3.19.73 \(\int \frac {A+B x}{(d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^{3/2}} \, dx\) [1873]

Optimal. Leaf size=414 \[ \frac {-4 b B d+9 A b e-5 a B e}{4 b (b d-a e)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A b-a B}{2 b (b d-a e) (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (4 b B d-9 A b e+5 a B e) (a+b x)}{20 b (b d-a e)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (4 b B d-9 A b e+5 a B e) (a+b x)}{12 (b d-a e)^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 b e (4 b B d-9 A b e+5 a B e) (a+b x)}{4 (b d-a e)^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 b^{3/2} e (4 b B d-9 A b e+5 a B e) (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 (b d-a e)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]

[Out]

1/4*(9*A*b*e-5*B*a*e-4*B*b*d)/b/(-a*e+b*d)^2/(e*x+d)^(5/2)/((b*x+a)^2)^(1/2)+1/2*(-A*b+B*a)/b/(-a*e+b*d)/(b*x+
a)/(e*x+d)^(5/2)/((b*x+a)^2)^(1/2)-7/20*e*(-9*A*b*e+5*B*a*e+4*B*b*d)*(b*x+a)/b/(-a*e+b*d)^3/(e*x+d)^(5/2)/((b*
x+a)^2)^(1/2)-7/12*e*(-9*A*b*e+5*B*a*e+4*B*b*d)*(b*x+a)/(-a*e+b*d)^4/(e*x+d)^(3/2)/((b*x+a)^2)^(1/2)+7/4*b^(3/
2)*e*(-9*A*b*e+5*B*a*e+4*B*b*d)*(b*x+a)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/(-a*e+b*d)^(11/2)/((b*
x+a)^2)^(1/2)-7/4*b*e*(-9*A*b*e+5*B*a*e+4*B*b*d)*(b*x+a)/(-a*e+b*d)^5/(e*x+d)^(1/2)/((b*x+a)^2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.25, antiderivative size = 414, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {784, 79, 44, 53, 65, 214} \begin {gather*} -\frac {7 b e (a+b x) (5 a B e-9 A b e+4 b B d)}{4 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^5}-\frac {7 e (a+b x) (5 a B e-9 A b e+4 b B d)}{12 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}-\frac {5 a B e-9 A b e+4 b B d}{4 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}-\frac {7 e (a+b x) (5 a B e-9 A b e+4 b B d)}{20 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}-\frac {A b-a B}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}+\frac {7 b^{3/2} e (a+b x) (5 a B e-9 A b e+4 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{11/2}} \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)^(3/2)),x]

[Out]

-1/4*(4*b*B*d - 9*A*b*e + 5*a*B*e)/(b*(b*d - a*e)^2*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (A*b - a*
B)/(2*b*(b*d - a*e)*(a + b*x)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (7*e*(4*b*B*d - 9*A*b*e + 5*a*B
*e)*(a + b*x))/(20*b*(b*d - a*e)^3*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (7*e*(4*b*B*d - 9*A*b*e +
5*a*B*e)*(a + b*x))/(12*(b*d - a*e)^4*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (7*b*e*(4*b*B*d - 9*A*b
*e + 5*a*B*e)*(a + b*x))/(4*(b*d - a*e)^5*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*b^(3/2)*e*(4*b*B*d
- 9*A*b*e + 5*a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*(b*d - a*e)^(11/2)*Sqrt[a
^2 + 2*a*b*x + b^2*x^2])

Rule 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 53

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 65

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 79

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] || L
tQ[p, n]))))

