3.19.82 \(\int \frac {A+B x}{(d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\) [1882]
Optimal. Leaf size=496 \[ -\frac {21 e^2 (8 b B d-11 A b e+3 a B e)}{64 b (b d-a e)^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A b-a B}{4 b (b d-a e) (a+b x)^3 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {8 b B d-11 A b e+3 a B e}{24 b (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e (8 b B d-11 A b e+3 a B e)}{32 b (b d-a e)^3 (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^3 (8 b B d-11 A b e+3 a B e) (a+b x)}{64 b (b d-a e)^5 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 (8 b B d-11 A b e+3 a B e) (a+b x)}{64 (b d-a e)^6 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {105 \sqrt {b} e^3 (8 b B d-11 A b e+3 a B e) (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 (b d-a e)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
[Out]
-21/64*e^2*(-11*A*b*e+3*B*a*e+8*B*b*d)/b/(-a*e+b*d)^4/(e*x+d)^(3/2)/((b*x+a)^2)^(1/2)+1/4*(-A*b+B*a)/b/(-a*e+b
*d)/(b*x+a)^3/(e*x+d)^(3/2)/((b*x+a)^2)^(1/2)+1/24*(11*A*b*e-3*B*a*e-8*B*b*d)/b/(-a*e+b*d)^2/(b*x+a)^2/(e*x+d)
^(3/2)/((b*x+a)^2)^(1/2)+3/32*e*(-11*A*b*e+3*B*a*e+8*B*b*d)/b/(-a*e+b*d)^3/(b*x+a)/(e*x+d)^(3/2)/((b*x+a)^2)^(
1/2)-35/64*e^3*(-11*A*b*e+3*B*a*e+8*B*b*d)*(b*x+a)/b/(-a*e+b*d)^5/(e*x+d)^(3/2)/((b*x+a)^2)^(1/2)+105/64*e^3*(
-11*A*b*e+3*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))*b^(1/2)/(-a*e+b*d)^(13/2)/(
(b*x+a)^2)^(1/2)-105/64*e^3*(-11*A*b*e+3*B*a*e+8*B*b*d)*(b*x+a)/(-a*e+b*d)^6/(e*x+d)^(1/2)/((b*x+a)^2)^(1/2)
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Rubi [A]
time = 0.36, antiderivative size = 496, normalized size of antiderivative = 1.00, number of steps
used = 9, 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 {105 e^3 (a+b x) (3 a B e-11 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)^6}-\frac {35 e^3 (a+b x) (3 a B e-11 A b e+8 b B d)}{64 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5}+\frac {105 \sqrt {b} e^3 (a+b x) (3 a B e-11 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)^{13/2}}-\frac {21 e^2 (3 a B e-11 A b e+8 b B d)}{64 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}+\frac {3 e (3 a B e-11 A b e+8 b B d)}{32 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}-\frac {3 a B e-11 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)^{3/2} (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)^{3/2} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(A + B*x)/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]
[Out]
(-21*e^2*(8*b*B*d - 11*A*b*e + 3*a*B*e))/(64*b*(b*d - a*e)^4*(d + e*x)^(3/2)*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)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (8*b*B*d - 11*A*b*e
+ 3*a*B*e)/(24*b*(b*d - a*e)^2*(a + b*x)^2*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*e*(8*b*B*d - 11
*A*b*e + 3*a*B*e))/(32*b*(b*d - a*e)^3*(a + b*x)*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*e^3*(8*b
*B*d - 11*A*b*e + 3*a*B*e)*(a + b*x))/(64*b*(b*d - a*e)^5*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (10
5*e^3*(8*b*B*d - 11*A*b*e + 3*a*B*e)*(a + b*x))/(64*(b*d - a*e)^6*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
+ (105*Sqrt[b]*e^3*(8*b*B*d - 11*A*b*e + 3*a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])
