3.19.30 \(\int \frac {(A+B x) \sqrt {d+e x}}{(a^2+2 a b x+b^2 x^2)^3} \, dx\) [1830]

Optimal. Leaf size=313 \[ -\frac {(10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {e (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{240 b^2 (b d-a e)^2 (a+b x)^3}+\frac {e^2 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{192 b^2 (b d-a e)^3 (a+b x)^2}-\frac {e^3 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{128 b^2 (b d-a e)^4 (a+b x)}-\frac {(A b-a B) (d+e x)^{3/2}}{5 b (b d-a e) (a+b x)^5}+\frac {e^4 (10 b B d-7 A b e-3 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{5/2} (b d-a e)^{9/2}} \]

[Out]

-1/5*(A*b-B*a)*(e*x+d)^(3/2)/b/(-a*e+b*d)/(b*x+a)^5+1/128*e^4*(-7*A*b*e-3*B*a*e+10*B*b*d)*arctanh(b^(1/2)*(e*x
+d)^(1/2)/(-a*e+b*d)^(1/2))/b^(5/2)/(-a*e+b*d)^(9/2)-1/40*(-7*A*b*e-3*B*a*e+10*B*b*d)*(e*x+d)^(1/2)/b^2/(-a*e+
b*d)/(b*x+a)^4-1/240*e*(-7*A*b*e-3*B*a*e+10*B*b*d)*(e*x+d)^(1/2)/b^2/(-a*e+b*d)^2/(b*x+a)^3+1/192*e^2*(-7*A*b*
e-3*B*a*e+10*B*b*d)*(e*x+d)^(1/2)/b^2/(-a*e+b*d)^3/(b*x+a)^2-1/128*e^3*(-7*A*b*e-3*B*a*e+10*B*b*d)*(e*x+d)^(1/
2)/b^2/(-a*e+b*d)^4/(b*x+a)

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Rubi [A]
time = 0.20, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {27, 79, 43, 44, 65, 214} \begin {gather*} \frac {e^4 (-3 a B e-7 A b e+10 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{5/2} (b d-a e)^{9/2}}-\frac {e^3 \sqrt {d+e x} (-3 a B e-7 A b e+10 b B d)}{128 b^2 (a+b x) (b d-a e)^4}+\frac {e^2 \sqrt {d+e x} (-3 a B e-7 A b e+10 b B d)}{192 b^2 (a+b x)^2 (b d-a e)^3}-\frac {e \sqrt {d+e x} (-3 a B e-7 A b e+10 b B d)}{240 b^2 (a+b x)^3 (b d-a e)^2}-\frac {\sqrt {d+e x} (-3 a B e-7 A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac {(d+e x)^{3/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \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)^3,x]

[Out]

-1/40*((10*b*B*d - 7*A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(b^2*(b*d - a*e)*(a + b*x)^4) - (e*(10*b*B*d - 7*A*b*e -
3*a*B*e)*Sqrt[d + e*x])/(240*b^2*(b*d - a*e)^2*(a + b*x)^3) + (e^2*(10*b*B*d - 7*A*b*e - 3*a*B*e)*Sqrt[d + e*x
])/(192*b^2*(b*d - a*e)^3*(a + b*x)^2) - (e^3*(10*b*B*d - 7*A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(128*b^2*(b*d - a*
e)^4*(a + b*x)) - ((A*b - a*B)*(d + e*x)^(3/2))/(5*b*(b*d - a*e)*(a + b*x)^5) + (e^4*(10*b*B*d - 7*A*b*e - 3*a
*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(5/2)*(b*d - a*e)^(9/2))

