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

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 = 0.025061, 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, Rules used = {78, 45, 37} \[ -\frac{4 \sqrt{x} (4 A b-a B)}{3 a^3 \sqrt{a+b x}}-\frac{2 \sqrt{x} (4 A b-a B)}{3 a^2 (a+b x)^{3/2}}-\frac{2 A}{a \sqrt{x} (a+b x)^{3/2}} \]

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])

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 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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^{3/2} (a+b x)^{5/2}} \, dx &=-\frac{2 A}{a \sqrt{x} (a+b x)^{3/2}}+\frac{\left (2 \left (-2 A b+\frac{a B}{2}\right )\right ) \int \frac{1}{\sqrt{x} (a+b x)^{5/2}} \, dx}{a}\\ &=-\frac{2 A}{a \sqrt{x} (a+b x)^{3/2}}-\frac{2 (4 A b-a B) \sqrt{x}}{3 a^2 (a+b x)^{3/2}}-\frac{(2 (4 A b-a B)) \int \frac{1}{\sqrt{x} (a+b x)^{3/2}} \, dx}{3 a^2}\\ &=-\frac{2 A}{a \sqrt{x} (a+b x)^{3/2}}-\frac{2 (4 A b-a B) \sqrt{x}}{3 a^2 (a+b x)^{3/2}}-\frac{4 (4 A b-a B) \sqrt{x}}{3 a^3 \sqrt{a+b x}}\\ \end{align*}

Mathematica [A]  time = 0.0207292, 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 = 0.004, size = 53, normalized size = 0.7 \begin{align*} -{\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}}}} \end{align*}

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.80233, size = 167, normalized size = 2.06 \begin{align*} -\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 )} \sqrt{b x + a} \sqrt{x}}{3 \,{\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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 [B]  time = 1.3309, size = 285, normalized size = 3.52 \begin{align*} -\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 |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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