3.19.81 \(\int \frac {A+B x}{(d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\) [1881]
Optimal. Leaf size=424 \[ -\frac {35 e^2 (8 b B d-9 A b e+a B e)}{192 b (b d-a e)^4 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A b-a B}{4 b (b d-a e) (a+b x)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {8 b B d-9 A b e+a B e}{24 b (b d-a e)^2 (a+b x)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (8 b B d-9 A b e+a B e)}{96 b (b d-a e)^3 (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^3 (8 b B d-9 A b e+a B e) (a+b x)}{64 b (b d-a e)^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^3 (8 b B d-9 A b e+a B e) (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 \sqrt {b} (b d-a e)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
[Out]
35/64*e^3*(-9*A*b*e+B*a*e+8*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
^(1/2)/((b*x+a)^2)^(1/2)-35/192*e^2*(-9*A*b*e+B*a*e+8*B*b*d)/b/(-a*e+b*d)^4/(e*x+d)^(1/2)/((b*x+a)^2)^(1/2)+1/
4*(-A*b+B*a)/b/(-a*e+b*d)/(b*x+a)^3/(e*x+d)^(1/2)/((b*x+a)^2)^(1/2)+1/24*(9*A*b*e-B*a*e-8*B*b*d)/b/(-a*e+b*d)^
2/(b*x+a)^2/(e*x+d)^(1/2)/((b*x+a)^2)^(1/2)+7/96*e*(-9*A*b*e+B*a*e+8*B*b*d)/b/(-a*e+b*d)^3/(b*x+a)/(e*x+d)^(1/
2)/((b*x+a)^2)^(1/2)-35/64*e^3*(-9*A*b*e+B*a*e+8*B*b*d)*(b*x+a)/b/(-a*e+b*d)^5/(e*x+d)^(1/2)/((b*x+a)^2)^(1/2)
________________________________________________________________________________________
Rubi [A]
time = 0.29, antiderivative size = 424, 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 {35 e^3 (a+b x) (a B e-9 A b e+8 b B d)}{64 b \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^5}+\frac {35 e^3 (a+b x) (a B e-9 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 {b} \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{11/2}}-\frac {35 e^2 (a B e-9 A b e+8 b B d)}{192 b \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^4}+\frac {7 e (a B e-9 A b e+8 b B d)}{96 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^3}-\frac {a B e-9 A b e+8 b B d}{24 b (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (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} \sqrt {d+e x} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(A + B*x)/((d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]
[Out]
(-35*e^2*(8*b*B*d - 9*A*b*e + a*B*e))/(192*b*(b*d - a*e)^4*Sqrt[d + e*x]*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*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (8*b*B*d - 9*A*b*e + a*B*e
)/(24*b*(b*d - a*e)^2*(a + b*x)^2*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*e*(8*b*B*d - 9*A*b*e + a*B
*e))/(96*b*(b*d - a*e)^3*(a + b*x)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^3*(8*b*B*d - 9*A*b*e +
a*B*e)*(a + b*x))/(64*b*(b*d - a*e)^5*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (35*e^3*(8*b*B*d - 9*A*b
*e + a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*Sqrt[b]*(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)^{3/2} \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)^{3/2}} \, 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 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (b^2 (8 b B d-9 A b e+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)^{3/2}} \, 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 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {8 b B d-9 A b e+a B e}{24 b (b d-a e)^2 (a+b x)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (7 b e (8 b B d-9 A b e+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)^{3/2}} \, 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 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {8 b B d-9 A b e+a B e}{24 b (b d-a e)^2 (a+b x)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (8 b B d-9 A b e+a B e)}{96 b (b d-a e)^3 (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (35 e^2 (8 b B d-9 A b e+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)^{3/2}} \, dx}{192 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {35 e^2 (8 b B d-9 A b e+a B e)}{192 b (b d-a e)^4 \sqrt {d+e x} \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 