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 784

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 )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {A+B x}{\left (a b+b^2 x\right )^3 (d+e x)^{7/2}} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {A b-a B}{2 b (b d-a e) (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left ((4 b B d-9 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}{4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {4 b B d-9 A b e+5 a B e}{4 b (b d-a e)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A b-a B}{2 b (b d-a e) (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (7 e (4 b B d-9 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}{8 b (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {4 b B d-9 A b e+5 a B e}{4 b (b d-a e)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A b-a B}{2 b (b d-a e) (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (4 b B d-9 A b e+5 a B e) (a+b x)}{20 b (b d-a e)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (7 e (4 b B d-9 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}{8 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {4 b B d-9 A b e+5 a B e}{4 b (b d-a e)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A b-a B}{2 b (b d-a e) (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (4 b B d-9 A b e+5 a B e) (a+b x)}{20 b (b d-a e)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (4 b B d-9 A b e+5 a B e) (a+b x)}{12 (b d-a e)^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (7 b e (4 b B d-9 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}{8 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {4 b B d-9 A b e+5 a B e}{4 b (b d-a e)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A b-a B}{2 b (b d-a e) (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (4 b B d-9 A b e+5 a B e) (a+b x)}{20 b (b d-a e)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (4 b B d-9 A b e+5 a B e) (a+b x)}{12 (b d-a e)^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 b e (4 b B d-9 A b e+5 a B e) (a+b x)}{4 (b d-a e)^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (7 b^2 e (4 b B d-9 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}{8 (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {4 b B d-9 A b e+5 a B e}{4 b (b d-a e)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A b-a B}{2 b (b d-a e) (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (4 b B d-9 A b e+5 a B e) (a+b x)}{20 b (b d-a e)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (4 b B d-9 A b e+5 a B e) (a+b x)}{12 (b d-a e)^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 b e (4 b B d-9 A b e+5 a B e) (a+b x)}{4 (b d-a e)^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (7 b^2 (4 b B d-9 A b e+5 a B e) \left (a b+b^2 x\right )\right ) \text {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 )}{4 (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {4 b B d-9 A b e+5 a B e}{4 b (b d-a e)^2 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A b-a B}{2 b (b d-a e) (a+b x) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (4 b B d-9 A b e+5 a B e) (a+b x)}{20 b (b d-a e)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 e (4 b B d-9 A b e+5 a B e) (a+b x)}{12 (b d-a e)^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {7 b e (4 b B d-9 A b e+5 a B e) (a+b x)}{4 (b d-a e)^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 b^{3/2} e (4 b B d-9 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 )}{4 (b d-a e)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 1.53, size = 427, normalized size = 1.03 \begin {gather*} \frac {e (a+b x)^3 \left (\frac {3 A \left (-8 a^4 e^4+8 a^3 b e^3 (7 d+3 e x)-24 a^2 b^2 e^2 \left (12 d^2+17 d e x+7 e^2 x^2\right )-a b^3 e \left (85 d^3+831 d^2 e x+1239 d e^2 x^2+525 e^3 x^3\right )+b^4 \left (10 d^4-45 d^3 e x-483 d^2 e^2 x^2-735 d e^3 x^3-315 e^4 x^4\right )\right )+B \left (-8 a^4 e^3 (2 d+5 e x)+8 a^3 b e^2 \left (34 d^2+81 d e x+35 e^2 x^2\right )+4 b^4 d x \left (15 d^3+161 d^2 e x+245 d e^2 x^2+105 e^3 x^3\right )+a^2 b^2 e \left (659 d^3+1929 d^2 e x+2289 d e^2 x^2+875 e^3 x^3\right )+a b^3 \left (30 d^4+1183 d^3 e x+2457 d^2 e^2 x^2+1925 d e^3 x^3+525 e^4 x^4\right )\right )}{e (-b d+a e)^5 (a+b x)^2 (d+e x)^{5/2}}+\frac {105 b^{3/2} (4 b B d-9 A b e+5 a B e) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{11/2}}\right )}{60 \left ((a+b x)^2\right )^{3/2}} \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)^(3/2)),x]

[Out]