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 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 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 )^3} \, dx &=\int \frac {(A+B x) \sqrt {d+e x}}{(a+b x)^6} \, dx\\ &=-\frac {(A b-a B) (d+e x)^{3/2}}{5 b (b d-a e) (a+b x)^5}+\frac {(10 b B d-7 A b e-3 a B e) \int \frac {\sqrt {d+e x}}{(a+b x)^5} \, dx}{10 b (b d-a e)}\\ &=-\frac {(10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{3/2}}{5 b (b d-a e) (a+b x)^5}+\frac {(e (10 b B d-7 A b e-3 a B e)) \int \frac {1}{(a+b x)^4 \sqrt {d+e x}} \, dx}{80 b^2 (b d-a e)}\\ &=-\frac {(10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {e (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{240 b^2 (b d-a e)^2 (a+b x)^3}-\frac {(A b-a B) (d+e x)^{3/2}}{5 b (b d-a e) (a+b x)^5}-\frac {\left (e^2 (10 b B d-7 A b e-3 a B e)\right ) \int \frac {1}{(a+b x)^3 \sqrt {d+e x}} \, dx}{96 b^2 (b d-a e)^2}\\ &=-\frac {(10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {e (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{240 b^2 (b d-a e)^2 (a+b x)^3}+\frac {e^2 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{192 b^2 (b d-a e)^3 (a+b x)^2}-\frac {(A b-a B) (d+e x)^{3/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (e^3 (10 b B d-7 A b e-3 a B e)\right ) \int \frac {1}{(a+b x)^2 \sqrt {d+e x}} \, dx}{128 b^2 (b d-a e)^3}\\ &=-\frac {(10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {e (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{240 b^2 (b d-a e)^2 (a+b x)^3}+\frac {e^2 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{192 b^2 (b d-a e)^3 (a+b x)^2}-\frac {e^3 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{128 b^2 (b d-a e)^4 (a+b x)}-\frac {(A b-a B) (d+e x)^{3/2}}{5 b (b d-a e) (a+b x)^5}-\frac {\left (e^4 (10 b B d-7 A b e-3 a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 b^2 (b d-a e)^4}\\ &=-\frac {(10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {e (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{240 b^2 (b d-a e)^2 (a+b x)^3}+\frac {e^2 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{192 b^2 (b d-a e)^3 (a+b x)^2}-\frac {e^3 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{128 b^2 (b d-a e)^4 (a+b x)}-\frac {(A b-a B) (d+e x)^{3/2}}{5 b (b d-a e) (a+b x)^5}-\frac {\left (e^3 (10 b B d-7 A b e-3 a B e)\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{128 b^2 (b d-a e)^4}\\ &=-\frac {(10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {e (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{240 b^2 (b d-a e)^2 (a+b x)^3}+\frac {e^2 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{192 b^2 (b d-a e)^3 (a+b x)^2}-\frac {e^3 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{128 b^2 (b d-a e)^4 (a+b x)}-\frac {(A b-a B) (d+e x)^{3/2}}{5 b (b d-a e) (a+b x)^5}+\frac {e^4 (10 b B d-7 A b e-3 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{5/2} (b d-a e)^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 2.64, size = 425, normalized size = 1.36 \begin {gather*} \frac {\frac {\sqrt {b} \sqrt {d+e x} \left (B \left (-45 a^5 e^4+30 a^4 b e^3 (4 d-7 e x)+2 a^3 b^2 e^2 \left (-218 d^2+409 d e x+192 e^2 x^2\right )-10 b^5 d x \left (48 d^3+8 d^2 e x-10 d e^2 x^2+15 e^3 x^3\right )+2 a^2 b^3 e \left (176 d^3-1178 d^2 e x-709 d e^2 x^2+105 e^3 x^3\right )+a b^4 \left (-96 d^4+1808 d^3 e x+484 d^2 e^2 x^2-730 d e^3 x^3+45 e^4 x^4\right )\right )+A b \left (-105 a^4 e^4+10 a^3 b e^3 (121 d+79 e x)+2 a^2 b^2 e^2 \left (-1052 d^2-289 d e x+448 e^2 x^2\right )+2 a b^3 e \left (744 d^3+128 d^2 e x-161 d e^2 x^2+245 e^3 x^3\right )+b^4 \left (-384 d^4-48 d^3 e x+56 d^2 e^2 x^2-70 d e^3 x^3+105 e^4 x^4\right )\right )\right )}{(b d-a e)^4 (a+b x)^5}+\frac {15 e^4 (-10 b B d+7 A b e+3 a B e) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{9/2}}}{1920 b^{5/2}} \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)^3,x]

[Out]

