3.17.12 \(\int \frac {A+B x}{(d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^3} \, dx\)
Optimal. Leaf size=352 \[ \frac {63 e^4 (a B e-11 A b e+10 b B d)}{128 b \sqrt {d+e x} (b d-a e)^6}-\frac {63 e^4 (a B e-11 A b e+10 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 \sqrt {b} (b d-a e)^{13/2}}+\frac {21 e^3 (a B e-11 A b e+10 b B d)}{128 b (a+b x) \sqrt {d+e x} (b d-a e)^5}-\frac {21 e^2 (a B e-11 A b e+10 b B d)}{320 b (a+b x)^2 \sqrt {d+e x} (b d-a e)^4}+\frac {3 e (a B e-11 A b e+10 b B d)}{80 b (a+b x)^3 \sqrt {d+e x} (b d-a e)^3}-\frac {a B e-11 A b e+10 b B d}{40 b (a+b x)^4 \sqrt {d+e x} (b d-a e)^2}-\frac {A b-a B}{5 b (a+b x)^5 \sqrt {d+e x} (b d-a e)} \]
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Rubi [A] time = 0.38, antiderivative size = 352, normalized size of antiderivative = 1.00,
number of steps used = 9, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used =
{27, 78, 51, 63, 208} \begin {gather*} \frac {63 e^4 (a B e-11 A b e+10 b B d)}{128 b \sqrt {d+e x} (b d-a e)^6}+\frac {21 e^3 (a B e-11 A b e+10 b B d)}{128 b (a+b x) \sqrt {d+e x} (b d-a e)^5}-\frac {21 e^2 (a B e-11 A b e+10 b B d)}{320 b (a+b x)^2 \sqrt {d+e x} (b d-a e)^4}-\frac {63 e^4 (a B e-11 A b e+10 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 \sqrt {b} (b d-a e)^{13/2}}+\frac {3 e (a B e-11 A b e+10 b B d)}{80 b (a+b x)^3 \sqrt {d+e x} (b d-a e)^3}-\frac {a B e-11 A b e+10 b B d}{40 b (a+b x)^4 \sqrt {d+e x} (b d-a e)^2}-\frac {A b-a B}{5 b (a+b x)^5 \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)^3),x]
[Out]
(63*e^4*(10*b*B*d - 11*A*b*e + a*B*e))/(128*b*(b*d - a*e)^6*Sqrt[d + e*x]) - (A*b - a*B)/(5*b*(b*d - a*e)*(a +
b*x)^5*Sqrt[d + e*x]) - (10*b*B*d - 11*A*b*e + a*B*e)/(40*b*(b*d - a*e)^2*(a + b*x)^4*Sqrt[d + e*x]) + (3*e*(
10*b*B*d - 11*A*b*e + a*B*e))/(80*b*(b*d - a*e)^3*(a + b*x)^3*Sqrt[d + e*x]) - (21*e^2*(10*b*B*d - 11*A*b*e +
a*B*e))/(320*b*(b*d - a*e)^4*(a + b*x)^2*Sqrt[d + e*x]) + (21*e^3*(10*b*B*d - 11*A*b*e + a*B*e))/(128*b*(b*d -
a*e)^5*(a + b*x)*Sqrt[d + e*x]) - (63*e^4*(10*b*B*d - 11*A*b*e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[
b*d - a*e]])/(128*Sqrt[b]*(b*d - a*e)^(13/2))
Rule 27
Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Rule 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]
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 )^3} \, dx &=\int \frac {A+B x}{(a+b x)^6 (d+e x)^{3/2}} \, dx\\ &=-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 \sqrt {d+e x}}+\frac {(10 b B d-11 A b e+a B e) \int \frac {1}{(a+b x)^5 (d+e x)^{3/2}} \, dx}{10 b (b d-a e)}\\ &=-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 \sqrt {d+e x}}-\frac {10 b B d-11 A b e+a B e}{40 b (b d-a e)^2 (a+b x)^4 \sqrt {d+e x}}-\frac {(9 e (10 b B d-11 A b e+a B e)) \int \frac {1}{(a+b x)^4 (d+e x)^{3/2}} \, dx}{80 b (b d-a e)^2}\\ &=-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 \sqrt {d+e x}}-\frac {10 b B d-11 A b e+a B e}{40 b (b d-a e)^2 (a+b x)^4 \sqrt {d+e x}}+\frac {3 e (10 b B d-11 A b e+a B e)}{80 b (b d-a e)^3 (a+b x)^3 \sqrt {d+e x}}+\frac {\left (21 e^2 (10 b B d-11 A b e+a B e)\right ) \int \frac {1}{(a+b x)^3 (d+e x)^{3/2}} \, dx}{160 b (b d-a e)^3}\\ &=-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 \sqrt {d+e x}}-\frac {10 b B d-11 A b e+a B e}{40 b (b d-a e)^2 (a+b x)^4 \sqrt {d+e x}}+\frac {3 e (10 b B d-11 A b e+a B e)}{80 b (b d-a e)^3 (a+b x)^3 \sqrt {d+e x}}-\frac {21 e^2 (10 b B d-11 A b e+a B e)}{320 b (b d-a