3.17.63 \(\int \frac {A+B x}{(d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)
Optimal. Leaf size=564 \[ -\frac {231 b e^3 (a+b x) (5 a B e-13 A b e+8 b B d)}{64 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^7}-\frac {77 e^3 (a+b x) (5 a B e-13 A b e+8 b B d)}{64 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^6}-\frac {231 e^3 (a+b x) (5 a B e-13 A b e+8 b B d)}{320 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^5}-\frac {33 e^2 (5 a B e-13 A b e+8 b B d)}{64 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^4}+\frac {11 e (5 a B e-13 A b e+8 b B d)}{96 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}-\frac {5 a B e-13 A b e+8 b B d}{24 b (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}-\frac {A b-a B}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}+\frac {231 b^{3/2} e^3 (a+b x) (5 a B e-13 A b e+8 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{15/2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.57, antiderivative size = 564, normalized size of antiderivative = 1.00,
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} \begin {gather*} -\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)} \end {gather*}
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 51
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]
Rule 63
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]
Rule 78
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ
[p, n]))))
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
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]
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 = 0.10, size = 115, normalized size = 0.20 \begin {gather*} \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)} \end {gather*}
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))
________________________________________________________________________________________
IntegrateAlgebraic [B] time = 79.19, size = 1217, normalized size = 2.16 \begin {gather*} \frac {(-a e-b x e) \left (-\frac {\left (-384 b^6 B d^7+384 A b^6 e d^6+2304 a b^5 B e d^6-1024 b^6 B (d+e x) d^6-2304 a A b^5 e^2 d^5-5760 a^2 b^4 B e^2 d^5-11264 b^6 B (d+e x)^2 d^5+1664 A b^6 e (d+e x) d^5+4480 a b^5 B e (d+e x) d^5+5760 a^2 A b^4 e^3 d^4+7680 a^3 b^3 B e^3 d^4+73656 b^6 B (d+e x)^3 d^4+18304 A b^6 e (d+e x)^2 d^4+38016 a b^5 B e (d+e x)^2 d^4-8320 a A b^5 e^2 (d+e x) d^4-7040 a^2 b^4 B e^2 (d+e x) d^4-7680 a^3 A b^3 e^4 d^3-5760 a^4 b^2 B e^4 d^3-134904 b^6 B (d+e x)^4 d^3-119691 A b^6 e (d+e x)^3 d^3-174933 a b^5 B e (d+e x)^3 d^3-73216 a A b^5 e^2 (d+e x)^2 d^3-39424 a^2 b^4 B e^2 (d+e x)^2 d^3+16640 a^2 A b^4 e^3 (d+e x) d^3+3840 a^3 b^3 B e^3 (d+e x) d^3+5760 a^4 A b^2 e^5 d^2+2304 a^5 b B e^5 d^2+101640 b^6 B (d+e x)^5 d^2+219219 A b^6 e (d+e x)^4 d^2+185493 a b^5 B e (d+e x)^4 d^2+359073 a A b^5 e^2 (d+e x)^3 d^2+82863 a^2 b^4 B e^2 (d+e x)^3 d^2+109824 a^2 A b^4 e^3 (d+e x)^2 d^2+2816 a^3 b^3 B e^3 (d+e x)^2 d^2-16640 a^3 A b^3 e^4 (d+e x) d^2+1280 a^4 b^2 B e^4 (d+e x) d^2-2304 a^5 A b e^6 d-384 a^6 B e^6 d-27720 b^6 B (d+e x)^6 d-165165 A b^6 e (d+e x)^5 d-38115 a b^5 B e (d+e x)^5 d-438438 a A b^5 e^2 (d+e x)^4 d+33726 a^2 b^4 B e^2 (d+e x)^4 d-359073 a^2 A b^4 e^3 (d+e x)^3 d+64449 a^3 b^3 B e^3 (d+e x)^3 d-73216 a^3 A b^3 e^4 (d+e x)^2 d+16896 a^4 b^2 B e^4 (d+e x)^2 