3.1883 \(\int \frac{A+B x}{(d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=564 \[ -\frac{231 b e^3 (a+b x) (5 a B e-13 A b e+8 b B d)}{64 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^7}-\frac{77 e^3 (a+b x) (5 a B e-13 A b e+8 b B d)}{64 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^6}-\frac{231 e^3 (a+b x) (5 a B e-13 A b e+8 b B d)}{320 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^5}-\frac{33 e^2 (5 a B e-13 A b e+8 b B d)}{64 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4}+\frac{231 b^{3/2} e^3 (a+b x) (5 a B e-13 A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{15/2}}+\frac{11 e (5 a B e-13 A b e+8 b B d)}{96 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}-\frac{5 a B e-13 A b e+8 b B d}{24 b (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}-\frac{A b-a B}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)} \]

[Out]

(-33*e^2*(8*b*B*d - 13*A*b*e + 5*a*B*e))/(64*b*(b*d - a*e)^4*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) -
(A*b - a*B)/(4*b*(b*d - a*e)*(a + b*x)^3*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (8*b*B*d - 13*A*b*e
+ 5*a*B*e)/(24*b*(b*d - a*e)^2*(a + b*x)^2*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (11*e*(8*b*B*d - 1
3*A*b*e + 5*a*B*e))/(96*b*(b*d - a*e)^3*(a + b*x)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (231*e^3*(8
*b*B*d - 13*A*b*e + 5*a*B*e)*(a + b*x))/(320*b*(b*d - a*e)^5*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) -
(77*e^3*(8*b*B*d - 13*A*b*e + 5*a*B*e)*(a + b*x))/(64*(b*d - a*e)^6*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x
^2]) - (231*b*e^3*(8*b*B*d - 13*A*b*e + 5*a*B*e)*(a + b*x))/(64*(b*d - a*e)^7*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2]) + (231*b^(3/2)*e^3*(8*b*B*d - 13*A*b*e + 5*a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[
b*d - a*e]])/(64*(b*d - a*e)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.572198, antiderivative size = 564, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {770, 78, 51, 63, 208} \[ -\frac{231 b e^3 (a+b x) (5 a B e-13 A b e+8 b B d)}{64 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^7}-\frac{77 e^3 (a+b x) (5 a B e-13 A b e+8 b B d)}{64 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^6}-\frac{231 e^3 (a+b x) (5 a B e-13 A b e+8 b B d)}{320 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^5}-\frac{33 e^2 (5 a B e-13 A b e+8 b B d)}{64 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4}+\frac{231 b^{3/2} e^3 (a+b x) (5 a B e-13 A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{15/2}}+\frac{11 e (5 a B e-13 A b e+8 b B d)}{96 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}-\frac{5 a B e-13 A b e+8 b B d}{24 b (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}-\frac{A b-a B}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(-33*e^2*(8*b*B*d - 13*A*b*e + 5*a*B*e))/(64*b*(b*d - a*e)^4*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) -
(A*b - a*B)/(4*b*(b*d - a*e)*(a + b*x)^3*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (8*b*B*d - 13*A*b*e
+ 5*a*B*e)/(24*b*(b*d - a*e)^2*(a + b*x)^2*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (11*e*(8*b*B*d - 1
3*A*b*e + 5*a*B*e))/(96*b*(b*d - a*e)^3*(a + b*x)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (231*e^3*(8
*b*B*d - 13*A*b*e + 5*a*B*e)*(a + b*x))/(320*b*(b*d - a*e)^5*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) -
(77*e^3*(8*b*B*d - 13*A*b*e + 5*a*B*e)*(a + b*x))/(64*(b*d - a*e)^6*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x
^2]) - (231*b*e^3*(8*b*B*d - 13*A*b*e + 5*a*B*e)*(a + b*x))/(64*(b*d - a*e)^7*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2]) + (231*b^(3/2)*e^3*(8*b*B*d - 13*A*b*e + 5*a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[
b*d - a*e]])/(64*(b*d - a*e)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

