3.18.26 \(\int \frac {\sqrt {d+e x}}{(a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\) [1726]

Optimal. Leaf size=280 \[ -\frac {5 e^3 \sqrt {d+e x}}{64 b (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\sqrt {d+e x}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e \sqrt {d+e x}}{24 b (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e^2 \sqrt {d+e x}}{96 b (b d-a e)^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e^4 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{3/2} (b d-a e)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]

[Out]

5/64*e^4*(b*x+a)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/b^(3/2)/(-a*e+b*d)^(7/2)/((b*x+a)^2)^(1/2)-5/
64*e^3*(e*x+d)^(1/2)/b/(-a*e+b*d)^3/((b*x+a)^2)^(1/2)-1/4*(e*x+d)^(1/2)/b/(b*x+a)^3/((b*x+a)^2)^(1/2)-1/24*e*(
e*x+d)^(1/2)/b/(-a*e+b*d)/(b*x+a)^2/((b*x+a)^2)^(1/2)+5/96*e^2*(e*x+d)^(1/2)/b/(-a*e+b*d)^2/(b*x+a)/((b*x+a)^2
)^(1/2)

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Rubi [A]
time = 0.12, antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {660, 43, 44, 65, 214} \begin {gather*} -\frac {5 e^3 \sqrt {d+e x}}{64 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac {5 e^2 \sqrt {d+e x}}{96 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac {e \sqrt {d+e x}}{24 b (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {\sqrt {d+e x}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e^4 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*e^3*Sqrt[d + e*x])/(64*b*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - Sqrt[d + e*x]/(4*b*(a + b*x)^3*Sqr
t[a^2 + 2*a*b*x + b^2*x^2]) - (e*Sqrt[d + e*x])/(24*b*(b*d - a*e)*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) +
(5*e^2*Sqrt[d + e*x])/(96*b*(b*d - a*e)^2*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*e^4*(a + b*x)*ArcTanh
[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(3/2)*(b*d - a*e)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

Rule 43

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, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

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*} \int \frac {\sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {\sqrt {d+e x}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (b^2 e \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^4 \sqrt {d+e x}} \, dx}{8 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {\sqrt {d+e x}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e \sqrt {d+e x}}{24 b (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (5 b e^2 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^3 \sqrt {d+e x}} \, dx}{48 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {\sqrt {d+e x}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e \sqrt {d+e x}}{24 b (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e^2 \sqrt {d+e x}}{96 b (b d-a e)^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (5 e^3 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^2 \sqrt {d+e x}} \, dx}{64 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {5 e^3 \sqrt {d+e x}}{64 b (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\sqrt {d+e x}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e \sqrt {d+e x}}{24 b (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e^2 \sqrt {d+e x}}{96 b (b d-a e)^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (5 e^4 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{128 b (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {5 e^3 \sqrt {d+e x}}{64 b (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\sqrt {d+e x}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e \sqrt {d+e x}}{24 b (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e^2 \sqrt {d+e x}}{96 b (b d-a e)^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (5 e^3 \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 )}{64 b (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {5 e^3 \sqrt {d+e x}}{64 b (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\sqrt {d+e x}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e \sqrt {d+e x}}{24 b (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e^2 \sqrt {d+e x}}{96 b (b d-a e)^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e^4 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{3/2} (b d-a e)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.90, size = 192, normalized size = 0.69 \begin {gather*} \frac {e^4 (a+b x)^5 \left (\frac {\sqrt {b} \sqrt {d+e x} \left (-15 a^3 e^3+a^2 b e^2 (118 d+73 e x)+a b^2 e \left (-136 d^2-36 d e x+55 e^2 x^2\right )+b^3 \left (48 d^3+8 d^2 e x-10 d e^2 x^2+15 e^3 x^3\right )\right )}{e^4 (-b d+a e)^3 (a+b x)^4}+\frac {15 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{7/2}}\right )}{192 b^{3/2} \left ((a+b x)^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e^4*(a + b*x)^5*((Sqrt[b]*Sqrt[d + e*x]*(-15*a^3*e^3 + a^2*b*e^2*(118*d + 73*e*x) + a*b^2*e*(-136*d^2 - 36*d*
e*x + 55*e^2*x^2) + b^3*(48*d^3 + 8*d^2*e*x - 10*d*e^2*x^2 + 15*e^3*x^3)))/(e^4*(-(b*d) + a*e)^3*(a + b*x)^4)
+ (15*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(-(b*d) + a*e)^(7/2)))/(192*b^(3/2)*((a + b*x)^2)^(5
/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(499\) vs. \(2(197)=394\).
time = 0.69, size = 500, normalized size = 1.79