d+8320 a^4 A b^2 e^5 (d+e x) d-2176 a^5 b B e^5 (d+e x) d+384 a^6 A e^7+45045 A b^6 e (d+e x)^6-17325 a b^5 B e (d+e x)^6+165165 a A b^5 e^2 (d+e x)^5-63525 a^2 b^4 B e^2 (d+e x)^5+219219 a^2 A b^4 e^3 (d+e x)^4-84315 a^3 b^3 B e^3 (d+e x)^4+119691 a^3 A b^3 e^4 (d+e x)^3-46035 a^4 b^2 B e^4 (d+e x)^3+18304 a^4 A b^2 e^5 (d+e x)^2-7040 a^5 b B e^5 (d+e x)^2-1664 a^5 A b e^6 (d+e x)+640 a^6 B e^6 (d+e x)\right ) e^3}{960 (b d-a e)^7 (d+e x)^{5/2} (b d-a e-b (d+e x))^4}-\frac {231 \left (-13 A b^{5/2} e^4+5 a b^{3/2} B e^4+8 b^{5/2} B d e^3\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {a e-b d} \sqrt {d+e x}}{b d-a e}\right )}{64 (b d-a e)^7 \sqrt {a e-b d}}\right )}{e \sqrt {\frac {(a e+b x e)^2}{e^2}}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(A + B*x)/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]
[Out]
((-(a*e) - b*e*x)*(-1/960*(e^3*(-384*b^6*B*d^7 + 384*A*b^6*d^6*e + 2304*a*b^5*B*d^6*e - 2304*a*A*b^5*d^5*e^2 -
5760*a^2*b^4*B*d^5*e^2 + 5760*a^2*A*b^4*d^4*e^3 + 7680*a^3*b^3*B*d^4*e^3 - 7680*a^3*A*b^3*d^3*e^4 - 5760*a^4*
b^2*B*d^3*e^4 + 5760*a^4*A*b^2*d^2*e^5 + 2304*a^5*b*B*d^2*e^5 - 2304*a^5*A*b*d*e^6 - 384*a^6*B*d*e^6 + 384*a^6
*A*e^7 - 1024*b^6*B*d^6*(d + e*x) + 1664*A*b^6*d^5*e*(d + e*x) + 4480*a*b^5*B*d^5*e*(d + e*x) - 8320*a*A*b^5*d
^4*e^2*(d + e*x) - 7040*a^2*b^4*B*d^4*e^2*(d + e*x) + 16640*a^2*A*b^4*d^3*e^3*(d + e*x) + 3840*a^3*b^3*B*d^3*e
^3*(d + e*x) - 16640*a^3*A*b^3*d^2*e^4*(d + e*x) + 1280*a^4*b^2*B*d^2*e^4*(d + e*x) + 8320*a^4*A*b^2*d*e^5*(d
+ e*x) - 2176*a^5*b*B*d*e^5*(d + e*x) - 1664*a^5*A*b*e^6*(d + e*x) + 640*a^6*B*e^6*(d + e*x) - 11264*b^6*B*d^5
*(d + e*x)^2 + 18304*A*b^6*d^4*e*(d + e*x)^2 + 38016*a*b^5*B*d^4*e*(d + e*x)^2 - 73216*a*A*b^5*d^3*e^2*(d + e*
x)^2 - 39424*a^2*b^4*B*d^3*e^2*(d + e*x)^2 + 109824*a^2*A*b^4*d^2*e^3*(d + e*x)^2 + 2816*a^3*b^3*B*d^2*e^3*(d
+ e*x)^2 - 73216*a^3*A*b^3*d*e^4*(d + e*x)^2 + 16896*a^4*b^2*B*d*e^4*(d + e*x)^2 + 18304*a^4*A*b^2*e^5*(d + e*
x)^2 - 7040*a^5*b*B*e^5*(d + e*x)^2 + 73656*b^6*B*d^4*(d + e*x)^3 - 119691*A*b^6*d^3*e*(d + e*x)^3 - 174933*a*
b^5*B*d^3*e*(d + e*x)^3 + 359073*a*A*b^5*d^2*e^2*(d + e*x)^3 + 82863*a^2*b^4*B*d^2*e^2*(d + e*x)^3 - 359073*a^
2*A*b^4*d*e^3*(d + e*x)^3 + 64449*a^3*b^3*B*d*e^3*(d + e*x)^3 + 119691*a^3*A*b^3*e^4*(d + e*x)^3 - 46035*a^4*b
^2*B*e^4*(d + e*x)^3 - 134904*b^6*B*d^3*(d + e*x)^4 + 219219*A*b^6*d^2*e*(d + e*x)^4 + 185493*a*b^5*B*d^2*e*(d
+ e*x)^4 - 438438*a*A*b^5*d*e^2*(d + e*x)^4 + 33726*a^2*b^4*B*d*e^2*(d + e*x)^4 + 219219*a^2*A*b^4*e^3*(d + e
*x)^4 - 84315*a^3*b^3*B*e^3*(d + e*x)^4 + 101640*b^6*B*d^2*(d + e*x)^5 - 165165*A*b^6*d*e*(d + e*x)^5 - 38115*
a*b^5*B*d*e*(d + e*x)^5 + 165165*a*A*b^5*e^2*(d + e*x)^5 - 63525*a^2*b^4*B*e^2*(d + e*x)^5 - 27720*b^6*B*d*(d
+ e*x)^6 + 45045*A*b^6*e*(d + e*x)^6 - 17325*a*b^5*B*e*(d + e*x)^6))/((b*d - a*e)^7*(d + e*x)^(5/2)*(b*d - a*e
- b*(d + e*x))^4) - (231*(8*b^(5/2)*B*d*e^3 - 13*A*b^(5/2)*e^4 + 5*a*b^(3/2)*B*e^4)*ArcTan[(Sqrt[b]*Sqrt[-(b*
d) + a*e]*Sqrt[d + e*x])/(b*d - a*e)])/(64*(b*d - a*e)^7*Sqrt[-(b*d) + a*e])))/(e*Sqrt[(a*e + b*e*x)^2/e^2])
________________________________________________________________________________________
fricas [B] time = 0.56, size = 4833, normalized size = 8.57
result too large to display
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)]
