3.1826 \(\int \frac {(A+B x) (d+e x)^{9/2}}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=352 \[ -\frac {63 e^4 (-11 a B e+A b e+10 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{13/2} \sqrt {b d-a e}}+\frac {63 e^4 \sqrt {d+e x} (-11 a B e+A b e+10 b B d)}{128 b^6 (b d-a e)}-\frac {21 e^3 (d+e x)^{3/2} (-11 a B e+A b e+10 b B d)}{128 b^5 (a+b x) (b d-a e)}-\frac {21 e^2 (d+e x)^{5/2} (-11 a B e+A b e+10 b B d)}{320 b^4 (a+b x)^2 (b d-a e)}-\frac {3 e (d+e x)^{7/2} (-11 a B e+A b e+10 b B d)}{80 b^3 (a+b x)^3 (b d-a e)}-\frac {(d+e x)^{9/2} (-11 a B e+A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac {(d+e x)^{11/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]

[Out]

-21/128*e^3*(A*b*e-11*B*a*e+10*B*b*d)*(e*x+d)^(3/2)/b^5/(-a*e+b*d)/(b*x+a)-21/320*e^2*(A*b*e-11*B*a*e+10*B*b*d
)*(e*x+d)^(5/2)/b^4/(-a*e+b*d)/(b*x+a)^2-3/80*e*(A*b*e-11*B*a*e+10*B*b*d)*(e*x+d)^(7/2)/b^3/(-a*e+b*d)/(b*x+a)
^3-1/40*(A*b*e-11*B*a*e+10*B*b*d)*(e*x+d)^(9/2)/b^2/(-a*e+b*d)/(b*x+a)^4-1/5*(A*b-B*a)*(e*x+d)^(11/2)/b/(-a*e+
b*d)/(b*x+a)^5-63/128*e^4*(A*b*e-11*B*a*e+10*B*b*d)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/b^(13/2)/(
-a*e+b*d)^(1/2)+63/128*e^4*(A*b*e-11*B*a*e+10*B*b*d)*(e*x+d)^(1/2)/b^6/(-a*e+b*d)

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Rubi [A]  time = 0.30, antiderivative size = 352, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {27, 78, 47, 50, 63, 208} \[ -\frac {21 e^2 (d+e x)^{5/2} (-11 a B e+A b e+10 b B d)}{320 b^4 (a+b x)^2 (b d-a e)}-\frac {21 e^3 (d+e x)^{3/2} (-11 a B e+A b e+10 b B d)}{128 b^5 (a+b x) (b d-a e)}+\frac {63 e^4 \sqrt {d+e x} (-11 a B e+A b e+10 b B d)}{128 b^6 (b d-a e)}-\frac {63 e^4 (-11 a B e+A b e+10 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{13/2} \sqrt {b d-a e}}-\frac {(d+e x)^{9/2} (-11 a B e+A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac {3 e (d+e x)^{7/2} (-11 a B e+A b e+10 b B d)}{80 b^3 (a+b x)^3 (b d-a e)}-\frac {(d+e x)^{11/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^(9/2))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(63*e^4*(10*b*B*d + A*b*e - 11*a*B*e)*Sqrt[d + e*x])/(128*b^6*(b*d - a*e)) - (21*e^3*(10*b*B*d + A*b*e - 11*a*
B*e)*(d + e*x)^(3/2))/(128*b^5*(b*d - a*e)*(a + b*x)) - (21*e^2*(10*b*B*d + A*b*e - 11*a*B*e)*(d + e*x)^(5/2))
/(320*b^4*(b*d - a*e)*(a + b*x)^2) - (3*e*(10*b*B*d + A*b*e - 11*a*B*e)*(d + e*x)^(7/2))/(80*b^3*(b*d - a*e)*(
a + b*x)^3) - ((10*b*B*d + A*b*e - 11*a*B*e)*(d + e*x)^(9/2))/(40*b^2*(b*d - a*e)*(a + b*x)^4) - ((A*b - a*B)*
(d + e*x)^(11/2))/(5*b*(b*d - a*e)*(a + b*x)^5) - (63*e^4*(10*b*B*d + A*b*e - 11*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[
d + e*x])/Sqrt[b*d - a*e]])/(128*b^(13/2)*Sqrt[b*d - a*e])