((Sqrt[b]*Sqrt[d + e*x]*(B*(-45*a^5*e^4 + 30*a^4*b*e^3*(4*d - 7*e*x) + 2*a^3*b^2*e^2*(-218*d^2 + 409*d*e*x + 1
92*e^2*x^2) - 10*b^5*d*x*(48*d^3 + 8*d^2*e*x - 10*d*e^2*x^2 + 15*e^3*x^3) + 2*a^2*b^3*e*(176*d^3 - 1178*d^2*e*
x - 709*d*e^2*x^2 + 105*e^3*x^3) + a*b^4*(-96*d^4 + 1808*d^3*e*x + 484*d^2*e^2*x^2 - 730*d*e^3*x^3 + 45*e^4*x^
4)) + A*b*(-105*a^4*e^4 + 10*a^3*b*e^3*(121*d + 79*e*x) + 2*a^2*b^2*e^2*(-1052*d^2 - 289*d*e*x + 448*e^2*x^2)
+ 2*a*b^3*e*(744*d^3 + 128*d^2*e*x - 161*d*e^2*x^2 + 245*e^3*x^3) + b^4*(-384*d^4 - 48*d^3*e*x + 56*d^2*e^2*x^
2 - 70*d*e^3*x^3 + 105*e^4*x^4))))/((b*d - a*e)^4*(a + b*x)^5) + (15*e^4*(-10*b*B*d + 7*A*b*e + 3*a*B*e)*ArcTa
n[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(-(b*d) + a*e)^(9/2))/(1920*b^(5/2))

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Maple [A]
time = 1.00, size = 389, normalized size = 1.24

method	result	size
derivativedivides	\(2 e^{4} \left (\frac {\frac {\left (7 A b e +3 B a e -10 B b d \right ) b^{2} \left (e x +d \right )^{\frac {9}{2}}}{256 e^{4} a^{4}-1024 a^{3} b d \,e^{3}+1536 a^{2} b^{2} d^{2} e^{2}-1024 a \,b^{3} d^{3} e +256 b^{4} d^{4}}+\frac {7 \left (7 A b e +3 B a e -10 B b d \right ) b \left (e x +d \right )^{\frac {7}{2}}}{384 \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}+\frac {\left (7 A b e +3 B a e -10 B b d \right ) \left (e x +d \right )^{\frac {5}{2}}}{30 a^{2} e^{2}-60 a b d e +30 b^{2} d^{2}}+\frac {\left (79 A b e -21 B a e -58 B b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{384 b \left (a e -b d \right )}-\frac {\left (7 A b e +3 B a e -10 B b d \right ) \sqrt {e x +d}}{256 b^{2}}}{\left (\left (e x +d \right ) b +a e -b d \right )^{5}}+\frac {\left (7 A b e +3 B a e -10 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{256 b^{2} \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \sqrt {b \left (a e -b d \right )}}\right )\)	\(389\)
default	\(2 e^{4} \left (\frac {\frac {\left (7 A b e +3 B a e -10 B b d \right ) b^{2} \left (e x +d \right )^{\frac {9}{2}}}{256 e^{4} a^{4}-1024 a^{3} b d \,e^{3}+1536 a^{2} b^{2} d^{2} e^{2}-1024 a \,b^{3} d^{3} e +256 b^{4} d^{4}}+\frac {7 \left (7 A b e +3 B a e -10 B b d \right ) b \left (e x +d \right )^{\frac {7}{2}}}{384 \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}+\frac {\left (7 A b e +3 B a e -10 B b d \right ) \left (e x +d \right )^{\frac {5}{2}}}{30 a^{2} e^{2}-60 a b d e +30 b^{2} d^{2}}+\frac {\left (79 A b e -21 B a e -58 B b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{384 b \left (a e -b d \right )}-\frac {\left (7 A b e +3 B a e -10 B b d \right ) \sqrt {e x +d}}{256 b^{2}}}{\left (\left (e x +d \right ) b +a e -b d \right )^{5}}+\frac {\left (7 A b e +3 B a e -10 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{256 b^{2} \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \sqrt {b \left (a e -b d \right )}}\right )\)	\(389\)

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)^3,x,method=_RETURNVERBOSE)

[Out]