e)^4 (a+b x)^2 \sqrt {d+e x}}-\frac {\left (21 e^3 (10 b B d-11 A b e+a B e)\right ) \int \frac {1}{(a+b x)^2 (d+e x)^{3/2}} \, dx}{128 b (b d-a e)^4}\\ &=-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 \sqrt {d+e x}}-\frac {10 b B d-11 A b e+a B e}{40 b (b d-a e)^2 (a+b x)^4 \sqrt {d+e x}}+\frac {3 e (10 b B d-11 A b e+a B e)}{80 b (b d-a e)^3 (a+b x)^3 \sqrt {d+e x}}-\frac {21 e^2 (10 b B d-11 A b e+a B e)}{320 b (b d-a e)^4 (a+b x)^2 \sqrt {d+e x}}+\frac {21 e^3 (10 b B d-11 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) \sqrt {d+e x}}+\frac {\left (63 e^4 (10 b B d-11 A b e+a B e)\right ) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{256 b (b d-a e)^5}\\ &=\frac {63 e^4 (10 b B d-11 A b e+a B e)}{128 b (b d-a e)^6 \sqrt {d+e x}}-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 \sqrt {d+e x}}-\frac {10 b B d-11 A b e+a B e}{40 b (b d-a e)^2 (a+b x)^4 \sqrt {d+e x}}+\frac {3 e (10 b B d-11 A b e+a B e)}{80 b (b d-a e)^3 (a+b x)^3 \sqrt {d+e x}}-\frac {21 e^2 (10 b B d-11 A b e+a B e)}{320 b (b d-a e)^4 (a+b x)^2 \sqrt {d+e x}}+\frac {21 e^3 (10 b B d-11 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) \sqrt {d+e x}}+\frac {\left (63 e^4 (10 b B d-11 A b e+a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 (b d-a e)^6}\\ &=\frac {63 e^4 (10 b B d-11 A b e+a B e)}{128 b (b d-a e)^6 \sqrt {d+e x}}-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 \sqrt {d+e x}}-\frac {10 b B d-11 A b e+a B e}{40 b (b d-a e)^2 (a+b x)^4 \sqrt {d+e x}}+\frac {3 e (10 b B d-11 A b e+a B e)}{80 b (b d-a e)^3 (a+b x)^3 \sqrt {d+e x}}-\frac {21 e^2 (10 b B d-11 A b e+a B e)}{320 b (b d-a e)^4 (a+b x)^2 \sqrt {d+e x}}+\frac {21 e^3 (10 b B d-11 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) \sqrt {d+e x}}+\frac {\left (63 e^3 (10 b B d-11 A b e+a B e)\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{128 (b d-a e)^6}\\ &=\frac {63 e^4 (10 b B d-11 A b e+a B e)}{128 b (b d-a e)^6 \sqrt {d+e x}}-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 \sqrt {d+e x}}-\frac {10 b B d-11 A b e+a B e}{40 b (b d-a e)^2 (a+b x)^4 \sqrt {d+e x}}+\frac {3 e (10 b B d-11 A b e+a B e)}{80 b (b d-a e)^3 (a+b x)^3 \sqrt {d+e x}}-\frac {21 e^2 (10 b B d-11 A b e+a B e)}{320 b (b d-a e)^4 (a+b x)^2 \sqrt {d+e x}}+\frac {21 e^3 (10 b B d-11 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) \sqrt {d+e x}}-\frac {63 e^4 (10 b B d-11 A b e+a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 \sqrt {b} (b d-a e)^{13/2}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 97, normalized size = 0.28 \begin {gather*} \frac {\frac {e^4 (a B e-11 A b e+10 b B d) \, _2F_1\left (-\frac {1}{2},5;\frac {1}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^5}+\frac {a B-A b}{(a+b x)^5}}{5 b \sqrt {d+e x} (b d-a e)} \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)^3),x]
[Out]
((-(A*b) + a*B)/(a + b*x)^5 + (e^4*(10*b*B*d - 11*A*b*e + a*B*e)*Hypergeometric2F1[-1/2, 5, 1/2, (b*(d + e*x))
/(b*d - a*e)])/(b*d - a*e)^5)/(5*b*(b*d - a*e)*Sqrt[d + e*x])
