3.1824 \(\int \frac{A+B x}{(d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^2} \, dx\)

Optimal. Leaf size=339 \[ -\frac{21 b^{3/2} e^2 (5 a B e-11 A b e+6 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{13/2}}+\frac{21 b e^2 (5 a B e-11 A b e+6 b B d)}{8 \sqrt{d+e x} (b d-a e)^6}+\frac{7 e^2 (5 a B e-11 A b e+6 b B d)}{8 (d+e x)^{3/2} (b d-a e)^5}+\frac{21 e^2 (5 a B e-11 A b e+6 b B d)}{40 b (d+e x)^{5/2} (b d-a e)^4}+\frac{3 e (5 a B e-11 A b e+6 b B d)}{8 b (a+b x) (d+e x)^{5/2} (b d-a e)^3}-\frac{5 a B e-11 A b e+6 b B d}{12 b (a+b x)^2 (d+e x)^{5/2} (b d-a e)^2}-\frac{A b-a B}{3 b (a+b x)^3 (d+e x)^{5/2} (b d-a e)} \]

[Out]

(21*e^2*(6*b*B*d - 11*A*b*e + 5*a*B*e))/(40*b*(b*d - a*e)^4*(d + e*x)^(5/2)) - (A*b - a*B)/(3*b*(b*d - a*e)*(a
+ b*x)^3*(d + e*x)^(5/2)) - (6*b*B*d - 11*A*b*e + 5*a*B*e)/(12*b*(b*d - a*e)^2*(a + b*x)^2*(d + e*x)^(5/2)) +
(3*e*(6*b*B*d - 11*A*b*e + 5*a*B*e))/(8*b*(b*d - a*e)^3*(a + b*x)*(d + e*x)^(5/2)) + (7*e^2*(6*b*B*d - 11*A*b
*e + 5*a*B*e))/(8*(b*d - a*e)^5*(d + e*x)^(3/2)) + (21*b*e^2*(6*b*B*d - 11*A*b*e + 5*a*B*e))/(8*(b*d - a*e)^6*
Sqrt[d + e*x]) - (21*b^(3/2)*e^2*(6*b*B*d - 11*A*b*e + 5*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e
]])/(8*(b*d - a*e)^(13/2))

________________________________________________________________________________________

Rubi [A]  time = 0.381144, antiderivative size = 339, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{21 b^{3/2} e^2 (5 a B e-11 A b e+6 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{13/2}}+\frac{21 b e^2 (5 a B e-11 A b e+6 b B d)}{8 \sqrt{d+e x} (b d-a e)^6}+\frac{7 e^2 (5 a B e-11 A b e+6 b B d)}{8 (d+e x)^{3/2} (b d-a e)^5}+\frac{21 e^2 (5 a B e-11 A b e+6 b B d)}{40 b (d+e x)^{5/2} (b d-a e)^4}+\frac{3 e (5 a B e-11 A b e+6 b B d)}{8 b (a+b x) (d+e x)^{5/2} (b d-a e)^3}-\frac{5 a B e-11 A b e+6 b B d}{12 b (a+b x)^2 (d+e x)^{5/2} (b d-a e)^2}-\frac{A b-a B}{3 b (a+b x)^3 (d+e x)^{5/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^2),x]

[Out]

(21*e^2*(6*b*B*d - 11*A*b*e + 5*a*B*e))/(40*b*(b*d - a*e)^4*(d + e*x)^(5/2)) - (A*b - a*B)/(3*b*(b*d - a*e)*(a
+ b*x)^3*(d + e*x)^(5/2)) - (6*b*B*d - 11*A*b*e + 5*a*B*e)/(12*b*(b*d - a*e)^2*(a + b*x)^2*(d + e*x)^(5/2)) +
(3*e*(6*b*B*d - 11*A*b*e + 5*a*B*e))/(8*b*(b*d - a*e)^3*(a + b*x)*(d + e*x)^(5/2)) + (7*e^2*(6*b*B*d - 11*A*b
*e + 5*a*B*e))/(8*(b*d - a*e)^5*(d + e*x)^(3/2)) + (21*b*e^2*(6*b*B*d - 11*A*b*e + 5*a*B*e))/(8*(b*d - a*e)^6*
Sqrt[d + e*x]) - (21*b^(3/2)*e^2*(6*b*B*d - 11*A*b*e + 5*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e
]])/(8*(b*d - a*e)^(13/2))

