3.17.13 \(\int \frac {A+B x}{(d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)^3} \, dx\)
Optimal. Leaf size=400 \[ \frac {231 e^4 (3 a B e-13 A b e+10 b B d)}{128 \sqrt {d+e x} (b d-a e)^7}+\frac {77 e^4 (3 a B e-13 A b e+10 b B d)}{128 b (d+e x)^{3/2} (b d-a e)^6}-\frac {231 \sqrt {b} e^4 (3 a B e-13 A b e+10 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{15/2}}+\frac {231 e^3 (3 a B e-13 A b e+10 b B d)}{640 b (a+b x) (d+e x)^{3/2} (b d-a e)^5}-\frac {33 e^2 (3 a B e-13 A b e+10 b B d)}{320 b (a+b x)^2 (d+e x)^{3/2} (b d-a e)^4}+\frac {11 e (3 a B e-13 A b e+10 b B d)}{240 b (a+b x)^3 (d+e x)^{3/2} (b d-a e)^3}-\frac {3 a B e-13 A b e+10 b B d}{40 b (a+b x)^4 (d+e x)^{3/2} (b d-a e)^2}-\frac {A b-a B}{5 b (a+b x)^5 (d+e x)^{3/2} (b d-a e)} \]
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Rubi [A] time = 0.45, antiderivative size = 400, normalized size of antiderivative = 1.00,
number of steps used = 10, 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 {231 e^4 (3 a B e-13 A b e+10 b B d)}{128 \sqrt {d+e x} (b d-a e)^7}+\frac {77 e^4 (3 a B e-13 A b e+10 b B d)}{128 b (d+e x)^{3/2} (b d-a e)^6}+\frac {231 e^3 (3 a B e-13 A b e+10 b B d)}{640 b (a+b x) (d+e x)^{3/2} (b d-a e)^5}-\frac {33 e^2 (3 a B e-13 A b e+10 b B d)}{320 b (a+b x)^2 (d+e x)^{3/2} (b d-a e)^4}-\frac {231 \sqrt {b} e^4 (3 a B e-13 A b e+10 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{15/2}}+\frac {11 e (3 a B e-13 A b e+10 b B d)}{240 b (a+b x)^3 (d+e x)^{3/2} (b d-a e)^3}-\frac {3 a B e-13 A b e+10 b B d}{40 b (a+b x)^4 (d+e x)^{3/2} (b d-a e)^2}-\frac {A b-a B}{5 b (a+b x)^5 (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)^3),x]
[Out]
(77*e^4*(10*b*B*d - 13*A*b*e + 3*a*B*e))/(128*b*(b*d - a*e)^6*(d + e*x)^(3/2)) - (A*b - a*B)/(5*b*(b*d - a*e)*
(a + b*x)^5*(d + e*x)^(3/2)) - (10*b*B*d - 13*A*b*e + 3*a*B*e)/(40*b*(b*d - a*e)^2*(a + b*x)^4*(d + e*x)^(3/2)
) + (11*e*(10*b*B*d - 13*A*b*e + 3*a*B*e))/(240*b*(b*d - a*e)^3*(a + b*x)^3*(d + e*x)^(3/2)) - (33*e^2*(10*b*B
*d - 13*A*b*e + 3*a*B*e))/(320*b*(b*d - a*e)^4*(a + b*x)^2*(d + e*x)^(3/2)) + (231*e^3*(10*b*B*d - 13*A*b*e +
3*a*B*e))/(640*b*(b*d - a*e)^5*(a + b*x)*(d + e*x)^(3/2)) + (231*e^4*(10*b*B*d - 13*A*b*e + 3*a*B*e))/(128*(b*
d - a*e)^7*Sqrt[d + e*x]) - (231*Sqrt[b]*e^4*(10*b*B*d - 13*A*b*e + 3*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/S
qrt[b*d - a*e]])/(128*(b*d - a*e)^(15/2))