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IntegrateAlgebraic [B] time = 4.87, size = 891, normalized size = 2.53 \begin {gather*} \frac {e^4 \left (1280 b^5 B d^6-1280 A b^5 e d^5-6400 a b^4 B e d^5-9650 b^5 B (d+e x) d^5+6400 a A b^4 e^2 d^4+12800 a^2 b^3 B e^2 d^4+23700 b^5 B (d+e x)^2 d^4+10615 A b^5 e (d+e x) d^4+37635 a b^4 B e (d+e x) d^4-12800 a^2 A b^3 e^3 d^3-12800 a^3 b^2 B e^3 d^3-26880 b^5 B (d+e x)^3 d^3-26070 A b^5 e (d+e x)^2 d^3-68730 a b^4 B e (d+e x)^2 d^3-42460 a A b^4 e^2 (d+e x) d^3-54040 a^2 b^3 B e^2 (d+e x) d^3+12800 a^3 A b^2 e^4 d^2+6400 a^4 b B e^4 d^2+14700 b^5 B (d+e x)^4 d^2+29568 A b^5 e (d+e x)^3 d^2+51072 a b^4 B e (d+e x)^3 d^2+78210 a A b^4 e^2 (d+e x)^2 d^2+63990 a^2 b^3 B e^2 (d+e x)^2 d^2+63690 a^2 A b^3 e^3 (d+e x) d^2+32810 a^3 b^2 B e^3 (d+e x) d^2-6400 a^4 A b e^5 d-1280 a^5 B e^5 d-3150 b^5 B (d+e x)^5 d-16170 A b^5 e (d+e x)^4 d-13230 a b^4 B e (d+e x)^4 d-59136 a A b^4 e^2 (d+e x)^3 d-21504 a^2 b^3 B e^2 (d+e x)^3 d-78210 a^2 A b^3 e^3 (d+e x)^2 d-16590 a^3 b^2 B e^3 (d+e x)^2 d-42460 a^3 A b^2 e^4 (d+e x) d-5790 a^4 b B e^4 (d+e x) d+1280 a^5 A e^6+3465 A b^5 e (d+e x)^5-315 a b^4 B e (d+e x)^5+16170 a A b^4 e^2 (d+e x)^4-1470 a^2 b^3 B e^2 (d+e x)^4+29568 a^2 A b^3 e^3 (d+e x)^3-2688 a^3 b^2 B e^3 (d+e x)^3+26070 a^3 A b^2 e^4 (d+e x)^2-2370 a^4 b B e^4 (d+e x)^2+10615 a^4 A b e^5 (d+e x)-965 a^5 B e^5 (d+e x)\right )}{640 (b d-a e)^6 \sqrt {d+e x} (b d-a e-b (d+e x))^5}-\frac {63 \left (-11 A b e^5+a B e^5+10 b B d e^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {a e-b d} \sqrt {d+e x}}{b d-a e}\right )}{128 \sqrt {b} (b d-a e)^6 \sqrt {a e-b d}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(A + B*x)/((d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]
[Out]
(e^4*(1280*b^5*B*d^6 - 1280*A*b^5*d^5*e - 6400*a*b^4*B*d^5*e + 6400*a*A*b^4*d^4*e^2 + 12800*a^2*b^3*B*d^4*e^2
- 12800*a^2*A*b^3*d^3*e^3 - 12800*a^3*b^2*B*d^3*e^3 + 12800*a^3*A*b^2*d^2*e^4 + 6400*a^4*b*B*d^2*e^4 - 6400*a^
4*A*b*d*e^5 - 1280*a^5*B*d*e^5 + 1280*a^5*A*e^6 - 9650*b^5*B*d^5*(d + e*x) + 10615*A*b^5*d^4*e*(d + e*x) + 376
35*a*b^4*B*d^4*e*(d + e*x) - 42460*a*A*b^4*d^3*e^2*(d + e*x) - 54040*a^2*b^3*B*d^3*e^2*(d + e*x) + 63690*a^2*A
*b^3*d^2*e^3*(d + e*x) + 32810*a^3*b^2*B*d^2*e^3*(d + e*x) - 42460*a^3*A*b^2*d*e^4*(d + e*x) - 5790*a^4*b*B*d*
e^4*(d + e*x) + 10615*a^4*A*b*e^5*(d + e*x) - 965*a^5*B*e^5*(d + e*x) + 23700*b^5*B*d^4*(d + e*x)^2 - 26070*A*
b^5*d^3*e*(d + e*x)^2 - 68730*a*b^4*B*d^3*e*(d + e*x)^2 + 78210*a*A*b^4*d^2*e^2*(d + e*x)^2 + 63990*a^2*b^3*B*
d^2*e^2*(d + e*x)^2 - 78210*a^2*A*b^3*d*e^3*(d + e*x)^2 - 16590*a^3*b^2*B*d*e^3*(d + e*x)^2 + 26070*a^3*A*b^2*
e^4*(d + e*x)^2 - 2370*a^4*b*B*e^4*(d + e*x)^2 - 26880*b^5*B*d^3*(d + e*x)^3 + 29568*A*b^5*d^2*e*(d + e*x)^3 +
51072*a*b^4*B*d^2*e*(d + e*x)^3 - 59136*a*A*b^4*d*e^2*(d + e*x)^3 - 21504*a^2*b^3*B*d*e^2*(d + e*x)^3 + 29568
*a^2*A*b^3*e^3*(d + e*x)^3 - 2688*a^3*b^2*B*e^3*(d + e*x)^3 + 14700*b^5*B*d^2*(d + e*x)^4 - 16170*A*b^5*d*e*(d
+ e*x)^4 - 13230*a*b^4*B*d*e*(d + e*x)^4 + 16170*a*A*b^4*e^2*(d + e*x)^4 - 1470*a^2*b^3*B*e^2*(d + e*x)^4 - 3
150*b^5*B*d*(d + e*x)^5 + 3465*A*b^5*e*(d + e*x)^5 - 315*a*b^4*B*e*(d + e*x)^5))/(640*(b*d - a*e)^6*Sqrt[d + e
*x]*(b*d - a*e - b*(d + e*x))^5) - (63*(10*b*B*d*e^4 - 11*A*b*e^5 + a*B*e^5)*ArcTan[(Sqrt[b]*Sqrt[-(b*d) + a*e
]*Sqrt[d + e*x])/(b*d - a*e)])/(128*Sqrt[b]*(b*d - a*e)^6*Sqrt[-(b*d) + a*e])
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fricas [B] time = 0.55, size = 3876, normalized size = 11.01
result too large to display
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)^3,x, algorithm="fricas")
[Out]
[-1/1280*(315*(10*B*a^5*b*d^2*e^4 + (B*a^6 - 11*A*a^5*b)*d*e^5 + (10*B*b^6*d*e^5 + (B*a*b^5 - 11*A*b^6)*e^6)*x
^6 + (10*B*b^6*d^2*e^4 + (51*B*a*b^5 - 11*A*b^6)*d*e^5 + 5*(B*a^2*b^4 - 11*A*a*b^5)*e^6)*x^5 + 5*(10*B*a*b^5*d
^2*e^4 + (21*B*a^2*b^4 - 11*A*a*b^5)*d*e^5 + 2*(B*a^3*b^3 - 11*A*a^2*b^4)*e^6)*x^4 + 10*(10*B*a^2*b^4*d^2*e^4