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 )^{5/2}} \, dx &=\frac{\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac{A+B x}{\left (a b+b^2 x\right )^5 (d+e x)^{7/2}} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{A b-a B}{4 b (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (b^2 (8 b B d-13 A b e+5 a B e) \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right )^4 (d+e x)^{7/2}} \, dx}{8 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{A b-a B}{4 b (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{8 b B d-13 A b e+5 a B e}{24 b (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (11 b e (8 b B d-13 A b e+5 a B e) \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right )^3 (d+e x)^{7/2}} \, dx}{48 (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{A b-a B}{4 b (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{8 b B d-13 A b e+5 a B e}{24 b (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{11 e (8 b B d-13 A b e+5 a B e)}{96 b (b d-a e)^3 (a+b x) (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (33 e^2 (8 b B d-13 A b e+5 a B e) \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right )^2 (d+e x)^{7/2}} \, dx}{64 (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{33 e^2 (8 b B d-13 A b e+5 a B e)}{64 b (b d-a e)^4 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A b-a B}{4 b (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{8 b B d-13 A b e+5 a B e}{24 b (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{11 e (8 b B d-13 A b e+5 a B e)}{96 b (b d-a e)^3 (a+b x) (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (231 e^3 (8 b B d-13 A b e+5 a B e) \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right ) (d+e x)^{7/2}} \, dx}{128 b (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{33 e^2 (8 b B d-13 A b e+5 a B e)}{64 b (b d-a e)^4 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A b-a B}{4 b (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{8 b B d-13 A b e+5 a B e}{24 b (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{11 e (8 b B d-13 A b e+5 a B e)}{96 b (b d-a e)^3 (a+b x) (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 e^3 (8 b B d-13 A b e+5 a B e) (a+b x)}{320 b (b d-a e)^5 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (231 e^3 (8 b B d-13 A b e+5 a B e) \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right ) (d+e x)^{5/2}} \, dx}{128 (b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{33 e^2 (8 b B d-13 A b e+5 a B e)}{64 b (b d-a e)^4 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A b-a B}{4 b (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{8 b B d-13 A b e+5 a B e}{24 b (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{11 e (8 b B d-13 A b e+5 a B e)}{96 b (b d-a e)^3 (a+b x) (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 e^3 (8 b B d-13 A b e+5 a B e) (a+b x)}{320 b (b d-a e)^5 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{77 e^3 (8 b B d-13 A b e+5 a B e) (a+b x)}{64 (b d-a e)^6 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (231 b e^3 (8 b B d-13 A b e+5 a B e) \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right ) (d+e x)^{3/2}} \, dx}{128 (b d-a e)^6 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{33 e^2 (8 b B d-13 A b e+5 a B e)}{64 b (b d-a e)^4 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A b-a B}{4 b (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{8 b B d-13 A b e+5 a B e}{24 b (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{11 e (8 b B d-13 A b e+5 a B e)}{96 b (b d-a e)^3 (a+b x) (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 e^3 (8 b B d-13 A b e+5 a B e) (a+b x)}{320 b (b d-a e)^5 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{77 e^3 (8 b B d-13 A b e+5 a B e) (a+b x)}{64 (b d-a e)^6 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 b e^3 (8 b B d-13 A b e+5 a B e) (a+b x)}{64 (b d-a e)^7 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (231 b^2 e^3 (8 b B d-13 A b e+5 a B e) \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right ) \sqrt{d+e x}} \, dx}{128 (b d-a e)^7 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{33 e^2 (8 b B d-13 A b e+5 a B e)}{64 b (b d-a e)^4 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A b-a B}{4 b (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{8 b B d-13 A b e+5 a B e}{24 b (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{11 e (8 b B d-13 A b e+5 a B e)}{96 b (b d-a e)^3 (a+b x) (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 e^3 (8 b B d-13 A b e+5 a B e) (a+b x)}{320 b (b d-a e)^5 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{77 e^3 (8 b B d-13 A b e+5 a B e) (a+b x)}{64 (b d-a e)^6 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 b e^3 (8 b B d-13 A b e+5 a B e) (a+b x)}{64 (b d-a e)^7 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (231 b^2 e^2 (8 b B d-13 A b e+5 a B e) \left (a b+b^2 x\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a b-\frac{b^2 d}{e}+\frac{b^2 x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{64 (b d-a e)^7 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{33 e^2 (8 b B d-13 A b e+5 a B e)}{64 b (b d-a e)^4 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A b-a B}{4 b (b d-a e) (a+b x)^3 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{8 b B d-13 A b e+5 a B e}{24 b (b d-a e)^2 (a+b x)^2 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{11 e (8 b B d-13 A b e+5 a B e)}{96 b (b d-a e)^3 (a+b x) (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 e^3 (8 b B d-13 A b e+5 a B e) (a+b x)}{320 b (b d-a e)^5 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{77 e^3 (8 b B d-13 A b e+5 a B e) (a+b x)}{64 (b d-a e)^6 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{231 b e^3 (8 b B d-13 A b e+5 a B e) (a+b x)}{64 (b d-a e)^7 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{231 b^{3/2} e^3 (8 b B d-13 A b e+5 a B e) (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 (b d-a e)^{15/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}

Mathematica [C]  time = 0.120768, size = 115, normalized size = 0.2 \[ \frac{\frac{e^3 (a+b x)^4 (-5 a B e+13 A b e-8 b B d) \, _2F_1\left (-\frac{5}{2},4;-\frac{3}{2};\frac{b (d+e x)}{b d-a e}\right )}{(b d-a e)^4}+5 a B-5 A b}{20 b (a+b x)^3 \sqrt{(a+b x)^2} (d+e x)^{5/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(-5*A*b + 5*a*B + (e^3*(-8*b*B*d + 13*A*b*e - 5*a*B*e)*(a + b*x)^4*Hypergeometric2F1[-5/2, 4, -3/2, (b*(d + e*
x))/(b*d - a*e)])/(b*d - a*e)^4)/(20*b*(b*d - a*e)*(a + b*x)^3*Sqrt[(a + b*x)^2]*(d + e*x)^(5/2))

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Maple [B]  time = 0.039, size = 2282, normalized size = 4.1 \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)^(5/2),x)

[Out]

