3.1834 \(\int \frac{A+B x}{(d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^3} \, dx\)
Optimal. Leaf size=435 \[ -\frac{3003 b^{3/2} e^4 (a B e-3 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 (b d-a e)^{17/2}}+\frac{3003 b e^4 (a B e-3 A b e+2 b B d)}{128 \sqrt{d+e x} (b d-a e)^8}+\frac{1001 e^4 (a B e-3 A b e+2 b B d)}{128 (d+e x)^{3/2} (b d-a e)^7}+\frac{3003 e^4 (a B e-3 A b e+2 b B d)}{640 b (d+e x)^{5/2} (b d-a e)^6}+\frac{429 e^3 (a B e-3 A b e+2 b B d)}{128 b (a+b x) (d+e x)^{5/2} (b d-a e)^5}-\frac{143 e^2 (a B e-3 A b e+2 b B d)}{192 b (a+b x)^2 (d+e x)^{5/2} (b d-a e)^4}+\frac{13 e (a B e-3 A b e+2 b B d)}{48 b (a+b x)^3 (d+e x)^{5/2} (b d-a e)^3}-\frac{a B e-3 A b e+2 b B d}{8 b (a+b x)^4 (d+e x)^{5/2} (b d-a e)^2}-\frac{A b-a B}{5 b (a+b x)^5 (d+e x)^{5/2} (b d-a e)} \]
[Out]
(3003*e^4*(2*b*B*d - 3*A*b*e + a*B*e))/(640*b*(b*d - a*e)^6*(d + e*x)^(5/2)) - (A*b - a*B)/(5*b*(b*d - a*e)*(a
+ b*x)^5*(d + e*x)^(5/2)) - (2*b*B*d - 3*A*b*e + a*B*e)/(8*b*(b*d - a*e)^2*(a + b*x)^4*(d + e*x)^(5/2)) + (13
*e*(2*b*B*d - 3*A*b*e + a*B*e))/(48*b*(b*d - a*e)^3*(a + b*x)^3*(d + e*x)^(5/2)) - (143*e^2*(2*b*B*d - 3*A*b*e
+ a*B*e))/(192*b*(b*d - a*e)^4*(a + b*x)^2*(d + e*x)^(5/2)) + (429*e^3*(2*b*B*d - 3*A*b*e + a*B*e))/(128*b*(b
*d - a*e)^5*(a + b*x)*(d + e*x)^(5/2)) + (1001*e^4*(2*b*B*d - 3*A*b*e + a*B*e))/(128*(b*d - a*e)^7*(d + e*x)^(
3/2)) + (3003*b*e^4*(2*b*B*d - 3*A*b*e + a*B*e))/(128*(b*d - a*e)^8*Sqrt[d + e*x]) - (3003*b^(3/2)*e^4*(2*b*B*
d - 3*A*b*e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*(b*d - a*e)^(17/2))
________________________________________________________________________________________
Rubi [A] time = 0.489587, antiderivative size = 435, normalized size of antiderivative = 1.,
number of steps used = 11, 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} \[ -\frac{3003 b^{3/2} e^4 (a B e-3 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 (b d-a e)^{17/2}}+\frac{3003 b e^4 (a B e-3 A b e+2 b B d)}{128 \sqrt{d+e x} (b d-a e)^8}+\frac{1001 e^4 (a B e-3 A b e+2 b B d)}{128 (d+e x)^{3/2} (b d-a e)^7}+\frac{3003 e^4 (a B e-3 A b e+2 b B d)}{640 b (d+e x)^{5/2} (b d-a e)^6}+\frac{429 e^3 (a B e-3 A b e+2 b B d)}{128 b (a+b x) (d+e x)^{5/2} (b d-a e)^5}-\frac{143 e^2 (a B e-3 A b e+2 b B d)}{192 b (a+b x)^2 (d+e x)^{5/2} (b d-a e)^4}+\frac{13 e (a B e-3 A b e+2 b B d)}{48 b (a+b x)^3 (d+e x)^{5/2} (b d-a e)^3}-\frac{a B e-3 A b e+2 b B d}{8 b (a+b x)^4 (d+e x)^{5/2} (b d-a e)^2}-\frac{A b-a B}{5 b (a+b x)^5 (d+e x)^{5/2} (b d-a e)} \]
Antiderivative was successfully verified.