________________________________________________________________________________________
giac [B] time = 0.85, size = 1527, normalized size = 2.71
result too large to display
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)
________________________________________________________________________________________
maple [B] time = 0.14, size = 2282, normalized size = 4.05
result too large to display
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*(520*A*((a*e-b*d)*b)^(1/2)*x*b^6*d^5*e-110880*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(5/
2)*x*a^3*b^4*d*e^3-110880*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(5/2)*x^3*a*b^6*d*e^3-166320*B
*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(5/2)*x^2*a^2*b^5*d*e^3+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+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+4504
5*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+25
6971*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-
121209*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^3-54
60*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+3160*B*((a*e-b*d)*b)^(1/2)*x*a*b^5*d^5*e-17325*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(5/2)*
x^4*a*b^6*e^4-27720*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(5/2)*x^4*b^7*d*e^3+180180*A*arctan(
(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(5/2)*x^3*a*b^6*e^4-69300*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1
/2)*b)*(e*x+d)^(5/2)*x^3*a^2*b^5*e^4+270270*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(5/2)*x^2*a^
2*b^5*e^4-103950*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(5/2)*x^2*a^3*b^4*e^4+180180*A*arctan((
e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(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^5-6
9300*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(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^4-27
720*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(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+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+45045*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^
(1/2)*b)*(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^6-27720*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+69069*A*((a*e-b*d)*b)^(1/2)*x^4*b^6*d^2*e^4+45045*A*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(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^3-17
325*B*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*(e*x+d)^(5/2)*a^5*b^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+32384*A*((a*e-b*d)*b)^(1/2)*a^4*b^2*d^2*e^4-3968*A*((a*e-b*d)*b)^(1/2)
*a^5*b*d*e^5)*(b*x+a)/(e*x+d)^(5/2)/((a*e-b*d)*b)^(1/2)/(a*e-b*d)^7/((b*x+a)^2)^(5/2)
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}
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)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+B\,x}{{\left (d+e\,x\right )}^{7/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A + B*x)/((d + e*x)^(7/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)),x)
[Out]
int((A + B*x)/((d + e*x)^(7/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)), x)
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
Timed out
________________________________________________________________________________________