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {8 b B d-9 A b e+a B e}{24 b (b d-a e)^2 (a+b x)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (8 b B d-9 A b e+a B e)}{96 b (b d-a e)^3 (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (35 e^3 (8 b B d-9 A b e+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 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {35 e^2 (8 b B d-9 A b e+a B e)}{192 b (b d-a e)^4 \sqrt {d+e x} \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 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {8 b B d-9 A b e+a B e}{24 b (b d-a e)^2 (a+b x)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (8 b B d-9 A b e+a B e)}{96 b (b d-a e)^3 (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^3 (8 b B d-9 A b e+a B e) (a+b x)}{64 b (b d-a e)^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (35 e^3 (8 b B d-9 A b e+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)^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {35 e^2 (8 b B d-9 A b e+a B e)}{192 b (b d-a e)^4 \sqrt {d+e x} \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 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {8 b B d-9 A b e+a B e}{24 b (b d-a e)^2 (a+b x)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (8 b B d-9 A b e+a B e)}{96 b (b d-a e)^3 (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^3 (8 b B d-9 A b e+a B e) (a+b x)}{64 b (b d-a e)^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (35 e^2 (8 b B d-9 A b e+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 )}{64 (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {35 e^2 (8 b B d-9 A b e+a B e)}{192 b (b d-a e)^4 \sqrt {d+e x} \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 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {8 b B d-9 A b e+a B e}{24 b (b d-a e)^2 (a+b x)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (8 b B d-9 A b e+a B e)}{96 b (b d-a e)^3 (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^3 (8 b B d-9 A b e+a B e) (a+b x)}{64 b (b d-a e)^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^3 (8 b B d-9 A b e+a B e) (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 \sqrt {b} (b d-a e)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 2.66, size = 425, normalized size = 1.00 \begin {gather*} \frac {e^3 (a+b x) \left (\frac {3 A \left (-128 a^4 e^4-a^3 b e^3 (325 d+837 e x)-3 a^2 b^2 e^2 \left (-70 d^2+185 d e x+511 e^2 x^2\right )-a b^3 e \left (88 d^3-156 d^2 e x+399 d e^2 x^2+1155 e^3 x^3\right )+b^4 \left (16 d^4-24 d^3 e x+42 d^2 e^2 x^2-105 d e^3 x^3-315 e^4 x^4\right )\right )+B \left (3 a^4 e^3 (221 d+93 e x)+a^3 b e^2 \left (370 d^2+2417 d e x+511 e^2 x^2\right )+8 b^4 d x \left (8 d^3-14 d^2 e x+35 d e^2 x^2+105 e^3 x^3\right )+a^2 b^2 e \left (-104 d^3+1428 d^2 e x+4221 d e^2 x^2+385 e^3 x^3\right )+a b^3 \left (16 d^4-408 d^3 e x+1050 d^2 e^2 x^2+3115 d e^3 x^3+105 e^4 x^4\right )\right )}{e^3 (-b d+a e)^5 (a+b x)^4 \sqrt {d+e x}}+\frac {105 (8 b B d-9 A b e+a B e) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{\sqrt {b} (-b d+a e)^{11/2}}\right )}{192 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)/((d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]
[Out]
(e^3*(a + b*x)*((3*A*(-128*a^4*e^4 - a^3*b*e^3*(325*d + 837*e*x) - 3*a^2*b^2*e^2*(-70*d^2 + 185*d*e*x + 511*e^
2*x^2) - a*b^3*e*(88*d^3 - 156*d^2*e*x + 399*d*e^2*x^2 + 1155*e^3*x^3) + b^4*(16*d^4 - 24*d^3*e*x + 42*d^2*e^2
*x^2 - 105*d*e^3*x^3 - 315*e^4*x^4)) + B*(3*a^4*e^3*(221*d + 93*e*x) + a^3*b*e^2*(370*d^2 + 2417*d*e*x + 511*e
^2*x^2) + 8*b^4*d*x*(8*d^3 - 14*d^2*e*x + 35*d*e^2*x^2 + 105*e^3*x^3) + a^2*b^2*e*(-104*d^3 + 1428*d^2*e*x + 4
221*d*e^2*x^2 + 385*e^3*x^3) + a*b^3*(16*d^4 - 408*d^3*e*x + 1050*d^2*e^2*x^2 + 3115*d*e^3*x^3 + 105*e^4*x^4))
)/(e^3*(-(b*d) + a*e)^5*(a + b*x)^4*Sqrt[d + e*x]) + (105*(8*b*B*d - 9*A*b*e + a*B*e)*ArcTan[(Sqrt[b]*Sqrt[d +
e*x])/Sqrt[-(b*d) + a*e]])/(Sqrt[b]*(-(b*d) + a*e)^(11/2))))/(192*Sqrt[(a + b*x)^2])
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1492\) vs.
