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

Optimal. Leaf size=313 \[ -\frac {7 e^4 (-a B e-9 A b e+10 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{3/2} (b d-a e)^{11/2}}+\frac {7 e^3 \sqrt {d+e x} (-a B e-9 A b e+10 b B d)}{128 b (a+b x) (b d-a e)^5}-\frac {7 e^2 \sqrt {d+e x} (-a B e-9 A b e+10 b B d)}{192 b (a+b x)^2 (b d-a e)^4}+\frac {7 e \sqrt {d+e x} (-a B e-9 A b e+10 b B d)}{240 b (a+b x)^3 (b d-a e)^3}-\frac {\sqrt {d+e x} (-a B e-9 A b e+10 b B d)}{40 b (a+b x)^4 (b d-a e)^2}-\frac {\sqrt {d+e x} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]

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Rubi [A]  time = 0.31, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {27, 78, 51, 63, 208} \begin {gather*} -\frac {7 e^4 (-a B e-9 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^{3/2} (b d-a e)^{11/2}}+\frac {7 e^3 \sqrt {d+e x} (-a B e-9 A b e+10 b B d)}{128 b (a+b x) (b d-a e)^5}-\frac {7 e^2 \sqrt {d+e x} (-a B e-9 A b e+10 b B d)}{192 b (a+b x)^2 (b d-a e)^4}+\frac {7 e \sqrt {d+e x} (-a B e-9 A b e+10 b B d)}{240 b (a+b x)^3 (b d-a e)^3}-\frac {\sqrt {d+e x} (-a B e-9 A b e+10 b B d)}{40 b (a+b x)^4 (b d-a e)^2}-\frac {\sqrt {d+e x} (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]

-((A*b - a*B)*Sqrt[d + e*x])/(5*b*(b*d - a*e)*(a + b*x)^5) - ((10*b*B*d - 9*A*b*e - a*B*e)*Sqrt[d + e*x])/(40*
b*(b*d - a*e)^2*(a + b*x)^4) + (7*e*(10*b*B*d - 9*A*b*e - a*B*e)*Sqrt[d + e*x])/(240*b*(b*d - a*e)^3*(a + b*x)
^3) - (7*e^2*(10*b*B*d - 9*A*b*e - a*B*e)*Sqrt[d + e*x])/(192*b*(b*d - a*e)^4*(a + b*x)^2) + (7*e^3*(10*b*B*d
- 9*A*b*e - a*B*e)*Sqrt[d + e*x])/(128*b*(b*d - a*e)^5*(a + b*x)) - (7*e^4*(10*b*B*d - 9*A*b*e - a*B*e)*ArcTan
h[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(3/2)*(b*d - a*e)^(11/2))

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 51

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 63

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 78

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] || LtQ
[p, n]))))