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 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 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 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)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{A+B x}{(a+b x)^4 (d+e x)^{7/2}} \, dx\\ &=-\frac{A b-a B}{3 b (b d-a e) (a+b x)^3 (d+e x)^{5/2}}+\frac{(6 b B d-11 A b e+5 a B e) \int \frac{1}{(a+b x)^3 (d+e x)^{7/2}} \, dx}{6 b (b d-a e)}\\ &=-\frac{A b-a B}{3 b (b d-a e) (a+b x)^3 (d+e x)^{5/2}}-\frac{6 b B d-11 A b e+5 a B e}{12 b (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}-\frac{(3 e (6 b B d-11 A b e+5 a B e)) \int \frac{1}{(a+b x)^2 (d+e x)^{7/2}} \, dx}{8 b (b d-a e)^2}\\ &=-\frac{A b-a B}{3 b (b d-a e) (a+b x)^3 (d+e x)^{5/2}}-\frac{6 b B d-11 A b e+5 a B e}{12 b (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}+\frac{3 e (6 b B d-11 A b e+5 a B e)}{8 b (b d-a e)^3 (a+b x) (d+e x)^{5/2}}+\frac{\left (21 e^2 (6 b B d-11 A b e+5 a B e)\right ) \int \frac{1}{(a+b x) (d+e x)^{7/2}} \, dx}{16 b (b d-a e)^3}\\ &=\frac{21 e^2 (6 b B d-11 A b e+5 a B e)}{40 b (b d-a e)^4 (d+e x)^{5/2}}-\frac{A b-a B}{3 b (b d-a e) (a+b x)^3 (d+e x)^{5/2}}-\frac{6 b B d-11 A b e+5 a B e}{12 b (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}+\frac{3 e (6 b B d-11 A b e+5 a B e)}{8 b (b d-a e)^3 (a+b x) (d+e x)^{5/2}}+\frac{\left (21 e^2 (6 b B d-11 A b e+5 a B e)\right ) \int \frac{1}{(a+b x) (d+e x)^{5/2}} \, dx}{16 (b d-a e)^4}\\ &=\frac{21 e^2 (6 b B d-11 A b e+5 a B e)}{40 b (b d-a e)^4 (d+e x)^{5/2}}-\frac{A b-a B}{3 b (b d-a e) (a+b x)^3 (d+e x)^{5/2}}-\frac{6 b B d-11 A b e+5 a B e}{12 b (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}+\frac{3 e (6 b B d-11 A b e+5 a B e)}{8 b (b d-a e)^3 (a+b x) (d+e x)^{5/2}}+\frac{7 e^2 (6 b B d-11 A b e+5 a B e)}{8 (b d-a e)^5 (d+e x)^{3/2}}+\frac{\left (21 b e^2 (6 b B d-11 A b e+5 a B e)\right ) \int \frac{1}{(a+b x) (d+e x)^{3/2}} \, dx}{16 (b d-a e)^5}\\ &=\frac{21 e^2 (6 b B d-11 A b e+5 a B e)}{40 b (b d-a e)^4 (d+e x)^{5/2}}-\frac{A b-a B}{3 b (b d-a e) (a+b x)^3 (d+e x)^{5/2}}-\frac{6 b B d-11 A b e+5 a B e}{12 b (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}+\frac{3 e (6 b B d-11 A b e+5 a B e)}{8 b (b d-a e)^3 (a+b x) (d+e x)^{5/2}}+\frac{7 e^2 (6 b B d-11 A b e+5 a B e)}{8 (b d-a e)^5 (d+e x)^{3/2}}+\frac{21 b e^2 (6 b B d-11 A b e+5 a B e)}{8 (b d-a e)^6 \sqrt{d+e x}}+\frac{\left (21 b^2 e^2 (6 b B d-11 A b e+5 a B e)\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{16 (b d-a e)^6}\\ &=\frac{21 e^2 (6 b B d-11 A b e+5 a B e)}{40 b (b d-a e)^4 (d+e x)^{5/2}}-\frac{A b-a B}{3 b (b d-a e) (a+b x)^3 (d+e x)^{5/2}}-\frac{6 b B d-11 A b e+5 a B e}{12 b (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}+\frac{3 e (6 b B d-11 A b e+5 a B e)}{8 b (b d-a e)^3 (a+b x) (d+e x)^{5/2}}+\frac{7 e^2 (6 b B d-11 A b e+5 a B e)}{8 (b d-a e)^5 (d+e x)^{3/2}}+\frac{21 b e^2 (6 b B d-11 A b e+5 a B e)}{8 (b d-a e)^6 \sqrt{d+e x}}+\frac{\left (21 b^2 e (6 b B d-11 A b e+5 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 )}{8 (b d-a e)^6}\\ &=\frac{21 e^2 (6 b B d-11 A b e+5 a B e)}{40 b (b d-a e)^4 (d+e x)^{5/2}}-\frac{A b-a B}{3 b (b d-a e) (a+b x)^3 (d+e x)^{5/2}}-\frac{6 b B d-11 A b e+5 a B e}{12 b (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2}}+\frac{3 e (6 b B d-11 A b e+5 a B e)}{8 b (b d-a e)^3 (a+b x) (d+e x)^{5/2}}+\frac{7 e^2 (6 b B d-11 A b e+5 a B e)}{8 (b d-a e)^5 (d+e x)^{3/2}}+\frac{21 b e^2 (6 b B d-11 A b e+5 a B e)}{8 (b d-a e)^6 \sqrt{d+e x}}-\frac{21 b^{3/2} e^2 (6 b B d-11 A b e+5 a B e) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{13/2}}\\ \end{align*}