[In]
Int[(A + B*x)/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]
[Out]
(3003*e^4*(2*b*B*d - 3*A*b*e + a*B*e))/(640*b*(b*d - a*e)^6*(d + e*x)^(5/2)) - (A*b - a*B)/(5*b*(b*d - a*e)*(a
+ b*x)^5*(d + e*x)^(5/2)) - (2*b*B*d - 3*A*b*e + a*B*e)/(8*b*(b*d - a*e)^2*(a + b*x)^4*(d + e*x)^(5/2)) + (13
*e*(2*b*B*d - 3*A*b*e + a*B*e))/(48*b*(b*d - a*e)^3*(a + b*x)^3*(d + e*x)^(5/2)) - (143*e^2*(2*b*B*d - 3*A*b*e
+ a*B*e))/(192*b*(b*d - a*e)^4*(a + b*x)^2*(d + e*x)^(5/2)) + (429*e^3*(2*b*B*d - 3*A*b*e + a*B*e))/(128*b*(b
*d - a*e)^5*(a + b*x)*(d + e*x)^(5/2)) + (1001*e^4*(2*b*B*d - 3*A*b*e + a*B*e))/(128*(b*d - a*e)^7*(d + e*x)^(
3/2)) + (3003*b*e^4*(2*b*B*d - 3*A*b*e + a*B*e))/(128*(b*d - a*e)^8*Sqrt[d + e*x]) - (3003*b^(3/2)*e^4*(2*b*B*
d - 3*A*b*e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*(b*d - a*e)^(17/2))
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 )^3} \, dx &=\int \frac{A+B x}{(a+b x)^6 (d+e x)^{7/2}} \, dx\\ &=-\frac{A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac{(2 b B d-3 A b e+a B e) \int \frac{1}{(a+b x)^5 (d+e x)^{7/2}} \, dx}{2 b (b d-a e)}\\ &=-\frac{A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac{2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}-\frac{(13 e (2 b B d-3 A b e+a B e)) \int \frac{1}{(a+b x)^4 (d+e x)^{7/2}} \, dx}{16 b (b d-a e)^2}\\ &=-\frac{A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac{2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac{13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}+\frac{\left (143 e^2 (2 b B d-3 A b e+a B e)\right ) \int \frac{1}{(a+b x)^3 (d+e x)^{7/2}} \, dx}{96 b (b d-a e)^3}\\ &=-\frac{A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac{2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac{13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac{143 e^2 (2 b B d-3 A b e+a B e)}{192 b (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}-\frac{\left (429 e^3 (2 b B d-3 A b e+a B e)\right ) \int \frac{1}{(a+b x)^2 (d+e x)^{7/2}} \, dx}{128 b (b d-a e)^4}\\ &=-\frac{A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac{2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac{13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac{143 e^2 (2 b B d-3 A b e+a B e)}{192 b (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}+\frac{429 e^3 (2 b B d-3 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) (d+e x)^{5/2}}+\frac{\left (3003 e^4 (2 b B d-3 A b e+a B e)\right ) \int \frac{1}{(a+b x) (d+e x)^{7/2}} \, dx}{256 b (b d-a e)^5}\\ &=\frac{3003 e^4 (2 b B d-3 A b e+a B e)}{640 b (b d-a e)^6 (d+e x)^{5/2}}-\frac{A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac{2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac{13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac{143 e^2 (2 b B d-3 A b e+a B e)}{192 b (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}+\frac{429 e^3 (2 b B d-3 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) (d+e x)^{5/2}}+\frac{\left (3003 e^4 (2 b B d-3 A b e+a B e)\right ) \int \frac{1}{(a+b x) (d+e x)^{5/2}} \, dx}{256 (b d-a e)^6}\\ &=\frac{3003 e^4 (2 b B d-3 A b e+a B e)}{640 b (b d-a e)^6 (d+e x)^{5/2}}-\frac{A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac{2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac{13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac{143 