3.21.78 \(\int \frac {(a+b x) \sqrt {d+e x}}{(a^2+2 a b x+b^2 x^2)^2} \, dx\) [2078]

Optimal. Leaf size=110 \[ -\frac {\sqrt {d+e x}}{2 b (a+b x)^2}-\frac {e \sqrt {d+e x}}{4 b (b d-a e) (a+b x)}+\frac {e^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{3/2} (b d-a e)^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {27, 43, 44, 65, 214} \begin {gather*} \frac {e^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{3/2} (b d-a e)^{3/2}}-\frac {e \sqrt {d+e x}}{4 b (a+b x) (b d-a e)}-\frac {\sqrt {d+e x}}{2 b (a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*Sqrt[d + e*x]/(b*(a + b*x)^2) - (e*Sqrt[d + e*x])/(4*b*(b*d - a*e)*(a + b*x)) + (e^2*ArcTanh[(Sqrt[b]*Sqr
t[d + e*x])/Sqrt[b*d - a*e]])/(4*b^(3/2)*(b*d - a*e)^(3/2))

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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]

Rubi steps

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

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Mathematica [A]
time = 0.10, size = 99, normalized size = 0.90 \begin {gather*} \frac {\frac {\sqrt {b} \sqrt {d+e x} (2 b d-a e+b e x)}{(-b d+a e) (a+b x)^2}+\frac {e^2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{3/2}}}{4 b^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.04, size = 106, normalized size = 0.96

