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

Optimal. Leaf size=81 \[ -\frac{4 \sqrt{x} (4 A b-a B)}{3 a^3 \sqrt{a+b x}}-\frac{2 \sqrt{x} (4 A b-a B)}{3 a^2 (a+b x)^{3/2}}-\frac{2 A}{a \sqrt{x} (a+b x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0972848, antiderivative size = 81, 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 \[ -\frac{4 \sqrt{x} (4 A b-a B)}{3 a^3 \sqrt{a+b x}}-\frac{2 \sqrt{x} (4 A b-a B)}{3 a^2 (a+b x)^{3/2}}-\frac{2 A}{a \sqrt{x} (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.00388, size = 76, normalized size = 0.94 \[ - \frac{2 A}{a \sqrt{x} \left (a + b x\right )^{\frac{3}{2}}} - \frac{2 \sqrt{x} \left (4 A b - B a\right )}{3 a^{2} \left (a + b x\right )^{\frac{3}{2}}} - \frac{4 \sqrt{x} \left (4 A b - B a\right )}{3 a^{3} \sqrt{a + b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0657162, size = 54, normalized size = 0.67 \[ \frac{-6 a^2 (A-B x)+4 a b x (B x-6 A)-16 A b^2 x^2}{3 a^3 \sqrt{x} (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-16*A*b^2*x^2 - 6*a^2*(A - B*x) + 4*a*b*x*(-6*A + B*x))/(3*a^3*Sqrt[x]*(a + b*x
)^(3/2))

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Maple [A]  time = 0.006, size = 53, normalized size = 0.7 \[ -{\frac{16\,A{b}^{2}{x}^{2}-4\,B{x}^{2}ab+24\,aAbx-6\,{a}^{2}Bx+6\,A{a}^{2}}{3\,{a}^{3}}{\frac{1}{\sqrt{x}}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(8*A*b^2*x^2-2*B*a*b*x^2+12*A*a*b*x-3*B*a^2*x+3*A*a^2)/x^(1/2)/(b*x+a)^(3/2
)/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)^(5/2)*x^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233514, size = 82, normalized size = 1.01 \[ -\frac{2 \,{\left (3 \, A a^{2} - 2 \,{\left (B a b - 4 \, A b^{2}\right )} x^{2} - 3 \,{\left (B a^{2} - 4 \, A a b\right )} x\right )}}{3 \,{\left (a^{3} b x + a^{4}\right )} \sqrt{b x + a} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.250108, size = 285, normalized size = 3.52 \[ -\frac{2 \, \sqrt{b x + a} A b^{2}}{\sqrt{{\left (b x + a\right )} b - a b} a^{3}{\left | b \right |}} + \frac{4 \,{\left (6 \, B a^{2}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{5}{2}} - 3 \, A{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac{5}{2}} + 2 \, B a^{3} b^{\frac{7}{2}} - 12 \, A a{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{7}{2}} - 5 \, A a^{2} b^{\frac{9}{2}}\right )}}{3 \,{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} a^{2}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(b*x + a)*A*b^2/(sqrt((b*x + a)*b - a*b)*a^3*abs(b)) + 4/3*(6*B*a^2*(sqrt
(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2*b^(5/2) - 3*A*(sqrt(b*x + a)*sqrt
(b) - sqrt((b*x + a)*b - a*b))^4*b^(5/2) + 2*B*a^3*b^(7/2) - 12*A*a*(sqrt(b*x +
a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2*b^(7/2) - 5*A*a^2*b^(9/2))/(((sqrt(b*x +
a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)^3*a^2*abs(b))