/(64*(b*d - a*e)^(13/2)*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)^{5/2} \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)^{5/2}} \, 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)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (b^2 (8 b B d-11 A b e+3 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)^{5/2}} \, 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)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {8 b B d-11 A b e+3 a B e}{24 b (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (3 b e (8 b B d-11 A b e+3 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)^{5/2}} \, dx}{16 (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)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {8 b B d-11 A b e+3 a B e}{24 b (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e (8 b B d-11 A b e+3 a B e)}{32 b (b d-a e)^3 (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (21 e^2 (8 b B d-11 A b e+3 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)^{5/2}} \, dx}{64 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {21 e^2 (8 b B d-11 A b e+3 a B e)}{64 b (b d-a e)^4 (d+e x)^{3/2} \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)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {8 b B d-11 A b e+3 a B e}{24 b (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e (8 b B d-11 A b e+3 a B e)}{32 b (b d-a e)^3 (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (105 e^3 (8 b B d-11 A b e+3 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 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {21 e^2 (8 b B d-11 A b e+3 a B e)}{64 b (b d-a e)^4 (d+e x)^{3/2} \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)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {8 b B d-11 A b e+3 a B e}{24 b (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e (8 b B d-11 A b e+3 a B e)}{32 b (b d-a e)^3 (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^3 (8 b B d-11 A b e+3 a B e) (a+b x)}{64 b (b d-a e)^5 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (105 e^3 (8 b B d-11 A b e+3 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)^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {21 e^2 (8 b B d-11 A b e+3 a B e)}{64 b (b d-a e)^4 (d+e x)^{3/2} \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)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {8 b B d-11 A b e+3 a B e}{24 b (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e (8 b B d-11 A b e+3 a B e)}{32 b (b d-a e)^3 (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^3 (8 b B d-11 A b e+3 a B e) (a+b x)}{64 b (b d-a e)^5 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 (8 b B d-11 A b e+3 a B e) (a+b x)}{64 (b d-a e)^6 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (105 b e^3 (8 b B d-11 A b e+3 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)^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {21 e^2 (8 b B d-11 A b e+3 a B e)}{64 b (b d-a e)^4 (d+e x)^{3/2} \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)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {8 b B d-11 A b e+3 a B e}{24 b (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e (8 b B d-11 A b e+3 a B e)}{32 b (b d-a e)^3 (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^3 (8 b B d-11 A b e+3 a B e) (a+b x)}{64 b (b d-a e)^5 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 (8 b B d-11 A b e+3 a B e) (a+b x)}{64 (b d-a e)^6 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (105 b e^2 (8 b B d-11 A b e+3 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)^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {21 