-1/960*(-27720*B*((a*e-b*d)*b)^(1/2)*x^6*b^6*d*e^5+165165*A*((a*e-b*d)*b)^(1/2)*x^5*a*b^5*e^6+105105*A*((a*e-b
*d)*b)^(1/2)*x^5*b^6*d*e^5-63525*B*((a*e-b*d)*b)^(1/2)*x^5*a^2*b^4*e^6-64680*B*((a*e-b*d)*b)^(1/2)*x^5*b^6*d^2
*e^4+219219*A*((a*e-b*d)*b)^(1/2)*x^4*a^2*b^4*e^6+45045*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^
(5/2)*a^4*b^3*e^4-84315*B*((a*e-b*d)*b)^(1/2)*x^4*a^3*b^3*e^6-42504*B*((a*e-b*d)*b)^(1/2)*x^4*b^6*d^3*e^3-1732
5*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(5/2)*a^5*b^2*e^4+119691*A*((a*e-b*d)*b)^(1/2)*x^3*a^3
*b^3*e^6+6435*A*((a*e-b*d)*b)^(1/2)*x^3*b^6*d^3*e^3-46035*B*((a*e-b*d)*b)^(1/2)*x^3*a^4*b^2*e^6-3960*B*((a*e-b
*d)*b)^(1/2)*x^3*b^6*d^4*e^2+18304*A*((a*e-b*d)*b)^(1/2)*x^2*a^4*b^2*e^6-1430*A*((a*e-b*d)*b)^(1/2)*x^2*b^6*d^
4*e^2-7040*B*((a*e-b*d)*b)^(1/2)*x^2*a^5*b*e^6+880*B*((a*e-b*d)*b)^(1/2)*x^2*b^6*d^5*e-1664*A*((a*e-b*d)*b)^(1
/2)*x*a^5*b*e^6+520*A*((a*e-b*d)*b)^(1/2)*x*b^6*d^5*e+45045*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x
+d)^(5/2)*x^4*b^7*e^4-17325*B*((a*e-b*d)*b)^(1/2)*x^6*a*b^5*e^6-3968*A*((a*e-b*d)*b)^(1/2)*a^5*b*d*e^5+32384*A
*((a*e-b*d)*b)^(1/2)*a^4*b^2*d^2*e^4+22155*A*((a*e-b*d)*b)^(1/2)*a^3*b^3*d^3*e^3-7630*A*((a*e-b*d)*b)^(1/2)*a^
2*b^4*d^4*e^2+1960*A*((a*e-b*d)*b)^(1/2)*a*b^5*d^5*e-6912*B*((a*e-b*d)*b)^(1/2)*a^5*b*d^2*e^4-33619*B*((a*e-b*
d)*b)^(1/2)*a^4*b^2*d^3*e^3-5530*B*((a*e-b*d)*b)^(1/2)*a^3*b^3*d^4*e^2+840*B*((a*e-b*d)*b)^(1/2)*a^2*b^4*d^5*e
+384*A*((a*e-b*d)*b)^(1/2)*a^6*e^6-240*A*((a*e-b*d)*b)^(1/2)*b^6*d^6+69069*A*((a*e-b*d)*b)^(1/2)*x^4*b^6*d^2*e
^4+45045*A*((a*e-b*d)*b)^(1/2)*x^6*b^6*e^6+640*B*((a*e-b*d)*b)^(1/2)*x*a^6*e^6-320*B*((a*e-b*d)*b)^(1/2)*x*b^6
*d^6+256*B*((a*e-b*d)*b)^(1/2)*a^6*d*e^5-80*B*((a*e-b*d)*b)^(1/2)*a*b^5*d^6-110880*B*arctan((e*x+d)^(1/2)*b/((
a*e-b*d)*b)^(1/2))*(e*x+d)^(5/2)*x*a^3*b^4*d*e^3-110880*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^
(5/2)*x^3*a*b^6*d*e^3-17325*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(5/2)*x^4*a*b^6*e^4-27720*B*
arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(5/2)*x^4*b^7*d*e^3+180180*A*arctan((e*x+d)^(1/2)*b/((a*e-