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)^{5/2} \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)^{5/2}} \, dx\\ &=-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{3/2}}+\frac {(10 b B d-13 A b e+3 a B e) \int \frac {1}{(a+b x)^5 (d+e x)^{5/2}} \, dx}{10 b (b d-a e)}\\ &=-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{3/2}}-\frac {10 b B d-13 A b e+3 a B e}{40 b (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}-\frac {(11 e (10 b B d-13 A b e+3 a B e)) \int \frac {1}{(a+b x)^4 (d+e x)^{5/2}} \, dx}{80 b (b d-a e)^2}\\ &=-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{3/2}}-\frac {10 b B d-13 A b e+3 a B e}{40 b (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}+\frac {11 e (10 b B d-13 A b e+3 a B e)}{240 b (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}+\frac {\left (33 e^2 (10 b B d-13 A b e+3 a B e)\right ) \int \frac {1}{(a+b x)^3 (d+e x)^{5/2}} \, dx}{160 b (b d-a e)^3}\\ &=-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{3/2}}-\frac {10 b B d-13 A b e+3 a B e}{40 b (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}+\frac {11 e (10 b B d-13 A b e+3 a B e)}{240 b (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}-\frac {33 e^2 (10 b B d-13 A b e+3 a B e)}{320 b (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}-\frac {\left (231 e^3 (10 b B d-13 A b e+3 a B e)\right ) \int \frac {1}{(a+b x)^2 (d+e x)^{5/2}} \, dx}{640 b (b d-a e)^4}\\ &=-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{3/2}}-\frac {10 b B d-13 A b e+3 a B e}{40 b (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}+\frac {11 e (10 b B d-13 A b e+3 a B e)}{240 b (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}-\frac {33 e^2 (10 b B d-13 A b e+3 a B e)}{320 b (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}+\frac {231 e^3 (10 b B d-13 A b e+3 a B e)}{640 b (b d-a e)^5 (a+b x) (d+e x)^{3/2}}+\frac {\left (231 e^4 (10 b B d-13 A b e+3 a B e)\right ) \int \frac {1}{(a+b x) (d+e x)^{5/2}} \, dx}{256 b (b d-a e)^5}\\ &=\frac {77 e^4 (10 b B d-13 A b e+3 a B e)}{128 b (b d-a e)^6 (d+e x)^{3/2}}-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{3/2}}-\frac {10 b B d-13 A b e+3 a B e}{40 b (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}+\frac {11 e (10 b B d-13 A b e+3 a B e)}{240 b (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}-\frac {33 e^2 (10 b B d-13 A b e+3 a B e)}{320 b (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}+\frac {231 e^3 (10 b B d-13 A b e+3 a B e)}{640 b (b d-a e)^5 (a+b x) (d+e x)^{3/2}}+\frac {\left (231 e^4 (10 b B d-13 A b e+3 a B e)\right ) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{256 (b d-a e)^6}\\ &=\frac {77 e^4 (10 b B d-13 A b e+3 a B e)}{128 b (b d-a e)^6 (d+e x)^{3/2}}-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{3/2}}-\frac {10 b B d-13 A b e+3 a B e}{40 b (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}+\frac {11 e (10 b B d-13 A b e+3 a B e)}{240 b (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}-\frac {33 e^2 (10 b B d-13 A b e+3 a B e)}{320 b (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}+\frac {231 e^3 (10 b B d-13 A b e+3 a B e)}{640 b (b d-a e)^5 (a+b x) (d+e x)^{3/2}}+\frac {231 e^4 (10 b B d-13 A b e+3 a B e)}{128 (b d-a e)^7 \sqrt {d+e x}}+\frac {\left (231 b e^4 (10 b B d-13 A b e+3 a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 (b d-a e)^7}\\ &=\frac {77 e^4 (10 b B d-13 A b e+3 a B e)}{128 b (b d-a e)^6 (d+e x)^{3/2}}-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{3/2}}-\frac {10 b B d-13 A b e+3 a B e}{40 b (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}+\frac {11 e (10 b B d-13 A b e+3 a B e)}{240 b (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}-\frac {33 e^2 (10 b B d-13 A b e+3 a B e)}{320 b (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}+\frac {231 e^3 (10 b B d-13 A b e+3 a B e)}{640 b (b d-a e)^5 (a+b x) (d+e x)^{3/2}}+\frac {231 e^4 (10 b B d-13 A b e+3 a B e)}{128 (b d-a e)^7 \sqrt {d+e x}}+\frac {\left (231 b e^3 (10 b B d-13 A b e+3 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)^7}\\ &=\frac {77 e^4 (10 b B d-13 A b e+3 a B e)}{128 b (b d-a e)^6 (d+e x)^{3/2}}-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{3/2}}-\frac {10 b B d-13 A b e+3 a B e}{40 b (b d-a e)^2 (a+b x)^4 (d+e x)^{3/2}}+\frac {11 e (10 b B d-13 A b e+3 a B e)}{240 b (b d-a e)^3 (a+b x)^3 (d+e x)^{3/2}}-\frac {33 e^2 (10 b B d-13 A b e+3 a B e)}{320 b (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}+\frac {231 e^3 (10 b B d-13 A b e+3 a B e)}{640 b (b d-a e)^5 (a+b x) (d+e x)^{3/2}}+\frac {231 e^4 (10 b B d-13 A b e+3 a B e)}{128 (b d-a e)^7 \sqrt {d+e x}}-\frac {231 \sqrt {b} e^4 (10 b B d-13 A b e+3 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{15/2}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 99, normalized size = 0.25 \begin {gather*} \frac {\frac {e^4 (3 a B e-13 A b e+10 b B d) \, _2F_1\left (-\frac {3}{2},5;-\frac {1}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^5}+\frac {3 a B-3 A b}{(a+b x)^5}}{15 b (d+e x)^{3/2} (b d-a e)} \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)^3),x]