e^2 (8 b B d-11 A b e+3 a B e)}{64 b (b d-a e)^4 (d+e x)^{3/2} \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)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {8 b B d-11 A b e+3 a B e}{24 b (b d-a e)^2 (a+b x)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e (8 b B d-11 A b e+3 a B e)}{32 b (b d-a e)^3 (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^3 (8 b B d-11 A b e+3 a B e) (a+b x)}{64 b (b d-a e)^5 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 (8 b B d-11 A b e+3 a B e) (a+b x)}{64 (b d-a e)^6 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {105 \sqrt {b} e^3 (8 b B d-11 A b e+3 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)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 3.11, size = 559, normalized size = 1.13 \begin {gather*} \frac {e^3 (a+b x) \left (\frac {-B \left (128 a^5 e^4 (2 d+3 e x)+a^4 b e^3 \left (2639 d^2+4510 d e x+2511 e^2 x^2\right )+a^3 b^2 e^2 \left (690 d^3+10331 d^2 e x+12960 d e^2 x^2+4599 e^3 x^3\right )+8 b^5 d x \left (8 d^4-18 d^3 e x+63 d^2 e^2 x^2+420 d e^3 x^3+315 e^4 x^4\right )+a^2 b^3 e \left (-136 d^4+2556 d^3 e x+17433 d^2 e^2 x^2+16926 d e^3 x^3+3465 e^4 x^4\right )+a b^4 \left (16 d^5-520 d^4 e x+1890 d^3 e^2 x^2+12621 d^2 e^3 x^3+10500 d e^4 x^4+945 e^5 x^5\right )\right )+A \left (-128 a^5 e^5+128 a^4 b e^4 (16 d+11 e x)+a^3 b^2 e^3 \left (2295 d^2+12782 d e x+9207 e^2 x^2\right )+a^2 b^3 e^2 \left (-1030 d^3+3795 d^2 e x+22968 d e^2 x^2+16863 e^3 x^3\right )+a b^4 e \left (328 d^4-748 d^3 e x+2673 d^2 e^2 x^2+17094 d e^3 x^3+12705 e^4 x^4\right )+b^5 \left (-48 d^5+88 d^4 e x-198 d^3 e^2 x^2+693 d^2 e^3 x^3+4620 d e^4 x^4+3465 e^5 x^5\right )\right )}{e^3 (b d-a e)^6 (a+b x)^4 (d+e x)^{3/2}}-\frac {315 \sqrt {b} (8 b B d-11 A b e+3 a B e) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{13/2}}\right )}{192 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]
[Out]
(e^3*(a + b*x)*((-(B*(128*a^5*e^4*(2*d + 3*e*x) + a^4*b*e^3*(2639*d^2 + 4510*d*e*x + 2511*e^2*x^2) + a^3*b^2*e
^2*(690*d^3 + 10331*d^2*e*x + 12960*d*e^2*x^2 + 4599*e^3*x^3) + 8*b^5*d*x*(8*d^4 - 18*d^3*e*x + 63*d^2*e^2*x^2
+ 420*d*e^3*x^3 + 315*e^4*x^4) + a^2*b^3*e*(-136*d^4 + 2556*d^3*e*x + 17433*d^2*e^2*x^2 + 16926*d*e^3*x^3 + 3
465*e^4*x^4) + a*b^4*(16*d^5 - 520*d^4*e*x + 1890*d^3*e^2*x^2 + 12621*d^2*e^3*x^3 + 10500*d*e^4*x^4 + 945*e^5*
x^5))) + A*(-128*a^5*e^5 + 128*a^4*b*e^4*(16*d + 11*e*x) + a^3*b^2*e^3*(2295*d^2 + 12782*d*e*x + 9207*e^2*x^2)
+ a^2*b^3*e^2*(-1030*d^3 + 3795*d^2*e*x + 22968*d*e^2*x^2 + 16863*e^3*x^3) + a*b^4*e*(328*d^4 - 748*d^3*e*x +
2673*d^2*e^2*x^2 + 17094*d*e^3*x^3 + 12705*e^4*x^4) + b^5*(-48*d^5 + 88*d^4*e*x - 198*d^3*e^2*x^2 + 693*d^2*e
^3*x^3 + 4620*d*e^4*x^4 + 3465*e^5*x^5)))/(e^3*(b*d - a*e)^6*(a + b*x)^4*(d + e*x)^(3/2)) - (315*Sqrt[b]*(8*b*
B*d - 11*A*b*e + 3*a*B*e)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(-(b*d) + a*e)^(13/2)))/(192*Sqr
t[(a + b*x)^2])
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1859\) vs.
\(2(383)=766\).
time = 1.02, size = 1860, normalized size = 3.75
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\(\text {Expression too large to display}\) |
\(1860\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)/(e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x,method=_RETURNVERBOSE)
[Out]
-1/192*(15120*B*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*(e*x+d)^(3/2)*a^2*b^4*d*e^3*x^2+10080*B*arctan(b*(