b*d)*b)^(1/2))*(e*x+d)^(5/2)*x^3*a*b^6*e^4-69300*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(5/2)*x
^3*a^2*b^5*e^4+270270*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(5/2)*x^2*a^2*b^5*e^4-103950*B*arc
tan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(5/2)*x^2*a^3*b^4*e^4+180180*A*arctan((e*x+d)^(1/2)*b/((a*e-b
*d)*b)^(1/2))*(e*x+d)^(5/2)*x*a^3*b^4*e^4-142065*B*((a*e-b*d)*b)^(1/2)*x^5*a*b^5*d*e^5-69300*B*arctan((e*x+d)^
(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(5/2)*x*a^4*b^3*e^4+387387*A*((a*e-b*d)*b)^(1/2)*x^4*a*b^5*d*e^5-283899*B
*((a*e-b*d)*b)^(1/2)*x^4*a^2*b^4*d*e^5-264957*B*((a*e-b*d)*b)^(1/2)*x^4*a*b^5*d^2*e^4-27720*B*arctan((e*x+d)^(
1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(5/2)*a^4*b^3*d*e^3+517803*A*((a*e-b*d)*b)^(1/2)*x^3*a^2*b^4*d*e^5+256971*
A*((a*e-b*d)*b)^(1/2)*x^3*a*b^5*d^2*e^4-272811*B*((a*e-b*d)*b)^(1/2)*x^3*a^3*b^3*d*e^5-417483*B*((a*e-b*d)*b)^
(1/2)*x^3*a^2*b^4*d^2*e^4-160611*B*((a*e-b*d)*b)^(1/2)*x^3*a*b^5*d^3*e^3+285857*A*((a*e-b*d)*b)^(1/2)*x^2*a^3*
b^3*d*e^5+347919*A*((a*e-b*d)*b)^(1/2)*x^2*a^2*b^4*d^2*e^4+25025*A*((a*e-b*d)*b)^(1/2)*x^2*a*b^5*d^3*e^3-12120
9*B*((a*e-b*d)*b)^(1/2)*x^2*a^4*b^2*d*e^5-309727*B*((a*e-b*d)*b)^(1/2)*x^2*a^3*b^3*d^2*e^4-223729*B*((a*e-b*d)
*b)^(1/2)*x^2*a^2*b^4*d^3*e^3-14850*B*((a*e-b*d)*b)^(1/2)*x^2*a*b^5*d^4*e^2+44928*A*((a*e-b*d)*b)^(1/2)*x*a^4*
b^2*d*e^5+196001*A*((a*e-b*d)*b)^(1/2)*x*a^3*b^3*d^2*e^4+35945*A*((a*e-b*d)*b)^(1/2)*x*a^2*b^4*d^3*e^3-5460*A*
((a*e-b*d)*b)^(1/2)*x*a*b^5*d^4*e^2-16256*B*((a*e-b*d)*b)^(1/2)*x*a^5*b*d*e^5-103033*B*((a*e-b*d)*b)^(1/2)*x*a
^4*b^2*d^2*e^4-134441*B*((a*e-b*d)*b)^(1/2)*x*a^3*b^3*d^3*e^3-20020*B*((a*e-b*d)*b)^(1/2)*x*a^2*b^4*d^4*e^2+31
60*B*((a*e-b*d)*b)^(1/2)*x*a*b^5*d^5*e-166320*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*(e*x+d)^(5/2)*x^2*
a^2*b^5*d*e^3)*(b*x+a)/((a*e-b*d)*b)^(1/2)/(e*x+d)^(5/2)/(a*e-b*d)^7/((b*x+a)^2)^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B x + A}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{5}{2}}{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \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)^(5/2),x, algorithm="maxima")