(e*(a + b*x)^3*((3*A*(-8*a^4*e^4 + 8*a^3*b*e^3*(7*d + 3*e*x) - 24*a^2*b^2*e^2*(12*d^2 + 17*d*e*x + 7*e^2*x^2)
- a*b^3*e*(85*d^3 + 831*d^2*e*x + 1239*d*e^2*x^2 + 525*e^3*x^3) + b^4*(10*d^4 - 45*d^3*e*x - 483*d^2*e^2*x^2 -
735*d*e^3*x^3 - 315*e^4*x^4)) + B*(-8*a^4*e^3*(2*d + 5*e*x) + 8*a^3*b*e^2*(34*d^2 + 81*d*e*x + 35*e^2*x^2) +
4*b^4*d*x*(15*d^3 + 161*d^2*e*x + 245*d*e^2*x^2 + 105*e^3*x^3) + a^2*b^2*e*(659*d^3 + 1929*d^2*e*x + 2289*d*e^
2*x^2 + 875*e^3*x^3) + a*b^3*(30*d^4 + 1183*d^3*e*x + 2457*d^2*e^2*x^2 + 1925*d*e^3*x^3 + 525*e^4*x^4)))/(e*(-
(b*d) + a*e)^5*(a + b*x)^2*(d + e*x)^(5/2)) + (105*b^(3/2)*(4*b*B*d - 9*A*b*e + 5*a*B*e)*ArcTan[(Sqrt[b]*Sqrt[
d + e*x])/Sqrt[-(b*d) + a*e]])/(-(b*d) + a*e)^(11/2)))/(60*((a + b*x)^2)^(3/2))

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1229\) vs. \(2(316)=632\).
time = 0.99, size = 1230, normalized size = 2.97

method	result	size
default	\(-\frac {\left (-840 B \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) \left (e x +d \right )^{\frac {5}{2}} a \,b^{4} d e x -525 B \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) \left (e x +d \right )^{\frac {5}{2}} a \,b^{4} e^{2} x^{2}-420 B \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) \left (e x +d \right )^{\frac {5}{2}} b^{5} d e \,x^{2}+24 A \sqrt {b \left (a e -b d \right )}\, a^{4} e^{4}-30 A \sqrt {b \left (a e -b d \right )}\, b^{4} d^{4}-648 B \sqrt {b \left (a e -b d \right )}\, a^{3} b d \,e^{3} x -1929 B \sqrt {b \left (a e -b d \right )}\, a^{2} b^{2} d^{2} e^{2} x -1183 B \sqrt {b \left (a e -b d \right )}\, a \,b^{3} d^{3} e x +945 A \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) \left (e x +d \right )^{\frac {5}{2}} b^{5} e^{2} x^{2}+945 A \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) \left (e x +d \right )^{\frac {5}{2}} a^{2} b^{3} e^{2}-525 B \sqrt {b \left (a e -b d \right )}\, a \,b^{3} e^{4} x^{4}-420 B \sqrt {b \left (a e -b d \right )}\, b^{4} d \,e^{3} x^{4}-525 B \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) \left (e x +d \right )^{\frac {5}{2}} a^{3} b^{2} e^{2}+1575 A \sqrt {b \left (a e -b d \right )}\, a \,b^{3} e^{4} x^{3}+2205 A \sqrt {b \left (a e -b d \right )}\, b^{4} d \,e^{3} x^{3}-875 B \sqrt {b \left (a e -b d \right )}\, a^{2} b^{2} e^{4} x^{3}-980 B \sqrt {b \left (a e -b d \right )}\, b^{4} d^{2} e^{2} x^{3}+504 A \sqrt {b \left (a e -b d \right )}\, a^{2} b^{2} e^{4} x^{2}+1449 A \sqrt {b \left (a e -b d \right )}\, b^{4} d^{2} e^{2} x^{2}-280 B \sqrt {b \left (a e -b d \right )}\, a^{3} b \,e^{4} x^{2}-644 B \sqrt {b \left (a e -b d \right )}\, b^{4} d^{3} e \,x^{2}-72 A \sqrt {b \left (a e -b d \right )}\, a^{3} b \,e^{4} x +135 A \sqrt {b \left (a e -b d \right )}\, b^{4} d^{3} e x -168 A \sqrt {b \left (a e -b d \right )}\, a^{3} b d \,e^{3}+864 A \sqrt {b \left (a e -b d \right )}\, a^{2} b^{2} d^{2} e^{2}+255 A \sqrt {b \left (a e -b d \right )}\, a \,b^{3} d^{3} e -272 B \sqrt {b \left (a e -b d \right )}\, a^{3} b \,d^{2} e^{2}-659 B \sqrt {b \left (a e -b d \right )}\, a^{2} b^{2} d^{3} e +945 A \sqrt {b \left (a e -b d \right )}\, b^{4} e^{4} x^{4}+40 B \sqrt {b \left (a e -b d \right )}\, a^{4} e^{4} x -60 B \sqrt {b \left (a e -b d \right )}\, b^{4} d^{4} x +16 B \sqrt {b \left (a e -b d \right )}\, a^{4} d \,e^{3}-30 B \sqrt {b \left (a e -b d \right )}\, a \,b^{3} d^{4}+1890 A \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) \left (e x +d \right )^{\frac {5}{2}} a \,b^{4} e^{2} x -1050 B \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) \left (e x +d \right )^{\frac {5}{2}} a^{2} b^{3} e^{2} x -420 B \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) \left (e x +d \right )^{\frac {5}{2}} a^{2} b^{3} d e -1925 B \sqrt {b \left (a e -b d \right )}\, a \,b^{3} d \,e^{3} x^{3}+3717 A \sqrt {b \left (a e -b d \right )}\, a \,b^{3} d \,e^{3} x^{2}-2289 B \sqrt {b \left (a e -b d \right )}\, a^{2} b^{2} d \,e^{3} x^{2}-2457 B \sqrt {b \left (a e -b d \right )}\, a \,b^{3} d^{2} e^{2} x^{2}+1224 A \sqrt {b \left (a e -b d \right )}\, a^{2} b^{2} d \,e^{3} x +2493 A \sqrt {b \left (a e -b d \right )}\, a \,b^{3} d^{2} e^{2} x \right ) \left (b x +a \right )}{60 \sqrt {b \left (a e -b d \right )}\, \left (e x +d \right )^{\frac {5}{2}} \left (a e -b d \right )^{5} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\)	\(1230\)