2*e^4*((1/256*(7*A*b*e+3*B*a*e-10*B*b*d)*b^2/(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)*(
e*x+d)^(9/2)+7/384*(7*A*b*e+3*B*a*e-10*B*b*d)*b/(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)*(e*x+d)^(7/2)+1/
30*(7*A*b*e+3*B*a*e-10*B*b*d)/(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e*x+d)^(5/2)+1/384*(79*A*b*e-21*B*a*e-58*B*b*d)/b/(
a*e-b*d)*(e*x+d)^(3/2)-1/256*(7*A*b*e+3*B*a*e-10*B*b*d)/b^2*(e*x+d)^(1/2))/((e*x+d)*b+a*e-b*d)^5+1/256*(7*A*b*
e+3*B*a*e-10*B*b*d)/b^2/(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)/(b*(a*e-b*d))^(1/2)*ar
ctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(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)^3,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. 1229 vs. \(2 (302) = 604\).
time = 1.24, size = 2472, normalized size = 7.90 \begin {gather*} \text {Too large to display} \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)^3,x, algorithm="fricas")

[Out]

[1/3840*(15*sqrt(b^2*d - a*b*e)*((3*B*a^6 + 7*A*a^5*b + (3*B*a*b^5 + 7*A*b^6)*x^5 + 5*(3*B*a^2*b^4 + 7*A*a*b^5
)*x^4 + 10*(3*B*a^3*b^3 + 7*A*a^2*b^4)*x^3 + 10*(3*B*a^4*b^2 + 7*A*a^3*b^3)*x^2 + 5*(3*B*a^5*b + 7*A*a^4*b^2)*
x)*e^5 - 10*(B*b^6*d*x^5 + 5*B*a*b^5*d*x^4 + 10*B*a^2*b^4*d*x^3 + 10*B*a^3*b^3*d*x^2 + 5*B*a^4*b^2*d*x + B*a^5
*b*d)*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*(480*B*b^7*d^5*x + 9
6*(B*a*b^6 + 4*A*b^7)*d^5 - (45*B*a^6*b + 105*A*a^5*b^2 - 15*(3*B*a^2*b^5 + 7*A*a*b^6)*x^4 - 70*(3*B*a^3*b^4 +
7*A*a^2*b^5)*x^3 - 128*(3*B*a^4*b^3 + 7*A*a^3*b^4)*x^2 + 10*(21*B*a^5*b^2 - 79*A*a^4*b^3)*x)*e^5 - (15*(13*B*
a*b^6 + 7*A*b^7)*d*x^4 + 20*(47*B*a^2*b^5 + 28*A*a*b^6)*d*x^3 + 2*(901*B*a^3*b^4 + 609*A*a^2*b^5)*d*x^2 - 4*(2
57*B*a^4*b^3 - 342*A*a^3*b^4)*d*x - 5*(33*B*a^5*b^2 + 263*A*a^4*b^3)*d)*e^4 + 2*(75*B*b^7*d^2*x^4 + 5*(83*B*a*
b^6 + 7*A*b^7)*d^2*x^3 + 3*(317*B*a^2*b^5 + 63*A*a*b^6)*d^2*x^2 - 3*(529*B*a^3*b^4 - 139*A*a^2*b^5)*d^2*x - (2
78*B*a^4*b^3 + 1657*A*a^3*b^4)*d^2)*e^3 - 4*(25*B*b^7*d^3*x^3 + (141*B*a*b^6 + 14*A*b^7)*d^3*x^2 - (1041*B*a^2
*b^5 - 76*A*a*b^6)*d^3*x - (197*B*a^3*b^4 + 898*A*a^2*b^5)*d^3)*e^2 + 16*(5*B*b^7*d^4*x^2 - (143*B*a*b^6 - 3*A
*b^7)*d^4*x - (28*B*a^2*b^5 + 117*A*a*b^6)*d^4)*e)*sqrt(x*e + d))/(b^13*d^5*x^5 + 5*a*b^12*d^5*x^4 + 10*a^2*b^
11*d^5*x^3 + 10*a^3*b^10*d^5*x^2 + 5*a^4*b^9*d^5*x + a^5*b^8*d^5 - (a^5*b^8*x^5 + 5*a^6*b^7*x^4 + 10*a^7*b^6*x
^3 + 10*a^8*b^5*x^2 + 5*a^9*b^4*x + a^10*b^3)*e^5 + 5*(a^4*b^9*d*x^5 + 5*a^5*b^8*d*x^4 + 10*a^6*b^7*d*x^3 + 10
*a^7*b^6*d*x^2 + 5*a^8*b^5*d*x + a^9*b^4*d)*e^4 - 10*(a^3*b^10*d^2*x^5 + 5*a^4*b^9*d^2*x^4 + 10*a^5*b^8*d^2*x^