e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*(e*x+d)^(3/2)*a^3*b^3*d*e^3*x+10080*B*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(
1/2))*(e*x+d)^(3/2)*a*b^5*d*e^3*x^3-22968*A*(b*(a*e-b*d))^(1/2)*a^2*b^3*d*e^4*x^2-2673*A*(b*(a*e-b*d))^(1/2)*a
*b^4*d^2*e^3*x^2+12960*B*(b*(a*e-b*d))^(1/2)*a^3*b^2*d*e^4*x^2+17433*B*(b*(a*e-b*d))^(1/2)*a^2*b^3*d^2*e^3*x^2
+1890*B*(b*(a*e-b*d))^(1/2)*a*b^4*d^3*e^2*x^2-12782*A*(b*(a*e-b*d))^(1/2)*a^3*b^2*d*e^4*x-3795*A*(b*(a*e-b*d))
^(1/2)*a^2*b^3*d^2*e^3*x+748*A*(b*(a*e-b*d))^(1/2)*a*b^4*d^3*e^2*x+4510*B*(b*(a*e-b*d))^(1/2)*a^4*b*d*e^4*x+10
331*B*(b*(a*e-b*d))^(1/2)*a^3*b^2*d^2*e^3*x-3465*A*(b*(a*e-b*d))^(1/2)*b^5*e^5*x^5+384*B*(b*(a*e-b*d))^(1/2)*a
^5*e^5*x+64*B*(b*(a*e-b*d))^(1/2)*b^5*d^5*x+256*B*(b*(a*e-b*d))^(1/2)*a^5*d*e^4+16*B*(b*(a*e-b*d))^(1/2)*a*b^4
*d^5-13860*A*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*(e*x+d)^(3/2)*a*b^5*e^4*x^3+3780*B*arctan(b*(e*x+d)^(
1/2)/(b*(a*e-b*d))^(1/2))*(e*x+d)^(3/2)*a^2*b^4*e^4*x^3-20790*A*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*(e
*x+d)^(3/2)*a^2*b^4*e^4*x^2+5670*B*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*(e*x+d)^(3/2)*a^3*b^3*e^4*x^2+9
45*B*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*(e*x+d)^(3/2)*a*b^5*e^4*x^4+2520*B*arctan(b*(e*x+d)^(1/2)/(b*
(a*e-b*d))^(1/2))*(e*x+d)^(3/2)*b^6*d*e^3*x^4+128*A*(b*(a*e-b*d))^(1/2)*a^5*e^5+48*A*(b*(a*e-b*d))^(1/2)*b^5*d
^5+1030*A*(b*(a*e-b*d))^(1/2)*a^2*b^3*d^3*e^2-328*A*(b*(a*e-b*d))^(1/2)*a*b^4*d^4*e+2639*B*(b*(a*e-b*d))^(1/2)
*a^4*b*d^2*e^3+690*B*(b*(a*e-b*d))^(1/2)*a^3*b^2*d^3*e^2-136*B*(b*(a*e-b*d))^(1/2)*a^2*b^3*d^4*e-3465*A*arctan
(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*(e*x+d)^(3/2)*b^6*e^4*x^4+945*B*(b*(a*e-b*d))^(1/2)*a*b^4*e^5*x^5+2520*B
*(b*(a*e-b*d))^(1/2)*b^5*d*e^4*x^5-12705*A*(b*(a*e-b*d))^(1/2)*a*b^4*e^5*x^4-4620*A*(b*(a*e-b*d))^(1/2)*b^5*d*
e^4*x^4+3465*B*(b*(a*e-b*d))^(1/2)*a^2*b^3*e^5*x^4+3360*B*(b*(a*e-b*d))^(1/2)*b^5*d^2*e^3*x^4-16863*A*(b*(a*e-
b*d))^(1/2)*a^2*b^3*e^5*x^3-693*A*(b*(a*e-b*d))^(1/2)*b^5*d^2*e^3*x^3-3465*A*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*
d))^(1/2))*(e*x+d)^(3/2)*a^4*b^2*e^4+4599*B*(b*(a*e-b*d))^(1/2)*a^3*b^2*e^5*x^3+504*B*(b*(a*e-b*d))^(1/2)*b^5*
d^3*e^2*x^3+945*B*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*(e*x+d)^(3/2)*a^5*b*e^4-9207*A*(b*(a*e-b*d))^(1/
2)*a^3*b^2*e^5*x^2+198*A*(b*(a*e-b*d))^(1/2)*b^5*d^3*e^2*x^2+2511*B*(b*(a*e-b*d))^(1/2)*a^4*b*e^5*x^2-144*B*(b
*(a*e-b*d))^(1/2)*b^5*d^4*e*x^2-1408*A*(b*(a*e-b*d))^(1/2)*a^4*b*e^5*x-88*A*(b*(a*e-b*d))^(1/2)*b^5*d^4*e*x-20
48*A*(b*(a*e-b*d))^(1/2)*a^4*b*d*e^4-2295*A*(b*(a*e-b*d))^(1/2)*a^3*b^2*d^2*e^3+2556*B*(b*(a*e-b*d))^(1/2)*a^2
*b^3*d^3*e^2*x-520*B*(b*(a*e-b*d))^(1/2)*a*b^4*d^4*e*x-13860*A*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))*(e*
x+d)^(3/2)*a^3*b^3*e^4*x+10500*B*(b*(a*e-b*d))^(1/2)*a*b^4*d*e^4*x^4+3780*B*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d
))^(1/2))*(e*x+d)^(3/2)*a^4*b^2*e^4*x-17094*A*(b*(a*e-b*d))^(1/2)*a*b^4*d*e^4*x^3+16926*B*(b*(a*e-b*d))^(1/2)*
a^2*b^3*d*e^4*x^3+12621*B*(b*(a*e-b*d))^(1/2)*a*b^4*d^2*e^3*x^3+2520*B*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1
/2))*(e*x+d)^(3/2)*a^4*b^2*d*e^3)*(b*x+a)/(e*x+d)^(3/2)/(b*(a*e-b*d))^(1/2)/(a*e-b*d)^6/((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)^(5/2)/(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)^(5/2)), x)
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1751 vs.