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, 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}{(a+b x)^6 \sqrt {d+e x}} \, dx\\ &=-\frac {(A b-a B) \sqrt {d+e x}}{5 b (b d-a e) (a+b x)^5}+\frac {(10 b B d-9 A b e-a B e) \int \frac {1}{(a+b x)^5 \sqrt {d+e x}} \, dx}{10 b (b d-a e)}\\ &=-\frac {(A b-a B) \sqrt {d+e x}}{5 b (b d-a e) (a+b x)^5}-\frac {(10 b B d-9 A b e-a B e) \sqrt {d+e x}}{40 b (b d-a e)^2 (a+b x)^4}-\frac {(7 e (10 b B d-9 A b e-a B e)) \int \frac {1}{(a+b x)^4 \sqrt {d+e x}} \, dx}{80 b (b d-a e)^2}\\ &=-\frac {(A b-a B) \sqrt {d+e x}}{5 b (b d-a e) (a+b x)^5}-\frac {(10 b B d-9 A b e-a B e) \sqrt {d+e x}}{40 b (b d-a e)^2 (a+b x)^4}+\frac {7 e (10 b B d-9 A b e-a B e) \sqrt {d+e x}}{240 b (b d-a e)^3 (a+b x)^3}+\frac {\left (7 e^2 (10 b B d-9 A b e-a B e)\right ) \int \frac {1}{(a+b x)^3 \sqrt {d+e x}} \, dx}{96 b (b d-a e)^3}\\ &=-\frac {(A b-a B) \sqrt {d+e x}}{5 b (b d-a e) (a+b x)^5}-\frac {(10 b B d-9 A b e-a B e) \sqrt {d+e x}}{40 b (b d-a e)^2 (a+b x)^4}+\frac {7 e (10 b B d-9 A b e-a B e) \sqrt {d+e x}}{240 b (b d-a e)^3 (a+b x)^3}-\frac {7 e^2 (10 b B d-9 A b e-a B e) \sqrt {d+e x}}{192 b (b d-a e)^4 (a+b x)^2}-\frac {\left (7 e^3 (10 b B d-9 A b e-a B e)\right ) \int \frac {1}{(a+b x)^2 \sqrt {d+e x}} \, dx}{128 b (b d-a e)^4}\\ &=-\frac {(A b-a B) \sqrt {d+e x}}{5 b (b d-a e) (a+b x)^5}-\frac {(10 b B d-9 A b e-a B e) \sqrt {d+e x}}{40 b (b d-a e)^2 (a+b x)^4}+\frac {7 e (10 b B d-9 A b e-a B e) \sqrt {d+e x}}{240 b (b d-a e)^3 (a+b x)^3}-\frac {7 e^2 (10 b B d-9 A b e-a B e) \sqrt {d+e x}}{192 b (b d-a e)^4 (a+b x)^2}+\frac {7 e^3 (10 b B d-9 A b e-a B e) \sqrt {d+e x}}{128 b (b d-a e)^5 (a+b x)}+\frac {\left (7 e^4 (10 b B d-9 A b e-a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 b (b d-a e)^5}\\ &=-\frac {(A b-a B) \sqrt {d+e x}}{5 b (b d-a e) (a+b x)^5}-\frac {(10 b B d-9 A b e-a B e) \sqrt {d+e x}}{40 b (b d-a e)^2 (a+b x)^4}+\frac {7 e (10 b B d-9 A b e-a B e) \sqrt {d+e x}}{240 b (b d-a e)^3 (a+b x)^3}-\frac {7 e^2 (10 b B d-9 A b e-a B e) \sqrt {d+e x}}{192 b (b d-a e)^4 (a+b x)^2}+\frac {7 e^3 (10 b B d-9 A b e-a B e) \sqrt {d+e x}}{128 b (b d-a e)^5 (a+b x)}+\frac {\left (7 e^3 (10 b B d-9 A b e-a B e)\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{128 b (b d-a e)^5}\\ &=-\frac {(A b-a B) \sqrt {d+e x}}{5 b (b d-a e) (a+b x)^5}-\frac {(10 b B d-9 A b e-a B e) \sqrt {d+e x}}{40 b (b d-a e)^2 (a+b x)^4}+\frac {7 e (10 b B d-9 A b e-a B e) \sqrt {d+e x}}{240 b (b d-a e)^3 (a+b x)^3}-\frac {7 e^2 (10 b B d-9 A b e-a B e) \sqrt {d+e x}}{192 b (b d-a e)^4 (a+b x)^2}+\frac {7 e^3 (10 b B d-9 A b e-a B e) \sqrt {d+e x}}{128 b (b d-a e)^5 (a+b x)}-\frac {7 e^4 (10 b B d-9 A b e-a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{3/2} (b d-a e)^{11/2}}\\ \end {align*}

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Mathematica [C]  time = 0.05, size = 97, normalized size = 0.31 \begin {gather*} \frac {\sqrt {d+e x} \left (\frac {e^4 (a B e+9 A b e-10 b B d) \, _2F_1\left (\frac {1}{2},5;\frac {3}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^5}+\frac {a B-A b}{(a+b x)^5}\right )}{5 b (b d-a e)} \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[d + e*x]*((-(A*b) + a*B)/(a + b*x)^5 + (e^4*(-10*b*B*d + 9*A*b*e + a*B*e)*Hypergeometric2F1[1/2, 5, 3/2,
(b*(d + e*x))/(b*d - a*e)])/(b*d - a*e)^5))/(5*b*(b*d - a*e))