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)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/60*(-840*B*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*(e*x+d)^(5/2)*a*b^4*d*e*x-525*B*arctan(b*(e*x+d)^(1/
2)/(b*(a*e-b*d))^(1/2))*(e*x+d)^(5/2)*a*b^4*e^2*x^2-420*B*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*(e*x+d)^
(5/2)*b^5*d*e*x^2+24*A*(b*(a*e-b*d))^(1/2)*a^4*e^4-30*A*(b*(a*e-b*d))^(1/2)*b^4*d^4-648*B*(b*(a*e-b*d))^(1/2)*
a^3*b*d*e^3*x-1929*B*(b*(a*e-b*d))^(1/2)*a^2*b^2*d^2*e^2*x-1183*B*(b*(a*e-b*d))^(1/2)*a*b^3*d^3*e*x+945*A*arct
an(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*(e*x+d)^(5/2)*b^5*e^2*x^2+945*A*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(
1/2))*(e*x+d)^(5/2)*a^2*b^3*e^2-525*B*(b*(a*e-b*d))^(1/2)*a*b^3*e^4*x^4-420*B*(b*(a*e-b*d))^(1/2)*b^4*d*e^3*x^
4-525*B*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*(e*x+d)^(5/2)*a^3*b^2*e^2+1575*A*(b*(a*e-b*d))^(1/2)*a*b^3
*e^4*x^3+2205*A*(b*(a*e-b*d))^(1/2)*b^4*d*e^3*x^3-875*B*(b*(a*e-b*d))^(1/2)*a^2*b^2*e^4*x^3-980*B*(b*(a*e-b*d)
)^(1/2)*b^4*d^2*e^2*x^3+504*A*(b*(a*e-b*d))^(1/2)*a^2*b^2*e^4*x^2+1449*A*(b*(a*e-b*d))^(1/2)*b^4*d^2*e^2*x^2-2
80*B*(b*(a*e-b*d))^(1/2)*a^3*b*e^4*x^2-644*B*(b*(a*e-b*d))^(1/2)*b^4*d^3*e*x^2-72*A*(b*(a*e-b*d))^(1/2)*a^3*b*
e^4*x+135*A*(b*(a*e-b*d))^(1/2)*b^4*d^3*e*x-168*A*(b*(a*e-b*d))^(1/2)*a^3*b*d*e^3+864*A*(b*(a*e-b*d))^(1/2)*a^
2*b^2*d^2*e^2+255*A*(b*(a*e-b*d))^(1/2)*a*b^3*d^3*e-272*B*(b*(a*e-b*d))^(1/2)*a^3*b*d^2*e^2-659*B*(b*(a*e-b*d)
)^(1/2)*a^2*b^2*d^3*e+945*A*(b*(a*e-b*d))^(1/2)*b^4*e^4*x^4+40*B*(b*(a*e-b*d))^(1/2)*a^4*e^4*x-60*B*(b*(a*e-b*
d))^(1/2)*b^4*d^4*x+16*B*(b*(a*e-b*d))^(1/2)*a^4*d*e^3-30*B*(b*(a*e-b*d))^(1/2)*a*b^3*d^4+1890*A*arctan(b*(e*x
+d)^(1/2)/(b*(a*e-b*d))^(1/2))*(e*x+d)^(5/2)*a*b^4*e^2*x-1050*B*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*(e
*x+d)^(5/2)*a^2*b^3*e^2*x-420*B*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*(e*x+d)^(5/2)*a^2*b^3*d*e-1925*B*(
b*(a*e-b*d))^(1/2)*a*b^3*d*e^3*x^3+3717*A*(b*(a*e-b*d))^(1/2)*a*b^3*d*e^3*x^2-2289*B*(b*(a*e-b*d))^(1/2)*a^2*b
^2*d*e^3*x^2-2457*B*(b*(a*e-b*d))^(1/2)*a*b^3*d^2*e^2*x^2+1224*A*(b*(a*e-b*d))^(1/2)*a^2*b^2*d*e^3*x+2493*A*(b
*(a*e-b*d))^(1/2)*a*b^3*d^2*e^2*x)*(b*x+a)/(b*(a*e-b*d))^(1/2)/(e*x+d)^(5/2)/(a*e-b*d)^5/((b*x+a)^2)^(3/2)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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)^(3/2),x, algorithm="maxima")