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 47

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},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && 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) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {(A+B x) (d+e x)^{9/2}}{(a+b x)^6} \, dx\\ &=-\frac {(A b-a B) (d+e x)^{11/2}}{5 b (b d-a e) (a+b x)^5}+\frac {(10 b B d+A b e-11 a B e) \int \frac {(d+e x)^{9/2}}{(a+b x)^5} \, dx}{10 b (b d-a e)}\\ &=-\frac {(10 b B d+A b e-11 a B e) (d+e x)^{9/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{11/2}}{5 b (b d-a e) (a+b x)^5}+\frac {(9 e (10 b B d+A b e-11 a B e)) \int \frac {(d+e x)^{7/2}}{(a+b x)^4} \, dx}{80 b^2 (b d-a e)}\\ &=-\frac {3 e (10 b B d+A b e-11 a B e) (d+e x)^{7/2}}{80 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d+A b e-11 a B e) (d+e x)^{9/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{11/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (21 e^2 (10 b B d+A b e-11 a B e)\right ) \int \frac {(d+e x)^{5/2}}{(a+b x)^3} \, dx}{160 b^3 (b d-a e)}\\ &=-\frac {21 e^2 (10 b B d+A b e-11 a B e) (d+e x)^{5/2}}{320 b^4 (b d-a e) (a+b x)^2}-\frac {3 e (10 b B d+A b e-11 a B e) (d+e x)^{7/2}}{80 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d+A b e-11 a B e) (d+e x)^{9/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{11/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (21 e^3 (10 b B d+A b e-11 a B e)\right ) \int \frac {(d+e x)^{3/2}}{(a+b x)^2} \, dx}{128 b^4 (b d-a e)}\\ &=-\frac {21 e^3 (10 b B d+A b e-11 a B e) (d+e x)^{3/2}}{128 b^5 (b d-a e) (a+b x)}-\frac {21 e^2 (10 b B d+A b e-11 a B e) (d+e x)^{5/2}}{320 b^4 (b d-a e) (a+b x)^2}-\frac {3 e (10 b B d+A b e-11 a B e) (d+e x)^{7/2}}{80 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d+A b e-11 a B e) (d+e x)^{9/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{11/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (63 e^4 (10 b B d+A b e-11 a B e)\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{256 b^5 (b d-a e)}\\ &=\frac {63 e^4 (10 b B d+A b e-11 a B e) \sqrt {d+e x}}{128 b^6 (b d-a e)}-\frac {21 e^3 (10 b B d+A b e-11 a B e) (d+e x)^{3/2}}{128 b^5 (b d-a e) (a+b x)}-\frac {21 e^2 (10 b B d+A b e-11 a B e) (d+e x)^{5/2}}{320 b^4 (b d-a e) (a+b x)^2}-\frac {3 e (10 b B d+A b e-11 a B e) (d+e x)^{7/2}}{80 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d+A b e-11 a B e) (d+e x)^{9/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{11/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (63 e^4 (10 b B d+A b e-11 a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 b^6}\\ &=\frac {63 e^4 (10 b B d+A b e-11 a B e) \sqrt {d+e x}}{128 b^6 (b d-a e)}-\frac {21 e^3 (10 b B d+A b e-11 a B e) (d+e x)^{3/2}}{128 b^5 (b d-a e) (a+b x)}-\frac {21 e^2 (10 b B d+A b e-11 a B e) (d+e x)^{5/2}}{320 b^4 (b d-a e) (a+b x)^2}-\frac {3 e (10 b B d+A b e-11 a B e) (d+e x)^{7/2}}{80 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d+A b e-11 a B e) (d+e x)^{9/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{11/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (63 e^3 (10 b B d+A b e-11 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^6}\\ &=\frac {63 e^4 (10 b B d+A b e-11 a B e) \sqrt {d+e x}}{128 b^6 (b d-a e)}-\frac {21 e^3 (10 b B d+A b e-11 a B e) (d+e x)^{3/2}}{128 b^5 (b d-a e) (a+b x)}-\frac {21 e^2 (10 b B d+A b e-11 a B e) (d+e x)^{5/2}}{320 b^4 (b d-a e) (a+b x)^2}-\frac {3 e (10 b B d+A b e-11 a B e) (d+e x)^{7/2}}{80 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d+A b e-11 a B e) (d+e x)^{9/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{11/2}}{5 b (b d-a e) (a+b x)^5}-\frac {63 e^4 (10 b B d+A b e-11 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{13/2} \sqrt {b d-a e}}\\ \end {align*}

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Mathematica [C]  time = 0.07, size = 99, normalized size = 0.28 \[ \frac {(d+e x)^{11/2} \left (\frac {11 (a B-A b)}{(a+b x)^5}-\frac {e^4 (-11 a B e+A b e+10 b B d) \, _2F_1\left (5,\frac {11}{2};\frac {13}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^5}\right )}{55 b (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^(9/2))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

((d + e*x)^(11/2)*((11*(-(A*b) + a*B))/(a + b*x)^5 - (e^4*(10*b*B*d + A*b*e - 11*a*B*e)*Hypergeometric2F1[5, 1
1/2, 13/2, (b*(d + e*x))/(b*d - a*e)])/(b*d - a*e)^5))/(55*b*(b*d - a*e))

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fricas [B]  time = 0.64, size = 1955, normalized size = 5.55 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[1/1280*(315*(10*B*a^5*b*d*e^4 - (11*B*a^6 - A*a^5*b)*e^5 + (10*B*b^6*d*e^4 - (11*B*a*b^5 - A*b^6)*e^5)*x^5 +
5*(10*B*a*b^5*d*e^4 - (11*B*a^2*b^4 - A*a*b^5)*e^5)*x^4 + 10*(10*B*a^2*b^4*d*e^4 - (11*B*a^3*b^3 - A*a^2*b^4)*
e^5)*x^3 + 10*(10*B*a^3*b^3*d*e^4 - (11*B*a^4*b^2 - A*a^3*b^3)*e^5)*x^2 + 5*(10*B*a^4*b^2*d*e^4 - (11*B*a^5*b
- 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*(32*(B*a*b^6 + 4*A*b^7)*d^5 + 16*(4*B*a^2*b^5 + A*a*b^6)*d^4*e + 12*(13*B*a^3*b^4 + 2*A*a^2*b^5)*d^3
*e^2 + 42*(14*B*a^4*b^3 + A*a^3*b^4)*d^2*e^3 - 105*(41*B*a^5*b^2 - A*a^4*b^3)*d*e^4 + 315*(11*B*a^6*b - A*a^5*
b^2)*e^5 - 1280*(B*b^7*d*e^4 - B*a*b^6*e^5)*x^5 + 5*(650*B*b^7*d^2*e^3 - (2773*B*a*b^6 - 193*A*b^7)*d*e^4 + 19
3*(11*B*a^2*b^5 - A*a*b^6)*e^5)*x^4 + 10*(210*B*b^7*d^3*e^2 + (521*B*a*b^6 + 149*A*b^7)*d^2*e^3 - 2*(1669*B*a^