\(2(327)=654\).
time = 1.01, size = 1493, normalized size = 3.52
| | |
| method |
result |
size |
| | |
| default |
\(\text {Expression too large to display}\) |
\(1493\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)/(e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x,method=_RETURNVERBOSE)
[Out]
-1/192*(-3360*B*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1/2)*a*b^4*d*e^3*x^3-5040*B*arctan(b*(e*x
+d)^(1/2)/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1/2)*a^2*b^3*d*e^3*x^2-3360*B*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/
2))*(e*x+d)^(1/2)*a^3*b^2*d*e^3*x+384*A*(b*(a*e-b*d))^(1/2)*a^4*e^4-48*A*(b*(a*e-b*d))^(1/2)*b^4*d^4+945*A*arc
tan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1/2)*a^4*b*e^4-2417*B*(b*(a*e-b*d))^(1/2)*a^3*b*d*e^3*x-1428
*B*(b*(a*e-b*d))^(1/2)*a^2*b^2*d^2*e^2*x+408*B*(b*(a*e-b*d))^(1/2)*a*b^3*d^3*e*x-105*B*(b*(a*e-b*d))^(1/2)*a*b
^3*e^4*x^4-840*B*(b*(a*e-b*d))^(1/2)*b^4*d*e^3*x^4+3465*A*(b*(a*e-b*d))^(1/2)*a*b^3*e^4*x^3+315*A*(b*(a*e-b*d)
)^(1/2)*b^4*d*e^3*x^3-385*B*(b*(a*e-b*d))^(1/2)*a^2*b^2*e^4*x^3-280*B*(b*(a*e-b*d))^(1/2)*b^4*d^2*e^2*x^3+4599
*A*(b*(a*e-b*d))^(1/2)*a^2*b^2*e^4*x^2-126*A*(b*(a*e-b*d))^(1/2)*b^4*d^2*e^2*x^2-511*B*(b*(a*e-b*d))^(1/2)*a^3
*b*e^4*x^2+112*B*(b*(a*e-b*d))^(1/2)*b^4*d^3*e*x^2+2511*A*(b*(a*e-b*d))^(1/2)*a^3*b*e^4*x+72*A*(b*(a*e-b*d))^(
1/2)*b^4*d^3*e*x+975*A*(b*(a*e-b*d))^(1/2)*a^3*b*d*e^3-630*A*(b*(a*e-b*d))^(1/2)*a^2*b^2*d^2*e^2+264*A*(b*(a*e
-b*d))^(1/2)*a*b^3*d^3*e-370*B*(b*(a*e-b*d))^(1/2)*a^3*b*d^2*e^2+104*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^4-279*B*(b*(a*e-b*d))^(1/2)*a^4*e^4*x-64*B*(b*(a*e-b*d))^(1/2)*b^4*d^4*x-105*B*
arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1/2)*a^5*e^4-663*B*(b*(a*e-b*d))^(1/2)*a^4*d*e^3-16*B*(b*
(a*e-b*d))^(1/2)*a*b^3*d^4-3115*B*(b*(a*e-b*d))^(1/2)*a*b^3*d*e^3*x^3+1197*A*(b*(a*e-b*d))^(1/2)*a*b^3*d*e^3*x
^2-4221*B*(b*(a*e-b*d))^(1/2)*a^2*b^2*d*e^3*x^2-1050*B*(b*(a*e-b*d))^(1/2)*a*b^3*d^2*e^2*x^2+1665*A*(b*(a*e-b*