+ 11*(B*a^3*b^3 - A*a^2*b^4)*d*e^5 + (B*a^4*b^2 - 11*A*a^3*b^3)*e^6)*x^3 + 5*(20*B*a^3*b^3*d^2*e^4 + 2*(6*B*a^
4*b^2 - 11*A*a^3*b^3)*d*e^5 + (B*a^5*b - 11*A*a^4*b^2)*e^6)*x^2 + (50*B*a^4*b^2*d^2*e^4 + 5*(3*B*a^5*b - 11*A*
a^4*b^2)*d*e^5 + (B*a^6 - 11*A*a^5*b)*e^6)*x)*sqrt(b^2*d - a*b*e)*log((b*e*x + 2*b*d - a*e + 2*sqrt(b^2*d - a*
b*e)*sqrt(e*x + d))/(b*x + a)) - 2*(1280*A*a^6*b*e^6 - 32*(B*a*b^6 + 4*A*b^7)*d^6 + 16*(16*B*a^2*b^5 + 59*A*a*
b^6)*d^5*e - 4*(239*B*a^3*b^4 + 766*A*a^2*b^5)*d^4*e^2 + 14*(178*B*a^4*b^3 + 417*A*a^3*b^4)*d^3*e^3 + 5*(97*B*
a^5*b^2 - 1561*A*a^4*b^3)*d^2*e^4 - 5*(449*B*a^6*b - 587*A*a^5*b^2)*d*e^5 + 315*(10*B*b^7*d^2*e^4 - (9*B*a*b^6
+ 11*A*b^7)*d*e^5 - (B*a^2*b^5 - 11*A*a*b^6)*e^6)*x^5 + 105*(10*B*b^7*d^3*e^3 + (131*B*a*b^6 - 11*A*b^7)*d^2*
e^4 - (127*B*a^2*b^5 + 143*A*a*b^6)*d*e^5 - 14*(B*a^3*b^4 - 11*A*a^2*b^5)*e^6)*x^4 - 42*(10*B*b^7*d^4*e^2 - (1
29*B*a*b^6 + 11*A*b^7)*d^3*e^3 - 13*(41*B*a^2*b^5 - 11*A*a*b^6)*d^2*e^4 + 4*(147*B*a^3*b^4 + 143*A*a^2*b^5)*d*
e^5 + 64*(B*a^4*b^3 - 11*A*a^3*b^4)*e^6)*x^3 + 6*(40*B*b^7*d^5*e - 2*(183*B*a*b^6 + 22*A*b^7)*d^4*e^2 + (1883*
B*a^2*b^5 + 407*A*a*b^6)*d^3*e^3 + 88*(29*B*a^3*b^4 - 24*A*a^2*b^5)*d^2*e^4 - 2*(1857*B*a^4*b^3 + 1298*A*a^3*b
^4)*d*e^5 - 395*(B*a^5*b^2 - 11*A*a^4*b^3)*e^6)*x^2 - (160*B*b^7*d^6 - 16*(79*B*a*b^6 + 11*A*b^7)*d^5*e + 4*(1
163*B*a^2*b^5 + 352*A*a*b^6)*d^4*e^2 - 2*(5991*B*a^3*b^4 + 2629*A*a^2*b^5)*d^3*e^3 - 2*(1048*B*a^4*b^3 - 6853*
A*a^3*b^4)*d^2*e^4 + 5*(1913*B*a^5*b^2 + 187*A*a^4*b^3)*d*e^5 + 965*(B*a^6*b - 11*A*a^5*b^2)*e^6)*x)*sqrt(e*x
+ d))/(a^5*b^8*d^8 - 7*a^6*b^7*d^7*e + 21*a^7*b^6*d^6*e^2 - 35*a^8*b^5*d^5*e^3 + 35*a^9*b^4*d^4*e^4 - 21*a^10*
b^3*d^3*e^5 + 7*a^11*b^2*d^2*e^6 - a^12*b*d*e^7 + (b^13*d^7*e - 7*a*b^12*d^6*e^2 + 21*a^2*b^11*d^5*e^3 - 35*a^
3*b^10*d^4*e^4 + 35*a^4*b^9*d^3*e^5 - 21*a^5*b^8*d^2*e^6 + 7*a^6*b^7*d*e^7 - a^7*b^6*e^8)*x^6 + (b^13*d^8 - 2*
a*b^12*d^7*e - 14*a^2*b^11*d^6*e^2 + 70*a^3*b^10*d^5*e^3 - 140*a^4*b^9*d^4*e^4 + 154*a^5*b^8*d^3*e^5 - 98*a^6*
b^7*d^2*e^6 + 34*a^7*b^6*d*e^7 - 5*a^8*b^5*e^8)*x^5 + 5*(a*b^12*d^8 - 5*a^2*b^11*d^7*e + 7*a^3*b^10*d^6*e^2 +
7*a^4*b^9*d^5*e^3 - 35*a^5*b^8*d^4*e^4 + 49*a^6*b^7*d^3*e^5 - 35*a^7*b^6*d^2*e^6 + 13*a^8*b^5*d*e^7 - 2*a^9*b^
4*e^8)*x^4 + 10*(a^2*b^11*d^8 - 6*a^3*b^10*d^7*e + 14*a^4*b^9*d^6*e^2 - 14*a^5*b^8*d^5*e^3 + 14*a^7*b^6*d^3*e^
5 - 14*a^8*b^5*d^2*e^6 + 6*a^9*b^4*d*e^7 - a^10*b^3*e^8)*x^3 + 5*(2*a^3*b^10*d^8 - 13*a^4*b^9*d^7*e + 35*a^5*b
^8*d^6*e^2 - 49*a^6*b^7*d^5*e^3 + 35*a^7*b^6*d^4*e^4 - 7*a^8*b^5*d^3*e^5 - 7*a^9*b^4*d^2*e^6 + 5*a^10*b^3*d*e^
7 - a^11*b^2*e^8)*x^2 + (5*a^4*b^9*d^8 - 34*a^5*b^8*d^7*e + 98*a^6*b^7*d^6*e^2 - 154*a^7*b^6*d^5*e^3 + 140*a^8
*b^5*d^4*e^4 - 70*a^9*b^4*d^3*e^5 + 14*a^10*b^3*d^2*e^6 + 2*a^11*b^2*d*e^7 - a^12*b*e^8)*x), 1/640*(315*(10*B*
a^5*b*d^2*e^4 + (B*a^6 - 11*A*a^5*b)*d*e^5 + (10*B*b^6*d*e^5 + (B*a*b^5 - 11*A*b^6)*e^6)*x^6 + (10*B*b^6*d^2*e
^4 + (51*B*a*b^5 - 11*A*b^6)*d*e^5 + 5*(B*a^2*b^4 - 11*A*a*b^5)*e^6)*x^5 + 5*(10*B*a*b^5*d^2*e^4 + (21*B*a^2*b
^4 - 11*A*a*b^5)*d*e^5 + 2*(B*a^3*b^3 - 11*A*a^2*b^4)*e^6)*x^4 + 10*(10*B*a^2*b^4*d^2*e^4 + 11*(B*a^3*b^3 - A*
a^2*b^4)*d*e^5 + (B*a^4*b^2 - 11*A*a^3*b^3)*e^6)*x^3 + 5*(20*B*a^3*b^3*d^2*e^4 + 2*(6*B*a^4*b^2 - 11*A*a^3*b^3
)*d*e^5 + (B*a^5*b - 11*A*a^4*b^2)*e^6)*x^2 + (50*B*a^4*b^2*d^2*e^4 + 5*(3*B*a^5*b - 11*A*a^4*b^2)*d*e^5 + (B*
a^6 - 11*A*a^5*b)*e^6)*x)*sqrt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d)/(b*e*x + b*d)) + (128