2*b^5 - 44*A*a*b^6)*d*e^4 + 237*(11*B*a^3*b^4 - A*a^2*b^5)*e^5)*x^3 + 2*(440*B*b^7*d^4*e + 2*(353*B*a*b^6 + 34
2*A*b^7)*d^3*e^2 + 3*(919*B*a^2*b^5 + 61*A*a*b^6)*d^2*e^3 - 3*(6229*B*a^3*b^4 - 159*A*a^2*b^5)*d*e^4 + 1344*(1
1*B*a^4*b^3 - A*a^3*b^4)*e^5)*x^2 + 2*(80*B*b^7*d^5 + 8*(19*B*a*b^6 + 41*A*b^7)*d^4*e + 2*(187*B*a^2*b^5 + 28*
A*a*b^6)*d^3*e^2 + 9*(159*B*a^3*b^4 + 11*A*a^2*b^5)*d^2*e^3 - 42*(241*B*a^4*b^3 - 6*A*a^3*b^4)*d*e^4 + 735*(11
*B*a^5*b^2 - A*a^4*b^3)*e^5)*x)*sqrt(e*x + d))/(a^5*b^8*d - a^6*b^7*e + (b^13*d - a*b^12*e)*x^5 + 5*(a*b^12*d
- a^2*b^11*e)*x^4 + 10*(a^2*b^11*d - a^3*b^10*e)*x^3 + 10*(a^3*b^10*d - a^4*b^9*e)*x^2 + 5*(a^4*b^9*d - a^5*b^
8*e)*x), 1/640*(315*(10*B*a^5*b*d*e^4 - (11*B*a^6 - A*a^5*b)*e^5 + (10*B*b^6*d*e^4 - (11*B*a*b^5 - A*b^6)*e^5)
*x^5 + 5*(10*B*a*b^5*d*e^4 - (11*B*a^2*b^4 - A*a*b^5)*e^5)*x^4 + 10*(10*B*a^2*b^4*d*e^4 - (11*B*a^3*b^3 - A*a^
2*b^4)*e^5)*x^3 + 10*(10*B*a^3*b^3*d*e^4 - (11*B*a^4*b^2 - A*a^3*b^3)*e^5)*x^2 + 5*(10*B*a^4*b^2*d*e^4 - (11*B
*a^5*b - 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)) - (3
2*(B*a*b^6 + 4*A*b^7)*d^5 + 16*(4*B*a^2*b^5 + A*a*b^6)*d^4*e + 12*(13*B*a^3*b^4 + 2*A*a^2*b^5)*d^3*e^2 + 42*(1
4*B*a^4*b^3 + A*a^3*b^4)*d^2*e^3 - 105*(41*B*a^5*b^2 - A*a^4*b^3)*d*e^4 + 315*(11*B*a^6*b - A*a^5*b^2)*e^5 - 1
280*(B*b^7*d*e^4 - B*a*b^6*e^5)*x^5 + 5*(650*B*b^7*d^2*e^3 - (2773*B*a*b^6 - 193*A*b^7)*d*e^4 + 193*(11*B*a^2*
b^5 - A*a*b^6)*e^5)*x^4 + 10*(210*B*b^7*d^3*e^2 + (521*B*a*b^6 + 149*A*b^7)*d^2*e^3 - 2*(1669*B*a^2*b^5 - 44*A
*a*b^6)*d*e^4 + 237*(11*B*a^3*b^4 - A*a^2*b^5)*e^5)*x^3 + 2*(440*B*b^7*d^4*e + 2*(353*B*a*b^6 + 342*A*b^7)*d^3
*e^2 + 3*(919*B*a^2*b^5 + 61*A*a*b^6)*d^2*e^3 - 3*(6229*B*a^3*b^4 - 159*A*a^2*b^5)*d*e^4 + 1344*(11*B*a^4*b^3
- A*a^3*b^4)*e^5)*x^2 + 2*(80*B*b^7*d^5 + 8*(19*B*a*b^6 + 41*A*b^7)*d^4*e + 2*(187*B*a^2*b^5 + 28*A*a*b^6)*d^3
*e^2 + 9*(159*B*a^3*b^4 + 11*A*a^2*b^5)*d^2*e^3 - 42*(241*B*a^4*b^3 - 6*A*a^3*b^4)*d*e^4 + 735*(11*B*a^5*b^2 -
A*a^4*b^3)*e^5)*x)*sqrt(e*x + d))/(a^5*b^8*d - a^6*b^7*e + (b^13*d - a*b^12*e)*x^5 + 5*(a*b^12*d - a^2*b^11*e
)*x^4 + 10*(a^2*b^11*d - a^3*b^10*e)*x^3 + 10*(a^3*b^10*d - a^4*b^9*e)*x^2 + 5*(a^4*b^9*d - a^5*b^8*e)*x)]

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giac [B]  time = 0.30, size = 770, normalized size = 2.19 \[ \frac {2 \, \sqrt {x e + d} B e^{4}}{b^{6}} + \frac {63 \, {\left (10 \, B b d e^{4} - 11 \, B a e^{5} + A b e^{5}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{128 \, \sqrt {-b^{2} d + a b e} b^{6}} - \frac {3250 \, {\left (x e + d\right )}^{\frac {9}{2}} B b^{5} d e^{4} - 10900 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{5} d^{2} e^{4} + 14080 