[Out]

integrate((B*x + A)/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(e*x + d)^(7/2)), x)

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Fricas [B]  time = 2.43389, size = 10375, normalized size = 18.4 \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)^(5/2),x, algorithm="fricas")

[Out]

[1/1920*(3465*(8*B*a^4*b^2*d^4*e^3 + (5*B*a^5*b - 13*A*a^4*b^2)*d^3*e^4 + (8*B*b^6*d*e^6 + (5*B*a*b^5 - 13*A*b
^6)*e^7)*x^7 + (24*B*b^6*d^2*e^5 + (47*B*a*b^5 - 39*A*b^6)*d*e^6 + 4*(5*B*a^2*b^4 - 13*A*a*b^5)*e^7)*x^6 + 3*(
8*B*b^6*d^3*e^4 + (37*B*a*b^5 - 13*A*b^6)*d^2*e^5 + 4*(9*B*a^2*b^4 - 13*A*a*b^5)*d*e^6 + 2*(5*B*a^3*b^3 - 13*A
*a^2*b^4)*e^7)*x^5 + (8*B*b^6*d^4*e^3 + (101*B*a*b^5 - 13*A*b^6)*d^3*e^4 + 12*(17*B*a^2*b^4 - 13*A*a*b^5)*d^2*
e^5 + 2*(61*B*a^3*b^3 - 117*A*a^2*b^4)*d*e^6 + 4*(5*B*a^4*b^2 - 13*A*a^3*b^3)*e^7)*x^4 + (32*B*a*b^5*d^4*e^3 +
4*(41*B*a^2*b^4 - 13*A*a*b^5)*d^3*e^4 + 6*(31*B*a^3*b^3 - 39*A*a^2*b^4)*d^2*e^5 + 4*(17*B*a^4*b^2 - 39*A*a^3*
b^3)*d*e^6 + (5*B*a^5*b - 13*A*a^4*b^2)*e^7)*x^3 + 3*(16*B*a^2*b^4*d^4*e^3 + 2*(21*B*a^3*b^3 - 13*A*a^2*b^4)*d
^3*e^4 + 4*(7*B*a^4*b^2 - 13*A*a^3*b^3)*d^2*e^5 + (5*B*a^5*b - 13*A*a^4*b^2)*d*e^6)*x^2 + (32*B*a^3*b^3*d^4*e^
3 + 4*(11*B*a^4*b^2 - 13*A*a^3*b^3)*d^3*e^4 + 3*(5*B*a^5*b - 13*A*a^4*b^2)*d^2*e^5)*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*(384*A*a^6*e^6 - 80*(B
*a*b^5 + 3*A*b^6)*d^6 + 280*(3*B*a^2*b^4 + 7*A*a*b^5)*d^5*e - 70*(79*B*a^3*b^3 + 109*A*a^2*b^4)*d^4*e^2 - (336
19*B*a^4*b^2 - 22155*A*a^3*b^3)*d^3*e^3 - 128*(54*B*a^5*b - 253*A*a^4*b^2)*d^2*e^4 + 128*(2*B*a^6 - 31*A*a^5*b
)*d*e^5 - 3465*(8*B*b^6*d*e^5 + (5*B*a*b^5 - 13*A*b^6)*e^6)*x^6 - 1155*(56*B*b^6*d^2*e^4 + (123*B*a*b^5 - 91*A
*b^6)*d*e^5 + 11*(5*B*a^2*b^4 - 13*A*a*b^5)*e^6)*x^5 - 231*(184*B*b^6*d^3*e^3 + (1147*B*a*b^5 - 299*A*b^6)*d^2
*e^4 + (1229*B*a^2*b^4 - 1677*A*a*b^5)*d*e^5 + 73*(5*B*a^3*b^3 - 13*A*a^2*b^4)*e^6)*x^4 - 33*(120*B*b^6*d^4*e^
2 + (4867*B*a*b^5 - 195*A*b^6)*d^3*e^3 + (12651*B*a^2*b^4 - 7787*A*a*b^5)*d^2*e^4 + (8267*B*a^3*b^3 - 15691*A*
a^2*b^4)*d*e^5 + 279*(5*B*a^4*b^2 - 13*A*a^3*b^3)*e^6)*x^3 + 11*(80*B*b^6*d^5*e - 10*(135*B*a*b^5 + 13*A*b^6)*
d^4*e^2 - (20339*B*a^2*b^4 - 2275*A*a*b^5)*d^3*e^3 - (28157*B*a^3*b^3 - 31629*A*a^2*b^4)*d^2*e^4 - (11019*B*a^
4*b^2 - 25987*A*a^3*b^3)*d*e^5 - 128*(5*B*a^5*b - 13*A*a^4*b^2)*e^6)*x^2 - (320*B*b^6*d^6 - 40*(79*B*a*b^5 + 1
3*A*b^6)*d^5*e + 1820*(11*B*a^2*b^4 + 3*A*a*b^5)*d^4*e^2 + (134441*B*a^3*b^3 - 35945*A*a^2*b^4)*d^3*e^3 + (103
033*B*a^4*b^2 - 196001*A*a^3*b^3)*d^2*e^4 + 128*(127*B*a^5*b - 351*A*a^4*b^2)*d*e^5 - 128*(5*B*a^6 - 13*A*a^5*
b)*e^6)*x)*sqrt(e*x + d))/(a^4*b^7*d^10 - 7*a^5*b^6*d^9*e + 21*a^6*b^5*d^8*e^2 - 35*a^7*b^4*d^7*e^3 + 35*a^8*b