3 + 10*a^6*b^7*d^2*x^2 + 5*a^7*b^6*d^2*x + a^8*b^5*d^2)*e^3 + 10*(a^2*b^11*d^3*x^5 + 5*a^3*b^10*d^3*x^4 + 10*a
^4*b^9*d^3*x^3 + 10*a^5*b^8*d^3*x^2 + 5*a^6*b^7*d^3*x + a^7*b^6*d^3)*e^2 - 5*(a*b^12*d^4*x^5 + 5*a^2*b^11*d^4*
x^4 + 10*a^3*b^10*d^4*x^3 + 10*a^4*b^9*d^4*x^2 + 5*a^5*b^8*d^4*x + a^6*b^7*d^4)*e), 1/1920*(15*sqrt(-b^2*d + a
*b*e)*((3*B*a^6 + 7*A*a^5*b + (3*B*a*b^5 + 7*A*b^6)*x^5 + 5*(3*B*a^2*b^4 + 7*A*a*b^5)*x^4 + 10*(3*B*a^3*b^3 +
7*A*a^2*b^4)*x^3 + 10*(3*B*a^4*b^2 + 7*A*a^3*b^3)*x^2 + 5*(3*B*a^5*b + 7*A*a^4*b^2)*x)*e^5 - 10*(B*b^6*d*x^5 +
5*B*a*b^5*d*x^4 + 10*B*a^2*b^4*d*x^3 + 10*B*a^3*b^3*d*x^2 + 5*B*a^4*b^2*d*x + B*a^5*b*d)*e^4)*arctan(sqrt(-b^
2*d + a*b*e)*sqrt(x*e + d)/(b*x*e + b*d)) - (480*B*b^7*d^5*x + 96*(B*a*b^6 + 4*A*b^7)*d^5 - (45*B*a^6*b + 105*
A*a^5*b^2 - 15*(3*B*a^2*b^5 + 7*A*a*b^6)*x^4 - 70*(3*B*a^3*b^4 + 7*A*a^2*b^5)*x^3 - 128*(3*B*a^4*b^3 + 7*A*a^3
*b^4)*x^2 + 10*(21*B*a^5*b^2 - 79*A*a^4*b^3)*x)*e^5 - (15*(13*B*a*b^6 + 7*A*b^7)*d*x^4 + 20*(47*B*a^2*b^5 + 28
*A*a*b^6)*d*x^3 + 2*(901*B*a^3*b^4 + 609*A*a^2*b^5)*d*x^2 - 4*(257*B*a^4*b^3 - 342*A*a^3*b^4)*d*x - 5*(33*B*a^
5*b^2 + 263*A*a^4*b^3)*d)*e^4 + 2*(75*B*b^7*d^2*x^4 + 5*(83*B*a*b^6 + 7*A*b^7)*d^2*x^3 + 3*(317*B*a^2*b^5 + 63
*A*a*b^6)*d^2*x^2 - 3*(529*B*a^3*b^4 - 139*A*a^2*b^5)*d^2*x - (278*B*a^4*b^3 + 1657*A*a^3*b^4)*d^2)*e^3 - 4*(2
5*B*b^7*d^3*x^3 + (141*B*a*b^6 + 14*A*b^7)*d^3*x^2 - (1041*B*a^2*b^5 - 76*A*a*b^6)*d^3*x - (197*B*a^3*b^4 + 89
8*A*a^2*b^5)*d^3)*e^2 + 16*(5*B*b^7*d^4*x^2 - (143*B*a*b^6 - 3*A*b^7)*d^4*x - (28*B*a^2*b^5 + 117*A*a*b^6)*d^4
)*e)*sqrt(x*e + d))/(b^13*d^5*x^5 + 5*a*b^12*d^5*x^4 + 10*a^2*b^11*d^5*x^3 + 10*a^3*b^10*d^5*x^2 + 5*a^4*b^9*d
^5*x + a^5*b^8*d^5 - (a^5*b^8*x^5 + 5*a^6*b^7*x^4 + 10*a^7*b^6*x^3 + 10*a^8*b^5*x^2 + 5*a^9*b^4*x + a^10*b^3)*
e^5 + 5*(a^4*b^9*d*x^5 + 5*a^5*b^8*d*x^4 + 10*a^6*b^7*d*x^3 + 10*a^7*b^6*d*x^2 + 5*a^8*b^5*d*x + a^9*b^4*d)*e^
4 - 10*(a^3*b^10*d^2*x^5 + 5*a^4*b^9*d^2*x^4 + 10*a^5*b^8*d^2*x^3 + 10*a^6*b^7*d^2*x^2 + 5*a^7*b^6*d^2*x + a^8
*b^5*d^2)*e^3 + 10*(a^2*b^11*d^3*x^5 + 5*a^3*b^10*d^3*x^4 + 10*a^4*b^9*d^3*x^3 + 10*a^5*b^8*d^3*x^2 + 5*a^6*b^
7*d^3*x + a^7*b^6*d^3)*e^2 - 5*(a*b^12*d^4*x^5 + 5*a^2*b^11*d^4*x^4 + 10*a^3*b^10*d^4*x^3 + 10*a^4*b^9*d^4*x^2
+ 5*a^5*b^8*d^4*x + a^6*b^7*d^4)*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((B*x+A)*(e*x+d)**(1/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 857 vs. \(2 (302) = 604\).