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IntegrateAlgebraic [B]  time = 2.03, size = 676, normalized size = 2.16 \begin {gather*} \frac {e^4 \sqrt {d+e x} \left (-105 a^5 B e^5+2895 a^4 A b e^5+790 a^4 b B e^4 (d+e x)-2370 a^4 b B d e^4+7110 a^3 A b^2 e^4 (d+e x)-11580 a^3 A b^2 d e^4+10530 a^3 b^2 B d^2 e^3+896 a^3 b^2 B e^3 (d+e x)^2-10270 a^3 b^2 B d e^3 (d+e x)+17370 a^2 A b^3 d^2 e^3+8064 a^2 A b^3 e^3 (d+e x)^2-21330 a^2 A b^3 d e^3 (d+e x)-16320 a^2 b^3 B d^3 e^2+26070 a^2 b^3 B d^2 e^2 (d+e x)+490 a^2 b^3 B e^2 (d+e x)^3-10752 a^2 b^3 B d e^2 (d+e x)^2-11580 a A b^4 d^3 e^2+21330 a A b^4 d^2 e^2 (d+e x)+4410 a A b^4 e^2 (d+e x)^3-16128 a A b^4 d e^2 (d+e x)^2+11055 a b^4 B d^4 e-24490 a b^4 B d^3 e (d+e x)+18816 a b^4 B d^2 e (d+e x)^2+105 a b^4 B e (d+e x)^4-5390 a b^4 B d e (d+e x)^3+2895 A b^5 d^4 e-7110 A b^5 d^3 e (d+e x)+8064 A b^5 d^2 e (d+e x)^2+945 A b^5 e (d+e x)^4-4410 A b^5 d e (d+e x)^3-2790 b^5 B d^5+7900 b^5 B d^4 (d+e x)-8960 b^5 B d^3 (d+e x)^2+4900 b^5 B d^2 (d+e x)^3-1050 b^5 B d (d+e x)^4\right )}{1920 b (b d-a e)^5 (-a e-b (d+e x)+b d)^5}-\frac {7 \left (-a B e^5-9 A b e^5+10 b B d e^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{128 b^{3/2} (b d-a e)^5 \sqrt {a e-b d}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x)/(Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

(e^4*Sqrt[d + e*x]*(-2790*b^5*B*d^5 + 2895*A*b^5*d^4*e + 11055*a*b^4*B*d^4*e - 11580*a*A*b^4*d^3*e^2 - 16320*a
^2*b^3*B*d^3*e^2 + 17370*a^2*A*b^3*d^2*e^3 + 10530*a^3*b^2*B*d^2*e^3 - 11580*a^3*A*b^2*d*e^4 - 2370*a^4*b*B*d*
e^4 + 2895*a^4*A*b*e^5 - 105*a^5*B*e^5 + 7900*b^5*B*d^4*(d + e*x) - 7110*A*b^5*d^3*e*(d + e*x) - 24490*a*b^4*B
*d^3*e*(d + e*x) + 21330*a*A*b^4*d^2*e^2*(d + e*x) + 26070*a^2*b^3*B*d^2*e^2*(d + e*x) - 21330*a^2*A*b^3*d*e^3
*(d + e*x) - 10270*a^3*b^2*B*d*e^3*(d + e*x) + 7110*a^3*A*b^2*e^4*(d + e*x) + 790*a^4*b*B*e^4*(d + e*x) - 8960
*b^5*B*d^3*(d + e*x)^2 + 8064*A*b^5*d^2*e*(d + e*x)^2 + 18816*a*b^4*B*d^2*e*(d + e*x)^2 - 16128*a*A*b^4*d*e^2*
(d + e*x)^2 - 10752*a^2*b^3*B*d*e^2*(d + e*x)^2 + 8064*a^2*A*b^3*e^3*(d + e*x)^2 + 896*a^3*b^2*B*e^3*(d + e*x)
^2 + 4900*b^5*B*d^2*(d + e*x)^3 - 4410*A*b^5*d*e*(d + e*x)^3 - 5390*a*b^4*B*d*e*(d + e*x)^3 + 4410*a*A*b^4*e^2
*(d + e*x)^3 + 490*a^2*b^3*B*e^2*(d + e*x)^3 - 1050*b^5*B*d*(d + e*x)^4 + 945*A*b^5*e*(d + e*x)^4 + 105*a*b^4*
B*e*(d + e*x)^4))/(1920*b*(b*d - a*e)^5*(b*d - a*e - b*(d + e*x))^5) - (7*(10*b*B*d*e^4 - 9*A*b*e^5 - a*B*e^5)
*ArcTan[(Sqrt[b]*Sqrt[-(b*d) + a*e]*Sqrt[d + e*x])/(b*d - a*e)])/(128*b^(3/2)*(b*d - a*e)^5*Sqrt[-(b*d) + a*e]
)

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fricas [B]  time = 0.52, size = 2715, normalized size = 8.67

