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

Optimal. Leaf size=53 \[ \frac{2 \sqrt{a+b x} (2 A b-3 a B)}{3 a^2 \sqrt{x}}-\frac{2 A \sqrt{a+b x}}{3 a x^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0667485, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{2 \sqrt{a+b x} (2 A b-3 a B)}{3 a^2 \sqrt{x}}-\frac{2 A \sqrt{a+b x}}{3 a x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.31903, size = 49, normalized size = 0.92 \[ - \frac{2 A \sqrt{a + b x}}{3 a x^{\frac{3}{2}}} + \frac{4 \sqrt{a + b x} \left (A b - \frac{3 B a}{2}\right )}{3 a^{2} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**(5/2)/(b*x+a)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0436959, size = 35, normalized size = 0.66 \[ -\frac{2 \sqrt{a+b x} (a (A+3 B x)-2 A b x)}{3 a^2 x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 30, normalized size = 0.6 \[ -{\frac{-4\,Abx+6\,Bax+2\,Aa}{3\,{a}^{2}}\sqrt{bx+a}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^(5/2)/(b*x+a)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241817, size = 41, normalized size = 0.77 \[ -\frac{2 \,{\left (A a +{\left (3 \, B a - 2 \, A b\right )} x\right )} \sqrt{b x + a}}{3 \, a^{2} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 49.4042, size = 66, normalized size = 1.25 \[ - \frac{2 A \sqrt{b} \sqrt{\frac{a}{b x} + 1}}{3 a x} + \frac{4 A b^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}}{3 a^{2}} - \frac{2 B \sqrt{b} \sqrt{\frac{a}{b x} + 1}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/x**(5/2)/(b*x+a)**(1/2),x)

[Out]

-2*A*sqrt(b)*sqrt(a/(b*x) + 1)/(3*a*x) + 4*A*b**(3/2)*sqrt(a/(b*x) + 1)/(3*a**2)
- 2*B*sqrt(b)*sqrt(a/(b*x) + 1)/a

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GIAC/XCAS [A]  time = 0.222155, size = 107, normalized size = 2.02 \[ \frac{\sqrt{b x + a} b{\left (\frac{{\left (3 \, B a b^{2} - 2 \, A b^{3}\right )}{\left (b x + a\right )}}{a^{2} b^{6}} - \frac{3 \,{\left (B a^{2} b^{2} - A a b^{3}\right )}}{a^{2} b^{6}}\right )}}{48 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{3}{2}}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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