[Out]

integrate((B*x + A)/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(x*e + d)^(7/2)), x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1305 vs. \(2 (343) = 686\).
time = 3.11, size = 2621, normalized size = 6.33 \begin {gather*} \text {Too large to display} \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)^(3/2),x, algorithm="fricas")

[Out]

[1/120*(105*(((5*B*a*b^3 - 9*A*b^4)*x^5 + 2*(5*B*a^2*b^2 - 9*A*a*b^3)*x^4 + (5*B*a^3*b - 9*A*a^2*b^2)*x^3)*e^5
+ (4*B*b^4*d*x^5 + (23*B*a*b^3 - 27*A*b^4)*d*x^4 + 2*(17*B*a^2*b^2 - 27*A*a*b^3)*d*x^3 + 3*(5*B*a^3*b - 9*A*a
^2*b^2)*d*x^2)*e^4 + 3*(4*B*b^4*d^2*x^4 + (13*B*a*b^3 - 9*A*b^4)*d^2*x^3 + 2*(7*B*a^2*b^2 - 9*A*a*b^3)*d^2*x^2
+ (5*B*a^3*b - 9*A*a^2*b^2)*d^2*x)*e^3 + (12*B*b^4*d^3*x^3 + (29*B*a*b^3 - 9*A*b^4)*d^3*x^2 + 2*(11*B*a^2*b^2
- 9*A*a*b^3)*d^3*x + (5*B*a^3*b - 9*A*a^2*b^2)*d^3)*e^2 + 4*(B*b^4*d^4*x^2 + 2*B*a*b^3*d^4*x + B*a^2*b^2*d^4)
*e)*sqrt(b/(b*d - a*e))*log((2*b*d + 2*(b*d - a*e)*sqrt(x*e + d)*sqrt(b/(b*d - a*e)) + (b*x - a)*e)/(b*x + a))
- 2*(60*B*b^4*d^4*x + 30*(B*a*b^3 + A*b^4)*d^4 - (24*A*a^4 - 105*(5*B*a*b^3 - 9*A*b^4)*x^4 - 175*(5*B*a^2*b^2
- 9*A*a*b^3)*x^3 - 56*(5*B*a^3*b - 9*A*a^2*b^2)*x^2 + 8*(5*B*a^4 - 9*A*a^3*b)*x)*e^4 + (420*B*b^4*d*x^4 + 35*
(55*B*a*b^3 - 63*A*b^4)*d*x^3 + 21*(109*B*a^2*b^2 - 177*A*a*b^3)*d*x^2 + 72*(9*B*a^3*b - 17*A*a^2*b^2)*d*x - 8
*(2*B*a^4 - 21*A*a^3*b)*d)*e^3 + (980*B*b^4*d^2*x^3 + 63*(39*B*a*b^3 - 23*A*b^4)*d^2*x^2 + 3*(643*B*a^2*b^2 -
831*A*a*b^3)*d^2*x + 16*(17*B*a^3*b - 54*A*a^2*b^2)*d^2)*e^2 + (644*B*b^4*d^3*x^2 + (1183*B*a*b^3 - 135*A*b^4)
*d^3*x + (659*B*a^2*b^2 - 255*A*a*b^3)*d^3)*e)*sqrt(x*e + d))/(b^7*d^8*x^2 + 2*a*b^6*d^8*x + a^2*b^5*d^8 - (a^
5*b^2*x^5 + 2*a^6*b*x^4 + a^7*x^3)*e^8 + (5*a^4*b^3*d*x^5 + 7*a^5*b^2*d*x^4 - a^6*b*d*x^3 - 3*a^7*d*x^2)*e^7 -
(10*a^3*b^4*d^2*x^5 + 5*a^4*b^3*d^2*x^4 - 17*a^5*b^2*d^2*x^3 - 9*a^6*b*d^2*x^2 + 3*a^7*d^2*x)*e^6 + (10*a^2*b
^5*d^3*x^5 - 10*a^3*b^4*d^3*x^4 - 35*a^4*b^3*d^3*x^3 - a^5*b^2*d^3*x^2 + 13*a^6*b*d^3*x - a^7*d^3)*e^5 - 5*(a*
b^6*d^4*x^5 - 4*a^2*b^5*d^4*x^4 - 5*a^3*b^4*d^4*x^3 + 5*a^4*b^3*d^4*x^2 + 4*a^5*b^2*d^4*x - a^6*b*d^4)*e^4 + (
b^7*d^5*x^5 - 13*a*b^6*d^5*x^4 + a^2*b^5*d^5*x^3 + 35*a^3*b^4*d^5*x^2 + 10*a^4*b^3*d^5*x - 10*a^5*b^2*d^5)*e^3