[Out]
((-3*A*b + 3*a*B)/(a + b*x)^5 + (e^4*(10*b*B*d - 13*A*b*e + 3*a*B*e)*Hypergeometric2F1[-3/2, 5, -1/2, (b*(d +
e*x))/(b*d - a*e)])/(b*d - a*e)^5)/(15*b*(b*d - a*e)*(d + e*x)^(3/2))
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IntegrateAlgebraic [B] time = 5.49, size = 1185, normalized size = 2.96 \begin {gather*} -\frac {\left (-1280 b^6 B d^7+1280 A b^6 e d^6+7680 a b^5 B e d^6-12800 b^6 B (d+e x) d^6-7680 a A b^5 e^2 d^5-19200 a^2 b^4 B e^2 d^5+106150 b^6 B (d+e x)^2 d^5+16640 A b^6 e (d+e x) d^5+60160 a b^5 B e (d+e x) d^5+19200 a^2 A b^4 e^3 d^4+25600 a^3 b^3 B e^3 d^4-260700 b^6 B (d+e x)^3 d^4-137995 A b^6 e (d+e x)^2 d^4-392755 a b^5 B e (d+e x)^2 d^4-83200 a A b^5 e^2 (d+e x) d^4-108800 a^2 b^4 B e^2 (d+e x) d^4-25600 a^3 A b^3 e^4 d^3-19200 a^4 b^2 B e^4 d^3+295680 b^6 B (d+e x)^4 d^3+338910 A b^6 e (d+e x)^3 d^3+703890 a b^5 B e (d+e x)^3 d^3+551980 a A b^5 e^2 (d+e x)^2 d^3+509520 a^2 b^4 B e^2 (d+e x)^2 d^3+166400 a^2 A b^4 e^3 (d+e x) d^3+89600 a^3 b^3 B e^3 (d+e x) d^3+19200 a^4 A b^2 e^5 d^2+7680 a^5 b B e^5 d^2-161700 b^6 B (d+e x)^5 d^2-384384 A b^6 e (d+e x)^4 d^2-502656 a b^5 B e (d+e x)^4 d^2-1016730 a A b^5 e^2 (d+e x)^3 d^2-547470 a^2 b^4 B e^2 (d+e x)^3 d^2-827970 a^2 A b^4 e^3 (d+e x)^2 d^2-233530 a^3 b^3 B e^3 (d+e x)^2 d^2-166400 a^3 A b^3 e^4 (d+e x) d^2-25600 a^4 b^2 B e^4 (d+e x) d^2-7680 a^5 A b e^6 d-1280 a^6 B e^6 d+34650 b^6 B (d+e x)^6 d+210210 A b^6 e (d+e x)^5 d+113190 a b^5 B e (d+e x)^5 d+768768 a A b^5 e^2 (d+e x)^4 d+118272 a^2 b^4 B e^2 (d+e x)^4 d+1016730 a^2 A b^4 e^3 (d+e x)^3 d+26070 a^3 b^3 B e^3 (d+e x)^3 d+551980 a^3 A b^3 e^4 (d+e x)^2 d-21230 a^4 b^2 B e^4 (d+e x)^2 d+83200 a^4 A b^2 e^5 (d+e x) d-6400 a^5 b B e^5 (d+e x) d+1280 a^6 A e^7-45045 A b^6 e (d+e x)^6+10395 a b^5 B e (d+e x)^6-210210 a A b^5 e^2 (d+e x)^5+48510 a^2 b^4 B e^2 (d+e x)^5-384384 a^2 A b^4 e^3 (d+e x)^4+88704 a^3 b^3 B e^3 (d+e x)^4-338910 a^3 A b^3 e^4 (d+e x)^3+78210 a^4 b^2 B e^4 (d+e x)^3-137995 a^4 A b^2 e^5 (d+e x)^2+31845 a^5 b B e^5 (d+e x)^2-16640 a^5 A b e^6 (d+e x)+3840 a^6 B e^6 (d+e x)\right ) e^4}{1920 (b d-a e)^7 (d+e x)^{3/2} (b d-a e-b (d+e x))^5}-\frac {231 \left (-13 A b^{3/2} e^5+3 a \sqrt {b} B e^5+10 b^{3/2} 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 (b d-a e)^7 \sqrt {a e-b d}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(A + B*x)/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]