^3*d^6*e^4 - 21*a^9*b^2*d^5*e^5 + 7*a^10*b*d^4*e^6 - a^11*d^3*e^7 + (b^11*d^7*e^3 - 7*a*b^10*d^6*e^4 + 21*a^2*
b^9*d^5*e^5 - 35*a^3*b^8*d^4*e^6 + 35*a^4*b^7*d^3*e^7 - 21*a^5*b^6*d^2*e^8 + 7*a^6*b^5*d*e^9 - a^7*b^4*e^10)*x
^7 + (3*b^11*d^8*e^2 - 17*a*b^10*d^7*e^3 + 35*a^2*b^9*d^6*e^4 - 21*a^3*b^8*d^5*e^5 - 35*a^4*b^7*d^4*e^6 + 77*a
^5*b^6*d^3*e^7 - 63*a^6*b^5*d^2*e^8 + 25*a^7*b^4*d*e^9 - 4*a^8*b^3*e^10)*x^6 + 3*(b^11*d^9*e - 3*a*b^10*d^8*e^
2 - 5*a^2*b^9*d^7*e^3 + 35*a^3*b^8*d^6*e^4 - 63*a^4*b^7*d^5*e^5 + 49*a^5*b^6*d^4*e^6 - 7*a^6*b^5*d^3*e^7 - 15*
a^7*b^4*d^2*e^8 + 10*a^8*b^3*d*e^9 - 2*a^9*b^2*e^10)*x^5 + (b^11*d^10 + 5*a*b^10*d^9*e - 45*a^2*b^9*d^8*e^2 +
95*a^3*b^8*d^7*e^3 - 35*a^4*b^7*d^6*e^4 - 147*a^5*b^6*d^5*e^5 + 245*a^6*b^5*d^4*e^6 - 155*a^7*b^4*d^3*e^7 + 30
*a^8*b^3*d^2*e^8 + 10*a^9*b^2*d*e^9 - 4*a^10*b*e^10)*x^4 + (4*a*b^10*d^10 - 10*a^2*b^9*d^9*e - 30*a^3*b^8*d^8*
e^2 + 155*a^4*b^7*d^7*e^3 - 245*a^5*b^6*d^6*e^4 + 147*a^6*b^5*d^5*e^5 + 35*a^7*b^4*d^4*e^6 - 95*a^8*b^3*d^3*e^
7 + 45*a^9*b^2*d^2*e^8 - 5*a^10*b*d*e^9 - a^11*e^10)*x^3 + 3*(2*a^2*b^9*d^10 - 10*a^3*b^8*d^9*e + 15*a^4*b^7*d
^8*e^2 + 7*a^5*b^6*d^7*e^3 - 49*a^6*b^5*d^6*e^4 + 63*a^7*b^4*d^5*e^5 - 35*a^8*b^3*d^4*e^6 + 5*a^9*b^2*d^3*e^7
+ 3*a^10*b*d^2*e^8 - a^11*d*e^9)*x^2 + (4*a^3*b^8*d^10 - 25*a^4*b^7*d^9*e + 63*a^5*b^6*d^8*e^2 - 77*a^6*b^5*d^
7*e^3 + 35*a^7*b^4*d^6*e^4 + 21*a^8*b^3*d^5*e^5 - 35*a^9*b^2*d^4*e^6 + 17*a^10*b*d^3*e^7 - 3*a^11*d^2*e^8)*x),
1/960*(3465*(8*B*a^4*b^2*d^4*e^3 + (5*B*a^5*b - 13*A*a^4*b^2)*d^3*e^4 + (8*B*b^6*d*e^6 + (5*B*a*b^5 - 13*A*b^
6)*e^7)*x^7 + (24*B*b^6*d^2*e^5 + (47*B*a*b^5 - 39*A*b^6)*d*e^6 + 4*(5*B*a^2*b^4 - 13*A*a*b^5)*e^7)*x^6 + 3*(8
*B*b^6*d^3*e^4 + (37*B*a*b^5 - 13*A*b^6)*d^2*e^5 + 4*(9*B*a^2*b^4 - 13*A*a*b^5)*d*e^6 + 2*(5*B*a^3*b^3 - 13*A*
a^2*b^4)*e^7)*x^5 + (8*B*b^6*d^4*e^3 + (101*B*a*b^5 - 13*A*b^6)*d^3*e^4 + 12*(17*B*a^2*b^4 - 13*A*a*b^5)*d^2*e
^5 + 2*(61*B*a^3*b^3 - 117*A*a^2*b^4)*d*e^6 + 4*(5*B*a^4*b^2 - 13*A*a^3*b^3)*e^7)*x^4 + (32*B*a*b^5*d^4*e^3 +
4*(41*B*a^2*b^4 - 13*A*a*b^5)*d^3*e^4 + 6*(31*B*a^3*b^3 - 39*A*a^2*b^4)*d^2*e^5 + 4*(17*B*a^4*b^2 - 39*A*a^3*b
^3)*d*e^6 + (5*B*a^5*b - 13*A*a^4*b^2)*e^7)*x^3 + 3*(16*B*a^2*b^4*d^4*e^3 + 2*(21*B*a^3*b^3 - 13*A*a^2*b^4)*d^
3*e^4 + 4*(7*B*a^4*b^2 - 13*A*a^3*b^3)*d^2*e^5 + (5*B*a^5*b - 13*A*a^4*b^2)*d*e^6)*x^2 + (32*B*a^3*b^3*d^4*e^3
+ 4*(11*B*a^4*b^2 - 13*A*a^3*b^3)*d^3*e^4 + 3*(5*B*a^5*b - 13*A*a^4*b^2)*d^2*e^5)*x)*sqrt(-b/(b*d - a*e))*arc
tan(-(b*d - a*e)*sqrt(e*x + d)*sqrt(-b/(b*d - a*e))/(b*e*x + b*d)) + (384*A*a^6*e^6 - 80*(B*a*b^5 + 3*A*b^6)*d
^6 + 280*(3*B*a^2*b^4 + 7*A*a*b^5)*d^5*e - 70*(79*B*a^3*b^3 + 109*A*a^2*b^4)*d^4*e^2 - (33619*B*a^4*b^2 - 2215