d))^(1/2)*a^2*b^2*d*e^3*x-468*A*(b*(a*e-b*d))^(1/2)*a*b^3*d^2*e^2*x-105*B*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))
^(1/2))*(e*x+d)^(1/2)*a*b^4*e^4*x^4-840*B*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1/2)*b^5*d*e^3*
x^4+3780*A*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1/2)*a*b^4*e^4*x^3-420*B*arctan(b*(e*x+d)^(1/2
)/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1/2)*a^2*b^3*e^4*x^3+5670*A*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*(e*x+d
)^(1/2)*a^2*b^3*e^4*x^2-630*B*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1/2)*a^3*b^2*e^4*x^2+3780*A
*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1/2)*a^3*b^2*e^4*x-420*B*arctan(b*(e*x+d)^(1/2)/(b*(a*e-
b*d))^(1/2))*(e*x+d)^(1/2)*a^4*b*e^4*x-840*B*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1/2)*a^4*b*d
*e^3+945*A*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*(e*x+d)^(1/2)*b^5*e^4*x^4)*(b*x+a)/(b*(a*e-b*d))^(1/2)/
(e*x+d)^(1/2)/(a*e-b*d)^5/((b*x+a)^2)^(5/2)
________________________________________________________________________________________
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)^(3/2)/(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)*(x*e + d)^(3/2)), x)
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1439 vs.
\(2 (347) = 694\).
time = 1.43, size = 2893, normalized size = 6.82 \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)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")
[Out]
[1/384*(105*sqrt(b^2*d - a*b*e)*(((B*a*b^4 - 9*A*b^5)*x^5 + 4*(B*a^2*b^3 - 9*A*a*b^4)*x^4 + 6*(B*a^3*b^2 - 9*A
*a^2*b^3)*x^3 + 4*(B*a^4*b - 9*A*a^3*b^2)*x^2 + (B*a^5 - 9*A*a^4*b)*x)*e^5 + (8*B*b^5*d*x^5 + 3*(11*B*a*b^4 -
3*A*b^5)*d*x^4 + 4*(13*B*a^2*b^3 - 9*A*a*b^4)*d*x^3 + 2*(19*B*a^3*b^2 - 27*A*a^2*b^3)*d*x^2 + 12*(B*a^4*b - 3*
A*a^3*b^2)*d*x + (B*a^5 - 9*A*a^4*b)*d)*e^4 + 8*(B*b^5*d^2*x^4 + 4*B*a*b^4*d^2*x^3 + 6*B*a^2*b^3*d^2*x^2 + 4*B
*a^3*b^2*d^2*x + B*a^4*b*d^2)*e^3)*log((2*b*d + (b*x - a)*e + 2*sqrt(b^2*d - a*b*e)*sqrt(x*e + d))/(b*x + a))
- 2*(64*B*b^6*d^5*x + 16*(B*a*b^5 + 3*A*b^6)*d^5 + (384*A*a^5*b - 105*(B*a^2*b^4 - 9*A*a*b^5)*x^4 - 385*(B*a^3
