\(\int \frac {\sqrt {b x+c x^2}}{x^{7/2}} \, dx\) [79]

Optimal result

Integrand size = 19, antiderivative size = 86 \[ \int \frac {\sqrt {b x+c x^2}}{x^{7/2}} \, dx=-\frac {\sqrt {b x+c x^2}}{2 x^{5/2}}-\frac {c \sqrt {b x+c x^2}}{4 b x^{3/2}}+\frac {c^2 \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 b^{3/2}} \]

[Out]

1/4*c^2*arctanh((c*x^2+b*x)^(1/2)/b^(1/2)/x^(1/2))/b^(3/2)-1/2*(c*x^2+b*x)^(1/2)/x^(5/2)-1/4*c*(c*x^2+b*x)^(1/
2)/b/x^(3/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {676, 686, 674, 213} \[ \int \frac {\sqrt {b x+c x^2}}{x^{7/2}} \, dx=\frac {c^2 \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 b^{3/2}}-\frac {c \sqrt {b x+c x^2}}{4 b x^{3/2}}-\frac {\sqrt {b x+c x^2}}{2 x^{5/2}} \]

[In]

Int[Sqrt[b*x + c*x^2]/x^(7/2),x]

[Out]

-1/2*Sqrt[b*x + c*x^2]/x^(5/2) - (c*Sqrt[b*x + c*x^2])/(4*b*x^(3/2)) + (c^2*ArcTanh[Sqrt[b*x + c*x^2]/(Sqrt[b]
*Sqrt[x])])/(4*b^(3/2))

Rule 213

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

Rule 674

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2*e, Subst[Int[1/(
2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^
2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 676

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*x + c*x^2)^p/(e*(m + p + 1))), x] - Dist[c*(p/(e^2*(m + p + 1))), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2
)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[
p, 0] && (LtQ[m, -2] || EqQ[m + 2*p + 1, 0]) && NeQ[m + p + 1, 0] && IntegerQ[2*p]

Rule 686

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-e)*(d + e*x)^m*((a
+ b*x + c*x^2)^(p + 1)/((m + p + 1)*(2*c*d - b*e))), x] + Dist[c*((m + 2*p + 2)/((m + p + 1)*(2*c*d - b*e))),
Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, 0] && NeQ[m + p + 1, 0] && IntegerQ[2*p]

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {b x+c x^2}}{2 x^{5/2}}+\frac {1}{4} c \int \frac {1}{x^{3/2} \sqrt {b x+c x^2}} \, dx \\ & = -\frac {\sqrt {b x+c x^2}}{2 x^{5/2}}-\frac {c \sqrt {b x+c x^2}}{4 b x^{3/2}}-\frac {c^2 \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx}{8 b} \\ & = -\frac {\sqrt {b x+c x^2}}{2 x^{5/2}}-\frac {c \sqrt {b x+c x^2}}{4 b x^{3/2}}-\frac {c^2 \text {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )}{4 b} \\ & = -\frac {\sqrt {b x+c x^2}}{2 x^{5/2}}-\frac {c \sqrt {b x+c x^2}}{4 b x^{3/2}}+\frac {c^2 \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 b^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {b x+c x^2}}{x^{7/2}} \, dx=\frac {\sqrt {x (b+c x)} \left (-\sqrt {b} \sqrt {b+c x} (2 b+c x)+c^2 x^2 \text {arctanh}\left (\frac {\sqrt {b+c x}}{\sqrt {b}}\right )\right )}{4 b^{3/2} x^{5/2} \sqrt {b+c x}} \]

[In]

Integrate[Sqrt[b*x + c*x^2]/x^(7/2),x]

[Out]

(Sqrt[x*(b + c*x)]*(-(Sqrt[b]*Sqrt[b + c*x]*(2*b + c*x)) + c^2*x^2*ArcTanh[Sqrt[b + c*x]/Sqrt[b]]))/(4*b^(3/2)
*x^(5/2)*Sqrt[b + c*x])

Maple [A] (verified)

