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

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

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Rubi [A]  time = 0.02, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {78, 37} \begin {gather*} \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}} \end {gather*}

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 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]

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]))))

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*}

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Mathematica [A]  time = 0.02, size = 35, normalized size = 0.54 \begin {gather*} \frac {2 \sqrt {x} (3 a A+a B x+2 A b x)}{3 a^2 (a+b x)^{3/2}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.13, size = 43, normalized size = 0.66 \begin {gather*} \frac {2 \left (3 a A \sqrt {x}+a B x^{3/2}+2 A b x^{3/2}\right )}{3 a^2 (a+b x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.54, size = 50, normalized size = 0.77 \begin {gather*} \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 {gather*}

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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giac [B]  time = 1.63, size = 130, normalized size = 2.00 \begin {gather*} \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 {gather*}

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))

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maple [A]  time = 0.00, size = 30, normalized size = 0.46 \begin {gather*} \frac {2 \left (2 A x b +B a x +3 A a \right ) \sqrt {x}}{3 \left (b x +a \right )^{\frac {3}{2}} a^{2}} \end {gather*}

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 [B]  time = 0.96, size = 161, normalized size = 2.48 \begin {gather*} -\frac {2 \, \sqrt {b x^{2} + a x} B a}{3 \, {\left (a b^{3} x^{2} + 2 \, a^{2} b^{2} x + a^{3} b\right )}} - \frac {4 \, \sqrt {b x^{2} + a x} B a}{3 \, {\left (a^{2} b^{2} x + a^{3} b\right )}} + \frac {2 \, \sqrt {b x^{2} + a x} A}{3 \, {\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )}} + \frac {4 \, \sqrt {b x^{2} + a x} A}{3 \, {\left (a^{2} b x + a^{3}\right )}} + \frac {2 \, \sqrt {b x^{2} + a x} B}{a b^{2} x + a^{2} b} \end {gather*}

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]

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

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mupad [B]  time = 0.88, size = 64, normalized size = 0.98 \begin {gather*} \frac {\left (\frac {x^2\,\left (4\,A\,b+2\,B\,a\right )}{3\,a^2\,b^2}+\frac {2\,A\,x}{a\,b^2}\right )\,\sqrt {a+b\,x}}{x^{5/2}+\frac {2\,a\,x^{3/2}}{b}+\frac {a^2\,\sqrt {x}}{b^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 39.11, size = 139, normalized size = 2.14 \begin {gather*} 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 {gather*}

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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