0*A*a^6*b*e^6 - 32*(B*a*b^6 + 4*A*b^7)*d^6 + 16*(16*B*a^2*b^5 + 59*A*a*b^6)*d^5*e - 4*(239*B*a^3*b^4 + 766*A*a
^2*b^5)*d^4*e^2 + 14*(178*B*a^4*b^3 + 417*A*a^3*b^4)*d^3*e^3 + 5*(97*B*a^5*b^2 - 1561*A*a^4*b^3)*d^2*e^4 - 5*(
449*B*a^6*b - 587*A*a^5*b^2)*d*e^5 + 315*(10*B*b^7*d^2*e^4 - (9*B*a*b^6 + 11*A*b^7)*d*e^5 - (B*a^2*b^5 - 11*A*
a*b^6)*e^6)*x^5 + 105*(10*B*b^7*d^3*e^3 + (131*B*a*b^6 - 11*A*b^7)*d^2*e^4 - (127*B*a^2*b^5 + 143*A*a*b^6)*d*e
^5 - 14*(B*a^3*b^4 - 11*A*a^2*b^5)*e^6)*x^4 - 42*(10*B*b^7*d^4*e^2 - (129*B*a*b^6 + 11*A*b^7)*d^3*e^3 - 13*(41
*B*a^2*b^5 - 11*A*a*b^6)*d^2*e^4 + 4*(147*B*a^3*b^4 + 143*A*a^2*b^5)*d*e^5 + 64*(B*a^4*b^3 - 11*A*a^3*b^4)*e^6
)*x^3 + 6*(40*B*b^7*d^5*e - 2*(183*B*a*b^6 + 22*A*b^7)*d^4*e^2 + (1883*B*a^2*b^5 + 407*A*a*b^6)*d^3*e^3 + 88*(
29*B*a^3*b^4 - 24*A*a^2*b^5)*d^2*e^4 - 2*(1857*B*a^4*b^3 + 1298*A*a^3*b^4)*d*e^5 - 395*(B*a^5*b^2 - 11*A*a^4*b
^3)*e^6)*x^2 - (160*B*b^7*d^6 - 16*(79*B*a*b^6 + 11*A*b^7)*d^5*e + 4*(1163*B*a^2*b^5 + 352*A*a*b^6)*d^4*e^2 -
2*(5991*B*a^3*b^4 + 2629*A*a^2*b^5)*d^3*e^3 - 2*(1048*B*a^4*b^3 - 6853*A*a^3*b^4)*d^2*e^4 + 5*(1913*B*a^5*b^2
+ 187*A*a^4*b^3)*d*e^5 + 965*(B*a^6*b - 11*A*a^5*b^2)*e^6)*x)*sqrt(e*x + d))/(a^5*b^8*d^8 - 7*a^6*b^7*d^7*e +
21*a^7*b^6*d^6*e^2 - 35*a^8*b^5*d^5*e^3 + 35*a^9*b^4*d^4*e^4 - 21*a^10*b^3*d^3*e^5 + 7*a^11*b^2*d^2*e^6 - a^12
*b*d*e^7 + (b^13*d^7*e - 7*a*b^12*d^6*e^2 + 21*a^2*b^11*d^5*e^3 - 35*a^3*b^10*d^4*e^4 + 35*a^4*b^9*d^3*e^5 - 2
1*a^5*b^8*d^2*e^6 + 7*a^6*b^7*d*e^7 - a^7*b^6*e^8)*x^6 + (b^13*d^8 - 2*a*b^12*d^7*e - 14*a^2*b^11*d^6*e^2 + 70
*a^3*b^10*d^5*e^3 - 140*a^4*b^9*d^4*e^4 + 154*a^5*b^8*d^3*e^5 - 98*a^6*b^7*d^2*e^6 + 34*a^7*b^6*d*e^7 - 5*a^8*
b^5*e^8)*x^5 + 5*(a*b^12*d^8 - 5*a^2*b^11*d^7*e + 7*a^3*b^10*d^6*e^2 + 7*a^4*b^9*d^5*e^3 - 35*a^5*b^8*d^4*e^4
+ 49*a^6*b^7*d^3*e^5 - 35*a^7*b^6*d^2*e^6 + 13*a^8*b^5*d*e^7 - 2*a^9*b^4*e^8)*x^4 + 10*(a^2*b^11*d^8 - 6*a^3*b
^10*d^7*e + 14*a^4*b^9*d^6*e^2 - 14*a^5*b^8*d^5*e^3 + 14*a^7*b^6*d^3*e^5 - 14*a^8*b^5*d^2*e^6 + 6*a^9*b^4*d*e^
7 - a^10*b^3*e^8)*x^3 + 5*(2*a^3*b^10*d^8 - 13*a^4*b^9*d^7*e + 35*a^5*b^8*d^6*e^2 - 49*a^6*b^7*d^5*e^3 + 35*a^
7*b^6*d^4*e^4 - 7*a^8*b^5*d^3*e^5 - 7*a^9*b^4*d^2*e^6 + 5*a^10*b^3*d*e^7 - a^11*b^2*e^8)*x^2 + (5*a^4*b^9*d^8
- 34*a^5*b^8*d^7*e + 98*a^6*b^7*d^6*e^2 - 154*a^7*b^6*d^5*e^3 + 140*a^8*b^5*d^4*e^4 - 70*a^9*b^4*d^3*e^5 + 14*
a^10*b^3*d^2*e^6 + 2*a^11*b^2*d*e^7 - a^12*b*e^8)*x)]
________________________________________________________________________________________
giac [B] time = 0.37, size = 994, normalized size = 2.82 \begin {gather*} \frac {63 \, {\left (10 \, B b d e^{4} + B a e^{5} - 11 \, A b e^{5}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{128 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \sqrt {-b^{2} d + a b e}} + \frac {2 \, {\left (B d e^{4} - A e^{5}\right )}}{{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \sqrt {x e + d}} + \frac {1870 \, {\left (x e + d\right )}^{\frac {9}{2}} B b^{5} d e^{4} - 8300 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{5} d^{2} e^{4} + 14080 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{5} d^{3} e^{4} - 10900 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{5} d^{4} e^{4} + 3250 \, \sqrt {x e + d} B b^{5} d^{5} e^{4} + 315 \, {\left (x e + d\right )}^{\frac {9}{2}} B a b^{4} e^{5} - 2185 \, {\left (x e + d\right )}^{\frac {9}{2}} A b^{5} e^{5} + 6830 \, {\left (x e + d\right )}^{\frac {7}{2}} B a b^{4} d e^{5} + 9770 