\(\int \frac {\sqrt {a+b x} (A+B x)}{x^{5/2}} \, dx\) [484]

Optimal result

Integrand size = 20, antiderivative size = 69 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{5/2}} \, dx=-\frac {2 B \sqrt {a+b x}}{\sqrt {x}}-\frac {2 A (a+b x)^{3/2}}{3 a x^{3/2}}+2 \sqrt {b} B \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {79, 49, 65, 223, 212} \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{5/2}} \, dx=-\frac {2 A (a+b x)^{3/2}}{3 a x^{3/2}}+2 \sqrt {b} B \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )-\frac {2 B \sqrt {a+b x}}{\sqrt {x}} \]

[In]

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

[Out]

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

Rule 49

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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]))))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 A (a+b x)^{3/2}}{3 a x^{3/2}}+B \int \frac {\sqrt {a+b x}}{x^{3/2}} \, dx \\ & = -\frac {2 B \sqrt {a+b x}}{\sqrt {x}}-\frac {2 A (a+b x)^{3/2}}{3 a x^{3/2}}+(b B) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx \\ & = -\frac {2 B \sqrt {a+b x}}{\sqrt {x}}-\frac {2 A (a+b x)^{3/2}}{3 a x^{3/2}}+(2 b B) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {2 B \sqrt {a+b x}}{\sqrt {x}}-\frac {2 A (a+b x)^{3/2}}{3 a x^{3/2}}+(2 b B) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right ) \\ & = -\frac {2 B \sqrt {a+b x}}{\sqrt {x}}-\frac {2 A (a+b x)^{3/2}}{3 a x^{3/2}}+2 \sqrt {b} B \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{5/2}} \, dx=-\frac {2 \sqrt {a+b x} (a A+A b x+3 a B x)}{3 a x^{3/2}}-2 \sqrt {b} B \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.17 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.13

method	result	size
risch	\(-\frac {2 \sqrt {b x +a}\, \left (A b x +3 B a x +A a \right )}{3 x^{\frac {3}{2}} a}+\frac {B \sqrt {b}\, \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) \sqrt {x \left (b x +a \right )}}{\sqrt {x}\, \sqrt {b x +a}}\)	\(78\)
default	\(-\frac {\sqrt {b x +a}\, \left (-3 B \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a b \,x^{2}+2 A \,b^{\frac {3}{2}} x \sqrt {x \left (b x +a \right )}+6 B a x \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 A a \sqrt {x \left (b x +a \right )}\, \sqrt {b}\right )}{3 x^{\frac {3}{2}} \sqrt {x \left (b x +a \right )}\, a \sqrt {b}}\)	\(112\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.94 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{5/2}} \, dx=\left [\frac {3 \, B a \sqrt {b} x^{2} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (A a + {\left (3 \, B a + A b\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{3 \, a x^{2}}, -\frac {2 \, {\left (3 \, B a \sqrt {-b} x^{2} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (A a + {\left (3 \, B a + A b\right )} x\right )} \sqrt {b x + a} \sqrt {x}\right )}}{3 \, a x^{2}}\right ] \]

[In]

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

[Out]

[1/3*(3*B*a*sqrt(b)*x^2*log(2*b*x + 2*sqrt(b*x + a)*sqrt(b)*sqrt(x) + a) - 2*(A*a + (3*B*a + A*b)*x)*sqrt(b*x
+ a)*sqrt(x))/(a*x^2), -2/3*(3*B*a*sqrt(-b)*x^2*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) + (A*a + (3*B*a + A
*b)*x)*sqrt(b*x + a)*sqrt(x))/(a*x^2)]

Sympy [A] (verification not implemented)

Time = 1.97 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.70 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{5/2}} \, dx=- \frac {2 A \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{3 x} - \frac {2 A b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{3 a} - \frac {2 B \sqrt {a}}{\sqrt {x} \sqrt {1 + \frac {b x}{a}}} + 2 B \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )} - \frac {2 B b \sqrt {x}}{\sqrt {a} \sqrt {1 + \frac {b x}{a}}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{5/2}} \, dx=-{\left (\sqrt {b} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right ) + \frac {2 \, \sqrt {b x + a}}{\sqrt {x}}\right )} B - \frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}} A}{3 \, a x^{\frac {3}{2}}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 77.40 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.41 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{5/2}} \, dx=-\frac {2 \, {\left (3 \, B \sqrt {b} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} + \sqrt {{\left (b x + a\right )} b - a b} \right |}\right ) - \frac {{\left (3 \, B a b^{2} - \frac {{\left (3 \, B a b^{2} + A b^{3}\right )} {\left (b x + a\right )}}{a}\right )} \sqrt {b x + a}}{{\left ({\left (b x + a\right )} b - a b\right )}^{\frac {3}{2}}}\right )} b}{3 \, {\left | b \right |}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{5/2}} \, dx=\int \frac {\left (A+B\,x\right )\,\sqrt {a+b\,x}}{x^{5/2}} \,d x \]

[In]

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

[Out]

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