*b^3 - 9*A*a^2*b^4)*x^3 - 511*(B*a^4*b^2 - 9*A*a^3*b^3)*x^2 - 279*(B*a^5*b - 9*A*a^4*b^2)*x)*e^5 - (105*(7*B*a
*b^5 + 9*A*b^6)*d*x^4 + 210*(13*B*a^2*b^4 + 15*A*a*b^5)*d*x^3 + 14*(265*B*a^3*b^3 + 243*A*a^2*b^4)*d*x^2 + 2*(
1069*B*a^4*b^2 + 423*A*a^3*b^3)*d*x + 3*(221*B*a^5*b - 197*A*a^4*b^2)*d)*e^4 + (840*B*b^6*d^2*x^4 + 315*(9*B*a
*b^5 - A*b^6)*d^2*x^3 + 21*(151*B*a^2*b^4 - 63*A*a*b^5)*d^2*x^2 + (989*B*a^3*b^3 - 2133*A*a^2*b^4)*d^2*x + (29
3*B*a^4*b^2 - 1605*A*a^3*b^3)*d^2)*e^3 + 2*(140*B*b^6*d^3*x^3 + 7*(83*B*a*b^5 + 9*A*b^6)*d^3*x^2 + 54*(17*B*a^
2*b^4 + 5*A*a*b^5)*d^3*x + 3*(79*B*a^3*b^3 + 149*A*a^2*b^4)*d^3)*e^2 - 8*(14*B*b^6*d^4*x^2 + (59*B*a*b^5 + 9*A
*b^6)*d^4*x + 3*(5*B*a^2*b^4 + 13*A*a*b^5)*d^4)*e)*sqrt(x*e + d))/(b^11*d^7*x^4 + 4*a*b^10*d^7*x^3 + 6*a^2*b^9
*d^7*x^2 + 4*a^3*b^8*d^7*x + a^4*b^7*d^7 + (a^6*b^5*x^5 + 4*a^7*b^4*x^4 + 6*a^8*b^3*x^3 + 4*a^9*b^2*x^2 + a^10
*b*x)*e^7 - (6*a^5*b^6*d*x^5 + 23*a^6*b^5*d*x^4 + 32*a^7*b^4*d*x^3 + 18*a^8*b^3*d*x^2 + 2*a^9*b^2*d*x - a^10*b
*d)*e^6 + 3*(5*a^4*b^7*d^2*x^5 + 18*a^5*b^6*d^2*x^4 + 22*a^6*b^5*d^2*x^3 + 8*a^7*b^4*d^2*x^2 - 3*a^8*b^3*d^2*x
- 2*a^9*b^2*d^2)*e^5 - 5*(4*a^3*b^8*d^3*x^5 + 13*a^4*b^7*d^3*x^4 + 12*a^5*b^6*d^3*x^3 - 2*a^6*b^5*d^3*x^2 - 8
*a^7*b^4*d^3*x - 3*a^8*b^3*d^3)*e^4 + 5*(3*a^2*b^9*d^4*x^5 + 8*a^3*b^8*d^4*x^4 + 2*a^4*b^7*d^4*x^3 - 12*a^5*b^
6*d^4*x^2 - 13*a^6*b^5*d^4*x - 4*a^7*b^4*d^4)*e^3 - 3*(2*a*b^10*d^5*x^5 + 3*a^2*b^9*d^5*x^4 - 8*a^3*b^8*d^5*x^
3 - 22*a^4*b^7*d^5*x^2 - 18*a^5*b^6*d^5*x - 5*a^6*b^5*d^5)*e^2 + (b^11*d^6*x^5 - 2*a*b^10*d^6*x^4 - 18*a^2*b^9
*d^6*x^3 - 32*a^3*b^8*d^6*x^2 - 23*a^4*b^7*d^6*x - 6*a^5*b^6*d^6)*e), -1/192*(105*sqrt(-b^2*d + a*b*e)*(((B*a*
b^4 - 9*A*b^5)*x^5 + 4*(B*a^2*b^3 - 9*A*a*b^4)*x^4 + 6*(B*a^3*b^2 - 9*A*a^2*b^3)*x^3 + 4*(B*a^4*b - 9*A*a^3*b^
2)*x^2 + (B*a^5 - 9*A*a^4*b)*x)*e^5 + (8*B*b^5*d*x^5 + 3*(11*B*a*b^4 - 3*A*b^5)*d*x^4 + 4*(13*B*a^2*b^3 - 9*A*
a*b^4)*d*x^3 + 2*(19*B*a^3*b^2 - 27*A*a^2*b^3)*d*x^2 + 12*(B*a^4*b - 3*A*a^3*b^2)*d*x + (B*a^5 - 9*A*a^4*b)*d)