+ (3*b^7*d^6*x^4 - 9*a*b^6*d^6*x^3 - 17*a^2*b^5*d^6*x^2 + 5*a^3*b^4*d^6*x + 10*a^4*b^3*d^6)*e^2 + (3*b^7*d^7*
x^3 + a*b^6*d^7*x^2 - 7*a^2*b^5*d^7*x - 5*a^3*b^4*d^7)*e), 1/60*(105*(((5*B*a*b^3 - 9*A*b^4)*x^5 + 2*(5*B*a^2*
b^2 - 9*A*a*b^3)*x^4 + (5*B*a^3*b - 9*A*a^2*b^2)*x^3)*e^5 + (4*B*b^4*d*x^5 + (23*B*a*b^3 - 27*A*b^4)*d*x^4 + 2
*(17*B*a^2*b^2 - 27*A*a*b^3)*d*x^3 + 3*(5*B*a^3*b - 9*A*a^2*b^2)*d*x^2)*e^4 + 3*(4*B*b^4*d^2*x^4 + (13*B*a*b^3
- 9*A*b^4)*d^2*x^3 + 2*(7*B*a^2*b^2 - 9*A*a*b^3)*d^2*x^2 + (5*B*a^3*b - 9*A*a^2*b^2)*d^2*x)*e^3 + (12*B*b^4*d
^3*x^3 + (29*B*a*b^3 - 9*A*b^4)*d^3*x^2 + 2*(11*B*a^2*b^2 - 9*A*a*b^3)*d^3*x + (5*B*a^3*b - 9*A*a^2*b^2)*d^3)*
e^2 + 4*(B*b^4*d^4*x^2 + 2*B*a*b^3*d^4*x + B*a^2*b^2*d^4)*e)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a*e)*sqrt(x*e
+ d)*sqrt(-b/(b*d - a*e))/(b*x*e + b*d)) - (60*B*b^4*d^4*x + 30*(B*a*b^3 + A*b^4)*d^4 - (24*A*a^4 - 105*(5*B*
a*b^3 - 9*A*b^4)*x^4 - 175*(5*B*a^2*b^2 - 9*A*a*b^3)*x^3 - 56*(5*B*a^3*b - 9*A*a^2*b^2)*x^2 + 8*(5*B*a^4 - 9*A
*a^3*b)*x)*e^4 + (420*B*b^4*d*x^4 + 35*(55*B*a*b^3 - 63*A*b^4)*d*x^3 + 21*(109*B*a^2*b^2 - 177*A*a*b^3)*d*x^2
+ 72*(9*B*a^3*b - 17*A*a^2*b^2)*d*x - 8*(2*B*a^4 - 21*A*a^3*b)*d)*e^3 + (980*B*b^4*d^2*x^3 + 63*(39*B*a*b^3 -
23*A*b^4)*d^2*x^2 + 3*(643*B*a^2*b^2 - 831*A*a*b^3)*d^2*x + 16*(17*B*a^3*b - 54*A*a^2*b^2)*d^2)*e^2 + (644*B*b
^4*d^3*x^2 + (1183*B*a*b^3 - 135*A*b^4)*d^3*x + (659*B*a^2*b^2 - 255*A*a*b^3)*d^3)*e)*sqrt(x*e + d))/(b^7*d^8*
x^2 + 2*a*b^6*d^8*x + a^2*b^5*d^8 - (a^5*b^2*x^5 + 2*a^6*b*x^4 + a^7*x^3)*e^8 + (5*a^4*b^3*d*x^5 + 7*a^5*b^2*d
*x^4 - a^6*b*d*x^3 - 3*a^7*d*x^2)*e^7 - (10*a^3*b^4*d^2*x^5 + 5*a^4*b^3*d^2*x^4 - 17*a^5*b^2*d^2*x^3 - 9*a^6*b
*d^2*x^2 + 3*a^7*d^2*x)*e^6 + (10*a^2*b^5*d^3*x^5 - 10*a^3*b^4*d^3*x^4 - 35*a^4*b^3*d^3*x^3 - a^5*b^2*d^3*x^2
+ 13*a^6*b*d^3*x - a^7*d^3)*e^5 - 5*(a*b^6*d^4*x^5 - 4*a^2*b^5*d^4*x^4 - 5*a^3*b^4*d^4*x^3 + 5*a^4*b^3*d^4*x^2
+ 4*a^5*b^2*d^4*x - a^6*b*d^4)*e^4 + (b^7*d^5*x^5 - 13*a*b^6*d^5*x^4 + a^2*b^5*d^5*x^3 + 35*a^3*b^4*d^5*x^2 +
10*a^4*b^3*d^5*x - 10*a^5*b^2*d^5)*e^3 + (3*b^7*d^6*x^4 - 9*a*b^6*d^6*x^3 - 17*a^2*b^5*d^6*x^2 + 5*a^3*b^4*d^
6*x + 10*a^4*b^3*d^6)*e^2 + (3*b^7*d^7*x^3 + a*b^6*d^7*x^2 - 7*a^2*b^5*d^7*x - 5*a^3*b^4*d^7)*e)]