e^2 (2 b B d-3 A b e+a B e)}{192 b (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}+\frac{429 e^3 (2 b B d-3 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) (d+e x)^{5/2}}+\frac{1001 e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^7 (d+e x)^{3/2}}+\frac{\left (3003 b e^4 (2 b B d-3 A b e+a B e)\right ) \int \frac{1}{(a+b x) (d+e x)^{3/2}} \, dx}{256 (b d-a e)^7}\\ &=\frac{3003 e^4 (2 b B d-3 A b e+a B e)}{640 b (b d-a e)^6 (d+e x)^{5/2}}-\frac{A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac{2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac{13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac{143 e^2 (2 b B d-3 A b e+a B e)}{192 b (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}+\frac{429 e^3 (2 b B d-3 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) (d+e x)^{5/2}}+\frac{1001 e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^7 (d+e x)^{3/2}}+\frac{3003 b e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^8 \sqrt{d+e x}}+\frac{\left (3003 b^2 e^4 (2 b B d-3 A b e+a B e)\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{256 (b d-a e)^8}\\ &=\frac{3003 e^4 (2 b B d-3 A b e+a B e)}{640 b (b d-a e)^6 (d+e x)^{5/2}}-\frac{A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac{2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac{13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac{143 e^2 (2 b B d-3 A b e+a B e)}{192 b (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}+\frac{429 e^3 (2 b B d-3 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) (d+e x)^{5/2}}+\frac{1001 e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^7 (d+e x)^{3/2}}+\frac{3003 b e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^8 \sqrt{d+e x}}+\frac{\left (3003 b^2 e^3 (2 b B d-3 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 d-a e)^8}\\ &=\frac{3003 e^4 (2 b B d-3 A b e+a B e)}{640 b (b d-a e)^6 (d+e x)^{5/2}}-\frac{A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac{2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac{13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac{143 e^2 (2 b B d-3 A b e+a B e)}{192 b (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}+\frac{429 e^3 (2 b B d-3 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) (d+e x)^{5/2}}+\frac{1001 e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^7 (d+e x)^{3/2}}+\frac{3003 b e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^8 \sqrt{d+e x}}-\frac{3003 b^{3/2} e^4 (2 b B d-3 A b e+a B e) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 (b d-a e)^{17/2}}\\ \end{align*}
Mathematica [C] time = 0.0862663, size = 100, normalized size = 0.23 \[ \frac{\frac{5 (a B-A b)}{(a+b x)^5}-\frac{5 e^4 (-a B e+3 A b e-2 b B d) \, _2F_1\left (-\frac{5}{2},5;-\frac{3}{2};\frac{b (d+e x)}{b d-a e}\right )}{(b d-a e)^5}}{25 b (d+e x)^{5/2} (b d-a e)} \]
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]
[Out]
((5*(-(A*b) + a*B))/(a + b*x)^5 - (5*e^4*(-2*b*B*d + 3*A*b*e - a*B*e)*Hypergeometric2F1[-5/2, 5, -3/2, (b*(d +
e*x))/(b*d - a*e)])/(b*d - a*e)^5)/(25*b*(b*d - a*e)*(d + e*x)^(5/2))
________________________________________________________________________________________
Maple [B] time = 0.043, size = 1735, normalized size = 4. \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)^3,x)