[Out]
-1/1920*(e^4*(-1280*b^6*B*d^7 + 1280*A*b^6*d^6*e + 7680*a*b^5*B*d^6*e - 7680*a*A*b^5*d^5*e^2 - 19200*a^2*b^4*B
*d^5*e^2 + 19200*a^2*A*b^4*d^4*e^3 + 25600*a^3*b^3*B*d^4*e^3 - 25600*a^3*A*b^3*d^3*e^4 - 19200*a^4*b^2*B*d^3*e
^4 + 19200*a^4*A*b^2*d^2*e^5 + 7680*a^5*b*B*d^2*e^5 - 7680*a^5*A*b*d*e^6 - 1280*a^6*B*d*e^6 + 1280*a^6*A*e^7 -
12800*b^6*B*d^6*(d + e*x) + 16640*A*b^6*d^5*e*(d + e*x) + 60160*a*b^5*B*d^5*e*(d + e*x) - 83200*a*A*b^5*d^4*e
^2*(d + e*x) - 108800*a^2*b^4*B*d^4*e^2*(d + e*x) + 166400*a^2*A*b^4*d^3*e^3*(d + e*x) + 89600*a^3*b^3*B*d^3*e
^3*(d + e*x) - 166400*a^3*A*b^3*d^2*e^4*(d + e*x) - 25600*a^4*b^2*B*d^2*e^4*(d + e*x) + 83200*a^4*A*b^2*d*e^5*
(d + e*x) - 6400*a^5*b*B*d*e^5*(d + e*x) - 16640*a^5*A*b*e^6*(d + e*x) + 3840*a^6*B*e^6*(d + e*x) + 106150*b^6
*B*d^5*(d + e*x)^2 - 137995*A*b^6*d^4*e*(d + e*x)^2 - 392755*a*b^5*B*d^4*e*(d + e*x)^2 + 551980*a*A*b^5*d^3*e^
2*(d + e*x)^2 + 509520*a^2*b^4*B*d^3*e^2*(d + e*x)^2 - 827970*a^2*A*b^4*d^2*e^3*(d + e*x)^2 - 233530*a^3*b^3*B
*d^2*e^3*(d + e*x)^2 + 551980*a^3*A*b^3*d*e^4*(d + e*x)^2 - 21230*a^4*b^2*B*d*e^4*(d + e*x)^2 - 137995*a^4*A*b
^2*e^5*(d + e*x)^2 + 31845*a^5*b*B*e^5*(d + e*x)^2 - 260700*b^6*B*d^4*(d + e*x)^3 + 338910*A*b^6*d^3*e*(d + e*
x)^3 + 703890*a*b^5*B*d^3*e*(d + e*x)^3 - 1016730*a*A*b^5*d^2*e^2*(d + e*x)^3 - 547470*a^2*b^4*B*d^2*e^2*(d +
e*x)^3 + 1016730*a^2*A*b^4*d*e^3*(d + e*x)^3 + 26070*a^3*b^3*B*d*e^3*(d + e*x)^3 - 338910*a^3*A*b^3*e^4*(d + e
*x)^3 + 78210*a^4*b^2*B*e^4*(d + e*x)^3 + 295680*b^6*B*d^3*(d + e*x)^4 - 384384*A*b^6*d^2*e*(d + e*x)^4 - 5026
56*a*b^5*B*d^2*e*(d + e*x)^4 + 768768*a*A*b^5*d*e^2*(d + e*x)^4 + 118272*a^2*b^4*B*d*e^2*(d + e*x)^4 - 384384*
a^2*A*b^4*e^3*(d + e*x)^4 + 88704*a^3*b^3*B*e^3*(d + e*x)^4 - 161700*b^6*B*d^2*(d + e*x)^5 + 210210*A*b^6*d*e*
(d + e*x)^5 + 113190*a*b^5*B*d*e*(d + e*x)^5 - 210210*a*A*b^5*e^2*(d + e*x)^5 + 48510*a^2*b^4*B*e^2*(d + e*x)^
5 + 34650*b^6*B*d*(d + e*x)^6 - 45045*A*b^6*e*(d + e*x)^6 + 10395*a*b^5*B*e*(d + e*x)^6))/((b*d - a*e)^7*(d +
e*x)^(3/2)*(b*d - a*e - b*(d + e*x))^5) - (231*(10*b^(3/2)*B*d*e^4 - 13*A*b^(3/2)*e^5 + 3*a*Sqrt[b]*B*e^5)*Arc