\(2 (407) = 814\).
time = 1.56, size = 3513, normalized size = 7.08 \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)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")
[Out]
[-1/384*(315*(((3*B*a*b^4 - 11*A*b^5)*x^6 + 4*(3*B*a^2*b^3 - 11*A*a*b^4)*x^5 + 6*(3*B*a^3*b^2 - 11*A*a^2*b^3)*
x^4 + 4*(3*B*a^4*b - 11*A*a^3*b^2)*x^3 + (3*B*a^5 - 11*A*a^4*b)*x^2)*e^6 + 2*(4*B*b^5*d*x^6 + (19*B*a*b^4 - 11
*A*b^5)*d*x^5 + 4*(9*B*a^2*b^3 - 11*A*a*b^4)*d*x^4 + 2*(17*B*a^3*b^2 - 33*A*a^2*b^3)*d*x^3 + 4*(4*B*a^4*b - 11
*A*a^3*b^2)*d*x^2 + (3*B*a^5 - 11*A*a^4*b)*d*x)*e^5 + (16*B*b^5*d^2*x^5 + (67*B*a*b^4 - 11*A*b^5)*d^2*x^4 + 4*
(27*B*a^2*b^3 - 11*A*a*b^4)*d^2*x^3 + 2*(41*B*a^3*b^2 - 33*A*a^2*b^3)*d^2*x^2 + 4*(7*B*a^4*b - 11*A*a^3*b^2)*d
^2*x + (3*B*a^5 - 11*A*a^4*b)*d^2)*e^4 + 8*(B*b^5*d^3*x^4 + 4*B*a*b^4*d^3*x^3 + 6*B*a^2*b^3*d^3*x^2 + 4*B*a^3*
b^2*d^3*x + B*a^4*b*d^3)*e^3)*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*(64*B*b^5*d^5*x + 16*(B*a*b^4 + 3*A*b^5)*d^5 + (128*A*a^5 + 315*(3*B*a*b^4 - 11
*A*b^5)*x^5 + 1155*(3*B*a^2*b^3 - 11*A*a*b^4)*x^4 + 1533*(3*B*a^3*b^2 - 11*A*a^2*b^3)*x^3 + 837*(3*B*a^4*b - 1
1*A*a^3*b^2)*x^2 + 128*(3*B*a^5 - 11*A*a^4*b)*x)*e^5 + 2*(1260*B*b^5*d*x^5 + 210*(25*B*a*b^4 - 11*A*b^5)*d*x^4
+ 21*(403*B*a^2*b^3 - 407*A*a*b^4)*d*x^3 + 36*(180*B*a^3*b^2 - 319*A*a^2*b^3)*d*x^2 + 11*(205*B*a^4*b - 581*A
*a^3*b^2)*d*x + 128*(B*a^5 - 8*A*a^4*b)*d)*e^4 + (3360*B*b^5*d^2*x^4 + 21*(601*B*a*b^4 - 33*A*b^5)*d^2*x^3 + 9
*(1937*B*a^2*b^3 - 297*A*a*b^4)*d^2*x^2 + (10331*B*a^3*b^2 - 3795*A*a^2*b^3)*d^2*x + (2639*B*a^4*b - 2295*A*a^
3*b^2)*d^2)*e^3 + 2*(252*B*b^5*d^3*x^3 + 9*(105*B*a*b^4 + 11*A*b^5)*d^3*x^2 + 2*(639*B*a^2*b^3 + 187*A*a*b^4)*
d^3*x + 5*(69*B*a^3*b^2 + 103*A*a^2*b^3)*d^3)*e^2 - 8*(18*B*b^5*d^4*x^2 + (65*B*a*b^4 + 11*A*b^5)*d^4*x + (17*
B*a^2*b^3 + 41*A*a*b^4)*d^4)*e)*sqrt(x*e + d))/(b^10*d^8*x^4 + 4*a*b^9*d^8*x^3 + 6*a^2*b^8*d^8*x^2 + 4*a^3*b^7
*d^8*x + a^4*b^6*d^8 + (a^6*b^4*x^6 + 4*a^7*b^3*x^5 + 6*a^8*b^2*x^4 + 4*a^9*b*x^3 + a^10*x^2)*e^8 - 2*(3*a^5*b
^5*d*x^6 + 11*a^6*b^4*d*x^5 + 14*a^7*b^3*d*x^4 + 6*a^8*b^2*d*x^3 - a^9*b*d*x^2 - a^10*d*x)*e^7 + (15*a^4*b^6*d
^2*x^6 + 48*a^5*b^5*d^2*x^5 + 43*a^6*b^4*d^2*x^4 - 8*a^7*b^3*d^2*x^3 - 27*a^8*b^2*d^2*x^2 - 8*a^9*b*d^2*x + a^
10*d^2)*e^6 - 2*(10*a^3*b^7*d^3*x^6 + 25*a^4*b^6*d^3*x^5 + 3*a^5*b^5*d^3*x^4 - 38*a^6*b^4*d^3*x^3 - 32*a^7*b^3
*d^3*x^2 - 3*a^8*b^2*d^3*x + 3*a^9*b*d^3)*e^5 + 5*(3*a^2*b^8*d^4*x^6 + 4*a^3*b^7*d^4*x^5 - 11*a^4*b^6*d^4*x^4
- 24*a^5*b^5*d^4*x^3 - 11*a^6*b^4*d^4*x^2 + 4*a^7*b^3*d^4*x + 3*a^8*b^2*d^4)*e^4 - 2*(3*a*b^9*d^5*x^6 - 3*a^2*