[Out]
12*e^5*b/(a*e-b*d)^8/(e*x+d)^(1/2)*a*B-3633/128*e^5/(a*e-b*d)^8*b^7/(b*e*x+a*e)^5*(e*x+d)^(9/2)*A-9009/128*e^5
/(a*e-b*d)^8*b^3/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*A-10/3*e^4/(a*e-b*d)^7/(e*x+d
)^(3/2)*B*b*d+30*e^4*b^2/(a*e-b*d)^8/(e*x+d)^(1/2)*B*d-2/5*e^5/(a*e-b*d)^6/(e*x+d)^(5/2)*A+4*e^5/(a*e-b*d)^7/(
e*x+d)^(3/2)*A*b-2/3*e^5/(a*e-b*d)^7/(e*x+d)^(3/2)*a*B-42*e^5*b^2/(a*e-b*d)^8/(e*x+d)^(1/2)*A+2/5*e^4/(a*e-b*d
)^6/(e*x+d)^(5/2)*B*d+1083/64*e^4/(a*e-b*d)^8*b^7/(b*e*x+a*e)^5*(e*x+d)^(9/2)*B*d+350/3*e^4/(a*e-b*d)^8*b^7/(b
*e*x+a*e)^5*(e*x+d)^(5/2)*B*d^3-8099/96*e^4/(a*e-b*d)^8*b^7/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*d^4+1477/64*e^4/(a*e
-b*d)^8*b^7/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*d^5-6941/96*e^4/(a*e-b*d)^8*b^7/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*d^2-10
01/5*e^7/(a*e-b*d)^8*b^5/(b*e*x+a*e)^5*(e*x+d)^(5/2)*A*a^2-1001/5*e^5/(a*e-b*d)^8*b^7/(b*e*x+a*e)^5*(e*x+d)^(5
/2)*A*d^2+2373/128*e^9/(a*e-b*d)^8*b^2/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^5+3003/128*e^5/(a*e-b*d)^8*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+1211/64*e^7/(a*e-b*d)^8*b^4/(b*e*x+a*e)^5*(e*x+d)^
(1/2)*B*a^3*d^2+1029/16*e^6/(a*e-b*d)^8*b^5/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^2*d^3-9443/128*e^5/(a*e-b*d)^8*b^6
/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a*d^4+5327/32*e^8/(a*e-b*d)^8*b^4/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a^3*d+28329/64*
e^7/(a*e-b*d)^8*b^5/(b*e*x+a*e)^5*(e*x+d)^(3/2)*A*a^2*d-28329/64*e^6/(a*e-b*d)^8*b^6/(b*e*x+a*e)^5*(e*x+d)^(3/
2)*A*a*d^2-20195/192*e^7/(a*e-b*d)^8*b^4/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a^3*d-4067/64*e^6/(a*e-b*d)^8*b^5/(b*e*
x+a*e)^5*(e*x+d)^(3/2)*B*a^2*d^2+36463/192*e^5/(a*e-b*d)^8*b^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a*d^3+4253/192*e^
5/(a*e-b*d)^8*b^6/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*a*d+2002/5*e^6/(a*e-b*d)^8*b^6/(b*e*x+a*e)^5*(e*x+d)^(5/2)*A*a
*d-749/5*e^5/(a*e-b*d)^8*b^6/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*a*d^2-252/5*e^6/(a*e-b*d)^8*b^5/(b*e*x+a*e)^5*(e*x+
d)^(5/2)*B*a^2*d-15981/64*e^7/(a*e-b*d)^8*b^5/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a^2*d^2+5327/32*e^6/(a*e-b*d)^8*b^
6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a*d^3-3269/64*e^8/(a*e-b*d)^8*b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^4*d+9443/64*
e^5/(a*e-b*d)^8*b^7/(b*e*x+a*e)^5*(e*x+d)^(3/2)*A*d^3+12131/192*e^8/(a*e-b*d)^8*b^3/(b*e*x+a*e)^5*(e*x+d)^(3/2
)*B*a^4-5327/128*e^9/(a*e-b*d)^8*b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a^4-5327/128*e^5/(a*e-b*d)^8*b^7/(b*e*x+a*e
)^5*(e*x+d)^(1/2)*A*d^4-7837/64*e^6/(a*e-b*d)^8*b^6/(b*e*x+a*e)^5*A*(e*x+d)^(7/2)*a+7837/64*e^5/(a*e-b*d)^8*b^
7/(b*e*x+a*e)^5*A*(e*x+d)^(7/2)*d+9629/192*e^6/(a*e-b*d)^8*b^5/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*a^2+1467/128*e^5/
(a*e-b*d)^8*b^6/(b*e*x+a*e)^5*(e*x+d)^(9/2)*B*a+3003/64*e^4/(a*e-b*d)^8*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+1253/15*e^7/(a*e-b*d)^8*b^4/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*a^3-9443/64*e^8/(a
*e-b*d)^8*b^4/(b*e*x+a*e)^5*(e*x+d)^(3/2)*A*a^3