Tan[(Sqrt[b]*Sqrt[-(b*d) + a*e]*Sqrt[d + e*x])/(b*d - a*e)])/(128*(b*d - a*e)^7*Sqrt[-(b*d) + a*e])
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fricas [B] time = 0.57, size = 4664, normalized size = 11.66
result too large to display
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)^3,x, algorithm="fricas")
[Out]
[1/3840*(3465*(10*B*a^5*b*d^3*e^4 + (3*B*a^6 - 13*A*a^5*b)*d^2*e^5 + (10*B*b^6*d*e^6 + (3*B*a*b^5 - 13*A*b^6)*
e^7)*x^7 + (20*B*b^6*d^2*e^5 + 2*(28*B*a*b^5 - 13*A*b^6)*d*e^6 + 5*(3*B*a^2*b^4 - 13*A*a*b^5)*e^7)*x^6 + (10*B
*b^6*d^3*e^4 + (103*B*a*b^5 - 13*A*b^6)*d^2*e^5 + 130*(B*a^2*b^4 - A*a*b^5)*d*e^6 + 10*(3*B*a^3*b^3 - 13*A*a^2
*b^4)*e^7)*x^5 + 5*(10*B*a*b^5*d^3*e^4 + (43*B*a^2*b^4 - 13*A*a*b^5)*d^2*e^5 + 4*(8*B*a^3*b^3 - 13*A*a^2*b^4)*
d*e^6 + 2*(3*B*a^4*b^2 - 13*A*a^3*b^3)*e^7)*x^4 + 5*(20*B*a^2*b^4*d^3*e^4 + 2*(23*B*a^3*b^3 - 13*A*a^2*b^4)*d^
2*e^5 + 2*(11*B*a^4*b^2 - 26*A*a^3*b^3)*d*e^6 + (3*B*a^5*b - 13*A*a^4*b^2)*e^7)*x^3 + (100*B*a^3*b^3*d^3*e^4 +
130*(B*a^4*b^2 - A*a^3*b^3)*d^2*e^5 + 10*(4*B*a^5*b - 13*A*a^4*b^2)*d*e^6 + (3*B*a^6 - 13*A*a^5*b)*e^7)*x^2 +
(50*B*a^4*b^2*d^3*e^4 + 5*(7*B*a^5*b - 13*A*a^4*b^2)*d^2*e^5 + 2*(3*B*a^6 - 13*A*a^5*b)*d*e^6)*x)*sqrt(b/(b*d
- a*e))*log((b*e*x + 2*b*d - a*e - 2*(b*d - a*e)*sqrt(e*x + d)*sqrt(b/(b*d - a*e)))/(b*x + a)) + 2*(1280*A*a^
6*e^6 - 96*(B*a*b^5 + 4*A*b^6)*d^6 + 16*(52*B*a^2*b^4 + 183*A*a*b^5)*d^5*e - 28*(127*B*a^3*b^3 + 358*A*a^2*b^4
)*d^4*e^2 + 70*(174*B*a^4*b^2 + 301*A*a^3*b^3)*d^3*e^3 + 5*(6625*B*a^5*b - 7119*A*a^4*b^2)*d^2*e^4 + 1280*(2*B
*a^6 - 19*A*a^5*b)*d*e^5 + 3465*(10*B*b^6*d*e^5 + (3*B*a*b^5 - 13*A*b^6)*e^6)*x^6 + 2310*(20*B*b^6*d^2*e^4 + 2
*(38*B*a*b^5 - 13*A*b^6)*d*e^5 + 7*(3*B*a^2*b^4 - 13*A*a*b^5)*e^6)*x^5 + 231*(30*B*b^6*d^3*e^3 + 13*(73*B*a*b^
5 - 3*A*b^6)*d^2*e^4 + 2*(781*B*a^2*b^4 - 611*A*a*b^5)*d*e^5 + 128*(3*B*a^3*b^3 - 13*A*a^2*b^4)*e^6)*x^4 - 66*
(30*B*b^6*d^4*e^2 - 3*(167*B*a*b^5 + 13*A*b^6)*d^3*e^3 - (6223*B*a^2*b^4 - 663*A*a*b^5)*d^2*e^4 - (5771*B*a^3*
b^3 - 7891*A*a^2*b^4)*d*e^5 - 395*(3*B*a^4*b^2 - 13*A*a^3*b^3)*e^6)*x^3 + 11*(80*B*b^6*d^5*e - 4*(209*B*a*b^5
+ 26*A*b^6)*d^4*e^2 + 2*(2811*B*a^2*b^4 + 559*A*a*b^5)*d^3*e^3 + 4*(8566*B*a^3*b^3 - 1911*A*a^2*b^4)*d^2*e^4 +
50*(388*B*a^4*b^2 - 845*A*a^3*b^3)*d*e^5 + 965*(3*B*a^5*b - 13*A*a^4*b^2)*e^6)*x^2 - 2*(240*B*b^6*d^6 - 8*(25
1*B*a*b^5 + 39*A*b^6)*d^5*e + 2*(4133*B*a^2*b^4 + 1352*A*a*b^5)*d^4*e^2 - 7*(3969*B*a^3*b^3 + 1651*A*a^2*b^4)*