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{5} d^{3} e^{4} - 8300 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{5} d^{4} e^{4} + 1870 \, \sqrt {x e + d} B b^{5} d^{5} e^{4} - 4215 \, {\left (x e + d\right )}^{\frac {9}{2}} B a b^{4} e^{5} + 965 \, {\left (x e + d\right )}^{\frac {9}{2}} A b^{5} e^{5} + 24170 \, {\left (x e + d\right )}^{\frac {7}{2}} B a b^{4} d e^{5} - 2370 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{5} d e^{5} - 44928 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{4} d^{2} e^{5} + 2688 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{5} d^{2} e^{5} + 34670 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{4} d^{3} e^{5} - 1470 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{5} d^{3} e^{5} - 9665 \, \sqrt {x e + d} B a b^{4} d^{4} e^{5} + 315 \, \sqrt {x e + d} A b^{5} d^{4} e^{5} - 13270 \, {\left (x e + d\right )}^{\frac {7}{2}} B a^{2} b^{3} e^{6} + 2370 \, {\left (x e + d\right )}^{\frac {7}{2}} A a b^{4} e^{6} + 47616 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} b^{3} d e^{6} - 5376 \, {\left (x e + d\right )}^{\frac {5}{2}} A a b^{4} d e^{6} - 54210 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{3} d^{2} e^{6} + 4410 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{4} d^{2} e^{6} + 19960 \, \sqrt {x e + d} B a^{2} b^{3} d^{3} e^{6} - 1260 \, \sqrt {x e + d} A a b^{4} d^{3} e^{6} - 16768 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{3} b^{2} e^{7} + 2688 \, {\left (x e + d\right )}^{\frac {5}{2}} A a^{2} b^{3} e^{7} + 37610 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b^{2} d e^{7} - 4410 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{3} d e^{7} - 20590 \, \sqrt {x e + d} B a^{3} b^{2} d^{2} e^{7} + 1890 \, \sqrt {x e + d} A a^{2} b^{3} d^{2} e^{7} - 9770 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{4} b e^{8} + 1470 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{3} b^{2} e^{8} + 10610 \, \sqrt {x e + d} B a^{4} b d e^{8} - 1260 \, \sqrt {x e + d} A a^{3} b^{2} d e^{8} - 2185 \, \sqrt {x e + d} B a^{5} e^{9} + 315 \, \sqrt {x e + d} A a^{4} b e^{9}}{640 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{5} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

2*sqrt(x*e + d)*B*e^4/b^6 + 63/128*(10*B*b*d*e^4 - 11*B*a*e^5 + A*b*e^5)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d +
a*b*e))/(sqrt(-b^2*d + a*b*e)*b^6) - 1/640*(3250*(x*e + d)^(9/2)*B*b^5*d*e^4 - 10900*(x*e + d)^(7/2)*B*b^5*d^2
*e^4 + 14080*(x*e + d)^(5/2)*B*b^5*d^3*e^4 - 8300*(x*e + d)^(3/2)*B*b^5*d^4*e^4 + 1870*sqrt(x*e + d)*B*b^5*d^5
*e^4 - 4215*(x*e + d)^(9/2)*B*a*b^4*e^5 + 965*(x*e + d)^(9/2)*A*b^5*e^5 + 24170*(x*e + d)^(7/2)*B*a*b^4*d*e^5