________________________________________________________________________________________
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)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 2.34149, size = 12957, normalized size = 29.79 \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)^3,x, algorithm="fricas")
[Out]
[-1/3840*(45045*(2*B*a^5*b^2*d^4*e^4 + (B*a^6*b - 3*A*a^5*b^2)*d^3*e^5 + (2*B*b^7*d*e^7 + (B*a*b^6 - 3*A*b^7)*
e^8)*x^8 + (6*B*b^7*d^2*e^6 + (13*B*a*b^6 - 9*A*b^7)*d*e^7 + 5*(B*a^2*b^5 - 3*A*a*b^6)*e^8)*x^7 + (6*B*b^7*d^3
*e^5 + 3*(11*B*a*b^6 - 3*A*b^7)*d^2*e^6 + 5*(7*B*a^2*b^5 - 9*A*a*b^6)*d*e^7 + 10*(B*a^3*b^4 - 3*A*a^2*b^5)*e^8
)*x^6 + (2*B*b^7*d^4*e^4 + (31*B*a*b^6 - 3*A*b^7)*d^3*e^5 + 15*(5*B*a^2*b^5 - 3*A*a*b^6)*d^2*e^6 + 10*(5*B*a^3
*b^4 - 9*A*a^2*b^5)*d*e^7 + 10*(B*a^4*b^3 - 3*A*a^3*b^4)*e^8)*x^5 + 5*(2*B*a*b^6*d^4*e^4 + (13*B*a^2*b^5 - 3*A
*a*b^6)*d^3*e^5 + 18*(B*a^3*b^4 - A*a^2*b^5)*d^2*e^6 + 2*(4*B*a^4*b^3 - 9*A*a^3*b^4)*d*e^7 + (B*a^5*b^2 - 3*A*
a^4*b^3)*e^8)*x^4 + (20*B*a^2*b^5*d^4*e^4 + 10*(7*B*a^3*b^4 - 3*A*a^2*b^5)*d^3*e^5 + 30*(2*B*a^4*b^3 - 3*A*a^3
*b^4)*d^2*e^6 + (17*B*a^5*b^2 - 45*A*a^4*b^3)*d*e^7 + (B*a^6*b - 3*A*a^5*b^2)*e^8)*x^3 + (20*B*a^3*b^4*d^4*e^4
+ 10*(4*B*a^4*b^3 - 3*A*a^3*b^4)*d^3*e^5 + 3*(7*B*a^5*b^2 - 15*A*a^4*b^3)*d^2*e^6 + 3*(B*a^6*b - 3*A*a^5*b^2)
*d*e^7)*x^2 + (10*B*a^4*b^3*d^4*e^4 + (11*B*a^5*b^2 - 15*A*a^4*b^3)*d^3*e^5 + 3*(B*a^6*b - 3*A*a^5*b^2)*d^2*e^
6)*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*(768*A*a^7*e^7 + 96*(B*a*b^6 + 4*A*b^7)*d^7 - 16*(62*B*a^2*b^5 + 213*A*a*b^6)*d^6*e + 28*(187*B*a^3*b^4
+ 498*A*a^2*b^5)*d^5*e^2 - 70*(332*B*a^4*b^3 + 519*A*a^3*b^4)*d^4*e^3 - (100363*B*a^5*b^2 - 79905*A*a^4*b^3)*
d^3*e^4 - 1024*(16*B*a^6*b - 87*A*a^5*b^2)*d^2*e^5 + 512*(B*a^7 - 18*A*a^6*b)*d*e^6 - 45045*(2*B*b^7*d*e^6 + (
B*a*b^6 - 3*A*b^7)*e^7)*x^7 - 105105*(2*B*b^7*d^2*e^5 + (5*B*a*b^6 - 3*A*b^7)*d*e^6 + 2*(B*a^2*b^5 - 3*A*a*b^6
)*e^7)*x^6 - 3003*(46*B*b^7*d^3*e^4 + 3*(117*B*a*b^6 - 23*A*b^7)*d^2*e^5 + 12*(35*B*a^2*b^5 - 41*A*a*b^6)*d*e^
6 + 128*(B*a^3*b^4 - 3*A*a^2*b^5)*e^7)*x^5 - 2145*(6*B*b^7*d^4*e^3 + (307*B*a*b^6 - 9*A*b^7)*d^3*e^4 + 12*(83*
B*a^2*b^5 - 38*A*a*b^6)*d^2*e^5 + 6*(123*B*a^3*b^4 - 211*A*a^2*b^5)*d*e^6 + 158*(B*a^4*b^3 - 3*A*a^3*b^4)*e^7)
*x^4 + 715*(4*B*b^7*d^5*e^2 - 2*(43*B*a*b^6 + 3*A*b^7)*d^4*e^3 - 4*(434*B*a^2*b^5 - 33*A*a*b^6)*d^3*e^4 - 2*(1
547*B*a^3*b^4 - 1269*A*a^2*b^5)*d^2*e^5 - 2*(755*B*a^4*b^3 - 1686*A*a^3*b^4)*d*e^6 - 193*(B*a^5*b^2 - 3*A*a^4*
b^3)*e^7)*x^3 - 65*(16*B*b^7*d^6*e - 12*(17*B*a*b^6 + 2*A*b^7)*d^5*e^2 + 6*(295*B*a^2*b^5 + 53*A*a*b^6)*d^4*e^
3 + 2*(8837*B*a^3*b^4 - 1407*A*a^2*b^5)*d^3*e^4 + 6*(3091*B*a^4*b^3 - 4184*A*a^3*b^4)*d^2*e^5 + 3*(1867*B*a^5*
b^2 - 5089*A*a^4*b^3)*d*e^6 + 256*(B*a^6*b - 3*A*a^5*b^2)*e^7)*x^2 + 5*(96*B*b^7*d^7 - 16*(59*B*a*b^6 + 9*A*b^
7)*d^6*e + 12*(395*B*a^2*b^5 + 124*A*a*b^6)*d^5*e^2 - 42*(491*B*a^3*b^4 + 187*A*a^2*b^5)*d^4*e^3 - 2*(51487*B*
a^4*b^3 - 17430*A*a^3*b^4)*d^3*e^4 - 3*(20687*B*a^5*b^2 - 45677*A*a^4*b^3)*d^2*e^5 - 1536*(5*B*a^6*b - 16*A*a^
5*b^2)*d*e^6 + 256*(B*a^7 - 3*A*a^6*b)*e^7)*x)*sqrt(e*x + d))/(a^5*b^8*d^11 - 8*a^6*b^7*d^10*e + 28*a^7*b^6*d^
9*e^2 - 56*a^8*b^5*d^8*e^3 + 70*a^9*b^4*d^7*e^4 - 56*a^10*b^3*d^6*e^5 + 28*a^11*b^2*d^5*e^6 - 8*a^12*b*d^4*e^7
+ a^13*d^3*e^8 + (b^13*d^8*e^3 - 8*a*b^12*d^7*e^4 + 28*a^2*b^11*d^6*e^5 - 56*a^3*b^10*d^5*e^6 + 70*a^4*b^9*d^