Mathematica [C]  time = 0.0895626, size = 100, normalized size = 0.29 \[ \frac{\frac{5 (a B-A b)}{(a+b x)^3}-\frac{e^2 (-5 a B e+11 A b e-6 b B d) \, _2F_1\left (-\frac{5}{2},3;-\frac{3}{2};\frac{b (d+e x)}{b d-a e}\right )}{(b d-a e)^3}}{15 b (d+e x)^{5/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^2),x]

[Out]

((5*(-(A*b) + a*B))/(a + b*x)^3 - (e^2*(-6*b*B*d + 11*A*b*e - 5*a*B*e)*Hypergeometric2F1[-5/2, 3, -3/2, (b*(d
+ e*x))/(b*d - a*e)])/(b*d - a*e)^3)/(15*b*(b*d - a*e)*(d + e*x)^(5/2))

________________________________________________________________________________________

Maple [B]  time = 0.032, size = 935, normalized size = 2.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

-71/8*e^3/(a*e-b*d)^6*b^5/(b*e*x+a*e)^3*(e*x+d)^(5/2)*A+12*e^2*b^2/(a*e-b*d)^6/(e*x+d)^(1/2)*B*d-2*e^2/(a*e-b*
d)^5/(e*x+d)^(3/2)*B*b*d+8*e^3*b/(a*e-b*d)^6/(e*x+d)^(1/2)*a*B-231/8*e^3/(a*e-b*d)^6*b^3/((a*e-b*d)*b)^(1/2)*a
rctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*A-2/5*e^3/(a*e-b*d)^4/(e*x+d)^(5/2)*A+2/5*e^2/(a*e-b*d)^4/(e*x+d)^(
5/2)*B*d-20*e^3*b^2/(a*e-b*d)^6/(e*x+d)^(1/2)*A+8/3*e^3/(a*e-b*d)^5/(e*x+d)^(3/2)*A*b-2/3*e^3/(a*e-b*d)^5/(e*x
+d)^(3/2)*a*B+59/3*e^3/(a*e-b*d)^6*b^5/(b*e*x+a*e)^3*A*(e*x+d)^(3/2)*d+35/3*e^4/(a*e-b*d)^6*b^3/(b*e*x+a*e)^3*
B*(e*x+d)^(3/2)*a^2-89/8*e^5/(a*e-b*d)^6*b^3/(b*e*x+a*e)^3*(e*x+d)^(1/2)*A*a^2-89/8*e^3/(a*e-b*d)^6*b^5/(b*e*x
+a*e)^3*(e*x+d)^(1/2)*A*d^2+55/8*e^5/(a*e-b*d)^6*b^2/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*a^3+105/8*e^3/(a*e-b*d)^6*b
^2/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*a*B+63/4*e^2/(a*e-b*d)^6*b^3/((a*e-b*d)*b)^
(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*B*d+17/4*e^2/(a*e-b*d)^6*b^5/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*d
^3-8*e^2/(a*e-b*d)^6*b^5/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*d^2+15/4*e^2/(a*e-b*d)^6*b^5/(b*e*x+a*e)^3*(e*x+d)^(5/2
)*B*d+41/8*e^3/(a*e-b*d)^6*b^4/(b*e*x+a*e)^3*(e*x+d)^(5/2)*B*a-59/3*e^4/(a*e-b*d)^6*b^4/(b*e*x+a*e)^3*A*(e*x+d
)^(3/2)*a-19/2*e^4/(a*e-b*d)^6*b^3/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*a^2*d-13/8*e^3/(a*e-b*d)^6*b^4/(b*e*x+a*e)^3*
(e*x+d)^(1/2)*B*a*d^2+89/4*e^4/(a*e-b*d)^6*b^4/(b*e*x+a*e)^3*(e*x+d)^(1/2)*A*a*d-11/3*e^3/(a*e-b*d)^6*b^4/(b*e
*x+a*e)^3*B*(e*x+d)^(3/2)*a*d