time = 1.24, size = 857, normalized size = 2.74 \begin {gather*} -\frac {{\left (10 \, B b d e^{4} - 3 \, B a e^{5} - 7 \, A b e^{5}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{128 \, {\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} \sqrt {-b^{2} d + a b e}} - \frac {150 \, {\left (x e + d\right )}^{\frac {9}{2}} B b^{5} d e^{4} - 700 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{5} d^{2} e^{4} + 1280 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{5} d^{3} e^{4} - 580 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{5} d^{4} e^{4} - 150 \, \sqrt {x e + d} B b^{5} d^{5} e^{4} - 45 \, {\left (x e + d\right )}^{\frac {9}{2}} B a b^{4} e^{5} - 105 \, {\left (x e + d\right )}^{\frac {9}{2}} A b^{5} e^{5} + 910 \, {\left (x e + d\right )}^{\frac {7}{2}} B a b^{4} d e^{5} + 490 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{5} d e^{5} - 2944 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{4} d^{2} e^{5} - 896 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{5} d^{2} e^{5} + 1530 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{4} d^{3} e^{5} + 790 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{5} d^{3} e^{5} + 645 \, \sqrt {x e + d} B a b^{4} d^{4} e^{5} + 105 \, \sqrt {x e + d} A b^{5} d^{4} e^{5} - 210 \, {\left (x e + d\right )}^{\frac {7}{2}} B a^{2} b^{3} e^{6} - 490 \, {\left (x e + d\right )}^{\frac {7}{2}} A a b^{4} e^{6} + 2048 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} b^{3} d e^{6} + 1792 \, {\left (x e + d\right )}^{\frac {5}{2}} A a b^{4} d e^{6} - 1110 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{3} d^{2} e^{6} - 2370 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{4} d^{2} e^{6} - 1080 \, \sqrt {x e + d} B a^{2} b^{3} d^{3} e^{6} - 420 \, \sqrt {x e + d} A a b^{4} d^{3} e^{6} - 384 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{3} b^{2} e^{7} - 896 \, {\left (x e + d\right )}^{\frac {5}{2}} A a^{2} b^{3} e^{7} - 50 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b^{2} d e^{7} + 2370 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{3} d e^{7} + 870 \, \sqrt {x e + d} B a^{3} b^{2} d^{2} e^{7} + 630 \, \sqrt {x e + d} A a^{2} b^{3} d^{2} e^{7} + 210 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{4} b e^{8} - 790 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{3} b^{2} e^{8} - 330 \, \sqrt {x e + d} B a^{4} b d e^{8} - 420 \, \sqrt {x e + d} A a^{3} b^{2} d e^{8} + 45 \, \sqrt {x e + d} B a^{5} e^{9} + 105 \, \sqrt {x e + d} A a^{4} b e^{9}}{1920 \, {\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{5}} \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)^3,x, algorithm="giac")