*e^4 + 8*(B*b^5*d^2*x^4 + 4*B*a*b^4*d^2*x^3 + 6*B*a^2*b^3*d^2*x^2 + 4*B*a^3*b^2*d^2*x + B*a^4*b*d^2)*e^3)*arct
an(sqrt(-b^2*d + a*b*e)*sqrt(x*e + d)/(b*x*e + b*d)) + (64*B*b^6*d^5*x + 16*(B*a*b^5 + 3*A*b^6)*d^5 + (384*A*a
^5*b - 105*(B*a^2*b^4 - 9*A*a*b^5)*x^4 - 385*(B*a^3*b^3 - 9*A*a^2*b^4)*x^3 - 511*(B*a^4*b^2 - 9*A*a^3*b^3)*x^2
- 279*(B*a^5*b - 9*A*a^4*b^2)*x)*e^5 - (105*(7*B*a*b^5 + 9*A*b^6)*d*x^4 + 210*(13*B*a^2*b^4 + 15*A*a*b^5)*d*x
^3 + 14*(265*B*a^3*b^3 + 243*A*a^2*b^4)*d*x^2 + 2*(1069*B*a^4*b^2 + 423*A*a^3*b^3)*d*x + 3*(221*B*a^5*b - 197*
A*a^4*b^2)*d)*e^4 + (840*B*b^6*d^2*x^4 + 315*(9*B*a*b^5 - A*b^6)*d^2*x^3 + 21*(151*B*a^2*b^4 - 63*A*a*b^5)*d^2
*x^2 + (989*B*a^3*b^3 - 2133*A*a^2*b^4)*d^2*x + (293*B*a^4*b^2 - 1605*A*a^3*b^3)*d^2)*e^3 + 2*(140*B*b^6*d^3*x
^3 + 7*(83*B*a*b^5 + 9*A*b^6)*d^3*x^2 + 54*(17*B*a^2*b^4 + 5*A*a*b^5)*d^3*x + 3*(79*B*a^3*b^3 + 149*A*a^2*b^4)
*d^3)*e^2 - 8*(14*B*b^6*d^4*x^2 + (59*B*a*b^5 + 9*A*b^6)*d^4*x + 3*(5*B*a^2*b^4 + 13*A*a*b^5)*d^4)*e)*sqrt(x*e
+ d))/(b^11*d^7*x^4 + 4*a*b^10*d^7*x^3 + 6*a^2*b^9*d^7*x^2 + 4*a^3*b^8*d^7*x + a^4*b^7*d^7 + (a^6*b^5*x^5 + 4
*a^7*b^4*x^4 + 6*a^8*b^3*x^3 + 4*a^9*b^2*x^2 + a^10*b*x)*e^7 - (6*a^5*b^6*d*x^5 + 23*a^6*b^5*d*x^4 + 32*a^7*b^
4*d*x^3 + 18*a^8*b^3*d*x^2 + 2*a^9*b^2*d*x - a^10*b*d)*e^6 + 3*(5*a^4*b^7*d^2*x^5 + 18*a^5*b^6*d^2*x^4 + 22*a^
6*b^5*d^2*x^3 + 8*a^7*b^4*d^2*x^2 - 3*a^8*b^3*d^2*x - 2*a^9*b^2*d^2)*e^5 - 5*(4*a^3*b^8*d^3*x^5 + 13*a^4*b^7*d
^3*x^4 + 12*a^5*b^6*d^3*x^3 - 2*a^6*b^5*d^3*x^2 - 8*a^7*b^4*d^3*x - 3*a^8*b^3*d^3)*e^4 + 5*(3*a^2*b^9*d^4*x^5
+ 8*a^3*b^8*d^4*x^4 + 2*a^4*b^7*d^4*x^3 - 12*a^5*b^6*d^4*x^2 - 13*a^6*b^5*d^4*x - 4*a^7*b^4*d^4)*e^3 - 3*(2*a*
b^10*d^5*x^5 + 3*a^2*b^9*d^5*x^4 - 8*a^3*b^8*d^5*x^3 - 22*a^4*b^7*d^5*x^2 - 18*a^5*b^6*d^5*x - 5*a^6*b^5*d^5)*
e^2 + (b^11*d^6*x^5 - 2*a*b^10*d^6*x^4 - 18*a^2*b^9*d^6*x^3 - 32*a^3*b^8*d^6*x^2 - 23*a^4*b^7*d^6*x - 6*a^5*b^
6*d^6)*e)]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
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)**(3/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 844 vs.