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.85747, size = 7695, normalized size = 22.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

[-1/240*(315*(6*B*a^3*b^2*d^4*e^2 + (5*B*a^4*b - 11*A*a^3*b^2)*d^3*e^3 + (6*B*b^5*d*e^5 + (5*B*a*b^4 - 11*A*b^
5)*e^6)*x^6 + 3*(6*B*b^5*d^2*e^4 + 11*(B*a*b^4 - A*b^5)*d*e^5 + (5*B*a^2*b^3 - 11*A*a*b^4)*e^6)*x^5 + 3*(6*B*b
^5*d^3*e^3 + (23*B*a*b^4 - 11*A*b^5)*d^2*e^4 + 3*(7*B*a^2*b^3 - 11*A*a*b^4)*d*e^5 + (5*B*a^3*b^2 - 11*A*a^2*b^
3)*e^6)*x^4 + (6*B*b^5*d^4*e^2 + (59*B*a*b^4 - 11*A*b^5)*d^3*e^3 + 99*(B*a^2*b^3 - A*a*b^4)*d^2*e^4 + 3*(17*B*
a^3*b^2 - 33*A*a^2*b^3)*d*e^5 + (5*B*a^4*b - 11*A*a^3*b^2)*e^6)*x^3 + 3*(6*B*a*b^4*d^4*e^2 + (23*B*a^2*b^3 - 1
1*A*a*b^4)*d^3*e^3 + 3*(7*B*a^3*b^2 - 11*A*a^2*b^3)*d^2*e^4 + (5*B*a^4*b - 11*A*a^3*b^2)*d*e^5)*x^2 + 3*(6*B*a
^2*b^3*d^4*e^2 + 11*(B*a^3*b^2 - A*a^2*b^3)*d^3*e^3 + (5*B*a^4*b - 11*A*a^3*b^2)*d^2*e^4)*x)*sqrt(b/(b*d - a*e
))*log((b*e*x + 2*b*d - a*e + 2*(b*d - a*e)*sqrt(e*x + d)*sqrt(b/(b*d - a*e)))/(b*x + a)) + 2*(48*A*a^5*e^5 +
20*(B*a*b^4 + 2*A*b^5)*d^5 - 10*(26*B*a^2*b^3 + 31*A*a*b^4)*d^4*e - 3*(851*B*a^3*b^2 - 445*A*a^2*b^3)*d^3*e^2
- 16*(44*B*a^4*b - 173*A*a^3*b^2)*d^2*e^3 + 32*(B*a^5 - 13*A*a^4*b)*d*e^4 - 315*(6*B*b^5*d*e^4 + (5*B*a*b^4 -
11*A*b^5)*e^5)*x^5 - 105*(42*B*b^5*d^2*e^3 + (83*B*a*b^4 - 77*A*b^5)*d*e^4 + 8*(5*B*a^2*b^3 - 11*A*a*b^4)*e^5)
*x^4 - 21*(138*B*b^5*d^3*e^2 + (679*B*a*b^4 - 253*A*b^5)*d^2*e^3 + 2*(334*B*a^2*b^3 - 517*A*a*b^4)*d*e^4 + 33*
(5*B*a^3*b^2 - 11*A*a^2*b^3)*e^5)*x^3 - 9*(30*B*b^5*d^4*e + (901*B*a*b^4 - 55*A*b^5)*d^3*e^2 + 2*(914*B*a^2*b^
3 - 803*A*a*b^4)*d^2*e^3 + 3*(337*B*a^3*b^2 - 671*A*a^2*b^3)*d*e^4 + 16*(5*B*a^4*b - 11*A*a^3*b^2)*e^5)*x^2 +
(60*B*b^5*d^5 - 10*(73*B*a*b^4 + 11*A*b^5)*d^4*e - 2*(3682*B*a^2*b^3 - 715*A*a*b^4)*d^3*e^2 - 3*(2569*B*a^3*b^
2 - 4103*A*a^2*b^3)*d^2*e^3 - 32*(52*B*a^4*b - 121*A*a^3*b^2)*d*e^4 + 16*(5*B*a^5 - 11*A*a^4*b)*e^5)*x)*sqrt(e