4*e^7 - 56*a^5*b^8*d^3*e^8 + 28*a^6*b^7*d^2*e^9 - 8*a^7*b^6*d*e^10 + a^8*b^5*e^11)*x^8 + (3*b^13*d^9*e^2 - 19*
a*b^12*d^8*e^3 + 44*a^2*b^11*d^7*e^4 - 28*a^3*b^10*d^6*e^5 - 70*a^4*b^9*d^5*e^6 + 182*a^5*b^8*d^4*e^7 - 196*a^
6*b^7*d^3*e^8 + 116*a^7*b^6*d^2*e^9 - 37*a^8*b^5*d*e^10 + 5*a^9*b^4*e^11)*x^7 + (3*b^13*d^10*e - 9*a*b^12*d^9*
e^2 - 26*a^2*b^11*d^8*e^3 + 172*a^3*b^10*d^7*e^4 - 350*a^4*b^9*d^6*e^5 + 322*a^5*b^8*d^5*e^6 - 56*a^6*b^7*d^4*
e^7 - 164*a^7*b^6*d^3*e^8 + 163*a^8*b^5*d^2*e^9 - 65*a^9*b^4*d*e^10 + 10*a^10*b^3*e^11)*x^6 + (b^13*d^11 + 7*a
*b^12*d^10*e - 62*a^2*b^11*d^9*e^2 + 134*a^3*b^10*d^8*e^3 - 10*a^4*b^9*d^7*e^4 - 406*a^5*b^8*d^6*e^5 + 728*a^6
*b^7*d^5*e^6 - 568*a^7*b^6*d^4*e^7 + 161*a^8*b^5*d^3*e^8 + 55*a^9*b^4*d^2*e^9 - 50*a^10*b^3*d*e^10 + 10*a^11*b
^2*e^11)*x^5 + 5*(a*b^12*d^11 - 2*a^2*b^11*d^10*e - 14*a^3*b^10*d^9*e^2 + 65*a^4*b^9*d^8*e^3 - 106*a^5*b^8*d^7
*e^4 + 56*a^6*b^7*d^6*e^5 + 56*a^7*b^6*d^5*e^6 - 106*a^8*b^5*d^4*e^7 + 65*a^9*b^4*d^3*e^8 - 14*a^10*b^3*d^2*e^
9 - 2*a^11*b^2*d*e^10 + a^12*b*e^11)*x^4 + (10*a^2*b^11*d^11 - 50*a^3*b^10*d^10*e + 55*a^4*b^9*d^9*e^2 + 161*a
^5*b^8*d^8*e^3 - 568*a^6*b^7*d^7*e^4 + 728*a^7*b^6*d^6*e^5 - 406*a^8*b^5*d^5*e^6 - 10*a^9*b^4*d^4*e^7 + 134*a^
10*b^3*d^3*e^8 - 62*a^11*b^2*d^2*e^9 + 7*a^12*b*d*e^10 + a^13*e^11)*x^3 + (10*a^3*b^10*d^11 - 65*a^4*b^9*d^10*
e + 163*a^5*b^8*d^9*e^2 - 164*a^6*b^7*d^8*e^3 - 56*a^7*b^6*d^7*e^4 + 322*a^8*b^5*d^6*e^5 - 350*a^9*b^4*d^5*e^6
+ 172*a^10*b^3*d^4*e^7 - 26*a^11*b^2*d^3*e^8 - 9*a^12*b*d^2*e^9 + 3*a^13*d*e^10)*x^2 + (5*a^4*b^9*d^11 - 37*a
^5*b^8*d^10*e + 116*a^6*b^7*d^9*e^2 - 196*a^7*b^6*d^8*e^3 + 182*a^8*b^5*d^7*e^4 - 70*a^9*b^4*d^6*e^5 - 28*a^10
*b^3*d^5*e^6 + 44*a^11*b^2*d^4*e^7 - 19*a^12*b*d^3*e^8 + 3*a^13*d^2*e^9)*x), -1/1920*(45045*(2*B*a^5*b^2*d^4*e
^4 + (B*a^6*b - 3*A*a^5*b^2)*d^3*e^5 + (2*B*b^7*d*e^7 + (B*a*b^6 - 3*A*b^7)*e^8)*x^8 + (6*B*b^7*d^2*e^6 + (13*
B*a*b^6 - 9*A*b^7)*d*e^7 + 5*(B*a^2*b^5 - 3*A*a*b^6)*e^8)*x^7 + (6*B*b^7*d^3*e^5 + 3*(11*B*a*b^6 - 3*A*b^7)*d^
2*e^6 + 5*(7*B*a^2*b^5 - 9*A*a*b^6)*d*e^7 + 10*(B*a^3*b^4 - 3*A*a^2*b^5)*e^8)*x^6 + (2*B*b^7*d^4*e^4 + (31*B*a
*b^6 - 3*A*b^7)*d^3*e^5 + 15*(5*B*a^2*b^5 - 3*A*a*b^6)*d^2*e^6 + 10*(5*B*a^3*b^4 - 9*A*a^2*b^5)*d*e^7 + 10*(B*
a^4*b^3 - 3*A*a^3*b^4)*e^8)*x^5 + 5*(2*B*a*b^6*d^4*e^4 + (13*B*a^2*b^5 - 3*A*a*b^6)*d^3*e^5 + 18*(B*a^3*b^4 -
A*a^2*b^5)*d^2*e^6 + 2*(4*B*a^4*b^3 - 9*A*a^3*b^4)*d*e^7 + (B*a^5*b^2 - 3*A*a^4*b^3)*e^8)*x^4 + (20*B*a^2*b^5*
d^4*e^4 + 10*(7*B*a^3*b^4 - 3*A*a^2*b^5)*d^3*e^5 + 30*(2*B*a^4*b^3 - 3*A*a^3*b^4)*d^2*e^6 + (17*B*a^5*b^2 - 45
*A*a^4*b^3)*d*e^7 + (B*a^6*b - 3*A*a^5*b^2)*e^8)*x^3 + (20*B*a^3*b^4*d^4*e^4 + 10*(4*B*a^4*b^3 - 3*A*a^3*b^4)*
d^3*e^5 + 3*(7*B*a^5*b^2 - 15*A*a^4*b^3)*d^2*e^6 + 3*(B*a^6*b - 3*A*a^5*b^2)*d*e^7)*x^2 + (10*B*a^4*b^3*d^4*e^
4 + (11*B*a^5*b^2 - 15*A*a^4*b^3)*d^3*e^5 + 3*(B*a^6*b - 3*A*a^5*b^2)*d^2*e^6)*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)) + (768*A*a^7*e^7 + 96*(B*a*b^6 + 4*A*b^7)*d^7 -
16*(62*B*a^2*b^5 + 213*A*a*b^6)*d^6*e + 28*(187*B*a^3*b^4 + 498*A*a^2*b^5)*d^5*e^2 - 70*(332*B*a^4*b^3 + 519*
A*a^3*b^4)*d^4*e^3 - (100363*B*a^5*b^2 - 79905*A*a^4*b^3)*d^3*e^4 - 1024*(16*B*a^6*b - 87*A*a^5*b^2)*d^2*e^5 +
512*(B*a^7 - 18*A*a^6*b)*d*e^6 - 45045*(2*B*b^7*d*e^6 + (B*a*b^6 - 3*A*b^7)*e^7)*x^7 - 105105*(2*B*b^7*d^2*e^
