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

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 = 0.0151402, 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, Rules used = {78, 37} \[ \frac{2 \sqrt{x} (a B+2 A b)}{3 a^2 b \sqrt{a+b x}}+\frac{2 \sqrt{x} (A b-a B)}{3 a b (a+b x)^{3/2}} \]

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])

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{A+B x}{\sqrt{x} (a+b x)^{5/2}} \, dx &=\frac{2 (A b-a B) \sqrt{x}}{3 a b (a+b x)^{3/2}}+\frac{(2 A b+a B) \int \frac{1}{\sqrt{x} (a+b x)^{3/2}} \, dx}{3 a b}\\ &=\frac{2 (A b-a B) \sqrt{x}}{3 a b (a+b x)^{3/2}}+\frac{2 (2 A b+a B) \sqrt{x}}{3 a^2 b \sqrt{a+b x}}\\ \end{align*}

Mathematica [A]  time = 0.0146887, 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 = 0.005, size = 30, normalized size = 0.5 \begin{align*}{\frac{4\,Abx+2\,Bax+6\,Aa}{3\,{a}^{2}}\sqrt{x} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.00805, size = 115, normalized size = 1.77 \begin{align*} \frac{2 \,{\left (3 \, A a +{\left (B a + 2 \, A b\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{3 \,{\left (a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a)^(5/2)/x^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 72.9026, size = 139, normalized size = 2.14 \begin{align*} A \left (\frac{6 a}{3 a^{3} \sqrt{b} \sqrt{\frac{a}{b x} + 1} + 3 a^{2} b^{\frac{3}{2}} x \sqrt{\frac{a}{b x} + 1}} + \frac{4 b x}{3 a^{3} \sqrt{b} \sqrt{\frac{a}{b x} + 1} + 3 a^{2} b^{\frac{3}{2}} x \sqrt{\frac{a}{b x} + 1}}\right ) + \frac{2 B x^{\frac{3}{2}}}{3 a^{\frac{5}{2}} \sqrt{1 + \frac{b x}{a}} + 3 a^{\frac{3}{2}} b x \sqrt{1 + \frac{b x}{a}}} \end{align*}

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]

A*(6*a/(3*a**3*sqrt(b)*sqrt(a/(b*x) + 1) + 3*a**2*b**(3/2)*x*sqrt(a/(b*x) + 1)) + 4*b*x/(3*a**3*sqrt(b)*sqrt(a
/(b*x) + 1) + 3*a**2*b**(3/2)*x*sqrt(a/(b*x) + 1))) + 2*B*x**(3/2)/(3*a**(5/2)*sqrt(1 + b*x/a) + 3*a**(3/2)*b*
x*sqrt(1 + b*x/a))

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Giac [B]  time = 1.23685, size = 176, normalized size = 2.71 \begin{align*} \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 |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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