b^8*d^5*x^5 - 32*a^3*b^7*d^5*x^4 - 38*a^4*b^6*d^5*x^3 + 3*a^5*b^5*d^5*x^2 + 25*a^6*b^4*d^5*x + 10*a^7*b^3*d^5)
*e^3 + (b^10*d^6*x^6 - 8*a*b^9*d^6*x^5 - 27*a^2*b^8*d^6*x^4 - 8*a^3*b^7*d^6*x^3 + 43*a^4*b^6*d^6*x^2 + 48*a^5*
b^5*d^6*x + 15*a^6*b^4*d^6)*e^2 + 2*(b^10*d^7*x^5 + a*b^9*d^7*x^4 - 6*a^2*b^8*d^7*x^3 - 14*a^3*b^7*d^7*x^2 - 1
1*a^4*b^6*d^7*x - 3*a^5*b^5*d^7)*e), 1/192*(315*(((3*B*a*b^4 - 11*A*b^5)*x^6 + 4*(3*B*a^2*b^3 - 11*A*a*b^4)*x^
5 + 6*(3*B*a^3*b^2 - 11*A*a^2*b^3)*x^4 + 4*(3*B*a^4*b - 11*A*a^3*b^2)*x^3 + (3*B*a^5 - 11*A*a^4*b)*x^2)*e^6 +
2*(4*B*b^5*d*x^6 + (19*B*a*b^4 - 11*A*b^5)*d*x^5 + 4*(9*B*a^2*b^3 - 11*A*a*b^4)*d*x^4 + 2*(17*B*a^3*b^2 - 33*A
*a^2*b^3)*d*x^3 + 4*(4*B*a^4*b - 11*A*a^3*b^2)*d*x^2 + (3*B*a^5 - 11*A*a^4*b)*d*x)*e^5 + (16*B*b^5*d^2*x^5 + (
67*B*a*b^4 - 11*A*b^5)*d^2*x^4 + 4*(27*B*a^2*b^3 - 11*A*a*b^4)*d^2*x^3 + 2*(41*B*a^3*b^2 - 33*A*a^2*b^3)*d^2*x
^2 + 4*(7*B*a^4*b - 11*A*a^3*b^2)*d^2*x + (3*B*a^5 - 11*A*a^4*b)*d^2)*e^4 + 8*(B*b^5*d^3*x^4 + 4*B*a*b^4*d^3*x
^3 + 6*B*a^2*b^3*d^3*x^2 + 4*B*a^3*b^2*d^3*x + B*a^4*b*d^3)*e^3)*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)) - (64*B*b^5*d^5*x + 16*(B*a*b^4 + 3*A*b^5)*d^5 + (128*A*a^5 + 31
5*(3*B*a*b^4 - 11*A*b^5)*x^5 + 1155*(3*B*a^2*b^3 - 11*A*a*b^4)*x^4 + 1533*(3*B*a^3*b^2 - 11*A*a^2*b^3)*x^3 + 8
37*(3*B*a^4*b - 11*A*a^3*b^2)*x^2 + 128*(3*B*a^5 - 11*A*a^4*b)*x)*e^5 + 2*(1260*B*b^5*d*x^5 + 210*(25*B*a*b^4
- 11*A*b^5)*d*x^4 + 21*(403*B*a^2*b^3 - 407*A*a*b^4)*d*x^3 + 36*(180*B*a^3*b^2 - 319*A*a^2*b^3)*d*x^2 + 11*(20
5*B*a^4*b - 581*A*a^3*b^2)*d*x + 128*(B*a^5 - 8*A*a^4*b)*d)*e^4 + (3360*B*b^5*d^2*x^4 + 21*(601*B*a*b^4 - 33*A
*b^5)*d^2*x^3 + 9*(1937*B*a^2*b^3 - 297*A*a*b^4)*d^2*x^2 + (10331*B*a^3*b^2 - 3795*A*a^2*b^3)*d^2*x + (2639*B*
a^4*b - 2295*A*a^3*b^2)*d^2)*e^3 + 2*(252*B*b^5*d^3*x^3 + 9*(105*B*a*b^4 + 11*A*b^5)*d^3*x^2 + 2*(639*B*a^2*b^
3 + 187*A*a*b^4)*d^3*x + 5*(69*B*a^3*b^2 + 103*A*a^2*b^3)*d^3)*e^2 - 8*(18*B*b^5*d^4*x^2 + (65*B*a*b^4 + 11*A*
b^5)*d^4*x + (17*B*a^2*b^3 + 41*A*a*b^4)*d^4)*e)*sqrt(x*e + d))/(b^10*d^8*x^4 + 4*a*b^9*d^8*x^3 + 6*a^2*b^8*d^
8*x^2 + 4*a^3*b^7*d^8*x + a^4*b^6*d^8 + (a^6*b^4*x^6 + 4*a^7*b^3*x^5 + 6*a^8*b^2*x^4 + 4*a^9*b*x^3 + a^10*x^2)
*e^8 - 2*(3*a^5*b^5*d*x^6 + 11*a^6*b^4*d*x^5 + 14*a^7*b^3*d*x^4 + 6*a^8*b^2*d*x^3 - a^9*b*d*x^2 - a^10*d*x)*e^
7 + (15*a^4*b^6*d^2*x^6 + 48*a^5*b^5*d^2*x^5 + 43*a^6*b^4*d^2*x^4 - 8*a^7*b^3*d^2*x^3 - 27*a^8*b^2*d^2*x^2 - 8
*a^9*b*d^2*x + a^10*d^2)*e^6 - 2*(10*a^3*b^7*d^3*x^6 + 25*a^4*b^6*d^3*x^5 + 3*a^5*b^5*d^3*x^4 - 38*a^6*b^4*d^3