Time = 2.11 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.81

method	result	size
risch	\(-\frac {\left (c x +b \right ) \left (c x +2 b \right )}{4 x^{\frac {3}{2}} b \sqrt {x \left (c x +b \right )}}+\frac {c^{2} \operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right ) \sqrt {c x +b}\, \sqrt {x}}{4 b^{\frac {3}{2}} \sqrt {x \left (c x +b \right )}}\)	\(70\)
default	\(\frac {\sqrt {x \left (c x +b \right )}\, \left (\operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right ) c^{2} x^{2}-c x \sqrt {b}\, \sqrt {c x +b}-2 b^{\frac {3}{2}} \sqrt {c x +b}\right )}{4 b^{\frac {3}{2}} x^{\frac {5}{2}} \sqrt {c x +b}}\)	\(71\)

[In]

int((c*x^2+b*x)^(1/2)/x^(7/2),x,method=_RETURNVERBOSE)

[Out]

-1/4*(c*x+b)*(c*x+2*b)/x^(3/2)/b/(x*(c*x+b))^(1/2)+1/4*c^2/b^(3/2)*arctanh((c*x+b)^(1/2)/b^(1/2))*(c*x+b)^(1/2
)*x^(1/2)/(x*(c*x+b))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.72 \[ \int \frac {\sqrt {b x+c x^2}}{x^{7/2}} \, dx=\left [\frac {\sqrt {b} c^{2} x^{3} \log \left (-\frac {c x^{2} + 2 \, b x + 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) - 2 \, {\left (b c x + 2 \, b^{2}\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{8 \, b^{2} x^{3}}, -\frac {\sqrt {-b} c^{2} x^{3} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) + {\left (b c x + 2 \, b^{2}\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{4 \, b^{2} x^{3}}\right ] \]

[In]

integrate((c*x^2+b*x)^(1/2)/x^(7/2),x, algorithm="fricas")

[Out]

[1/8*(sqrt(b)*c^2*x^3*log(-(c*x^2 + 2*b*x + 2*sqrt(c*x^2 + b*x)*sqrt(b)*sqrt(x))/x^2) - 2*(b*c*x + 2*b^2)*sqrt
(c*x^2 + b*x)*sqrt(x))/(b^2*x^3), -1/4*(sqrt(-b)*c^2*x^3*arctan(sqrt(-b)*sqrt(x)/sqrt(c*x^2 + b*x)) + (b*c*x +
2*b^2)*sqrt(c*x^2 + b*x)*sqrt(x))/(b^2*x^3)]

Sympy [F]

\[ \int \frac {\sqrt {b x+c x^2}}{x^{7/2}} \, dx=\int \frac {\sqrt {x \left (b + c x\right )}}{x^{\frac {7}{2}}}\, dx \]

[In]

integrate((c*x**2+b*x)**(1/2)/x**(7/2),x)

[Out]

Integral(sqrt(x*(b + c*x))/x**(7/2), x)

Maxima [F]

\[ \int \frac {\sqrt {b x+c x^2}}{x^{7/2}} \, dx=\int { \frac {\sqrt {c x^{2} + b x}}{x^{\frac {7}{2}}} \,d x } \]

[In]

integrate((c*x^2+b*x)^(1/2)/x^(7/2),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^2 + b*x)/x^(7/2), x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {b x+c x^2}}{x^{7/2}} \, dx=-\frac {\frac {c^{3} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b} + \frac {{\left (c x + b\right )}^{\frac {3}{2}} c^{3} + \sqrt {c x + b} b c^{3}}{b c^{2} x^{2}}}{4 \, c} \]

[In]

integrate((c*x^2+b*x)^(1/2)/x^(7/2),x, algorithm="giac")

[Out]

-1/4*(c^3*arctan(sqrt(c*x + b)/sqrt(-b))/(sqrt(-b)*b) + ((c*x + b)^(3/2)*c^3 + sqrt(c*x + b)*b*c^3)/(b*c^2*x^2
))/c

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {b x+c x^2}}{x^{7/2}} \, dx=\int \frac {\sqrt {c\,x^2+b\,x}}{x^{7/2}} \,d x \]

[In]

int((b*x + c*x^2)^(1/2)/x^(7/2),x)

[Out]

int((b*x + c*x^2)^(1/2)/x^(7/2), x)