method	result	size
default	\(\frac {\left (b x +a \right ) \left (15 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) b^{4} e^{4} x^{4}+60 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a \,b^{3} e^{4} x^{3}+15 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {7}{2}} b^{3}+90 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{2} b^{2} e^{4} x^{2}+55 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {5}{2}} a \,b^{2} e -55 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {5}{2}} b^{3} d +60 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{3} b \,e^{4} x +73 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {3}{2}} a^{2} b \,e^{2}-146 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {3}{2}} a \,b^{2} d e +73 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {3}{2}} b^{3} d^{2}+15 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{4} e^{4}-15 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a^{3} e^{3}+45 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a^{2} b d \,e^{2}-45 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a \,b^{2} d^{2} e +15 \sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, b^{3} d^{3}\right )}{192 \sqrt {b \left (a e -b d \right )}\, b \left (a e -b d \right ) \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\)	\(500\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*(b*x+a)*(15*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*b^4*e^4*x^4+60*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*
d))^(1/2))*a*b^3*e^4*x^3+15*(b*(a*e-b*d))^(1/2)*(e*x+d)^(7/2)*b^3+90*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2
))*a^2*b^2*e^4*x^2+55*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*a*b^2*e-55*(b*(a*e-b*d))^(1/2)*(e*x+d)^(5/2)*b^3*d+60*
arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*a^3*b*e^4*x+73*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a^2*b*e^2-146*(b*
(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*a*b^2*d*e+73*(b*(a*e-b*d))^(1/2)*(e*x+d)^(3/2)*b^3*d^2+15*arctan(b*(e*x+d)^(1/2
)/(b*(a*e-b*d))^(1/2))*a^4*e^4-15*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a^3*e^3+45*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/
2)*a^2*b*d*e^2-45*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*a*b^2*d^2*e+15*(b*(a*e-b*d))^(1/2)*(e*x+d)^(1/2)*b^3*d^3)/
(b*(a*e-b*d))^(1/2)/b/(a*e-b*d)/(a^2*e^2-2*a*b*d*e+b^2*d^2)/((b*x+a)^2)^(5/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 558 vs. \(2 (205) = 410\).
time = 2.60, size = 1131, normalized size = 4.04 \begin {gather*} \left [-\frac {15 \, {\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \sqrt {b^{2} d - a b e} e^{4} \log \left (\frac {2 \, b d + {\left (b x - a\right )} e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {x e + d}}{b x + a}\right ) + 2 \, {\left (48 \, b^{5} d^{4} - {\left (15 \, a b^{4} x^{3} + 55 \, a^{2} b^{3} x^{2} + 73 \, a^{3} b^{2} x - 15 \, a^{4} b\right )} e^{4} + {\left (15 \, b^{5} d x^{3} + 65 \, a b^{4} d x^{2} + 109 \, a^{2} b^{3} d x - 133 \, a^{3} b^{2} d\right )} e^{3} - 2 \, {\left (5 \, b^{5} d^{2} x^{2} + 22 \, a b^{4} d^{2} x - 127 \, a^{2} b^{3} d^{2}\right )} e^{2} + 8 \, {\left (b^{5} d^{3} x - 23 \, a b^{4} d^{3}\right )} e\right )} \sqrt {x e + d}}{384 \, {\left (b^{10} d^{4} x^{4} + 4 \, a b^{9} d^{4} x^{3} + 6 \, a^{2} b^{8} d^{4} x^{2} + 4 \, a^{3} b^{7} d^{4} x + a^{4} b^{6} d^{4} + {\left (a^{4} b^{6} x^{4} + 4 \, a^{5} b^{5} x^{3} + 6 \, a^{6} b^{4} x^{2} + 4 \, a^{7} b^{3} x + a^{8} b^{2}\right )} e^{4} - 4 \, {\left (a^{3} b^{7} d x^{4} + 4 \, a^{4} b^{6} d x^{3} + 6 \, a^{5} b^{5} d x^{2} + 4 \, a^{6} b^{4} d x + a^{7} b^{3} d\right )} e^{3} + 6 \, {\left (a^{2} b^{8} d^{2} x^{4} + 4 \, a^{3} b^{7} d^{2} x^{3} + 6 \, a^{4} b^{6} d^{2} x^{2} + 4 \, a^{5} b^{5} d^{2} x + a^{6} b^{4} d^{2}\right )} e^{2} - 4 \, {\left (a b^{9} d^{3} x^{4} + 4 \, a^{2} b^{8} d^{3} x^{3} + 6 \, a^{3} b^{7} d^{3} x^{2} + 4 \, a^{4} b^{6} d^{3} x + a^{5} b^{5} d^{3}\right )} e\right )}}, -\frac {15 \, {\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {x e + d}}{b x e + b d}\right ) e^{4} + {\left (48 \, b^{5} d^{4} - {\left (15 \, a b^{4} x^{3} + 55 \, a^{2} b^{3} x^{2} + 73 \, a^{3} b^{2} x - 15 \, a^{4} b\right )} e^{4} + {\left (15 \, b^{5} d x^{3} + 65 \, a b^{4} d x^{2} + 109 \, a^{2} b^{3} d x - 133 \, a^{3} b^{2} d\right )} e^{3} - 2 \, {\left (5 \, b^{5} d^{2} x^{2} + 22 \, a b^{4} d^{2} x - 127 \, a^{2} b^{3} d^{2}\right )} e^{2} + 8 \, {\left (b^{5} d^{3} x - 23 \, a b^{4} d^{3}\right )} e\right )} \sqrt {x e + d}}{192 \, {\left (b^{10} d^{4} x^{4} + 4 \, a b^{9} d^{4} x^{3} + 6 \, a^{2} b^{8} d^{4} x^{2} + 4 \, a^{3} b^{7} d^{4} x + a^{4} b^{6} d^{4} + {\left (a^{4} b^{6} x^{4} + 4 \, a^{5} b^{5} x^{3} + 6 \, a^{6} b^{4} x^{2} + 4 \, a^{7} b^{3} x + a^{8} b^{2}\right )} e^{4} - 4 \, {\left (a^{3} b^{7} d x^{4} + 4 \, a^{4} b^{6} d x^{3} + 6 \, a^{5} b^{5} d x^{2} + 4 \, a^{6} b^{4} d x + a^{7} b^{3} d\right )} e^{3} + 6 \, {\left (a^{2} b^{8} d^{2} x^{4} + 4 \, a^{3} b^{7} d^{2} x^{3} + 6 \, a^{4} b^{6} d^{2} x^{2} + 4 \, a^{5} b^{5} d^{2} x + a^{6} b^{4} d^{2}\right )} e^{2} - 4 \, {\left (a b^{9} d^{3} x^{4} + 4 \, a^{2} b^{8} d^{3} x^{3} + 6 \, a^{3} b^{7} d^{3} x^{2} + 4 \, a^{4} b^{6} d^{3} x + a^{5} b^{5} d^{3}\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/384*(15*(b^4*x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x + a^4)*sqrt(b^2*d - a*b*e)*e^4*log((2*b*d + (b*
x - a)*e - 2*sqrt(b^2*d - a*b*e)*sqrt(x*e + d))/(b*x + a)) + 2*(48*b^5*d^4 - (15*a*b^4*x^3 + 55*a^2*b^3*x^2 +
73*a^3*b^2*x - 15*a^4*b)*e^4 + (15*b^5*d*x^3 + 65*a*b^4*d*x^2 + 109*a^2*b^3*d*x - 133*a^3*b^2*d)*e^3 - 2*(5*b^
5*d^2*x^2 + 22*a*b^4*d^2*x - 127*a^2*b^3*d^2)*e^2 + 8*(b^5*d^3*x - 23*a*b^4*d^3)*e)*sqrt(x*e + d))/(b^10*d^4*x
^4 + 4*a*b^9*d^4*x^3 + 6*a^2*b^8*d^4*x^2 + 4*a^3*b^7*d^4*x + a^4*b^6*d^4 + (a^4*b^6*x^4 + 4*a^5*b^5*x^3 + 6*a^
6*b^4*x^2 + 4*a^7*b^3*x + a^8*b^2)*e^4 - 4*(a^3*b^7*d*x^4 + 4*a^4*b^6*d*x^3 + 6*a^5*b^5*d*x^2 + 4*a^6*b^4*d*x
+ a^7*b^3*d)*e^3 + 6*(a^2*b^8*d^2*x^4 + 4*a^3*b^7*d^2*x^3 + 6*a^4*b^6*d^2*x^2 + 4*a^5*b^5*d^2*x + a^6*b^4*d^2)
*e^2 - 4*(a*b^9*d^3*x^4 + 4*a^2*b^8*d^3*x^3 + 6*a^3*b^7*d^3*x^2 + 4*a^4*b^6*d^3*x + a^5*b^5*d^3)*e), -1/192*(1
