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

Optimal. Leaf size=65 \[ \frac{2 \sqrt{x} (a B+2 A b)}{3 a^2 b \sqrt{a+b x}}+\frac{2 \sqrt{x} (A b-a B)}{3 a b (a+b x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0698676, antiderivative size = 65, 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 \[ \frac{2 \sqrt{x} (a B+2 A b)}{3 a^2 b \sqrt{a+b x}}+\frac{2 \sqrt{x} (A b-a B)}{3 a b (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.80071, size = 56, normalized size = 0.86 \[ \frac{2 \sqrt{x} \left (A b - B a\right )}{3 a b \left (a + b x\right )^{\frac{3}{2}}} + \frac{4 \sqrt{x} \left (A b + \frac{B a}{2}\right )}{3 a^{2} b \sqrt{a + b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.043678, size = 35, normalized size = 0.54 \[ \frac{2 \sqrt{x} (3 a A+a B x+2 A b x)}{3 a^2 (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 30, normalized size = 0.5 \[{\frac{4\,Abx+2\,Bax+6\,Aa}{3\,{a}^{2}}\sqrt{x} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222385, size = 57, normalized size = 0.88 \[ \frac{2 \,{\left (3 \, A a x +{\left (B a + 2 \, A b\right )} x^{2}\right )}}{3 \,{\left (a^{2} b x + a^{3}\right )} \sqrt{b x + a} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(3*A*a*x + (B*a + 2*A*b)*x^2)/((a^2*b*x + a^3)*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)/(b*x+a)**(5/2)/x**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.231524, size = 176, normalized size = 2.71 \[ \frac{4 \,{\left (3 \, B{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} \sqrt{b} + B a^{2} b^{\frac{5}{2}} + 6 \, A{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{5}{2}} + 2 \, A a b^{\frac{7}{2}}\right )}}{3 \,{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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