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

[1/3840*(105*(10*B*a^5*b*d*e^4 - (B*a^6 + 9*A*a^5*b)*e^5 + (10*B*b^6*d*e^4 - (B*a*b^5 + 9*A*b^6)*e^5)*x^5 + 5*
(10*B*a*b^5*d*e^4 - (B*a^2*b^4 + 9*A*a*b^5)*e^5)*x^4 + 10*(10*B*a^2*b^4*d*e^4 - (B*a^3*b^3 + 9*A*a^2*b^4)*e^5)
*x^3 + 10*(10*B*a^3*b^3*d*e^4 - (B*a^4*b^2 + 9*A*a^3*b^3)*e^5)*x^2 + 5*(10*B*a^4*b^2*d*e^4 - (B*a^5*b + 9*A*a^
4*b^2)*e^5)*x)*sqrt(b^2*d - a*b*e)*log((b*e*x + 2*b*d - a*e - 2*sqrt(b^2*d - a*b*e)*sqrt(e*x + d))/(b*x + a))
- 2*(96*(B*a*b^6 + 4*A*b^7)*d^5 - 16*(38*B*a^2*b^5 + 147*A*a*b^6)*d^4*e + 12*(139*B*a^3*b^4 + 506*A*a^2*b^5)*d
^3*e^2 - 6*(456*B*a^4*b^3 + 1429*A*a^3*b^4)*d^2*e^3 + 5*(295*B*a^5*b^2 + 1473*A*a^4*b^3)*d*e^4 + 15*(7*B*a^6*b
- 193*A*a^5*b^2)*e^5 - 105*(10*B*b^7*d^2*e^3 - (11*B*a*b^6 + 9*A*b^7)*d*e^4 + (B*a^2*b^5 + 9*A*a*b^6)*e^5)*x^
4 + 70*(10*B*b^7*d^3*e^2 - 9*(9*B*a*b^6 + A*b^7)*d^2*e^3 + 6*(13*B*a^2*b^5 + 12*A*a*b^6)*d*e^4 - 7*(B*a^3*b^4
+ 9*A*a^2*b^5)*e^5)*x^3 - 14*(40*B*b^7*d^4*e - 2*(137*B*a*b^6 + 18*A*b^7)*d^3*e^2 + 3*(299*B*a^2*b^5 + 81*A*a*
b^6)*d^2*e^3 - (727*B*a^3*b^4 + 783*A*a^2*b^5)*d*e^4 + 64*(B*a^4*b^3 + 9*A*a^3*b^4)*e^5)*x^2 + 2*(240*B*b^7*d^
5 - 8*(193*B*a*b^6 + 27*A*b^7)*d^4*e + 2*(2161*B*a^2*b^5 + 684*A*a*b^6)*d^3*e^2 - 3*(2419*B*a^3*b^4 + 1251*A*a
^2*b^5)*d^2*e^3 + 2*(2317*B*a^4*b^3 + 3078*A*a^3*b^4)*d*e^4 - 395*(B*a^5*b^2 + 9*A*a^4*b^3)*e^5)*x)*sqrt(e*x +
d))/(a^5*b^8*d^6 - 6*a^6*b^7*d^5*e + 15*a^7*b^6*d^4*e^2 - 20*a^8*b^5*d^3*e^3 + 15*a^9*b^4*d^2*e^4 - 6*a^10*b^
3*d*e^5 + a^11*b^2*e^6 + (b^13*d^6 - 6*a*b^12*d^5*e + 15*a^2*b^11*d^4*e^2 - 20*a^3*b^10*d^3*e^3 + 15*a^4*b^9*d
^2*e^4 - 6*a^5*b^8*d*e^5 + a^6*b^7*e^6)*x^5 + 5*(a*b^12*d^6 - 6*a^2*b^11*d^5*e + 15*a^3*b^10*d^4*e^2 - 20*a^4*
b^9*d^3*e^3 + 15*a^5*b^8*d^2*e^4 - 6*a^6*b^7*d*e^5 + a^7*b^6*e^6)*x^4 + 10*(a^2*b^11*d^6 - 6*a^3*b^10*d^5*e +
15*a^4*b^9*d^4*e^2 - 20*a^5*b^8*d^3*e^3 + 15*a^6*b^7*d^2*e^4 - 6*a^7*b^6*d*e^5 + a^8*b^5*e^6)*x^3 + 10*(a^3*b^
10*d^6 - 6*a^4*b^9*d^5*e + 15*a^5*b^8*d^4*e^2 - 20*a^6*b^7*d^3*e^3 + 15*a^7*b^6*d^2*e^4 - 6*a^8*b^5*d*e^5 + a^
9*b^4*e^6)*x^2 + 5*(a^4*b^9*d^6 - 6*a^5*b^8*d^5*e + 15*a^6*b^7*d^4*e^2 - 20*a^7*b^6*d^3*e^3 + 15*a^8*b^5*d^2*e
