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

Optimal. Leaf size=83 \[ \frac{4 \sqrt{a+b x} (4 A b-3 a B)}{3 a^3 \sqrt{x}}-\frac{2 (4 A b-3 a B)}{3 a^2 \sqrt{x} \sqrt{a+b x}}-\frac{2 A}{3 a x^{3/2} \sqrt{a+b x}} \]

[Out]

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

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Rubi [A]  time = 0.0257422, antiderivative size = 83, 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} \[ \frac{4 \sqrt{a+b x} (4 A b-3 a B)}{3 a^3 \sqrt{x}}-\frac{2 (4 A b-3 a B)}{3 a^2 \sqrt{x} \sqrt{a+b x}}-\frac{2 A}{3 a x^{3/2} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.01778, size = 54, normalized size = 0.65 \[ -\frac{2 \left (a^2 (A+3 B x)+2 a b x (3 B x-2 A)-8 A b^2 x^2\right )}{3 a^3 x^{3/2} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(-8*A*b^2*x^2 + 2*a*b*x*(-2*A + 3*B*x) + a^2*(A + 3*B*x)))/(3*a^3*x^(3/2)*Sqrt[a + b*x])

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Maple [A]  time = 0.004, size = 52, normalized size = 0.6 \begin{align*} -{\frac{-16\,A{b}^{2}{x}^{2}+12\,B{x}^{2}ab-8\,aAbx+6\,{a}^{2}Bx+2\,A{a}^{2}}{3\,{a}^{3}}{x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{bx+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(-8*A*b^2*x^2+6*B*a*b*x^2-4*A*a*b*x+3*B*a^2*x+A*a^2)/x^(3/2)/(b*x+a)^(1/2)/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^(5/2)/(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.90889, size = 149, normalized size = 1.8 \begin{align*} -\frac{2 \,{\left (A a^{2} + 2 \,{\left (3 \, B a b - 4 \, A b^{2}\right )} x^{2} +{\left (3 \, B a^{2} - 4 \, A a b\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{3 \,{\left (a^{3} b x^{3} + a^{4} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(A*a^2 + 2*(3*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*x^3 + a^4*x^2)

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Sympy [B]  time = 117.854, size = 265, normalized size = 3.19 \begin{align*} A \left (- \frac{2 a^{3} b^{\frac{9}{2}} \sqrt{\frac{a}{b x} + 1}}{3 a^{5} b^{4} x + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} + \frac{6 a^{2} b^{\frac{11}{2}} x \sqrt{\frac{a}{b x} + 1}}{3 a^{5} b^{4} x + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} + \frac{24 a b^{\frac{13}{2}} x^{2} \sqrt{\frac{a}{b x} + 1}}{3 a^{5} b^{4} x + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} + \frac{16 b^{\frac{15}{2}} x^{3} \sqrt{\frac{a}{b x} + 1}}{3 a^{5} b^{4} x + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}}\right ) + B \left (- \frac{2}{a \sqrt{b} x \sqrt{\frac{a}{b x} + 1}} - \frac{4 \sqrt{b}}{a^{2} \sqrt{\frac{a}{b x} + 1}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*(-2*a**3*b**(9/2)*sqrt(a/(b*x) + 1)/(3*a**5*b**4*x + 6*a**4*b**5*x**2 + 3*a**3*b**6*x**3) + 6*a**2*b**(11/2)
*x*sqrt(a/(b*x) + 1)/(3*a**5*b**4*x + 6*a**4*b**5*x**2 + 3*a**3*b**6*x**3) + 24*a*b**(13/2)*x**2*sqrt(a/(b*x)
+ 1)/(3*a**5*b**4*x + 6*a**4*b**5*x**2 + 3*a**3*b**6*x**3) + 16*b**(15/2)*x**3*sqrt(a/(b*x) + 1)/(3*a**5*b**4*
x + 6*a**4*b**5*x**2 + 3*a**3*b**6*x**3)) + B*(-2/(a*sqrt(b)*x*sqrt(a/(b*x) + 1)) - 4*sqrt(b)/(a**2*sqrt(a/(b*
x) + 1)))

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Giac [B]  time = 2.46844, size = 200, normalized size = 2.41 \begin{align*} \frac{\sqrt{b x + a}{\left (\frac{{\left (3 \, B a^{3} b^{3}{\left | b \right |} - 5 \, A a^{2} b^{4}{\left | b \right |}\right )}{\left (b x + a\right )}}{a^{2} b^{6}} - \frac{3 \,{\left (B a^{4} b^{3}{\left | b \right |} - 2 \, A a^{3} b^{4}{\left | b \right |}\right )}}{a^{2} b^{6}}\right )}}{48 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{3}{2}}} - \frac{4 \,{\left (B a b^{\frac{5}{2}} - A b^{\frac{7}{2}}\right )}}{{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} a^{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)^(3/2),x, algorithm="giac")

[Out]

1/48*sqrt(b*x + a)*((3*B*a^3*b^3*abs(b) - 5*A*a^2*b^4*abs(b))*(b*x + a)/(a^2*b^6) - 3*(B*a^4*b^3*abs(b) - 2*A*
a^3*b^4*abs(b))/(a^2*b^6))/((b*x + a)*b - a*b)^(3/2) - 4*(B*a*b^(5/2) - A*b^(7/2))/(((sqrt(b*x + a)*sqrt(b) -
sqrt((b*x + a)*b - a*b))^2 + a*b)*a^2*abs(b))