- 2370*(x*e + d)^(7/2)*A*b^5*d*e^5 - 44928*(x*e + d)^(5/2)*B*a*b^4*d^2*e^5 + 2688*(x*e + d)^(5/2)*A*b^5*d^2*e^
5 + 34670*(x*e + d)^(3/2)*B*a*b^4*d^3*e^5 - 1470*(x*e + d)^(3/2)*A*b^5*d^3*e^5 - 9665*sqrt(x*e + d)*B*a*b^4*d^
4*e^5 + 315*sqrt(x*e + d)*A*b^5*d^4*e^5 - 13270*(x*e + d)^(7/2)*B*a^2*b^3*e^6 + 2370*(x*e + d)^(7/2)*A*a*b^4*e
^6 + 47616*(x*e + d)^(5/2)*B*a^2*b^3*d*e^6 - 5376*(x*e + d)^(5/2)*A*a*b^4*d*e^6 - 54210*(x*e + d)^(3/2)*B*a^2*
b^3*d^2*e^6 + 4410*(x*e + d)^(3/2)*A*a*b^4*d^2*e^6 + 19960*sqrt(x*e + d)*B*a^2*b^3*d^3*e^6 - 1260*sqrt(x*e + d
)*A*a*b^4*d^3*e^6 - 16768*(x*e + d)^(5/2)*B*a^3*b^2*e^7 + 2688*(x*e + d)^(5/2)*A*a^2*b^3*e^7 + 37610*(x*e + d)
^(3/2)*B*a^3*b^2*d*e^7 - 4410*(x*e + d)^(3/2)*A*a^2*b^3*d*e^7 - 20590*sqrt(x*e + d)*B*a^3*b^2*d^2*e^7 + 1890*s
qrt(x*e + d)*A*a^2*b^3*d^2*e^7 - 9770*(x*e + d)^(3/2)*B*a^4*b*e^8 + 1470*(x*e + d)^(3/2)*A*a^3*b^2*e^8 + 10610
*sqrt(x*e + d)*B*a^4*b*d*e^8 - 1260*sqrt(x*e + d)*A*a^3*b^2*d*e^8 - 2185*sqrt(x*e + d)*B*a^5*e^9 + 315*sqrt(x*
e + d)*A*a^4*b*e^9)/(((x*e + d)*b - b*d + a*e)^5*b^6)

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maple [B]  time = 0.08, size = 1173, normalized size = 3.33 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

2*e^4*B/b^6*(e*x+d)^(1/2)+63/32*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a^3*d+42/5*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)
^(5/2)*A*a*d+351/5*e^5/b^2/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*a*d^2+2059/64*e^7/b^4/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a
^3*d^2-189/64*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a^2*d^2+63/32*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a*d^3-
1061/64*e^8/b^5/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^4*d-499/16*e^6/b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^2*d^3+1933/
128*e^5/b^2/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a*d^4-2417/64*e^5/b^2/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*a*d+441/64*e^7/b
^3/(b*e*x+a*e)^5*A*(e*x+d)^(3/2)*a^2*d-372/5*e^6/b^3/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*a^2*d-3467/64*e^5/b^2/(b*e*
x+a*e)^5*B*(e*x+d)^(3/2)*a*d^3-441/64*e^6/b^2/(b*e*x+a*e)^5*A*(e*x+d)^(3/2)*a*d^2-3761/64*e^7/b^4/(b*e*x+a*e)^
5*B*(e*x+d)^(3/2)*a^3*d+5421/64*e^6/b^3/(b*e*x+a*e)^5*B*(e*x+d)^(3/2)*a^2*d^2+63/128*e^5/b^5/((a*e-b*d)*b)^(1/
2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*A-193/128*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(9/2)*A-237/64*e^6/b^2/(b
*e*x+a*e)^5*A*(e*x+d)^(7/2)*a+237/64*e^5/b/(b*e*x+a*e)^5*A*(e*x+d)^(7/2)*d+437/128*e^9/b^6/(b*e*x+a*e)^5*(e*x+
d)^(1/2)*B*a^5-693/128*e^5/b^6/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a*B+1327/64*e^6