5*(b^4*x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x + a^4)*sqrt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*s
qrt(x*e + d)/(b*x*e + b*d))*e^4 + (48*b^5*d^4 - (15*a*b^4*x^3 + 55*a^2*b^3*x^2 + 73*a^3*b^2*x - 15*a^4*b)*e^4
+ (15*b^5*d*x^3 + 65*a*b^4*d*x^2 + 109*a^2*b^3*d*x - 133*a^3*b^2*d)*e^3 - 2*(5*b^5*d^2*x^2 + 22*a*b^4*d^2*x -
127*a^2*b^3*d^2)*e^2 + 8*(b^5*d^3*x - 23*a*b^4*d^3)*e)*sqrt(x*e + d))/(b^10*d^4*x^4 + 4*a*b^9*d^4*x^3 + 6*a^2*
b^8*d^4*x^2 + 4*a^3*b^7*d^4*x + a^4*b^6*d^4 + (a^4*b^6*x^4 + 4*a^5*b^5*x^3 + 6*a^6*b^4*x^2 + 4*a^7*b^3*x + a^8
*b^2)*e^4 - 4*(a^3*b^7*d*x^4 + 4*a^4*b^6*d*x^3 + 6*a^5*b^5*d*x^2 + 4*a^6*b^4*d*x + a^7*b^3*d)*e^3 + 6*(a^2*b^8
*d^2*x^4 + 4*a^3*b^7*d^2*x^3 + 6*a^4*b^6*d^2*x^2 + 4*a^5*b^5*d^2*x + a^6*b^4*d^2)*e^2 - 4*(a*b^9*d^3*x^4 + 4*a
^2*b^8*d^3*x^3 + 6*a^3*b^7*d^3*x^2 + 4*a^4*b^6*d^3*x + a^5*b^5*d^3)*e)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]
time = 1.38, size = 361, normalized size = 1.29 \begin {gather*} -\frac {5 \, \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{4}}{64 \, {\left (b^{4} d^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a b^{3} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{2} d e^{2} \mathrm {sgn}\left (b x + a\right ) - a^{3} b e^{3} \mathrm {sgn}\left (b x + a\right )\right )} \sqrt {-b^{2} d + a b e}} - \frac {15 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{3} e^{4} - 55 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{3} d e^{4} + 73 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} d^{2} e^{4} + 15 \, \sqrt {x e + d} b^{3} d^{3} e^{4} + 55 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{2} e^{5} - 146 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{2} d e^{5} - 45 \, \sqrt {x e + d} a b^{2} d^{2} e^{5} + 73 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b e^{6} + 45 \, \sqrt {x e + d} a^{2} b d e^{6} - 15 \, \sqrt {x e + d} a^{3} e^{7}}{192 \, {\left (b^{4} d^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a b^{3} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{2} d e^{2} \mathrm {sgn}\left (b x + a\right ) - a^{3} b e^{3} \mathrm {sgn}\left (b x + a\right )\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/64*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^4/((b^4*d^3*sgn(b*x + a) - 3*a*b^3*d^2*e*sgn(b*x + a) + 3
*a^2*b^2*d*e^2*sgn(b*x + a) - a^3*b*e^3*sgn(b*x + a))*sqrt(-b^2*d + a*b*e)) - 1/192*(15*(x*e + d)^(7/2)*b^3*e^
4 - 55*(x*e + d)^(5/2)*b^3*d*e^4 + 73*(x*e + d)^(3/2)*b^3*d^2*e^4 + 15*sqrt(x*e + d)*b^3*d^3*e^4 + 55*(x*e + d
)^(5/2)*a*b^2*e^5 - 146*(x*e + d)^(3/2)*a*b^2*d*e^5 - 45*sqrt(x*e + d)*a*b^2*d^2*e^5 + 73*(x*e + d)^(3/2)*a^2*
b*e^6 + 45*sqrt(x*e + d)*a^2*b*d*e^6 - 15*sqrt(x*e + d)*a^3*e^7)/((b^4*d^3*sgn(b*x + a) - 3*a*b^3*d^2*e*sgn(b*
x + a) + 3*a^2*b^2*d*e^2*sgn(b*x + a) - a^3*b*e^3*sgn(b*x + a))*((x*e + d)*b - b*d + a*e)^4)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {d+e\,x}}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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