*x + d))/(a^3*b^6*d^9 - 6*a^4*b^5*d^8*e + 15*a^5*b^4*d^7*e^2 - 20*a^6*b^3*d^6*e^3 + 15*a^7*b^2*d^5*e^4 - 6*a^8
*b*d^4*e^5 + a^9*d^3*e^6 + (b^9*d^6*e^3 - 6*a*b^8*d^5*e^4 + 15*a^2*b^7*d^4*e^5 - 20*a^3*b^6*d^3*e^6 + 15*a^4*b
^5*d^2*e^7 - 6*a^5*b^4*d*e^8 + a^6*b^3*e^9)*x^6 + 3*(b^9*d^7*e^2 - 5*a*b^8*d^6*e^3 + 9*a^2*b^7*d^5*e^4 - 5*a^3
*b^6*d^4*e^5 - 5*a^4*b^5*d^3*e^6 + 9*a^5*b^4*d^2*e^7 - 5*a^6*b^3*d*e^8 + a^7*b^2*e^9)*x^5 + 3*(b^9*d^8*e - 3*a
*b^8*d^7*e^2 - 2*a^2*b^7*d^6*e^3 + 19*a^3*b^6*d^5*e^4 - 30*a^4*b^5*d^4*e^5 + 19*a^5*b^4*d^3*e^6 - 2*a^6*b^3*d^
2*e^7 - 3*a^7*b^2*d*e^8 + a^8*b*e^9)*x^4 + (b^9*d^9 + 3*a*b^8*d^8*e - 30*a^2*b^7*d^7*e^2 + 62*a^3*b^6*d^6*e^3
- 36*a^4*b^5*d^5*e^4 - 36*a^5*b^4*d^4*e^5 + 62*a^6*b^3*d^3*e^6 - 30*a^7*b^2*d^2*e^7 + 3*a^8*b*d*e^8 + a^9*e^9)
*x^3 + 3*(a*b^8*d^9 - 3*a^2*b^7*d^8*e - 2*a^3*b^6*d^7*e^2 + 19*a^4*b^5*d^6*e^3 - 30*a^5*b^4*d^5*e^4 + 19*a^6*b
^3*d^4*e^5 - 2*a^7*b^2*d^3*e^6 - 3*a^8*b*d^2*e^7 + a^9*d*e^8)*x^2 + 3*(a^2*b^7*d^9 - 5*a^3*b^6*d^8*e + 9*a^4*b
^5*d^7*e^2 - 5*a^5*b^4*d^6*e^3 - 5*a^6*b^3*d^5*e^4 + 9*a^7*b^2*d^4*e^5 - 5*a^8*b*d^3*e^6 + a^9*d^2*e^7)*x), -1
/120*(315*(6*B*a^3*b^2*d^4*e^2 + (5*B*a^4*b - 11*A*a^3*b^2)*d^3*e^3 + (6*B*b^5*d*e^5 + (5*B*a*b^4 - 11*A*b^5)*
e^6)*x^6 + 3*(6*B*b^5*d^2*e^4 + 11*(B*a*b^4 - A*b^5)*d*e^5 + (5*B*a^2*b^3 - 11*A*a*b^4)*e^6)*x^5 + 3*(6*B*b^5*
d^3*e^3 + (23*B*a*b^4 - 11*A*b^5)*d^2*e^4 + 3*(7*B*a^2*b^3 - 11*A*a*b^4)*d*e^5 + (5*B*a^3*b^2 - 11*A*a^2*b^3)*
e^6)*x^4 + (6*B*b^5*d^4*e^2 + (59*B*a*b^4 - 11*A*b^5)*d^3*e^3 + 99*(B*a^2*b^3 - A*a*b^4)*d^2*e^4 + 3*(17*B*a^3
*b^2 - 33*A*a^2*b^3)*d*e^5 + (5*B*a^4*b - 11*A*a^3*b^2)*e^6)*x^3 + 3*(6*B*a*b^4*d^4*e^2 + (23*B*a^2*b^3 - 11*A
*a*b^4)*d^3*e^3 + 3*(7*B*a^3*b^2 - 11*A*a^2*b^3)*d^2*e^4 + (5*B*a^4*b - 11*A*a^3*b^2)*d*e^5)*x^2 + 3*(6*B*a^2*
b^3*d^4*e^2 + 11*(B*a^3*b^2 - A*a^2*b^3)*d^3*e^3 + (5*B*a^4*b - 11*A*a^3*b^2)*d^2*e^4)*x)*sqrt(-b/(b*d - a*e))
*arctan(-(b*d - a*e)*sqrt(e*x + d)*sqrt(-b/(b*d - a*e))/(b*e*x + b*d)) + (48*A*a^5*e^5 + 20*(B*a*b^4 + 2*A*b^5
