\(\int \frac {1}{(d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx\) [1709]

Optimal result

Integrand size = 30, antiderivative size = 168 \[ \int \frac {1}{(d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {2 (a+b x)}{3 (b d-a e) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 b (a+b x)}{(b d-a e)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b^{3/2} (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]

[Out]

2/3*(b*x+a)/(-a*e+b*d)/(e*x+d)^(3/2)/((b*x+a)^2)^(1/2)-2*b^(3/2)*(b*x+a)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b
*d)^(1/2))/(-a*e+b*d)^(5/2)/((b*x+a)^2)^(1/2)+2*b*(b*x+a)/(-a*e+b*d)^2/(e*x+d)^(1/2)/((b*x+a)^2)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {660, 53, 65, 214} \[ \int \frac {1}{(d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {2 b^{3/2} (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{5/2}}+\frac {2 b (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^2}+\frac {2 (a+b x)}{3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)} \]

[In]

Int[1/((d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

(2*(a + b*x))/(3*(b*d - a*e)*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*b*(a + b*x))/((b*d - a*e)^2*S
qrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (2*b^(3/2)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d -
a*e]])/((b*d - a*e)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

Rule 53

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*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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 214

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

Rule 660

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

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.67 \[ \int \frac {1}{(d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {2 (a+b x) \left (\sqrt {-b d+a e} (4 b d-a e+3 b e x)+3 b^{3/2} (d+e x)^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )\right )}{3 (-b d+a e)^{5/2} \sqrt {(a+b x)^2} (d+e x)^{3/2}} \]

[In]

Integrate[1/((d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

(2*(a + b*x)*(Sqrt[-(b*d) + a*e]*(4*b*d - a*e + 3*b*e*x) + 3*b^(3/2)*(d + e*x)^(3/2)*ArcTan[(Sqrt[b]*Sqrt[d +
e*x])/Sqrt[-(b*d) + a*e]]))/(3*(-(b*d) + a*e)^(5/2)*Sqrt[(a + b*x)^2]*(d + e*x)^(3/2))

Maple [A] (verified)

Time = 2.58 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.77

method	result	size
default	\(-\frac {2 \left (b x +a \right ) \left (-3 b^{2} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) \left (e x +d \right )^{\frac {3}{2}}-3 \sqrt {\left (a e -b d \right ) b}\, b e x +\sqrt {\left (a e -b d \right ) b}\, a e -4 \sqrt {\left (a e -b d \right ) b}\, b d \right )}{3 \sqrt {\left (b x +a \right )^{2}}\, \left (a e -b d \right )^{2} \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}}\)	\(129\)

[In]

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

[Out]

-2/3*(b*x+a)*(-3*b^2*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*(e*x+d)^(3/2)-3*((a*e-b*d)*b)^(1/2)*b*e*x+((a
*e-b*d)*b)^(1/2)*a*e-4*((a*e-b*d)*b)^(1/2)*b*d)/((b*x+a)^2)^(1/2)/(a*e-b*d)^2/(e*x+d)^(3/2)/((a*e-b*d)*b)^(1/2
)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 398, normalized size of antiderivative = 2.37 \[ \int \frac {1}{(d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\left [\frac {3 \, {\left (b e^{2} x^{2} + 2 \, b d e x + b d^{2}\right )} \sqrt {\frac {b}{b d - a e}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, {\left (b d - a e\right )} \sqrt {e x + d} \sqrt {\frac {b}{b d - a e}}}{b x + a}\right ) + 2 \, {\left (3 \, b e x + 4 \, b d - a e\right )} \sqrt {e x + d}}{3 \, {\left (b^{2} d^{4} - 2 \, a b d^{3} e + a^{2} d^{2} e^{2} + {\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} x^{2} + 2 \, {\left (b^{2} d^{3} e - 2 \, a b d^{2} e^{2} + a^{2} d e^{3}\right )} x\right )}}, -\frac {2 \, {\left (3 \, {\left (b e^{2} x^{2} + 2 \, b d e x + b d^{2}\right )} \sqrt {-\frac {b}{b d - a e}} \arctan \left (-\frac {{\left (b d - a e\right )} \sqrt {e x + d} \sqrt {-\frac {b}{b d - a e}}}{b e x + b d}\right ) - {\left (3 \, b e x + 4 \, b d - a e\right )} \sqrt {e x + d}\right )}}{3 \, {\left (b^{2} d^{4} - 2 \, a b d^{3} e + a^{2} d^{2} e^{2} + {\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} x^{2} + 2 \, {\left (b^{2} d^{3} e - 2 \, a b d^{2} e^{2} + a^{2} d e^{3}\right )} x\right )}}\right ] \]

[In]

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

[Out]

[1/3*(3*(b*e^2*x^2 + 2*b*d*e*x + b*d^2)*sqrt(b/(b*d - a*e))*log((b*e*x + 2*b*d - a*e - 2*(b*d - a*e)*sqrt(e*x
+ d)*sqrt(b/(b*d - a*e)))/(b*x + a)) + 2*(3*b*e*x + 4*b*d - a*e)*sqrt(e*x + d))/(b^2*d^4 - 2*a*b*d^3*e + a^2*d
^2*e^2 + (b^2*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*x^2 + 2*(b^2*d^3*e - 2*a*b*d^2*e^2 + a^2*d*e^3)*x), -2/3*(3*(b*
e^2*x^2 + 2*b*d*e*x + b*d^2)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a*e)*sqrt(e*x + d)*sqrt(-b/(b*d - a*e))/(b*e*
x + b*d)) - (3*b*e*x + 4*b*d - a*e)*sqrt(e*x + d))/(b^2*d^4 - 2*a*b*d^3*e + a^2*d^2*e^2 + (b^2*d^2*e^2 - 2*a*b
*d*e^3 + a^2*e^4)*x^2 + 2*(b^2*d^3*e - 2*a*b*d^2*e^2 + a^2*d*e^3)*x)]

Sympy [F]

\[ \int \frac {1}{(d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {1}{\left (d + e x\right )^{\frac {5}{2}} \sqrt {\left (a + b x\right )^{2}}}\, dx \]

[In]

integrate(1/(e*x+d)**(5/2)/((b*x+a)**2)**(1/2),x)

[Out]

Integral(1/((d + e*x)**(5/2)*sqrt((a + b*x)**2)), x)

Maxima [F]

\[ \int \frac {1}{(d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int { \frac {1}{\sqrt {{\left (b x + a\right )}^{2}} {\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

integrate(1/(sqrt((b*x + a)^2)*(e*x + d)^(5/2)), x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.71 \[ \int \frac {1}{(d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {2}{3} \, {\left (\frac {3 \, b^{2} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \sqrt {-b^{2} d + a b e}} + \frac {3 \, {\left (e x + d\right )} b + b d - a e}{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}}}\right )} \mathrm {sgn}\left (b x + a\right ) \]

[In]

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

[Out]

2/3*(3*b^2*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^2*d^2 - 2*a*b*d*e + a^2*e^2)*sqrt(-b^2*d + a*b*e))
+ (3*(e*x + d)*b + b*d - a*e)/((b^2*d^2 - 2*a*b*d*e + a^2*e^2)*(e*x + d)^(3/2)))*sgn(b*x + a)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {1}{\sqrt {{\left (a+b\,x\right )}^2}\,{\left (d+e\,x\right )}^{5/2}} \,d x \]

[In]

int(1/(((a + b*x)^2)^(1/2)*(d + e*x)^(5/2)),x)

[Out]

int(1/(((a + b*x)^2)^(1/2)*(d + e*x)^(5/2)), x)