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x}{\left (d + e x\right )^{\frac {7}{2}} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \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)**(3/2),x)

[Out]

Integral((A + B*x)/((d + e*x)**(7/2)*((a + b*x)**2)**(3/2)), x)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 717 vs. \(2 (343) = 686\).
time = 1.04, size = 717, normalized size = 1.73 \begin {gather*} -\frac {7 \, {\left (4 \, B b^{3} d e + 5 \, B a b^{2} e^{2} - 9 \, A b^{3} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{4 \, {\left (b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) - a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} \sqrt {-b^{2} d + a b e}} - \frac {4 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{4} d e - 4 \, \sqrt {x e + d} B b^{4} d^{2} e + 11 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{3} e^{2} - 15 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{4} e^{2} - 9 \, \sqrt {x e + d} B a b^{3} d e^{2} + 17 \, \sqrt {x e + d} A b^{4} d e^{2} + 13 \, \sqrt {x e + d} B a^{2} b^{2} e^{3} - 17 \, \sqrt {x e + d} A a b^{3} e^{3}}{4 \, {\left (b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) - a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{2}} - \frac {2 \, {\left (45 \, {\left (x e + d\right )}^{2} B b^{2} d e + 10 \, {\left (x e + d\right )} B b^{2} d^{2} e + 3 \, B b^{2} d^{3} e + 45 \, {\left (x e + d\right )}^{2} B a b e^{2} - 90 \, {\left (x e + d\right )}^{2} A b^{2} e^{2} - 5 \, {\left (x e + d\right )} B a b d e^{2} - 15 \, {\left (x e + d\right )} A b^{2} d e^{2} - 6 \, B a b d^{2} e^{2} - 3 \, A b^{2} d^{2} e^{2} - 5 \, {\left (x e + d\right )} B a^{2} e^{3} + 15 \, {\left (x e + d\right )} A a b e^{3} + 3 \, B a^{2} d e^{3} + 6 \, A a b d e^{3} - 3 \, A a^{2} e^{4}\right )}}{15 \, {\left (b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) - a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} {\left (x e + d\right )}^{\frac {5}{2}}} \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)^(3/2),x, algorithm="giac")

[Out]

-7/4*(4*B*b^3*d*e + 5*B*a*b^2*e^2 - 9*A*b^3*e^2)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^5*d^5*sgn(b*
x + a) - 5*a*b^4*d^4*e*sgn(b*x + a) + 10*a^2*b^3*d^3*e^2*sgn(b*x + a) - 10*a^3*b^2*d^2*e^3*sgn(b*x + a) + 5*a^
4*b*d*e^4*sgn(b*x + a) - a^5*e^5*sgn(b*x + a))*sqrt(-b^2*d + a*b*e)) - 1/4*(4*(x*e + d)^(3/2)*B*b^4*d*e - 4*sq
rt(x*e + d)*B*b^4*d^2*e + 11*(x*e + d)^(3/2)*B*a*b^3*e^2 - 15*(x*e + d)^(3/2)*A*b^4*e^2 - 9*sqrt(x*e + d)*B*a*
b^3*d*e^2 + 17*sqrt(x*e + d)*A*b^4*d*e^2 + 13*sqrt(x*e + d)*B*a^2*b^2*e^3 - 17*sqrt(x*e + d)*A*a*b^3*e^3)/((b^
5*d^5*sgn(b*x + a) - 5*a*b^4*d^4*e*sgn(b*x + a) + 10*a^2*b^3*d^3*e^2*sgn(b*x + a) - 10*a^3*b^2*d^2*e^3*sgn(b*x
+ a) + 5*a^4*b*d*e^4*sgn(b*x + a) - a^5*e^5*sgn(b*x + a))*((x*e + d)*b - b*d + a*e)^2) - 2/15*(45*(x*e + d)^2
*B*b^2*d*e + 10*(x*e + d)*B*b^2*d^2*e + 3*B*b^2*d^3*e + 45*(x*e + d)^2*B*a*b*e^2 - 90*(x*e + d)^2*A*b^2*e^2 -
5*(x*e + d)*B*a*b*d*e^2 - 15*(x*e + d)*A*b^2*d*e^2 - 6*B*a*b*d^2*e^2 - 3*A*b^2*d^2*e^2 - 5*(x*e + d)*B*a^2*e^3
+ 15*(x*e + d)*A*a*b*e^3 + 3*B*a^2*d*e^3 + 6*A*a*b*d*e^3 - 3*A*a^2*e^4)/((b^5*d^5*sgn(b*x + a) - 5*a*b^4*d^4*
e*sgn(b*x + a) + 10*a^2*b^3*d^3*e^2*sgn(b*x + a) - 10*a^3*b^2*d^2*e^3*sgn(b*x + a) + 5*a^4*b*d*e^4*sgn(b*x + a
) - a^5*e^5*sgn(b*x + a))*(x*e + d)^(5/2))

________________________________________________________________________________________

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 )}^{3/2}} \,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)^(3/2)),x)

[Out]

int((A + B*x)/((d + e*x)^(7/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2)), x)

________________________________________________________________________________________