5 + (5*B*a*b^6 - 3*A*b^7)*d*e^6 + 2*(B*a^2*b^5 - 3*A*a*b^6)*e^7)*x^6 - 3003*(46*B*b^7*d^3*e^4 + 3*(117*B*a*b^6
- 23*A*b^7)*d^2*e^5 + 12*(35*B*a^2*b^5 - 41*A*a*b^6)*d*e^6 + 128*(B*a^3*b^4 - 3*A*a^2*b^5)*e^7)*x^5 - 2145*(6
*B*b^7*d^4*e^3 + (307*B*a*b^6 - 9*A*b^7)*d^3*e^4 + 12*(83*B*a^2*b^5 - 38*A*a*b^6)*d^2*e^5 + 6*(123*B*a^3*b^4 -
211*A*a^2*b^5)*d*e^6 + 158*(B*a^4*b^3 - 3*A*a^3*b^4)*e^7)*x^4 + 715*(4*B*b^7*d^5*e^2 - 2*(43*B*a*b^6 + 3*A*b^
7)*d^4*e^3 - 4*(434*B*a^2*b^5 - 33*A*a*b^6)*d^3*e^4 - 2*(1547*B*a^3*b^4 - 1269*A*a^2*b^5)*d^2*e^5 - 2*(755*B*a
^4*b^3 - 1686*A*a^3*b^4)*d*e^6 - 193*(B*a^5*b^2 - 3*A*a^4*b^3)*e^7)*x^3 - 65*(16*B*b^7*d^6*e - 12*(17*B*a*b^6
+ 2*A*b^7)*d^5*e^2 + 6*(295*B*a^2*b^5 + 53*A*a*b^6)*d^4*e^3 + 2*(8837*B*a^3*b^4 - 1407*A*a^2*b^5)*d^3*e^4 + 6*
(3091*B*a^4*b^3 - 4184*A*a^3*b^4)*d^2*e^5 + 3*(1867*B*a^5*b^2 - 5089*A*a^4*b^3)*d*e^6 + 256*(B*a^6*b - 3*A*a^5
*b^2)*e^7)*x^2 + 5*(96*B*b^7*d^7 - 16*(59*B*a*b^6 + 9*A*b^7)*d^6*e + 12*(395*B*a^2*b^5 + 124*A*a*b^6)*d^5*e^2
- 42*(491*B*a^3*b^4 + 187*A*a^2*b^5)*d^4*e^3 - 2*(51487*B*a^4*b^3 - 17430*A*a^3*b^4)*d^3*e^4 - 3*(20687*B*a^5*
b^2 - 45677*A*a^4*b^3)*d^2*e^5 - 1536*(5*B*a^6*b - 16*A*a^5*b^2)*d*e^6 + 256*(B*a^7 - 3*A*a^6*b)*e^7)*x)*sqrt(
e*x + d))/(a^5*b^8*d^11 - 8*a^6*b^7*d^10*e + 28*a^7*b^6*d^9*e^2 - 56*a^8*b^5*d^8*e^3 + 70*a^9*b^4*d^7*e^4 - 56
*a^10*b^3*d^6*e^5 + 28*a^11*b^2*d^5*e^6 - 8*a^12*b*d^4*e^7 + a^13*d^3*e^8 + (b^13*d^8*e^3 - 8*a*b^12*d^7*e^4 +
28*a^2*b^11*d^6*e^5 - 56*a^3*b^10*d^5*e^6 + 70*a^4*b^9*d^4*e^7 - 56*a^5*b^8*d^3*e^8 + 28*a^6*b^7*d^2*e^9 - 8*
a^7*b^6*d*e^10 + a^8*b^5*e^11)*x^8 + (3*b^13*d^9*e^2 - 19*a*b^12*d^8*e^3 + 44*a^2*b^11*d^7*e^4 - 28*a^3*b^10*d
^6*e^5 - 70*a^4*b^9*d^5*e^6 + 182*a^5*b^8*d^4*e^7 - 196*a^6*b^7*d^3*e^8 + 116*a^7*b^6*d^2*e^9 - 37*a^8*b^5*d*e
^10 + 5*a^9*b^4*e^11)*x^7 + (3*b^13*d^10*e - 9*a*b^12*d^9*e^2 - 26*a^2*b^11*d^8*e^3 + 172*a^3*b^10*d^7*e^4 - 3
50*a^4*b^9*d^6*e^5 + 322*a^5*b^8*d^5*e^6 - 56*a^6*b^7*d^4*e^7 - 164*a^7*b^6*d^3*e^8 + 163*a^8*b^5*d^2*e^9 - 65
*a^9*b^4*d*e^10 + 10*a^10*b^3*e^11)*x^6 + (b^13*d^11 + 7*a*b^12*d^10*e - 62*a^2*b^11*d^9*e^2 + 134*a^3*b^10*d^
8*e^3 - 10*a^4*b^9*d^7*e^4 - 406*a^5*b^8*d^6*e^5 + 728*a^6*b^7*d^5*e^6 - 568*a^7*b^6*d^4*e^7 + 161*a^8*b^5*d^3
*e^8 + 55*a^9*b^4*d^2*e^9 - 50*a^10*b^3*d*e^10 + 10*a^11*b^2*e^11)*x^5 + 5*(a*b^12*d^11 - 2*a^2*b^11*d^10*e -
14*a^3*b^10*d^9*e^2 + 65*a^4*b^9*d^8*e^3 - 106*a^5*b^8*d^7*e^4 + 56*a^6*b^7*d^6*e^5 + 56*a^7*b^6*d^5*e^6 - 106
*a^8*b^5*d^4*e^7 + 65*a^9*b^4*d^3*e^8 - 14*a^10*b^3*d^2*e^9 - 2*a^11*b^2*d*e^10 + a^12*b*e^11)*x^4 + (10*a^2*b
^11*d^11 - 50*a^3*b^10*d^10*e + 55*a^4*b^9*d^9*e^2 + 161*a^5*b^8*d^8*e^3 - 568*a^6*b^7*d^7*e^4 + 728*a^7*b^6*d
^6*e^5 - 406*a^8*b^5*d^5*e^6 - 10*a^9*b^4*d^4*e^7 + 134*a^10*b^3*d^3*e^8 - 62*a^11*b^2*d^2*e^9 + 7*a^12*b*d*e^
10 + a^13*e^11)*x^3 + (10*a^3*b^10*d^11 - 65*a^4*b^9*d^10*e + 163*a^5*b^8*d^9*e^2 - 164*a^6*b^7*d^8*e^3 - 56*a
^7*b^6*d^7*e^4 + 322*a^8*b^5*d^6*e^5 - 350*a^9*b^4*d^5*e^6 + 172*a^10*b^3*d^4*e^7 - 26*a^11*b^2*d^3*e^8 - 9*a^
12*b*d^2*e^9 + 3*a^13*d*e^10)*x^2 + (5*a^4*b^9*d^11 - 37*a^5*b^8*d^10*e + 116*a^6*b^7*d^9*e^2 - 196*a^7*b^6*d^
8*e^3 + 182*a^8*b^5*d^7*e^4 - 70*a^9*b^4*d^6*e^5 - 28*a^10*b^3*d^5*e^6 + 44*a^11*b^2*d^4*e^7 - 19*a^12*b*d^3*e