[Out]

-1/128*(10*B*b*d*e^4 - 3*B*a*e^5 - 7*A*b*e^5)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^6*d^4 - 4*a*b^5
*d^3*e + 6*a^2*b^4*d^2*e^2 - 4*a^3*b^3*d*e^3 + a^4*b^2*e^4)*sqrt(-b^2*d + a*b*e)) - 1/1920*(150*(x*e + d)^(9/2
)*B*b^5*d*e^4 - 700*(x*e + d)^(7/2)*B*b^5*d^2*e^4 + 1280*(x*e + d)^(5/2)*B*b^5*d^3*e^4 - 580*(x*e + d)^(3/2)*B
*b^5*d^4*e^4 - 150*sqrt(x*e + d)*B*b^5*d^5*e^4 - 45*(x*e + d)^(9/2)*B*a*b^4*e^5 - 105*(x*e + d)^(9/2)*A*b^5*e^
5 + 910*(x*e + d)^(7/2)*B*a*b^4*d*e^5 + 490*(x*e + d)^(7/2)*A*b^5*d*e^5 - 2944*(x*e + d)^(5/2)*B*a*b^4*d^2*e^5
- 896*(x*e + d)^(5/2)*A*b^5*d^2*e^5 + 1530*(x*e + d)^(3/2)*B*a*b^4*d^3*e^5 + 790*(x*e + d)^(3/2)*A*b^5*d^3*e^
5 + 645*sqrt(x*e + d)*B*a*b^4*d^4*e^5 + 105*sqrt(x*e + d)*A*b^5*d^4*e^5 - 210*(x*e + d)^(7/2)*B*a^2*b^3*e^6 -
490*(x*e + d)^(7/2)*A*a*b^4*e^6 + 2048*(x*e + d)^(5/2)*B*a^2*b^3*d*e^6 + 1792*(x*e + d)^(5/2)*A*a*b^4*d*e^6 -
1110*(x*e + d)^(3/2)*B*a^2*b^3*d^2*e^6 - 2370*(x*e + d)^(3/2)*A*a*b^4*d^2*e^6 - 1080*sqrt(x*e + d)*B*a^2*b^3*d
^3*e^6 - 420*sqrt(x*e + d)*A*a*b^4*d^3*e^6 - 384*(x*e + d)^(5/2)*B*a^3*b^2*e^7 - 896*(x*e + d)^(5/2)*A*a^2*b^3
*e^7 - 50*(x*e + d)^(3/2)*B*a^3*b^2*d*e^7 + 2370*(x*e + d)^(3/2)*A*a^2*b^3*d*e^7 + 870*sqrt(x*e + d)*B*a^3*b^2
*d^2*e^7 + 630*sqrt(x*e + d)*A*a^2*b^3*d^2*e^7 + 210*(x*e + d)^(3/2)*B*a^4*b*e^8 - 790*(x*e + d)^(3/2)*A*a^3*b
^2*e^8 - 330*sqrt(x*e + d)*B*a^4*b*d*e^8 - 420*sqrt(x*e + d)*A*a^3*b^2*d*e^8 + 45*sqrt(x*e + d)*B*a^5*e^9 + 10
5*sqrt(x*e + d)*A*a^4*b*e^9)/((b^6*d^4 - 4*a*b^5*d^3*e + 6*a^2*b^4*d^2*e^2 - 4*a^3*b^3*d*e^3 + a^4*b^2*e^4)*((
x*e + d)*b - b*d + a*e)^5)