method	result	size
derivativedivides	\(2 e^{2} \left (\frac {\frac {\left (e x +d \right )^{\frac {3}{2}}}{8 a e -8 b d}-\frac {\sqrt {e x +d}}{8 b}}{\left (b \left (e x +d \right )+a e -b d \right )^{2}}+\frac {\arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \left (a e -b d \right ) b \sqrt {\left (a e -b d \right ) b}}\right )\)	\(106\)
default	\(2 e^{2} \left (\frac {\frac {\left (e x +d \right )^{\frac {3}{2}}}{8 a e -8 b d}-\frac {\sqrt {e x +d}}{8 b}}{\left (b \left (e x +d \right )+a e -b d \right )^{2}}+\frac {\arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \left (a e -b d \right ) b \sqrt {\left (a e -b d \right ) b}}\right )\)	\(106\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e^2*((1/8/(a*e-b*d)*(e*x+d)^(3/2)-1/8*(e*x+d)^(1/2)/b)/(b*(e*x+d)+a*e-b*d)^2+1/8/(a*e-b*d)/b/((a*e-b*d)*b)^(
1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*d-%e*a>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (96) = 192\).
time = 2.03, size = 451, normalized size = 4.10 \begin {gather*} \left [-\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {b^{2} d - a b e} e^{2} \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 (2 \, b^{3} d^{2} - {\left (a b^{2} x - a^{2} b\right )} e^{2} + {\left (b^{3} d x - 3 \, a b^{2} d\right )} e\right )} \sqrt {x e + d}}{8 \, {\left (b^{6} d^{2} x^{2} + 2 \, a b^{5} d^{2} x + a^{2} b^{4} d^{2} + {\left (a^{2} b^{4} x^{2} + 2 \, a^{3} b^{3} x + a^{4} b^{2}\right )} e^{2} - 2 \, {\left (a b^{5} d x^{2} + 2 \, a^{2} b^{4} d x + a^{3} b^{3} d\right )} e\right )}}, -\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\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^{2} + {\left (2 \, b^{3} d^{2} - {\left (a b^{2} x - a^{2} b\right )} e^{2} + {\left (b^{3} d x - 3 \, a b^{2} d\right )} e\right )} \sqrt {x e + d}}{4 \, {\left (b^{6} d^{2} x^{2} + 2 \, a b^{5} d^{2} x + a^{2} b^{4} d^{2} + {\left (a^{2} b^{4} x^{2} + 2 \, a^{3} b^{3} x + a^{4} b^{2}\right )} e^{2} - 2 \, {\left (a b^{5} d x^{2} + 2 \, a^{2} b^{4} d x + a^{3} b^{3} d\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/8*((b^2*x^2 + 2*a*b*x + a^2)*sqrt(b^2*d - a*b*e)*e^2*log((2*b*d + (b*x - a)*e - 2*sqrt(b^2*d - a*b*e)*sqrt
(x*e + d))/(b*x + a)) + 2*(2*b^3*d^2 - (a*b^2*x - a^2*b)*e^2 + (b^3*d*x - 3*a*b^2*d)*e)*sqrt(x*e + d))/(b^6*d^
2*x^2 + 2*a*b^5*d^2*x + a^2*b^4*d^2 + (a^2*b^4*x^2 + 2*a^3*b^3*x + a^4*b^2)*e^2 - 2*(a*b^5*d*x^2 + 2*a^2*b^4*d
*x + a^3*b^3*d)*e), -1/4*((b^2*x^2 + 2*a*b*x + a^2)*sqrt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(x*e
+ d)/(b*x*e + b*d))*e^2 + (2*b^3*d^2 - (a*b^2*x - a^2*b)*e^2 + (b^3*d*x - 3*a*b^2*d)*e)*sqrt(x*e + d))/(b^6*d^
2*x^2 + 2*a*b^5*d^2*x + a^2*b^4*d^2 + (a^2*b^4*x^2 + 2*a^3*b^3*x + a^4*b^2)*e^2 - 2*(a*b^5*d*x^2 + 2*a^2*b^4*d
*x + a^3*b^3*d)*e)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1658 vs. \(2 (88) = 176\).
time = 121.98, size = 1658, normalized size = 15.07 \begin {gather*} - \frac {10 a^{2} e^{4} \sqrt {d + e x}}{8 a^{4} b e^{4} - 16 a^{3} b^{2} d e^{3} + 16 a^{3} b^{2} e^{4} x - 48 a^{2} b^{3} d e^{3} x + 8 a^{2} b^{3} e^{2} \left (d + e x\right )^{2} + 16 a b^{4} d^{3} e + 48 a b^{4} d^{2} e^{2} x - 16 a b^{4} d e \left (d + e x\right )^{2} - 8 b^{5} d^{4} - 16 b^{5} d^{3} e x + 8 b^{5} d^{2} \left (d + e x\right )^{2}} + \frac {20 a d e^{3} \sqrt {d + e x}}{8 a^{4} e^{4} - 16 a^{3} b d e^{3} + 16 a^{3} b e^{4} x - 48 a^{2} b^{2} d e^{3} x + 8 a^{2} b^{2} e^{2} \left (d + e x\right )^{2} + 16 a b^{3} d^{3} e + 48 a b^{3} d^{2} e^{2} x - 16 a b^{3} d e \left (d + e x\right )^{2} - 8 b^{4} d^{4} - 16 b^{4} d^{3} e x + 8 b^{4} d^{2} \left (d + e x\right )^{2}} - \frac {6 a e^{3} \left (d + e x\right )^{\frac {3}{2}}}{8 a^{4} e^{4} - 16 a^{3} b d e^{3} + 16 a^{3} b e^{4} x - 48 a^{2} b^{2} d e^{3} x + 8 a^{2} b^{2} e^{2} \left (d + e x\right )^{2} + 16 a b^{3} d^{3} e + 48 a b^{3} d^{2} e^{2} x - 16 a b^{3} d e \left (d + e x\right )^{2} - 8 b^{4} d^{4} - 16 b^{4} d^{3} e x + 8 b^{4} d^{2} \left (d + e x\right )^{2}} + \frac {3 a e^{3} \sqrt {- \frac {1}{b \left (a e - b d\right )^{5}}} \log {\left (- a^{3} e^{3} \sqrt {- \frac {1}{b \left (a e - b d\right )^{5}}} + 3 a^{2} b d e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{5}}} - 3 a b^{2} d^{2} e \sqrt {- \frac {1}{b \left (a e - b d\right )^{5}}} + b^{3} d^{3} \sqrt {- \frac {1}{b \left (a e - b d\right )^{5}}} + \sqrt {d + e x} \right )}}{8 b} - \frac {3 a e^{3} \sqrt {- \frac {1}{b \left (a e - b d\right )^{5}}} \log {\left (a^{3} e^{3} \sqrt {- \frac {1}{b \left (a e - b d\right )^{5}}} - 3 a^{2} b d e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{5}}} + 3 a b^{2} d^{2} e \sqrt {- \frac {1}{b \left (a e - b d\right )^{5}}} - b^{3} d^{3} \sqrt {- \frac {1}{b \left (a e - b d\right )^{5}}} + \sqrt {d + e x} \right )}}{8 b} - \frac {10 b d^{2} e^{2} \sqrt {d + e x}}{8 a^{4} e^{4} - 16 a^{3} b d e^{3} + 16 a^{3} b e^{4} x - 48 a^{2} b^{2} d e^{3} x + 8 a^{2} b^{2} e^{2} \left (d + e x\right )^{2} + 16 a b^{3} d^{3} e + 48 a b^{3} d^{2} e^{2} x - 16 a b^{3} d e \left (d + e x\right )^{2} - 8 b^{4} d^{4} - 16 b^{4} d^{3} e x + 8 b^{4} d^{2} \left (d + e x\right )^{2}} + \frac {6 b d e^{2} \left (d + e x\right )^{\frac {3}{2}}}{8 a^{4} e^{4} - 16 a^{3} b d e^{3} + 16 a^{3} b e^{4} x - 48 a^{2} b^{2} d e^{3} x + 8 a^{2} b^{2} e^{2} \left (d + e x\right )^{2} + 16 a b^{3} d^{3} e + 48 a b^{3} d^{2} e^{2} x - 16 a b^{3} d e \left (d + e x\right )^{2} - 8 b^{4} d^{4} - 16 b^{4} d^{3} e x + 8 b^{4} d^{2} \left (d + e x\right )^{2}} - \frac {3 d e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{5}}} \log {\left (- a^{3} e^{3} \sqrt {- \frac {1}{b \left (a e - b d\right )^{5}}} + 3 a^{2} b d e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{5}}} - 3 a b^{2} d^{2} e \sqrt {- \frac {1}{b \left (a e - b d\right )^{5}}} + b^{3} d^{3} \sqrt {- \frac {1}{b \left (a e - b d\right )^{5}}} + \sqrt {d + e x} \right )}}{8} + \frac {3 d e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{5}}} \log {\left (a^{3} e^{3} \sqrt {- \frac {1}{b \left (a e - b d\right )^{5}}} - 3 a^{2} b d e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{5}}} + 3 a b^{2} d^{2} e \sqrt {- \frac {1}{b \left (a e - b d\right )^{5}}} - b^{3} d^{3} \sqrt {- \frac {1}{b \left (a e - b d\right )^{5}}} + \sqrt {d + e x} \right )}}{8} + \frac {2 e^{2} \sqrt {d + e x}}{2 a^{2} b e^{2} - 2 a b^{2} d e + 2 a b^{2} e^{2} x - 2 b^{3} d e x} - \frac {e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} \log {\left (- a^{2} e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + 2 a b d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} - b^{2} d^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + \sqrt {d + e x} \right )}}{2 b} + \frac {e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} \log {\left (a^{2} e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} - 2 a b d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + b^{2} d^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + \sqrt {d + e x} \right )}}{2 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-10*a**2*e**4*sqrt(d + e*x)/(8*a**4*b*e**4 - 16*a**3*b**2*d*e**3 + 16*a**3*b**2*e**4*x - 48*a**2*b**3*d*e**3*x
+ 8*a**2*b**3*e**2*(d + e*x)**2 + 16*a*b**4*d**3*e + 48*a*b**4*d**2*e**2*x - 16*a*b**4*d*e*(d + e*x)**2 - 8*b