5*A*a^3*b^3)*d^3*e^3 - 128*(54*B*a^5*b - 253*A*a^4*b^2)*d^2*e^4 + 128*(2*B*a^6 - 31*A*a^5*b)*d*e^5 - 3465*(8*B
*b^6*d*e^5 + (5*B*a*b^5 - 13*A*b^6)*e^6)*x^6 - 1155*(56*B*b^6*d^2*e^4 + (123*B*a*b^5 - 91*A*b^6)*d*e^5 + 11*(5
*B*a^2*b^4 - 13*A*a*b^5)*e^6)*x^5 - 231*(184*B*b^6*d^3*e^3 + (1147*B*a*b^5 - 299*A*b^6)*d^2*e^4 + (1229*B*a^2*
b^4 - 1677*A*a*b^5)*d*e^5 + 73*(5*B*a^3*b^3 - 13*A*a^2*b^4)*e^6)*x^4 - 33*(120*B*b^6*d^4*e^2 + (4867*B*a*b^5 -
195*A*b^6)*d^3*e^3 + (12651*B*a^2*b^4 - 7787*A*a*b^5)*d^2*e^4 + (8267*B*a^3*b^3 - 15691*A*a^2*b^4)*d*e^5 + 27
9*(5*B*a^4*b^2 - 13*A*a^3*b^3)*e^6)*x^3 + 11*(80*B*b^6*d^5*e - 10*(135*B*a*b^5 + 13*A*b^6)*d^4*e^2 - (20339*B*
a^2*b^4 - 2275*A*a*b^5)*d^3*e^3 - (28157*B*a^3*b^3 - 31629*A*a^2*b^4)*d^2*e^4 - (11019*B*a^4*b^2 - 25987*A*a^3
*b^3)*d*e^5 - 128*(5*B*a^5*b - 13*A*a^4*b^2)*e^6)*x^2 - (320*B*b^6*d^6 - 40*(79*B*a*b^5 + 13*A*b^6)*d^5*e + 18
20*(11*B*a^2*b^4 + 3*A*a*b^5)*d^4*e^2 + (134441*B*a^3*b^3 - 35945*A*a^2*b^4)*d^3*e^3 + (103033*B*a^4*b^2 - 196
001*A*a^3*b^3)*d^2*e^4 + 128*(127*B*a^5*b - 351*A*a^4*b^2)*d*e^5 - 128*(5*B*a^6 - 13*A*a^5*b)*e^6)*x)*sqrt(e*x
+ d))/(a^4*b^7*d^10 - 7*a^5*b^6*d^9*e + 21*a^6*b^5*d^8*e^2 - 35*a^7*b^4*d^7*e^3 + 35*a^8*b^3*d^6*e^4 - 21*a^9
*b^2*d^5*e^5 + 7*a^10*b*d^4*e^6 - a^11*d^3*e^7 + (b^11*d^7*e^3 - 7*a*b^10*d^6*e^4 + 21*a^2*b^9*d^5*e^5 - 35*a^
3*b^8*d^4*e^6 + 35*a^4*b^7*d^3*e^7 - 21*a^5*b^6*d^2*e^8 + 7*a^6*b^5*d*e^9 - a^7*b^4*e^10)*x^7 + (3*b^11*d^8*e^
2 - 17*a*b^10*d^7*e^3 + 35*a^2*b^9*d^6*e^4 - 21*a^3*b^8*d^5*e^5 - 35*a^4*b^7*d^4*e^6 + 77*a^5*b^6*d^3*e^7 - 63
*a^6*b^5*d^2*e^8 + 25*a^7*b^4*d*e^9 - 4*a^8*b^3*e^10)*x^6 + 3*(b^11*d^9*e - 3*a*b^10*d^8*e^2 - 5*a^2*b^9*d^7*e
^3 + 35*a^3*b^8*d^6*e^4 - 63*a^4*b^7*d^5*e^5 + 49*a^5*b^6*d^4*e^6 - 7*a^6*b^5*d^3*e^7 - 15*a^7*b^4*d^2*e^8 + 1
0*a^8*b^3*d*e^9 - 2*a^9*b^2*e^10)*x^5 + (b^11*d^10 + 5*a*b^10*d^9*e - 45*a^2*b^9*d^8*e^2 + 95*a^3*b^8*d^7*e^3
- 35*a^4*b^7*d^6*e^4 - 147*a^5*b^6*d^5*e^5 + 245*a^6*b^5*d^4*e^6 - 155*a^7*b^4*d^3*e^7 + 30*a^8*b^3*d^2*e^8 +
10*a^9*b^2*d*e^9 - 4*a^10*b*e^10)*x^4 + (4*a*b^10*d^10 - 10*a^2*b^9*d^9*e - 30*a^3*b^8*d^8*e^2 + 155*a^4*b^7*d
^7*e^3 - 245*a^5*b^6*d^6*e^4 + 147*a^6*b^5*d^5*e^5 + 35*a^7*b^4*d^4*e^6 - 95*a^8*b^3*d^3*e^7 + 45*a^9*b^2*d^2*
e^8 - 5*a^10*b*d*e^9 - a^11*e^10)*x^3 + 3*(2*a^2*b^9*d^10 - 10*a^3*b^8*d^9*e + 15*a^4*b^7*d^8*e^2 + 7*a^5*b^6*
d^7*e^3 - 49*a^6*b^5*d^6*e^4 + 63*a^7*b^4*d^5*e^5 - 35*a^8*b^3*d^4*e^6 + 5*a^9*b^2*d^3*e^7 + 3*a^10*b*d^2*e^8
- a^11*d*e^9)*x^2 + (4*a^3*b^8*d^10 - 25*a^4*b^7*d^9*e + 63*a^5*b^6*d^8*e^2 - 77*a^6*b^5*d^7*e^3 + 35*a^7*b^4*
d^6*e^4 + 21*a^8*b^3*d^5*e^5 - 35*a^9*b^2*d^4*e^6 + 17*a^10*b*d^3*e^7 - 3*a^11*d^2*e^8)*x)]

