]

-2/3*(A*a^2 + 2*(3*B*a*b - 4*A*b^2)*x^2 + (3*B*a^2 - 4*A*a*b)*x)/(sqrt(b*x + a)*
a^3*x^(3/2))

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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**(5/2)/(b*x+a)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.241426, size = 200, normalized size = 2.41 \[ \frac{\sqrt{b x + a}{\left (\frac{{\left (3 \, B a^{3} b^{3}{\left | b \right |} - 5 \, A a^{2} b^{4}{\left | b \right |}\right )}{\left (b x + a\right )}}{a^{2} b^{6}} - \frac{3 \,{\left (B a^{4} b^{3}{\left | b \right |} - 2 \, A a^{3} b^{4}{\left | b \right |}\right )}}{a^{2} b^{6}}\right )}}{48 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{3}{2}}} - \frac{4 \,{\left (B a b^{\frac{5}{2}} - A b^{\frac{7}{2}}\right )}}{{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} a^{2}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^(3/2)*x^(5/2)),x, algorithm="giac")

[Out]

1/48*sqrt(b*x + a)*((3*B*a^3*b^3*abs(b) - 5*A*a^2*b^4*abs(b))*(b*x + a)/(a^2*b^6
) - 3*(B*a^4*b^3*abs(b) - 2*A*a^3*b^4*abs(b))/(a^2*b^6))/((b*x + a)*b - a*b)^(3/
2) - 4*(B*a*b^(5/2) - A*b^(7/2))/(((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a
*b))^2 + a*b)*a^2*abs(b))