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

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*Sqrt[x])

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Rubi [A]  time = 0.0140899, 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, Rules used = {78, 37} \[ \frac{2 \sqrt{a+b x} (2 A b-3 a B)}{3 a^2 \sqrt{x}}-\frac{2 A \sqrt{a+b x}}{3 a x^{3/2}} \]

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

Mathematica [A]  time = 0.0123049, 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 = 0.005, size = 30, normalized size = 0.6 \begin{align*} -{\frac{-4\,Abx+6\,Bax+2\,Aa}{3\,{a}^{2}}\sqrt{bx+a}{x}^{-{\frac{3}{2}}}} \end{align*}

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.57789, size = 81, normalized size = 1.53 \begin{align*} -\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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(5/2)/(b*x+a)^(1/2),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 = 15.7828, size = 66, normalized size = 1.25 \begin{align*} - \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} \end{align*}

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 [A]  time = 1.32924, size = 107, normalized size = 2.02 \begin{align*} \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 |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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