)*d^5 - 10*(26*B*a^2*b^3 + 31*A*a*b^4)*d^4*e - 3*(851*B*a^3*b^2 - 445*A*a^2*b^3)*d^3*e^2 - 16*(44*B*a^4*b - 17
3*A*a^3*b^2)*d^2*e^3 + 32*(B*a^5 - 13*A*a^4*b)*d*e^4 - 315*(6*B*b^5*d*e^4 + (5*B*a*b^4 - 11*A*b^5)*e^5)*x^5 -
105*(42*B*b^5*d^2*e^3 + (83*B*a*b^4 - 77*A*b^5)*d*e^4 + 8*(5*B*a^2*b^3 - 11*A*a*b^4)*e^5)*x^4 - 21*(138*B*b^5*
d^3*e^2 + (679*B*a*b^4 - 253*A*b^5)*d^2*e^3 + 2*(334*B*a^2*b^3 - 517*A*a*b^4)*d*e^4 + 33*(5*B*a^3*b^2 - 11*A*a
^2*b^3)*e^5)*x^3 - 9*(30*B*b^5*d^4*e + (901*B*a*b^4 - 55*A*b^5)*d^3*e^2 + 2*(914*B*a^2*b^3 - 803*A*a*b^4)*d^2*
e^3 + 3*(337*B*a^3*b^2 - 671*A*a^2*b^3)*d*e^4 + 16*(5*B*a^4*b - 11*A*a^3*b^2)*e^5)*x^2 + (60*B*b^5*d^5 - 10*(7
3*B*a*b^4 + 11*A*b^5)*d^4*e - 2*(3682*B*a^2*b^3 - 715*A*a*b^4)*d^3*e^2 - 3*(2569*B*a^3*b^2 - 4103*A*a^2*b^3)*d
^2*e^3 - 32*(52*B*a^4*b - 121*A*a^3*b^2)*d*e^4 + 16*(5*B*a^5 - 11*A*a^4*b)*e^5)*x)*sqrt(e*x + d))/(a^3*b^6*d^9
- 6*a^4*b^5*d^8*e + 15*a^5*b^4*d^7*e^2 - 20*a^6*b^3*d^6*e^3 + 15*a^7*b^2*d^5*e^4 - 6*a^8*b*d^4*e^5 + a^9*d^3*
e^6 + (b^9*d^6*e^3 - 6*a*b^8*d^5*e^4 + 15*a^2*b^7*d^4*e^5 - 20*a^3*b^6*d^3*e^6 + 15*a^4*b^5*d^2*e^7 - 6*a^5*b^
4*d*e^8 + a^6*b^3*e^9)*x^6 + 3*(b^9*d^7*e^2 - 5*a*b^8*d^6*e^3 + 9*a^2*b^7*d^5*e^4 - 5*a^3*b^6*d^4*e^5 - 5*a^4*
b^5*d^3*e^6 + 9*a^5*b^4*d^2*e^7 - 5*a^6*b^3*d*e^8 + a^7*b^2*e^9)*x^5 + 3*(b^9*d^8*e - 3*a*b^8*d^7*e^2 - 2*a^2*
b^7*d^6*e^3 + 19*a^3*b^6*d^5*e^4 - 30*a^4*b^5*d^4*e^5 + 19*a^5*b^4*d^3*e^6 - 2*a^6*b^3*d^2*e^7 - 3*a^7*b^2*d*e
^8 + a^8*b*e^9)*x^4 + (b^9*d^9 + 3*a*b^8*d^8*e - 30*a^2*b^7*d^7*e^2 + 62*a^3*b^6*d^6*e^3 - 36*a^4*b^5*d^5*e^4
- 36*a^5*b^4*d^4*e^5 + 62*a^6*b^3*d^3*e^6 - 30*a^7*b^2*d^2*e^7 + 3*a^8*b*d*e^8 + a^9*e^9)*x^3 + 3*(a*b^8*d^9 -
3*a^2*b^7*d^8*e - 2*a^3*b^6*d^7*e^2 + 19*a^4*b^5*d^6*e^3 - 30*a^5*b^4*d^5*e^4 + 19*a^6*b^3*d^4*e^5 - 2*a^7*b^
2*d^3*e^6 - 3*a^8*b*d^2*e^7 + a^9*d*e^8)*x^2 + 3*(a^2*b^7*d^9 - 5*a^3*b^6*d^8*e + 9*a^4*b^5*d^7*e^2 - 5*a^5*b^
4*d^6*e^3 - 5*a^6*b^3*d^5*e^4 + 9*a^7*b^2*d^4*e^5 - 5*a^8*b*d^3*e^6 + a^9*d^2*e^7)*x)]