^4 - 6*a^9*b^4*d*e^5 + a^10*b^3*e^6)*x), 1/1920*(105*(10*B*a^5*b*d*e^4 - (B*a^6 + 9*A*a^5*b)*e^5 + (10*B*b^6*d
*e^4 - (B*a*b^5 + 9*A*b^6)*e^5)*x^5 + 5*(10*B*a*b^5*d*e^4 - (B*a^2*b^4 + 9*A*a*b^5)*e^5)*x^4 + 10*(10*B*a^2*b^
4*d*e^4 - (B*a^3*b^3 + 9*A*a^2*b^4)*e^5)*x^3 + 10*(10*B*a^3*b^3*d*e^4 - (B*a^4*b^2 + 9*A*a^3*b^3)*e^5)*x^2 + 5
*(10*B*a^4*b^2*d*e^4 - (B*a^5*b + 9*A*a^4*b^2)*e^5)*x)*sqrt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(e
*x + d)/(b*e*x + b*d)) - (96*(B*a*b^6 + 4*A*b^7)*d^5 - 16*(38*B*a^2*b^5 + 147*A*a*b^6)*d^4*e + 12*(139*B*a^3*b
^4 + 506*A*a^2*b^5)*d^3*e^2 - 6*(456*B*a^4*b^3 + 1429*A*a^3*b^4)*d^2*e^3 + 5*(295*B*a^5*b^2 + 1473*A*a^4*b^3)*
d*e^4 + 15*(7*B*a^6*b - 193*A*a^5*b^2)*e^5 - 105*(10*B*b^7*d^2*e^3 - (11*B*a*b^6 + 9*A*b^7)*d*e^4 + (B*a^2*b^5
+ 9*A*a*b^6)*e^5)*x^4 + 70*(10*B*b^7*d^3*e^2 - 9*(9*B*a*b^6 + A*b^7)*d^2*e^3 + 6*(13*B*a^2*b^5 + 12*A*a*b^6)*
d*e^4 - 7*(B*a^3*b^4 + 9*A*a^2*b^5)*e^5)*x^3 - 14*(40*B*b^7*d^4*e - 2*(137*B*a*b^6 + 18*A*b^7)*d^3*e^2 + 3*(29
9*B*a^2*b^5 + 81*A*a*b^6)*d^2*e^3 - (727*B*a^3*b^4 + 783*A*a^2*b^5)*d*e^4 + 64*(B*a^4*b^3 + 9*A*a^3*b^4)*e^5)*
x^2 + 2*(240*B*b^7*d^5 - 8*(193*B*a*b^6 + 27*A*b^7)*d^4*e + 2*(2161*B*a^2*b^5 + 684*A*a*b^6)*d^3*e^2 - 3*(2419
*B*a^3*b^4 + 1251*A*a^2*b^5)*d^2*e^3 + 2*(2317*B*a^4*b^3 + 3078*A*a^3*b^4)*d*e^4 - 395*(B*a^5*b^2 + 9*A*a^4*b^
3)*e^5)*x)*sqrt(e*x + d))/(a^5*b^8*d^6 - 6*a^6*b^7*d^5*e + 15*a^7*b^6*d^4*e^2 - 20*a^8*b^5*d^3*e^3 + 15*a^9*b^
4*d^2*e^4 - 6*a^10*b^3*d*e^5 + a^11*b^2*e^6 + (b^13*d^6 - 6*a*b^12*d^5*e + 15*a^2*b^11*d^4*e^2 - 20*a^3*b^10*d
^3*e^3 + 15*a^4*b^9*d^2*e^4 - 6*a^5*b^8*d*e^5 + a^6*b^7*e^6)*x^5 + 5*(a*b^12*d^6 - 6*a^2*b^11*d^5*e + 15*a^3*b
^10*d^4*e^2 - 20*a^4*b^9*d^3*e^3 + 15*a^5*b^8*d^2*e^4 - 6*a^6*b^7*d*e^5 + a^7*b^6*e^6)*x^4 + 10*(a^2*b^11*d^6
- 6*a^3*b^10*d^5*e + 15*a^4*b^9*d^4*e^2 - 20*a^5*b^8*d^3*e^3 + 15*a^6*b^7*d^2*e^4 - 6*a^7*b^6*d*e^5 + a^8*b^5*
e^6)*x^3 + 10*(a^3*b^10*d^6 - 6*a^4*b^9*d^5*e + 15*a^5*b^8*d^4*e^2 - 20*a^6*b^7*d^3*e^3 + 15*a^7*b^6*d^2*e^4 -
6*a^8*b^5*d*e^5 + a^9*b^4*e^6)*x^2 + 5*(a^4*b^9*d^6 - 6*a^5*b^8*d^5*e + 15*a^6*b^7*d^4*e^2 - 20*a^7*b^6*d^3*e
^3 + 15*a^8*b^5*d^2*e^4 - 6*a^9*b^4*d*e^5 + a^10*b^3*e^6)*x)]

