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

Optimal. Leaf size=40 \[ -\frac{2 (A b-a B)}{3 b^2 (a+b x)^{3/2}}-\frac{2 B}{b^2 \sqrt{a+b x}} \]

[Out]

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

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Rubi [A]  time = 0.0430979, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{2 (A b-a B)}{3 b^2 (a+b x)^{3/2}}-\frac{2 B}{b^2 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.50699, size = 39, normalized size = 0.98 \[ - \frac{2 B}{b^{2} \sqrt{a + b x}} - \frac{2 \left (A b - B a\right )}{3 b^{2} \left (a + b x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0298781, size = 29, normalized size = 0.72 \[ -\frac{2 (2 a B+A b+3 b B x)}{3 b^2 (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 26, normalized size = 0.7 \[ -{\frac{6\,bBx+2\,Ab+4\,Ba}{3\,{b}^{2}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(b*x+a)^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.35919, size = 38, normalized size = 0.95 \[ -\frac{2 \,{\left (3 \,{\left (b x + a\right )} B - B a + A b\right )}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.211941, size = 47, normalized size = 1.18 \[ -\frac{2 \,{\left (3 \, B b x + 2 \, B a + A b\right )}}{3 \,{\left (b^{3} x + a b^{2}\right )} \sqrt{b x + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.78701, size = 124, normalized size = 3.1 \[ \begin{cases} - \frac{2 A b}{3 a b^{2} \sqrt{a + b x} + 3 b^{3} x \sqrt{a + b x}} - \frac{4 B a}{3 a b^{2} \sqrt{a + b x} + 3 b^{3} x \sqrt{a + b x}} - \frac{6 B b x}{3 a b^{2} \sqrt{a + b x} + 3 b^{3} x \sqrt{a + b x}} & \text{for}\: b \neq 0 \\\frac{A x + \frac{B x^{2}}{2}}{a^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-2*A*b/(3*a*b**2*sqrt(a + b*x) + 3*b**3*x*sqrt(a + b*x)) - 4*B*a/(3*a
*b**2*sqrt(a + b*x) + 3*b**3*x*sqrt(a + b*x)) - 6*B*b*x/(3*a*b**2*sqrt(a + b*x)
+ 3*b**3*x*sqrt(a + b*x)), Ne(b, 0)), ((A*x + B*x**2/2)/a**(5/2), True))

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GIAC/XCAS [A]  time = 0.208154, size = 38, normalized size = 0.95 \[ -\frac{2 \,{\left (3 \,{\left (b x + a\right )} B - B a + A b\right )}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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