3.6.20 \(\int \frac {A+B x}{x^{5/2} \sqrt {a+b x}} \, dx\) [520]

Optimal. Leaf size=53 \[ -\frac {2 A \sqrt {a+b x}}{3 a x^{3/2}}+\frac {2 (2 A b-3 a B) \sqrt {a+b x}}{3 a^2 \sqrt {x}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 53, 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 = {79, 37} \begin {gather*} \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}} \end {gather*}

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 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 79

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] || L
tQ[p, n]))))

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*}

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Mathematica [A]
time = 0.09, size = 35, normalized size = 0.66 \begin {gather*} -\frac {2 \sqrt {a+b x} (a A-2 A b x+3 a B x)}{3 a^2 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.06, size = 30, normalized size = 0.57

method	result	size
gosper	\(-\frac {2 \sqrt {b x +a}\, \left (-2 A b x +3 B a x +A a \right )}{3 x^{\frac {3}{2}} a^{2}}\)	\(30\)
default	\(-\frac {2 \sqrt {b x +a}\, \left (-2 A b x +3 B a x +A a \right )}{3 x^{\frac {3}{2}} a^{2}}\)	\(30\)
risch	\(-\frac {2 \sqrt {b x +a}\, \left (-2 A b x +3 B a x +A a \right )}{3 x^{\frac {3}{2}} a^{2}}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 [A]
time = 0.29, size = 62, normalized size = 1.17 \begin {gather*} -\frac {2 \, \sqrt {b x^{2} + a x} B}{a x} + \frac {4 \, \sqrt {b x^{2} + a x} A b}{3 \, a^{2} x} - \frac {2 \, \sqrt {b x^{2} + a x} A}{3 \, a x^{2}} \end {gather*}

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]

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

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Fricas [A]
time = 0.93, size = 30, normalized size = 0.57 \begin {gather*} -\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 {gather*}

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 = 7.68, size = 66, normalized size = 1.25 \begin {gather*} - \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 {gather*}

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 = 1.08, size = 73, normalized size = 1.38 \begin {gather*} -\frac {2 \, \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}} - \frac {3 \, {\left (B a^{2} b^{2} - A a b^{3}\right )}}{a^{2}}\right )}}{3 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {3}{2}} {\left | b \right |}} \end {gather*}

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]

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

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Mupad [B]
time = 0.81, size = 34, normalized size = 0.64 \begin {gather*} -\frac {\left (\frac {2\,A}{3\,a}-\frac {x\,\left (4\,A\,b-6\,B\,a\right )}{3\,a^2}\right )\,\sqrt {a+b\,x}}{x^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(((2*A)/(3*a) - (x*(4*A*b - 6*B*a))/(3*a^2))*(a + b*x)^(1/2))/x^(3/2)

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