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giac [B]  time = 0.30, size = 881, normalized size = 2.81 \begin {gather*} \frac {7 \, {\left (10 \, B b d e^{4} - B a e^{5} - 9 \, 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^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )} \sqrt {-b^{2} d + a b e}} + \frac {1050 \, {\left (x e + d\right )}^{\frac {9}{2}} B b^{5} d e^{4} - 4900 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{5} d^{2} e^{4} + 8960 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{5} d^{3} e^{4} - 7900 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{5} d^{4} e^{4} + 2790 \, \sqrt {x e + d} B b^{5} d^{5} e^{4} - 105 \, {\left (x e + d\right )}^{\frac {9}{2}} B a b^{4} e^{5} - 945 \, {\left (x e + d\right )}^{\frac {9}{2}} A b^{5} e^{5} + 5390 \, {\left (x e + d\right )}^{\frac {7}{2}} B a b^{4} d e^{5} + 4410 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{5} d e^{5} - 18816 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{4} d^{2} e^{5} - 8064 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{5} d^{2} e^{5} + 24490 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{4} d^{3} e^{5} + 7110 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{5} d^{3} e^{5} - 11055 \, \sqrt {x e + d} B a b^{4} d^{4} e^{5} - 2895 \, \sqrt {x e + d} A b^{5} d^{4} e^{5} - 490 \, {\left (x e + d\right )}^{\frac {7}{2}} B a^{2} b^{3} e^{6} - 4410 \, {\left (x e + d\right )}^{\frac {7}{2}} A a b^{4} e^{6} + 10752 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} b^{3} d e^{6} + 16128 \, {\left (x e + d\right )}^{\frac {5}{2}} A a b^{4} d e^{6} - 26070 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{3} d^{2} e^{6} - 21330 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{4} d^{2} e^{6} + 16320 \, \sqrt {x e + d} B a^{2} b^{3} d^{3} e^{6} + 11580 \, \sqrt {x e + d} A a b^{4} d^{3} e^{6} - 896 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{3} b^{2} e^{7} - 8064 \, {\left (x e + d\right )}^{\frac {5}{2}} A a^{2} b^{3} e^{7} + 10270 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b^{2} d e^{7} + 21330 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{3} d e^{7} - 10530 \, \sqrt {x e + d} B a^{3} b^{2} d^{2} e^{7} - 17370 \, \sqrt {x e + d} A a^{2} b^{3} d^{2} e^{7} - 790 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{4} b e^{8} - 7110 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{3} b^{2} e^{8} + 2370 \, \sqrt {x e + d} B a^{4} b d e^{8} + 11580 \, \sqrt {x e + d} A a^{3} b^{2} d e^{8} + 105 \, \sqrt {x e + d} B a^{5} e^{9} - 2895 \, \sqrt {x e + d} A a^{4} b e^{9}}{1920 \, {\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\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)/(b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