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Mupad [B]
time = 2.24, size = 564, normalized size = 1.80 \begin {gather*} \frac {\frac {7\,{\left (d+e\,x\right )}^{7/2}\,\left (7\,A\,b^2\,e^5-10\,B\,d\,b^2\,e^4+3\,B\,a\,b\,e^5\right )}{192\,{\left (a\,e-b\,d\right )}^3}-\frac {\sqrt {d+e\,x}\,\left (7\,A\,b\,e^5+3\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{128\,b^2}+\frac {{\left (d+e\,x\right )}^{5/2}\,\left (7\,A\,b\,e^5+3\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{15\,{\left (a\,e-b\,d\right )}^2}+\frac {b^2\,{\left (d+e\,x\right )}^{9/2}\,\left (7\,A\,b\,e^5+3\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{128\,{\left (a\,e-b\,d\right )}^4}-\frac {{\left (d+e\,x\right )}^{3/2}\,\left (21\,B\,a\,e^5-79\,A\,b\,e^5+58\,B\,b\,d\,e^4\right )}{192\,b\,\left (a\,e-b\,d\right )}}{\left (d+e\,x\right )\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )-{\left (d+e\,x\right )}^2\,\left (-10\,a^3\,b^2\,e^3+30\,a^2\,b^3\,d\,e^2-30\,a\,b^4\,d^2\,e+10\,b^5\,d^3\right )+b^5\,{\left (d+e\,x\right )}^5-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^4+a^5\,e^5-b^5\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )-10\,a^2\,b^3\,d^3\,e^2+10\,a^3\,b^2\,d^2\,e^3+5\,a\,b^4\,d^4\,e-5\,a^4\,b\,d\,e^4}+\frac {e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^4\,\sqrt {d+e\,x}\,\left (7\,A\,b\,e+3\,B\,a\,e-10\,B\,b\,d\right )}{\sqrt {a\,e-b\,d}\,\left (7\,A\,b\,e^5+3\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}\right )\,\left (7\,A\,b\,e+3\,B\,a\,e-10\,B\,b\,d\right )}{128\,b^{5/2}\,{\left (a\,e-b\,d\right )}^{9/2}} \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)^3,x)

[Out]

((7*(d + e*x)^(7/2)*(7*A*b^2*e^5 + 3*B*a*b*e^5 - 10*B*b^2*d*e^4))/(192*(a*e - b*d)^3) - ((d + e*x)^(1/2)*(7*A*
b*e^5 + 3*B*a*e^5 - 10*B*b*d*e^4))/(128*b^2) + ((d + e*x)^(5/2)*(7*A*b*e^5 + 3*B*a*e^5 - 10*B*b*d*e^4))/(15*(a
*e - b*d)^2) + (b^2*(d + e*x)^(9/2)*(7*A*b*e^5 + 3*B*a*e^5 - 10*B*b*d*e^4))/(128*(a*e - b*d)^4) - ((d + e*x)^(
3/2)*(21*B*a*e^5 - 79*A*b*e^5 + 58*B*b*d*e^4))/(192*b*(a*e - b*d)))/((d + e*x)*(5*b^5*d^4 + 5*a^4*b*e^4 - 20*a
^3*b^2*d*e^3 + 30*a^2*b^3*d^2*e^2 - 20*a*b^4*d^3*e) - (d + e*x)^2*(10*b^5*d^3 - 10*a^3*b^2*e^3 + 30*a^2*b^3*d*
e^2 - 30*a*b^4*d^2*e) + b^5*(d + e*x)^5 - (5*b^5*d - 5*a*b^4*e)*(d + e*x)^4 + a^5*e^5 - b^5*d^5 + (d + e*x)^3*
(10*b^5*d^2 + 10*a^2*b^3*e^2 - 20*a*b^4*d*e) - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4
*b*d*e^4) + (e^4*atan((b^(1/2)*e^4*(d + e*x)^(1/2)*(7*A*b*e + 3*B*a*e - 10*B*b*d))/((a*e - b*d)^(1/2)*(7*A*b*e
^5 + 3*B*a*e^5 - 10*B*b*d*e^4)))*(7*A*b*e + 3*B*a*e - 10*B*b*d))/(128*b^(5/2)*(a*e - b*d)^(9/2))

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