^8 + 3*a^13*d^2*e^9)*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)**3,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.31683, size = 2263, normalized size = 5.2 \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)^3,x, algorithm="giac")
[Out]
3003/128*(2*B*b^3*d*e^4 + B*a*b^2*e^5 - 3*A*b^3*e^5)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^8*d^8 -
8*a*b^7*d^7*e + 28*a^2*b^6*d^6*e^2 - 56*a^3*b^5*d^5*e^3 + 70*a^4*b^4*d^4*e^4 - 56*a^5*b^3*d^3*e^5 + 28*a^6*b^2
*d^2*e^6 - 8*a^7*b*d*e^7 + a^8*e^8)*sqrt(-b^2*d + a*b*e)) + 1/1920*(90090*(x*e + d)^7*B*b^7*d*e^4 - 420420*(x*
e + d)^6*B*b^7*d^2*e^4 + 768768*(x*e + d)^5*B*b^7*d^3*e^4 - 677820*(x*e + d)^4*B*b^7*d^4*e^4 + 275990*(x*e + d
)^3*B*b^7*d^5*e^4 - 33280*(x*e + d)^2*B*b^7*d^6*e^4 - 2560*(x*e + d)*B*b^7*d^7*e^4 - 768*B*b^7*d^8*e^4 + 45045
*(x*e + d)^7*B*a*b^6*e^5 - 135135*(x*e + d)^7*A*b^7*e^5 + 210210*(x*e + d)^6*B*a*b^6*d*e^5 + 630630*(x*e + d)^
6*A*b^7*d*e^5 - 1153152*(x*e + d)^5*B*a*b^6*d^2*e^5 - 1153152*(x*e + d)^5*A*b^7*d^2*e^5 + 1694550*(x*e + d)^4*
B*a*b^6*d^3*e^5 + 1016730*(x*e + d)^4*A*b^7*d^3*e^5 - 965965*(x*e + d)^3*B*a*b^6*d^4*e^5 - 413985*(x*e + d)^3*
A*b^7*d^4*e^5 + 149760*(x*e + d)^2*B*a*b^6*d^5*e^5 + 49920*(x*e + d)^2*A*b^7*d^5*e^5 + 14080*(x*e + d)*B*a*b^6
*d^6*e^5 + 3840*(x*e + d)*A*b^7*d^6*e^5 + 5376*B*a*b^6*d^7*e^5 + 768*A*b^7*d^7*e^5 + 210210*(x*e + d)^6*B*a^2*
b^5*e^6 - 630630*(x*e + d)^6*A*a*b^6*e^6 + 2306304*(x*e + d)^5*A*a*b^6*d*e^6 - 1016730*(x*e + d)^4*B*a^2*b^5*d
^2*e^6 - 3050190*(x*e + d)^4*A*a*b^6*d^2*e^6 + 1103960*(x*e + d)^3*B*a^2*b^5*d^3*e^6 + 1655940*(x*e + d)^3*A*a
*b^6*d^3*e^6 - 249600*(x*e + d)^2*B*a^2*b^5*d^4*e^6 - 249600*(x*e + d)^2*A*a*b^6*d^4*e^6 - 30720*(x*e + d)*B*a
^2*b^5*d^5*e^6 - 23040*(x*e + d)*A*a*b^6*d^5*e^6 - 16128*B*a^2*b^5*d^6*e^6 - 5376*A*a*b^6*d^6*e^6 + 384384*(x*
e + d)^5*B*a^3*b^4*e^7 - 1153152*(x*e + d)^5*A*a^2*b^5*e^7 - 338910*(x*e + d)^4*B*a^3*b^4*d*e^7 + 3050190*(x*e
+ d)^4*A*a^2*b^5*d*e^7 - 275990*(x*e + d)^3*B*a^3*b^4*d^2*e^7 - 2483910*(x*e + d)^3*A*a^2*b^5*d^2*e^7 + 16640
0*(x*e + d)^2*B*a^3*b^4*d^3*e^7 + 499200*(x*e + d)^2*A*a^2*b^5*d^3*e^7 + 32000*(x*e + d)*B*a^3*b^4*d^4*e^7 + 5
7600*(x*e + d)*A*a^2*b^5*d^4*e^7 + 26880*B*a^3*b^4*d^5*e^7 + 16128*A*a^2*b^5*d^5*e^7 + 338910*(x*e + d)^4*B*a^
4*b^3*e^8 - 1016730*(x*e + d)^4*A*a^3*b^4*e^8 - 275990*(x*e + d)^3*B*a^4*b^3*d*e^8 + 1655940*(x*e + d)^3*A*a^3
*b^4*d*e^8 - 499200*(x*e + d)^2*A*a^3*b^4*d^2*e^8 - 12800*(x*e + d)*B*a^4*b^3*d^3*e^8 - 76800*(x*e + d)*A*a^3*
b^4*d^3*e^8 - 26880*B*a^4*b^3*d^4*e^8 - 26880*A*a^3*b^4*d^4*e^8 + 137995*(x*e + d)^3*B*a^5*b^2*e^9 - 413985*(x
*e + d)^3*A*a^4*b^3*e^9 - 49920*(x*e + d)^2*B*a^5*b^2*d*e^9 + 249600*(x*e + d)^2*A*a^4*b^3*d*e^9 - 3840*(x*e +
d)*B*a^5*b^2*d^2*e^9 + 57600*(x*e + d)*A*a^4*b^3*d^2*e^9 + 16128*B*a^5*b^2*d^3*e^9 + 26880*A*a^4*b^3*d^3*e^9
+ 16640*(x*e + d)^2*B*a^6*b*e^10 - 49920*(x*e + d)^2*A*a^5*b^2*e^10 + 5120*(x*e + d)*B*a^6*b*d*e^10 - 23040*(x
*e + d)*A*a^5*b^2*d*e^10 - 5376*B*a^6*b*d^2*e^10 - 16128*A*a^5*b^2*d^2*e^10 - 1280*(x*e + d)*B*a^7*e^11 + 3840
*(x*e + d)*A*a^6*b*e^11 + 768*B*a^7*d*e^11 + 5376*A*a^6*b*d*e^11 - 768*A*a^7*e^12)/((b^8*d^8 - 8*a*b^7*d^7*e +
28*a^2*b^6*d^6*e^2 - 56*a^3*b^5*d^5*e^3 + 70*a^4*b^4*d^4*e^4 - 56*a^5*b^3*d^3*e^5 + 28*a^6*b^2*d^2*e^6 - 8*a^
7*b*d*e^7 + a^8*e^8)*((x*e + d)^(3/2)*b - sqrt(x*e + d)*b*d + sqrt(x*e + d)*a*e)^5)