*x^3 - 32*a^7*b^3*d^3*x^2 - 3*a^8*b^2*d^3*x + 3...
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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)**(5/2)/(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. 965 vs.
\(2 (407) = 814\).
time = 1.43, size = 965, normalized size = 1.95 \begin {gather*} -\frac {105 \, {\left (8 \, B b^{2} d e^{3} + 3 \, B a b e^{4} - 11 \, A b^{2} e^{4}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, {\left (b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 6 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 6 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} \sqrt {-b^{2} d + a b e}} - \frac {2 \, {\left (12 \, {\left (x e + d\right )} B b d e^{3} + B b d^{2} e^{3} + 3 \, {\left (x e + d\right )} B a e^{4} - 15 \, {\left (x e + d\right )} A b e^{4} - B a d e^{4} - A b d e^{4} + A a e^{5}\right )}}{3 \, {\left (b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 6 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 6 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} {\left (x e + d\right )}^{\frac {3}{2}}} - \frac {984 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{5} d e^{3} - 3224 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{5} d^{2} e^{3} + 3560 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{5} d^{3} e^{3} - 1320 \, \sqrt {x e + d} B b^{5} d^{4} e^{3} + 561 \, {\left (x e + d\right )}^{\frac {7}{2}} B a b^{4} e^{4} - 1545 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{5} e^{4} + 1295 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{4} d e^{4} + 5153 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{5} d e^{4} - 4825 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{4} d^{2} e^{4} - 5855 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{5} d^{2} e^{4} + 2985 \, \sqrt {x e + d} B a b^{4} d^{3} e^{4} + 2295 \, \sqrt {x e + d} A b^{5} d^{3} e^{4} + 1929 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} b^{3} e^{5} - 5153 \, {\left (x e + d\right )}^{\frac {5}{2}} A a b^{4} e^{5} - 1030 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{3} d e^{5} + 11710 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{4} d e^{5} - 1035 \, \sqrt {x e + d} B a^{2} b^{3} d^{2} e^{5} - 6885 \, \sqrt {x e + d} A a b^{4} d^{2} e^{5} + 2295 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b^{2} e^{6} - 5855 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{3} e^{6} - 1605 \, \sqrt {x e + d} B a^{3} b^{2} d e^{6} + 6885 \, \sqrt {x e + d} A a^{2} b^{3} d e^{6} + 975 \, \sqrt {x e + d} B a^{4} b e^{7} - 2295 \, \sqrt {x e + d} A a^{3} b^{2} e^{7}}{192 \, {\left (b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 6 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 6 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{6} e^{6} \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)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")