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{5} d e^{5} - 25472 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{4} d^{2} e^{5} - 16768 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{5} d^{2} e^{5} + 30330 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{4} d^{3} e^{5} + 13270 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{5} d^{3} e^{5} - 12035 \, \sqrt {x e + d} B a b^{4} d^{4} e^{5} - 4215 \, \sqrt {x e + d} A b^{5} d^{4} e^{5} + 1470 \, {\left (x e + d\right )}^{\frac {7}{2}} B a^{2} b^{3} e^{6} - 9770 \, {\left (x e + d\right )}^{\frac {7}{2}} A a b^{4} e^{6} + 8704 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} b^{3} d e^{6} + 33536 \, {\left (x e + d\right )}^{\frac {5}{2}} A a b^{4} d e^{6} - 25590 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{3} d^{2} e^{6} - 39810 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{4} d^{2} e^{6} + 15640 \, \sqrt {x e + d} B a^{2} b^{3} d^{3} e^{6} + 16860 \, \sqrt {x e + d} A a b^{4} d^{3} e^{6} + 2688 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{3} b^{2} e^{7} - 16768 \, {\left (x e + d\right )}^{\frac {5}{2}} A a^{2} b^{3} e^{7} + 3790 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b^{2} d e^{7} + 39810 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{3} d e^{7} - 7210 \, \sqrt {x e + d} B a^{3} b^{2} d^{2} e^{7} - 25290 \, \sqrt {x e + d} A a^{2} b^{3} d^{2} e^{7} + 2370 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{4} b e^{8} - 13270 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{3} b^{2} e^{8} - 610 \, \sqrt {x e + d} B a^{4} b d e^{8} + 16860 \, \sqrt {x e + d} A a^{3} b^{2} d e^{8} + 965 \, \sqrt {x e + d} B a^{5} e^{9} - 4215 \, \sqrt {x e + d} A a^{4} b e^{9}}{640 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{5}} \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)^3,x, algorithm="giac")
[Out]
63/128*(10*B*b*d*e^4 + B*a*e^5 - 11*A*b*e^5)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^6*d^6 - 6*a*b^5*
d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6)*sqrt(-b^2*d +
a*b*e)) + 2*(B*d*e^4 - A*e^5)/((b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2
*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6)*sqrt(x*e + d)) + 1/640*(1870*(x*e + d)^(9/2)*B*b^5*d*e^4 - 8300*(x*e + d)^
(7/2)*B*b^5*d^2*e^4 + 14080*(x*e + d)^(5/2)*B*b^5*d^3*e^4 - 10900*(x*e + d)^(3/2)*B*b^5*d^4*e^4 + 3250*sqrt(x*
e + d)*B*b^5*d^5*e^4 + 315*(x*e + d)^(9/2)*B*a*b^4*e^5 - 2185*(x*e + d)^(9/2)*A*b^5*e^5 + 6830*(x*e + d)^(7/2)
*B*a*b^4*d*e^5 + 9770*(x*e + d)^(7/2)*A*b^5*d*e^5 - 25472*(x*e + d)^(5/2)*B*a*b^4*d^2*e^5 - 16768*(x*e + d)^(5
/2)*A*b^5*d^2*e^5 + 30330*(x*e + d)^(3/2)*B*a*b^4*d^3*e^5 + 13270*(x*e + d)^(3/2)*A*b^5*d^3*e^5 - 12035*sqrt(x
*e + d)*B*a*b^4*d^4*e^5 - 4215*sqrt(x*e + d)*A*b^5*d^4*e^5 + 1470*(x*e + d)^(7/2)*B*a^2*b^3*e^6 - 9770*(x*e +
d)^(7/2)*A*a*b^4*e^6 + 8704*(x*e + d)^(5/2)*B*a^2*b^3*d*e^6 + 33536*(x*e + d)^(5/2)*A*a*b^4*d*e^6 - 25590*(x*e
+ d)^(3/2)*B*a^2*b^3*d^2*e^6 - 39810*(x*e + d)^(3/2)*A*a*b^4*d^2*e^6 + 15640*sqrt(x*e + d)*B*a^2*b^3*d^3*e^6
+ 16860*sqrt(x*e + d)*A*a*b^4*d^3*e^6 + 2688*(x*e + d)^(5/2)*B*a^3*b^2*e^7 - 16768*(x*e + d)^(5/2)*A*a^2*b^3*e
^7 + 3790*(x*e + d)^(3/2)*B*a^3*b^2*d*e^7 + 39810*(x*e + d)^(3/2)*A*a^2*b^3*d*e^7 - 7210*sqrt(x*e + d)*B*a^3*b
^2*d^2*e^7 - 25290*sqrt(x*e + d)*A*a^2*b^3*d^2*e^7 + 2370*(x*e + d)^(3/2)*B*a^4*b*e^8 - 13270*(x*e + d)^(3/2)*
A*a^3*b^2*e^8 - 610*sqrt(x*e + d)*B*a^4*b*d*e^8 + 16860*sqrt(x*e + d)*A*a^3*b^2*d*e^8 + 965*sqrt(x*e + d)*B*a^