7/128*(10*B*b*d*e^4 - B*a*e^5 - 9*A*b*e^5)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^6*d^5 - 5*a*b^5*d^
4*e + 10*a^2*b^4*d^3*e^2 - 10*a^3*b^3*d^2*e^3 + 5*a^4*b^2*d*e^4 - a^5*b*e^5)*sqrt(-b^2*d + a*b*e)) + 1/1920*(1
050*(x*e + d)^(9/2)*B*b^5*d*e^4 - 4900*(x*e + d)^(7/2)*B*b^5*d^2*e^4 + 8960*(x*e + d)^(5/2)*B*b^5*d^3*e^4 - 79
00*(x*e + d)^(3/2)*B*b^5*d^4*e^4 + 2790*sqrt(x*e + d)*B*b^5*d^5*e^4 - 105*(x*e + d)^(9/2)*B*a*b^4*e^5 - 945*(x
*e + d)^(9/2)*A*b^5*e^5 + 5390*(x*e + d)^(7/2)*B*a*b^4*d*e^5 + 4410*(x*e + d)^(7/2)*A*b^5*d*e^5 - 18816*(x*e +
d)^(5/2)*B*a*b^4*d^2*e^5 - 8064*(x*e + d)^(5/2)*A*b^5*d^2*e^5 + 24490*(x*e + d)^(3/2)*B*a*b^4*d^3*e^5 + 7110*
(x*e + d)^(3/2)*A*b^5*d^3*e^5 - 11055*sqrt(x*e + d)*B*a*b^4*d^4*e^5 - 2895*sqrt(x*e + d)*A*b^5*d^4*e^5 - 490*(
x*e + d)^(7/2)*B*a^2*b^3*e^6 - 4410*(x*e + d)^(7/2)*A*a*b^4*e^6 + 10752*(x*e + d)^(5/2)*B*a^2*b^3*d*e^6 + 1612
8*(x*e + d)^(5/2)*A*a*b^4*d*e^6 - 26070*(x*e + d)^(3/2)*B*a^2*b^3*d^2*e^6 - 21330*(x*e + d)^(3/2)*A*a*b^4*d^2*
e^6 + 16320*sqrt(x*e + d)*B*a^2*b^3*d^3*e^6 + 11580*sqrt(x*e + d)*A*a*b^4*d^3*e^6 - 896*(x*e + d)^(5/2)*B*a^3*
b^2*e^7 - 8064*(x*e + d)^(5/2)*A*a^2*b^3*e^7 + 10270*(x*e + d)^(3/2)*B*a^3*b^2*d*e^7 + 21330*(x*e + d)^(3/2)*A
*a^2*b^3*d*e^7 - 10530*sqrt(x*e + d)*B*a^3*b^2*d^2*e^7 - 17370*sqrt(x*e + d)*A*a^2*b^3*d^2*e^7 - 790*(x*e + d)
^(3/2)*B*a^4*b*e^8 - 7110*(x*e + d)^(3/2)*A*a^3*b^2*e^8 + 2370*sqrt(x*e + d)*B*a^4*b*d*e^8 + 11580*sqrt(x*e +
d)*A*a^3*b^2*d*e^8 + 105*sqrt(x*e + d)*B*a^5*e^9 - 2895*sqrt(x*e + d)*A*a^4*b*e^9)/((b^6*d^5 - 5*a*b^5*d^4*e +
10*a^2*b^4*d^3*e^2 - 10*a^3*b^3*d^2*e^3 + 5*a^4*b^2*d*e^4 - a^5*b*e^5)*((x*e + d)*b - b*d + a*e)^5)

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maple [B]  time = 0.08, size = 1274, normalized size = 4.07

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(1/2),x)

[Out]

63/128*e^5/(b*e*x+a*e)^5*b^4/(a^5*e^5-5*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3-10*a^2*b^3*d^3*e^2+5*a*b^4*d^4*e-b^5*d^
5)*(e*x+d)^(9/2)*A+7/128*e^5/(b*e*x+a*e)^5*b^3/(a^5*e^5-5*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3-10*a^2*b^3*d^3*e^2+5*
a*b^4*d^4*e-b^5*d^5)*(e*x+d)^(9/2)*a*B-35/64*e^4/(b*e*x+a*e)^5*b^4/(a^5*e^5-5*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3-1
0*a^2*b^3*d^3*e^2+5*a*b^4*d^4*e-b^5*d^5)*(e*x+d)^(9/2)*B*d+147/64*e^5/(b*e*x+a*e)^5*b^3/(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)^(7/2)*A+49/192*e^5/(b*e*x+a*e)^5*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)^(7/2)*a*B-245/96*e^4/(b*e*x+a*e)^5*b^3/(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)^(7/2)*B*d+21/5*e^5/(b*e*x+a*e)^5*b^2/(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)^(5/2)*A+7/15*e^5/(b*e*x+a*e)^5*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)^(5/2)*a*B-14/3*e^4/(b*e*x+a*e)^5*b^2/(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)^(5/2)*B*d+2
37/64*e^5/(b*e*x+a*e)^5/(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e*x+d)^(3/2)*A*b+79/192*e^5/(b*e*x+a*e)^5/(a^2*e^2-2*a*b*
d*e+b^2*d^2)*(e*x+d)^(3/2)*a*B-395/96*e^4/(b*e*x+a*e)^5/(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e*x+d)^(3/2)*B*b*d+193/12
8*e^5/(b*e*x+a*e)^5/(a*e-b*d)*(e*x+d)^(1/2)*A-7/128*e^5/(b*e*x+a*e)^5/b/(a*e-b*d)*(e*x+d)^(1/2)*a*B-93/64*e^4/
(b*e*x+a*e)^5/(a*e-b*d)*(e*x+d)^(1/2)*B*d+63/128*e^5/(a^5*e^5-5*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3-10*a^2*b^3*d^3*
e^2+5*a*b^4*d^4*e-b^5*d^5)/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*A+7/128*e^5/b/(a^5*
e^5-5*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3-10*a^2*b^3*d^3*e^2+5*a*b^4*d^4*e-b^5*d^5)/((a*e-b*d)*b)^(1/2)*arctan((e*x
+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a*B-35/64*e^4/(a^5*e^5-5*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3-10*a^2*b^3*d^3*e^2+5*
a*b^4*d^4*e-b^5*d^5)/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*B*d