/b^3/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*a^2-147/64*e^8/b^4/(b*e*x+a*e)^5*A*(e*x+d)^(3/2)*a^3+147/64*e^5/b/(b*e*x+a*
e)^5*A*(e*x+d)^(3/2)*d^3+977/64*e^8/b^5/(b*e*x+a*e)^5*B*(e*x+d)^(3/2)*a^4+843/128*e^5/b^2/(b*e*x+a*e)^5*(e*x+d
)^(9/2)*B*a-21/5*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(5/2)*A*a^2-21/5*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(5/2)*A*d^2+131/5*
e^7/b^4/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*a^3-63/128*e^9/b^5/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a^4+415/32*e^4/b/(b*e*x
+a*e)^5*B*(e*x+d)^(3/2)*d^4+315/64*e^4/b^5/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*B*d
-325/64*e^4/b/(b*e*x+a*e)^5*(e*x+d)^(9/2)*B*d-22*e^4/b/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*d^3-187/64*e^4/b/(b*e*x+a
*e)^5*(e*x+d)^(1/2)*B*d^5+545/32*e^4/b/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*d^2-63/128*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(1
/2)*A*d^4

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(9/2)/(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(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 = 2.28, size = 838, normalized size = 2.38 \[ \frac {2\,B\,e^4\,\sqrt {d+e\,x}}{b^6}-\frac {{\left (d+e\,x\right )}^{5/2}\,\left (-\frac {131\,B\,a^3\,b^2\,e^7}{5}+\frac {372\,B\,a^2\,b^3\,d\,e^6}{5}+\frac {21\,A\,a^2\,b^3\,e^7}{5}-\frac {351\,B\,a\,b^4\,d^2\,e^5}{5}-\frac {42\,A\,a\,b^4\,d\,e^6}{5}+22\,B\,b^5\,d^3\,e^4+\frac {21\,A\,b^5\,d^2\,e^5}{5}\right )-{\left (d+e\,x\right )}^{3/2}\,\left (\frac {977\,B\,a^4\,b\,e^8}{64}-\frac {3761\,B\,a^3\,b^2\,d\,e^7}{64}-\frac {147\,A\,a^3\,b^2\,e^8}{64}+\frac {5421\,B\,a^2\,b^3\,d^2\,e^6}{64}+\frac {441\,A\,a^2\,b^3\,d\,e^7}{64}-\frac {3467\,B\,a\,b^4\,d^3\,e^5}{64}-\frac {441\,A\,a\,b^4\,d^2\,e^6}{64}+\frac {415\,B\,b^5\,d^4\,e^4}{32}+\frac {147\,A\,b^5\,d^3\,e^5}{64}\right )+{\left (d+e\,x\right )}^{9/2}\,\left (\frac {193\,A\,b^5\,e^5}{128}+\frac {325\,B\,d\,b^5\,e^4}{64}-\frac {843\,B\,a\,b^4\,e^5}{128}\right )+\sqrt {d+e\,x}\,\left (-\frac {437\,B\,a^5\,e^9}{128}+\frac {1061\,B\,a^4\,b\,d\,e^8}{64}+\frac {63\,A\,a^4\,b\,e^9}{128}-\frac {2059\,B\,a^3\,b^2\,d^2\,e^7}{64}-\frac {63\,A\,a^3\,b^2\,d\,e^8}{32}+\frac {499\,B\,a^2\,b^3\,d^3\,e^6}{16}+\frac {189\,A\,a^2\,b^3\,d^2\,e^7}{64}-\frac {1933\,B\,a\,b^4\,d^4\,e^5}{128}-\frac {63\,A\,a\,b^4\,d^3\,e^6}{32}+\frac {187\,B\,b^5\,d^5\,e^4}{64}+\frac {63\,A\,b^5\,d^4\,e^5}{128}\right )-{\left (d+e\,x\right )}^{7/2}\,\left (\frac {1327\,B\,a^2\,b^3\,e^6}{64}-\frac {2417\,B\,a\,b^4\,d\,e^5}{64}-\frac {237\,A\,a\,b^4\,e^6}{64}+\frac {545\,B\,b^5\,d^2\,e^4}{32}+\frac {237\,A\,b^5\,d\,e^5}{64}\right )}{\left (d+e\,x\right )\,\left (5\,a^4\,b^7\,e^4-20\,a^3\,b^8\,d\,e^3+30\,a^2\,b^9\,d^2\,e^2-20\,a\,b^{10}\,d^3\,e+5\,b^{11}\,d^4\right )-{\left (d+e\,x\right )}^2\,\left (-10\,a^3\,b^8\,e^3+30\,a^2\,b^9\,d\,e^2-30\,a\,b^{10}\,d^2\,e+10\,b^{11}\,d^3\right )+b^{11}\,{\left (d+e\,x\right )}^5-\left (5\,b^{11}\,d-5\,a\,b^{10}\,e\right )\,{\left (d+e\,x\right )}^4-b^{11}\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^9\,e^2-20\,a\,b^{10}\,d\,e+10\,b^{11}\,d^2\right )+a^5\,b^6\,e^5-5\,a^4\,b^7\,d\,e^4-10\,a^2\,b^9\,d^3\,e^2+10\,a^3\,b^8\,d^2\,e^3+5\,a\,b^{10}\,d^4\,e}+\frac {63\,e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^4\,\sqrt {d+e\,x}\,\left (A\,b\,e-11\,B\,a\,e+10\,B\,b\,d\right )}{\sqrt {a\,e-b\,d}\,\left (A\,b\,e^5-11\,B\,a\,e^5+10\,B\,b\,d\,e^4\right )}\right )\,\left (A\,b\,e-11\,B\,a\,e+10\,B\,b\,d\right )}{128\,b^{13/2}\,\sqrt {a\,e-b\,d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^(9/2))/(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

(2*B*e^4*(d + e*x)^(1/2))/b^6 - ((d + e*x)^(5/2)*((21*A*a^2*b^3*e^7)/5 - (131*B*a^3*b^2*e^7)/5 + (21*A*b^5*d^2
*e^5)/5 + 22*B*b^5*d^3*e^4 - (351*B*a*b^4*d^2*e^5)/5 + (372*B*a^2*b^3*d*e^6)/5 - (42*A*a*b^4*d*e^6)/5) - (d +
e*x)^(3/2)*((977*B*a^4*b*e^8)/64 - (147*A*a^3*b^2*e^8)/64 + (147*A*b^5*d^3*e^5)/64 + (415*B*b^5*d^4*e^4)/32 -
(441*A*a*b^4*d^2*e^6)/64 + (441*A*a^2*b^3*d*e^7)/64 - (3467*B*a*b^4*d^3*e^5)/64 - (3761*B*a^3*b^2*d*e^7)/64 +
(5421*B*a^2*b^3*d^2*e^6)/64) + (d + e*x)^(9/2)*((193*A*b^5*e^5)/128 - (843*B*a*b^4*e^5)/128 + (325*B*b^5*d*e^4
)/64) + (d + e*x)^(1/2)*((63*A*a^4*b*e^9)/128 - (437*B*a^5*e^9)/128 + (63*A*b^5*d^4*e^5)/128 + (187*B*b^5*d^5*
e^4)/64 - (63*A*a*b^4*d^3*e^6)/32 - (63*A*a^3*b^2*d*e^8)/32 - (1933*B*a*b^4*d^4*e^5)/128 + (189*A*a^2*b^3*d^2*
e^7)/64 + (499*B*a^2*b^3*d^3*e^6)/16 - (2059*B*a^3*b^2*d^2*e^7)/64 + (1061*B*a^4*b*d*e^8)/64) - (d + e*x)^(7/2
)*((237*A*b^5*d*e^5)/64 - (237*A*a*b^4*e^6)/64 + (1327*B*a^2*b^3*e^6)/64 + (545*B*b^5*d^2*e^4)/32 - (2417*B*a*
b^4*d*e^5)/64))/((d + e*x)*(5*b^11*d^4 + 5*a^4*b^7*e^4 - 20*a^3*b^8*d*e^3 + 30*a^2*b^9*d^2*e^2 - 20*a*b^10*d^3
*e) - (d + e*x)^2*(10*b^11*d^3 - 10*a^3*b^8*e^3 + 30*a^2*b^9*d*e^2 - 30*a*b^10*d^2*e) + b^11*(d + e*x)^5 - (5*
b^11*d - 5*a*b^10*e)*(d + e*x)^4 - b^11*d^5 + (d + e*x)^3*(10*b^11*d^2 + 10*a^2*b^9*e^2 - 20*a*b^10*d*e) + a^5
*b^6*e^5 - 5*a^4*b^7*d*e^4 - 10*a^2*b^9*d^3*e^2 + 10*a^3*b^8*d^2*e^3 + 5*a*b^10*d^4*e) + (63*e^4*atan((b^(1/2)
*e^4*(d + e*x)^(1/2)*(A*b*e - 11*B*a*e + 10*B*b*d))/((a*e - b*d)^(1/2)*(A*b*e^5 - 11*B*a*e^5 + 10*B*b*d*e^4)))
*(A*b*e - 11*B*a*e + 10*B*b*d))/(128*b^(13/2)*(a*e - b*d)^(1/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**(9/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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