5*e^9 - 4215*sqrt(x*e + d)*A*a^4*b*e^9)/((b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 +
15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6)*((x*e + d)*b - b*d + a*e)^5)
________________________________________________________________________________________
maple [B] time = 0.09, size = 1568, normalized size = 4.45
result too large to display
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)^3,x)
[Out]
843/32*e^6/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a*b^4*d^3+843/32*e^8/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(1
/2)*A*a^3*b^2*d+379/64*e^7/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a^3*b^2*d-2559/64*e^6/(a*e-b*d)^6/(b*e*x+
a*e)^5*(e*x+d)^(3/2)*B*a^2*b^3*d^2+3033/64*e^5/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a*b^4*d^3+3981/64*e^7
/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*A*a^2*b^3*d-3981/64*e^6/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*A*a*b
^4*d^2-199/5*e^5/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*a*b^4*d^2+262/5*e^6/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+
d)^(5/2)*A*a*b^4*d+683/64*e^5/(a*e-b*d)^6/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*a*b^4*d-61/64*e^8/(a*e-b*d)^6/(b*e*x+a
*e)^5*(e*x+d)^(1/2)*B*a^4*b*d-721/64*e^7/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^3*b^2*d^2+391/16*e^6/(a*e
-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^2*b^3*d^3-2529/64*e^7/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a^2*b^
3*d^2+68/5*e^6/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*a^2*b^3*d-2407/128*e^5/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x
+d)^(1/2)*B*a*b^4*d^4-2*e^5/(a*e-b*d)^6/(e*x+d)^(1/2)*A+2*e^4/(a*e-b*d)^6/(e*x+d)^(1/2)*B*d-437/128*e^5/(a*e-b
*d)^6/(b*e*x+a*e)^5*(e*x+d)^(9/2)*A*b^5-693/128*e^5/(a*e-b*d)^6/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e
-b*d)*b)^(1/2)*b)*A*b+63/128*e^5/(a*e-b*d)^6/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a
*B+193/128*e^9/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^5+1327/64*e^5/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(3/
2)*A*b^5*d^3+237/64*e^8/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a^4*b-843/128*e^9/(a*e-b*d)^6/(b*e*x+a*e)^5*
(e*x+d)^(1/2)*A*a^4*b-843/128*e^5/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*b^5*d^4-977/64*e^6/(a*e-b*d)^6/(b*
e*x+a*e)^5*A*(e*x+d)^(7/2)*a*b^4+977/64*e^5/(a*e-b*d)^6/(b*e*x+a*e)^5*A*(e*x+d)^(7/2)*b^5*d+147/64*e^6/(a*e-b*
d)^6/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*a^2*b^3+22*e^4/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*b^5*d^3-545/32*e^4
/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*b^5*d^4+325/64*e^4/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*b^5*d^
5-415/32*e^4/(a*e-b*d)^6/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*b^5*d^2+63/128*e^5/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(9
/2)*B*a*b^4-131/5*e^7/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(5/2)*A*a^2*b^3-131/5*e^5/(a*e-b*d)^6/(b*e*x+a*e)^5*(e
*x+d)^(5/2)*A*b^5*d^2+21/5*e^7/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*a^3*b^2-1327/64*e^8/(a*e-b*d)^6/(b*e*
x+a*e)^5*(e*x+d)^(3/2)*A*a^3*b^2+187/64*e^4/(a*e-b*d)^6/(b*e*x+a*e)^5*(e*x+d)^(9/2)*B*b^5*d+315/64*e^4/(a*e-b*
d)^6/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*B*b*d
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for
more details)Is a*e-b*d positive or negative?