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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)/(b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(1/2),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(a*e-b*d>0)', see `assume?` for
more details)Is a*e-b*d positive or negative?

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mupad [B]  time = 2.29, size = 567, normalized size = 1.81 \begin {gather*} \frac {\frac {49\,{\left (d+e\,x\right )}^{7/2}\,\left (9\,A\,b^3\,e^5-10\,B\,d\,b^3\,e^4+B\,a\,b^2\,e^5\right )}{192\,{\left (a\,e-b\,d\right )}^4}+\frac {79\,{\left (d+e\,x\right )}^{3/2}\,\left (9\,A\,b\,e^5+B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{192\,{\left (a\,e-b\,d\right )}^2}+\frac {7\,b\,{\left (d+e\,x\right )}^{5/2}\,\left (9\,A\,b\,e^5+B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{15\,{\left (a\,e-b\,d\right )}^3}+\frac {7\,b^3\,{\left (d+e\,x\right )}^{9/2}\,\left (9\,A\,b\,e^5+B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{128\,{\left (a\,e-b\,d\right )}^5}-\frac {\sqrt {d+e\,x}\,\left (7\,B\,a\,e^5-193\,A\,b\,e^5+186\,B\,b\,d\,e^4\right )}{128\,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 {7\,e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^4\,\sqrt {d+e\,x}\,\left (9\,A\,b\,e+B\,a\,e-10\,B\,b\,d\right )}{\sqrt {a\,e-b\,d}\,\left (9\,A\,b\,e^5+B\,a\,e^5-10\,B\,b\,d\,e^4\right )}\right )\,\left (9\,A\,b\,e+B\,a\,e-10\,B\,b\,d\right )}{128\,b^{3/2}\,{\left (a\,e-b\,d\right )}^{11/2}} \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]

((49*(d + e*x)^(7/2)*(9*A*b^3*e^5 + B*a*b^2*e^5 - 10*B*b^3*d*e^4))/(192*(a*e - b*d)^4) + (79*(d + e*x)^(3/2)*(
9*A*b*e^5 + B*a*e^5 - 10*B*b*d*e^4))/(192*(a*e - b*d)^2) + (7*b*(d + e*x)^(5/2)*(9*A*b*e^5 + B*a*e^5 - 10*B*b*
d*e^4))/(15*(a*e - b*d)^3) + (7*b^3*(d + e*x)^(9/2)*(9*A*b*e^5 + B*a*e^5 - 10*B*b*d*e^4))/(128*(a*e - b*d)^5)
- ((d + e*x)^(1/2)*(7*B*a*e^5 - 193*A*b*e^5 + 186*B*b*d*e^4))/(128*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) + (7*e^4*atan((b^(1/2)*e^4*(d + e*x)^(1/2)*(9*A*b*e + B*a*e - 10*B*b*d))/((a*e - b*d)
^(1/2)*(9*A*b*e^5 + B*a*e^5 - 10*B*b*d*e^4)))*(9*A*b*e + B*a*e - 10*B*b*d))/(128*b^(3/2)*(a*e - b*d)^(11/2))

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sympy [F(-1)]  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)/(b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**(1/2),x)

[Out]

Timed out

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