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.27828, size = 1052, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

21/8*(6*B*b^3*d*e^2 + 5*B*a*b^2*e^3 - 11*A*b^3*e^3)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^6*d^6 - 6
*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6)*sqrt(-b
^2*d + a*b*e)) + 2/15*(90*(x*e + d)^2*B*b^2*d*e^2 + 15*(x*e + d)*B*b^2*d^2*e^2 + 3*B*b^2*d^3*e^2 + 60*(x*e + d
)^2*B*a*b*e^3 - 150*(x*e + d)^2*A*b^2*e^3 - 10*(x*e + d)*B*a*b*d*e^3 - 20*(x*e + d)*A*b^2*d*e^3 - 6*B*a*b*d^2*
e^3 - 3*A*b^2*d^2*e^3 - 5*(x*e + d)*B*a^2*e^4 + 20*(x*e + d)*A*a*b*e^4 + 3*B*a^2*d*e^4 + 6*A*a*b*d*e^4 - 3*A*a
^2*e^5)/((b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e
^5 + a^6*e^6)*(x*e + d)^(5/2)) + 1/24*(90*(x*e + d)^(5/2)*B*b^5*d*e^2 - 192*(x*e + d)^(3/2)*B*b^5*d^2*e^2 + 10
2*sqrt(x*e + d)*B*b^5*d^3*e^2 + 123*(x*e + d)^(5/2)*B*a*b^4*e^3 - 213*(x*e + d)^(5/2)*A*b^5*e^3 - 88*(x*e + d)
^(3/2)*B*a*b^4*d*e^3 + 472*(x*e + d)^(3/2)*A*b^5*d*e^3 - 39*sqrt(x*e + d)*B*a*b^4*d^2*e^3 - 267*sqrt(x*e + d)*
A*b^5*d^2*e^3 + 280*(x*e + d)^(3/2)*B*a^2*b^3*e^4 - 472*(x*e + d)^(3/2)*A*a*b^4*e^4 - 228*sqrt(x*e + d)*B*a^2*
b^3*d*e^4 + 534*sqrt(x*e + d)*A*a*b^4*d*e^4 + 165*sqrt(x*e + d)*B*a^3*b^2*e^5 - 267*sqrt(x*e + d)*A*a^2*b^3*e^
5)/((b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 +
a^6*e^6)*((x*e + d)*b - b*d + a*e)^3)