**5*d**4 - 16*b**5*d**3*e*x + 8*b**5*d**2*(d + e*x)**2) + 20*a*d*e**3*sqrt(d + e*x)/(8*a**4*e**4 - 16*a**3*b*d
*e**3 + 16*a**3*b*e**4*x - 48*a**2*b**2*d*e**3*x + 8*a**2*b**2*e**2*(d + e*x)**2 + 16*a*b**3*d**3*e + 48*a*b**
3*d**2*e**2*x - 16*a*b**3*d*e*(d + e*x)**2 - 8*b**4*d**4 - 16*b**4*d**3*e*x + 8*b**4*d**2*(d + e*x)**2) - 6*a*
e**3*(d + e*x)**(3/2)/(8*a**4*e**4 - 16*a**3*b*d*e**3 + 16*a**3*b*e**4*x - 48*a**2*b**2*d*e**3*x + 8*a**2*b**2
*e**2*(d + e*x)**2 + 16*a*b**3*d**3*e + 48*a*b**3*d**2*e**2*x - 16*a*b**3*d*e*(d + e*x)**2 - 8*b**4*d**4 - 16*
b**4*d**3*e*x + 8*b**4*d**2*(d + e*x)**2) + 3*a*e**3*sqrt(-1/(b*(a*e - b*d)**5))*log(-a**3*e**3*sqrt(-1/(b*(a*
e - b*d)**5)) + 3*a**2*b*d*e**2*sqrt(-1/(b*(a*e - b*d)**5)) - 3*a*b**2*d**2*e*sqrt(-1/(b*(a*e - b*d)**5)) + b*
*3*d**3*sqrt(-1/(b*(a*e - b*d)**5)) + sqrt(d + e*x))/(8*b) - 3*a*e**3*sqrt(-1/(b*(a*e - b*d)**5))*log(a**3*e**
3*sqrt(-1/(b*(a*e - b*d)**5)) - 3*a**2*b*d*e**2*sqrt(-1/(b*(a*e - b*d)**5)) + 3*a*b**2*d**2*e*sqrt(-1/(b*(a*e
- b*d)**5)) - b**3*d**3*sqrt(-1/(b*(a*e - b*d)**5)) + sqrt(d + e*x))/(8*b) - 10*b*d**2*e**2*sqrt(d + e*x)/(8*a
**4*e**4 - 16*a**3*b*d*e**3 + 16*a**3*b*e**4*x - 48*a**2*b**2*d*e**3*x + 8*a**2*b**2*e**2*(d + e*x)**2 + 16*a*
b**3*d**3*e + 48*a*b**3*d**2*e**2*x - 16*a*b**3*d*e*(d + e*x)**2 - 8*b**4*d**4 - 16*b**4*d**3*e*x + 8*b**4*d**
2*(d + e*x)**2) + 6*b*d*e**2*(d + e*x)**(3/2)/(8*a**4*e**4 - 16*a**3*b*d*e**3 + 16*a**3*b*e**4*x - 48*a**2*b**
2*d*e**3*x + 8*a**2*b**2*e**2*(d + e*x)**2 + 16*a*b**3*d**3*e + 48*a*b**3*d**2*e**2*x - 16*a*b**3*d*e*(d + e*x
)**2 - 8*b**4*d**4 - 16*b**4*d**3*e*x + 8*b**4*d**2*(d + e*x)**2) - 3*d*e**2*sqrt(-1/(b*(a*e - b*d)**5))*log(-
a**3*e**3*sqrt(-1/(b*(a*e - b*d)**5)) + 3*a**2*b*d*e**2*sqrt(-1/(b*(a*e - b*d)**5)) - 3*a*b**2*d**2*e*sqrt(-1/
(b*(a*e - b*d)**5)) + b**3*d**3*sqrt(-1/(b*(a*e - b*d)**5)) + sqrt(d + e*x))/8 + 3*d*e**2*sqrt(-1/(b*(a*e - b*
d)**5))*log(a**3*e**3*sqrt(-1/(b*(a*e - b*d)**5)) - 3*a**2*b*d*e**2*sqrt(-1/(b*(a*e - b*d)**5)) + 3*a*b**2*d**
2*e*sqrt(-1/(b*(a*e - b*d)**5)) - b**3*d**3*sqrt(-1/(b*(a*e - b*d)**5)) + sqrt(d + e*x))/8 + 2*e**2*sqrt(d + e
*x)/(2*a**2*b*e**2 - 2*a*b**2*d*e + 2*a*b**2*e**2*x - 2*b**3*d*e*x) - e**2*sqrt(-1/(b*(a*e - b*d)**3))*log(-a*
*2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) + 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) - b**2*d**2*sqrt(-1/(b*(a*e - b*d)
**3)) + sqrt(d + e*x))/(2*b) + e**2*sqrt(-1/(b*(a*e - b*d)**3))*log(a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) - 2*
a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) + b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/(2*b)

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Giac [A]
time = 1.13, size = 132, normalized size = 1.20 \begin {gather*} -\frac {\arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{2}}{4 \, {\left (b^{2} d - a b e\right )} \sqrt {-b^{2} d + a b e}} - \frac {{\left (x e + d\right )}^{\frac {3}{2}} b e^{2} + \sqrt {x e + d} b d e^{2} - \sqrt {x e + d} a e^{3}}{4 \, {\left (b^{2} d - a b e\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.10, size = 135, normalized size = 1.23 \begin {gather*} \frac {e^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{4\,b^{3/2}\,{\left (a\,e-b\,d\right )}^{3/2}}-\frac {\frac {e^2\,\sqrt {d+e\,x}}{4\,b}-\frac {e^2\,{\left (d+e\,x\right )}^{3/2}}{4\,\left (a\,e-b\,d\right )}}{b^2\,{\left (d+e\,x\right )}^2-\left (2\,b^2\,d-2\,a\,b\,e\right )\,\left (d+e\,x\right )+a^2\,e^2+b^2\,d^2-2\,a\,b\,d\,e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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