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mupad [B] time = 2.84, size = 683, normalized size = 1.94 \begin {gather*} \frac {\frac {237\,{\left (d+e\,x\right )}^2\,\left (-11\,A\,b^2\,e^5+10\,B\,d\,b^2\,e^4+B\,a\,b\,e^5\right )}{64\,{\left (a\,e-b\,d\right )}^3}-\frac {2\,\left (A\,e^5-B\,d\,e^4\right )}{a\,e-b\,d}+\frac {147\,{\left (d+e\,x\right )}^4\,\left (-11\,A\,b^4\,e^5+10\,B\,d\,b^4\,e^4+B\,a\,b^3\,e^5\right )}{64\,{\left (a\,e-b\,d\right )}^5}+\frac {193\,\left (d+e\,x\right )\,\left (B\,a\,e^5-11\,A\,b\,e^5+10\,B\,b\,d\,e^4\right )}{128\,{\left (a\,e-b\,d\right )}^2}+\frac {21\,b^2\,{\left (d+e\,x\right )}^3\,\left (B\,a\,e^5-11\,A\,b\,e^5+10\,B\,b\,d\,e^4\right )}{5\,{\left (a\,e-b\,d\right )}^4}+\frac {63\,b^4\,{\left (d+e\,x\right )}^5\,\left (B\,a\,e^5-11\,A\,b\,e^5+10\,B\,b\,d\,e^4\right )}{128\,{\left (a\,e-b\,d\right )}^6}}{\sqrt {d+e\,x}\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )-{\left (d+e\,x\right )}^{5/2}\,\left (-10\,a^3\,b^2\,e^3+30\,a^2\,b^3\,d\,e^2-30\,a\,b^4\,d^2\,e+10\,b^5\,d^3\right )+{\left (d+e\,x\right )}^{3/2}\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )+b^5\,{\left (d+e\,x\right )}^{11/2}-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^{9/2}+{\left (d+e\,x\right )}^{7/2}\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )}+\frac {63\,e^4\,\mathrm {atan}\left (\frac {63\,\sqrt {b}\,e^4\,\sqrt {d+e\,x}\,\left (B\,a\,e-11\,A\,b\,e+10\,B\,b\,d\right )\,\left (a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6\right )}{{\left (a\,e-b\,d\right )}^{13/2}\,\left (63\,B\,a\,e^5-693\,A\,b\,e^5+630\,B\,b\,d\,e^4\right )}\right )\,\left (B\,a\,e-11\,A\,b\,e+10\,B\,b\,d\right )}{128\,\sqrt {b}\,{\left (a\,e-b\,d\right )}^{13/2}} \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)^3),x)
[Out]
((237*(d + e*x)^2*(B*a*b*e^5 - 11*A*b^2*e^5 + 10*B*b^2*d*e^4))/(64*(a*e - b*d)^3) - (2*(A*e^5 - B*d*e^4))/(a*e
- b*d) + (147*(d + e*x)^4*(B*a*b^3*e^5 - 11*A*b^4*e^5 + 10*B*b^4*d*e^4))/(64*(a*e - b*d)^5) + (193*(d + e*x)*
(B*a*e^5 - 11*A*b*e^5 + 10*B*b*d*e^4))/(128*(a*e - b*d)^2) + (21*b^2*(d + e*x)^3*(B*a*e^5 - 11*A*b*e^5 + 10*B*
b*d*e^4))/(5*(a*e - b*d)^4) + (63*b^4*(d + e*x)^5*(B*a*e^5 - 11*A*b*e^5 + 10*B*b*d*e^4))/(128*(a*e - b*d)^6))/
((d + e*x)^(1/2)*(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4)
- (d + e*x)^(5/2)*(10*b^5*d^3 - 10*a^3*b^2*e^3 + 30*a^2*b^3*d*e^2 - 30*a*b^4*d^2*e) + (d + e*x)^(3/2)*(5*b^5*
d^4 + 5*a^4*b*e^4 - 20*a^3*b^2*d*e^3 + 30*a^2*b^3*d^2*e^2 - 20*a*b^4*d^3*e) + b^5*(d + e*x)^(11/2) - (5*b^5*d
- 5*a*b^4*e)*(d + e*x)^(9/2) + (d + e*x)^(7/2)*(10*b^5*d^2 + 10*a^2*b^3*e^2 - 20*a*b^4*d*e)) + (63*e^4*atan((6
3*b^(1/2)*e^4*(d + e*x)^(1/2)*(B*a*e - 11*A*b*e + 10*B*b*d)*(a^6*e^6 + b^6*d^6 + 15*a^2*b^4*d^4*e^2 - 20*a^3*b
^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a*b^5*d^5*e - 6*a^5*b*d*e^5))/((a*e - b*d)^(13/2)*(63*B*a*e^5 - 693*A*b*e^
5 + 630*B*b*d*e^4)))*(B*a*e - 11*A*b*e + 10*B*b*d))/(128*b^(1/2)*(a*e - b*d)^(13/2))
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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)**(3/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
Timed out
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