[Out]
-105/64*(8*B*b^2*d*e^3 + 3*B*a*b*e^4 - 11*A*b^2*e^4)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^6*d^6*sg
n(b*x + a) - 6*a*b^5*d^5*e*sgn(b*x + a) + 15*a^2*b^4*d^4*e^2*sgn(b*x + a) - 20*a^3*b^3*d^3*e^3*sgn(b*x + a) +
15*a^4*b^2*d^2*e^4*sgn(b*x + a) - 6*a^5*b*d*e^5*sgn(b*x + a) + a^6*e^6*sgn(b*x + a))*sqrt(-b^2*d + a*b*e)) - 2
/3*(12*(x*e + d)*B*b*d*e^3 + B*b*d^2*e^3 + 3*(x*e + d)*B*a*e^4 - 15*(x*e + d)*A*b*e^4 - B*a*d*e^4 - A*b*d*e^4
+ A*a*e^5)/((b^6*d^6*sgn(b*x + a) - 6*a*b^5*d^5*e*sgn(b*x + a) + 15*a^2*b^4*d^4*e^2*sgn(b*x + a) - 20*a^3*b^3*
d^3*e^3*sgn(b*x + a) + 15*a^4*b^2*d^2*e^4*sgn(b*x + a) - 6*a^5*b*d*e^5*sgn(b*x + a) + a^6*e^6*sgn(b*x + a))*(x
*e + d)^(3/2)) - 1/192*(984*(x*e + d)^(7/2)*B*b^5*d*e^3 - 3224*(x*e + d)^(5/2)*B*b^5*d^2*e^3 + 3560*(x*e + d)^
(3/2)*B*b^5*d^3*e^3 - 1320*sqrt(x*e + d)*B*b^5*d^4*e^3 + 561*(x*e + d)^(7/2)*B*a*b^4*e^4 - 1545*(x*e + d)^(7/2
)*A*b^5*e^4 + 1295*(x*e + d)^(5/2)*B*a*b^4*d*e^4 + 5153*(x*e + d)^(5/2)*A*b^5*d*e^4 - 4825*(x*e + d)^(3/2)*B*a
*b^4*d^2*e^4 - 5855*(x*e + d)^(3/2)*A*b^5*d^2*e^4 + 2985*sqrt(x*e + d)*B*a*b^4*d^3*e^4 + 2295*sqrt(x*e + d)*A*
b^5*d^3*e^4 + 1929*(x*e + d)^(5/2)*B*a^2*b^3*e^5 - 5153*(x*e + d)^(5/2)*A*a*b^4*e^5 - 1030*(x*e + d)^(3/2)*B*a
^2*b^3*d*e^5 + 11710*(x*e + d)^(3/2)*A*a*b^4*d*e^5 - 1035*sqrt(x*e + d)*B*a^2*b^3*d^2*e^5 - 6885*sqrt(x*e + d)
*A*a*b^4*d^2*e^5 + 2295*(x*e + d)^(3/2)*B*a^3*b^2*e^6 - 5855*(x*e + d)^(3/2)*A*a^2*b^3*e^6 - 1605*sqrt(x*e + d
)*B*a^3*b^2*d*e^6 + 6885*sqrt(x*e + d)*A*a^2*b^3*d*e^6 + 975*sqrt(x*e + d)*B*a^4*b*e^7 - 2295*sqrt(x*e + d)*A*
a^3*b^2*e^7)/((b^6*d^6*sgn(b*x + a) - 6*a*b^5*d^5*e*sgn(b*x + a) + 15*a^2*b^4*d^4*e^2*sgn(b*x + a) - 20*a^3*b^
3*d^3*e^3*sgn(b*x + a) + 15*a^4*b^2*d^2*e^4*sgn(b*x + a) - 6*a^5*b*d*e^5*sgn(b*x + a) + a^6*e^6*sgn(b*x + a))*
((x*e + d)*b - b*d + a*e)^4)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+B\,x}{{\left (d+e\,x\right )}^{5/2}\,{\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)^(5/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)),x)
[Out]
int((A + B*x)/((d + e*x)^(5/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)), x)
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