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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)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.59543, size = 2061, normalized size = 3.65 \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)^(5/2),x, algorithm="giac")

[Out]

-231/64*(8*B*b^3*d*e^3 + 5*B*a*b^2*e^4 - 13*A*b^3*e^4)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^7*d^7*
sgn((x*e + d)*b*e - b*d*e + a*e^2) - 7*a*b^6*d^6*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 21*a^2*b^5*d^5*e^2*sgn
((x*e + d)*b*e - b*d*e + a*e^2) - 35*a^3*b^4*d^4*e^3*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 35*a^4*b^3*d^3*e^4*s
gn((x*e + d)*b*e - b*d*e + a*e^2) - 21*a^5*b^2*d^2*e^5*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 7*a^6*b*d*e^6*sgn(
(x*e + d)*b*e - b*d*e + a*e^2) - a^7*e^7*sgn((x*e + d)*b*e - b*d*e + a*e^2))*sqrt(-b^2*d + a*b*e)) - 2/15*(150
*(x*e + d)^2*B*b^2*d*e^3 + 20*(x*e + d)*B*b^2*d^2*e^3 + 3*B*b^2*d^3*e^3 + 75*(x*e + d)^2*B*a*b*e^4 - 225*(x*e
+ d)^2*A*b^2*e^4 - 15*(x*e + d)*B*a*b*d*e^4 - 25*(x*e + d)*A*b^2*d*e^4 - 6*B*a*b*d^2*e^4 - 3*A*b^2*d^2*e^4 - 5
*(x*e + d)*B*a^2*e^5 + 25*(x*e + d)*A*a*b*e^5 + 3*B*a^2*d*e^5 + 6*A*a*b*d*e^5 - 3*A*a^2*e^6)/((b^7*d^7*sgn((x*
e + d)*b*e - b*d*e + a*e^2) - 7*a*b^6*d^6*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 21*a^2*b^5*d^5*e^2*sgn((x*e +
d)*b*e - b*d*e + a*e^2) - 35*a^3*b^4*d^4*e^3*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 35*a^4*b^3*d^3*e^4*sgn((x*e
+ d)*b*e - b*d*e + a*e^2) - 21*a^5*b^2*d^2*e^5*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 7*a^6*b*d*e^6*sgn((x*e +
d)*b*e - b*d*e + a*e^2) - a^7*e^7*sgn((x*e + d)*b*e - b*d*e + a*e^2))*(x*e + d)^(5/2)) - 1/192*(1704*(x*e + d)
^(7/2)*B*b^6*d*e^3 - 5480*(x*e + d)^(5/2)*B*b^6*d^2*e^3 + 5912*(x*e + d)^(3/2)*B*b^6*d^3*e^3 - 2136*sqrt(x*e +
d)*B*b^6*d^4*e^3 + 1545*(x*e + d)^(7/2)*B*a*b^5*e^4 - 3249*(x*e + d)^(7/2)*A*b^6*e^4 + 327*(x*e + d)^(5/2)*B*
a*b^5*d*e^4 + 10633*(x*e + d)^(5/2)*A*b^6*d*e^4 - 5969*(x*e + d)^(3/2)*B*a*b^5*d^2*e^4 - 11767*(x*e + d)^(3/2)
*A*b^6*d^2*e^4 + 4113*sqrt(x*e + d)*B*a*b^5*d^3*e^4 + 4431*sqrt(x*e + d)*A*b^6*d^3*e^4 + 5153*(x*e + d)^(5/2)*
B*a^2*b^4*e^5 - 10633*(x*e + d)^(5/2)*A*a*b^5*e^5 - 5798*(x*e + d)^(3/2)*B*a^2*b^4*d*e^5 + 23534*(x*e + d)^(3/
2)*A*a*b^5*d*e^5 + 477*sqrt(x*e + d)*B*a^2*b^4*d^2*e^5 - 13293*sqrt(x*e + d)*A*a*b^5*d^2*e^5 + 5855*(x*e + d)^
(3/2)*B*a^3*b^3*e^6 - 11767*(x*e + d)^(3/2)*A*a^2*b^4*e^6 - 4749*sqrt(x*e + d)*B*a^3*b^3*d*e^6 + 13293*sqrt(x*
e + d)*A*a^2*b^4*d*e^6 + 2295*sqrt(x*e + d)*B*a^4*b^2*e^7 - 4431*sqrt(x*e + d)*A*a^3*b^3*e^7)/((b^7*d^7*sgn((x
*e + d)*b*e - b*d*e + a*e^2) - 7*a*b^6*d^6*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 21*a^2*b^5*d^5*e^2*sgn((x*e
+ d)*b*e - b*d*e + a*e^2) - 35*a^3*b^4*d^4*e^3*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 35*a^4*b^3*d^3*e^4*sgn((x*
e + d)*b*e - b*d*e + a*e^2) - 21*a^5*b^2*d^2*e^5*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 7*a^6*b*d*e^6*sgn((x*e +
d)*b*e - b*d*e + a*e^2) - a^7*e^7*sgn((x*e + d)*b*e - b*d*e + a*e^2))*((x*e + d)*b - b*d + a*e)^4)