\(2 (347) = 694\).
time = 1.32, size = 844, normalized size = 1.99 \begin {gather*} -\frac {35 \, {\left (8 \, B b d e^{3} + B a e^{4} - 9 \, A b e^{4}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, {\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 {2 \, {\left (B d e^{3} - A e^{4}\right )}}{{\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 {x e + d}} - \frac {456 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{4} d e^{3} - 1544 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{4} d^{2} e^{3} + 1784 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{4} d^{3} e^{3} - 696 \, \sqrt {x e + d} B b^{4} d^{4} e^{3} + 105 \, {\left (x e + d\right )}^{\frac {7}{2}} B a b^{3} e^{4} - 561 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{4} e^{4} + 1159 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{3} d e^{4} + 1929 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{4} d e^{4} - 3057 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{3} d^{2} e^{4} - 2295 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{4} d^{2} e^{4} + 1809 \, \sqrt {x e + d} B a b^{3} d^{3} e^{4} + 975 \, \sqrt {x e + d} A b^{4} d^{3} e^{4} + 385 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} b^{2} e^{5} - 1929 \, {\left (x e + d\right )}^{\frac {5}{2}} A a b^{3} e^{5} + 762 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{2} d e^{5} + 4590 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{3} d e^{5} - 1251 \, \sqrt {x e + d} B a^{2} b^{2} d^{2} e^{5} - 2925 \, \sqrt {x e + d} A a b^{3} d^{2} e^{5} + 511 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b e^{6} - 2295 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{2} e^{6} - 141 \, \sqrt {x e + d} B a^{3} b d e^{6} + 2925 \, \sqrt {x e + d} A a^{2} b^{2} d e^{6} + 279 \, \sqrt {x e + d} B a^{4} e^{7} - 975 \, \sqrt {x e + d} A a^{3} b e^{7}}{192 \, {\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 )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")
[Out]
-35/64*(8*B*b*d*e^3 + B*a*e^4 - 9*A*b*e^4)*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)) - 2*(B*d*e^3 - A*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))*sqrt(x*e + d)) - 1/192*(456*(x*e + d)^(7/2)*B*b^4*d*e^3 - 1544*(x*e + d)
^(5/2)*B*b^4*d^2*e^3 + 1784*(x*e + d)^(3/2)*B*b^4*d^3*e^3 - 696*sqrt(x*e + d)*B*b^4*d^4*e^3 + 105*(x*e + d)^(7
/2)*B*a*b^3*e^4 - 561*(x*e + d)^(7/2)*A*b^4*e^4 + 1159*(x*e + d)^(5/2)*B*a*b^3*d*e^4 + 1929*(x*e + d)^(5/2)*A*
b^4*d*e^4 - 3057*(x*e + d)^(3/2)*B*a*b^3*d^2*e^4 - 2295*(x*e + d)^(3/2)*A*b^4*d^2*e^4 + 1809*sqrt(x*e + d)*B*a
*b^3*d^3*e^4 + 975*sqrt(x*e + d)*A*b^4*d^3*e^4 + 385*(x*e + d)^(5/2)*B*a^2*b^2*e^5 - 1929*(x*e + d)^(5/2)*A*a*
b^3*e^5 + 762*(x*e + d)^(3/2)*B*a^2*b^2*d*e^5 + 4590*(x*e + d)^(3/2)*A*a*b^3*d*e^5 - 1251*sqrt(x*e + d)*B*a^2*
b^2*d^2*e^5 - 2925*sqrt(x*e + d)*A*a*b^3*d^2*e^5 + 511*(x*e + d)^(3/2)*B*a^3*b*e^6 - 2295*(x*e + d)^(3/2)*A*a^
2*b^2*e^6 - 141*sqrt(x*e + d)*B*a^3*b*d*e^6 + 2925*sqrt(x*e + d)*A*a^2*b^2*d*e^6 + 279*sqrt(x*e + d)*B*a^4*e^7
- 975*sqrt(x*e + d)*A*a^3*b*e^7)/((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)^4)
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+B\,x}{{\left (d+e\,x\right )}^{3/2}\,{\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)^(3/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)),x)
[Out]
int((A + B*x)/((d + e*x)^(3/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)), x)
________________________________________________________________________________________