d^3*e^3 - 5*(16657*B*a^4*b^2 - 7917*A*a^3*b^3)*d^2*e^4 - 5*(5729*B*a^5*b - 19279*A*a^4*b^2)*d*e^5 - 640*(3*B*a
^6 - 13*A*a^5*b)*e^6)*x)*sqrt(e*x + d))/(a^5*b^7*d^9 - 7*a^6*b^6*d^8*e + 21*a^7*b^5*d^7*e^2 - 35*a^8*b^4*d^6*e
^3 + 35*a^9*b^3*d^5*e^4 - 21*a^10*b^2*d^4*e^5 + 7*a^11*b*d^3*e^6 - a^12*d^2*e^7 + (b^12*d^7*e^2 - 7*a*b^11*d^6
*e^3 + 21*a^2*b^10*d^5*e^4 - 35*a^3*b^9*d^4*e^5 + 35*a^4*b^8*d^3*e^6 - 21*a^5*b^7*d^2*e^7 + 7*a^6*b^6*d*e^8 -
a^7*b^5*e^9)*x^7 + (2*b^12*d^8*e - 9*a*b^11*d^7*e^2 + 7*a^2*b^10*d^6*e^3 + 35*a^3*b^9*d^5*e^4 - 105*a^4*b^8*d^
4*e^5 + 133*a^5*b^7*d^3*e^6 - 91*a^6*b^6*d^2*e^7 + 33*a^7*b^5*d*e^8 - 5*a^8*b^4*e^9)*x^6 + (b^12*d^9 + 3*a*b^1
1*d^8*e - 39*a^2*b^10*d^7*e^2 + 105*a^3*b^9*d^6*e^3 - 105*a^4*b^8*d^5*e^4 - 21*a^5*b^7*d^4*e^5 + 147*a^6*b^6*d
^3*e^6 - 141*a^7*b^5*d^2*e^7 + 60*a^8*b^4*d*e^8 - 10*a^9*b^3*e^9)*x^5 + 5*(a*b^11*d^9 - 3*a^2*b^10*d^8*e - 5*a
^3*b^9*d^7*e^2 + 35*a^4*b^8*d^6*e^3 - 63*a^5*b^7*d^5*e^4 + 49*a^6*b^6*d^4*e^5 - 7*a^7*b^5*d^3*e^6 - 15*a^8*b^4
*d^2*e^7 + 10*a^9*b^3*d*e^8 - 2*a^10*b^2*e^9)*x^4 + 5*(2*a^2*b^10*d^9 - 10*a^3*b^9*d^8*e + 15*a^4*b^8*d^7*e^2
+ 7*a^5*b^7*d^6*e^3 - 49*a^6*b^6*d^5*e^4 + 63*a^7*b^5*d^4*e^5 - 35*a^8*b^4*d^3*e^6 + 5*a^9*b^3*d^2*e^7 + 3*a^1
0*b^2*d*e^8 - a^11*b*e^9)*x^3 + (10*a^3*b^9*d^9 - 60*a^4*b^8*d^8*e + 141*a^5*b^7*d^7*e^2 - 147*a^6*b^6*d^6*e^3
+ 21*a^7*b^5*d^5*e^4 + 105*a^8*b^4*d^4*e^5 - 105*a^9*b^3*d^3*e^6 + 39*a^10*b^2*d^2*e^7 - 3*a^11*b*d*e^8 - a^1
2*e^9)*x^2 + (5*a^4*b^8*d^9 - 33*a^5*b^7*d^8*e + 91*a^6*b^6*d^7*e^2 - 133*a^7*b^5*d^6*e^3 + 105*a^8*b^4*d^5*e^
4 - 35*a^9*b^3*d^4*e^5 - 7*a^10*b^2*d^3*e^6 + 9*a^11*b*d^2*e^7 - 2*a^12*d*e^8)*x), -1/1920*(3465*(10*B*a^5*b*d
^3*e^4 + (3*B*a^6 - 13*A*a^5*b)*d^2*e^5 + (10*B*b^6*d*e^6 + (3*B*a*b^5 - 13*A*b^6)*e^7)*x^7 + (20*B*b^6*d^2*e^
5 + 2*(28*B*a*b^5 - 13*A*b^6)*d*e^6 + 5*(3*B*a^2*b^4 - 13*A*a*b^5)*e^7)*x^6 + (10*B*b^6*d^3*e^4 + (103*B*a*b^5
- 13*A*b^6)*d^2*e^5 + 130*(B*a^2*b^4 - A*a*b^5)*d*e^6 + 10*(3*B*a^3*b^3 - 13*A*a^2*b^4)*e^7)*x^5 + 5*(10*B*a*
b^5*d^3*e^4 + (43*B*a^2*b^4 - 13*A*a*b^5)*d^2*e^5 + 4*(8*B*a^3*b^3 - 13*A*a^2*b^4)*d*e^6 + 2*(3*B*a^4*b^2 - 13
*A*a^3*b^3)*e^7)*x^4 + 5*(20*B*a^2*b^4*d^3*e^4 + 2*(23*B*a^3*b^3 - 13*A*a^2*b^4)*d^2*e^5 + 2*(11*B*a^4*b^2 - 2
6*A*a^3*b^3)*d*e^6 + (3*B*a^5*b - 13*A*a^4*b^2)*e^7)*x^3 + (100*B*a^3*b^3*d^3*e^4 + 130*(B*a^4*b^2 - A*a^3*b^3
)*d^2*e^5 + 10*(4*B*a^5*b - 13*A*a^4*b^2)*d*e^6 + (3*B*a^6 - 13*A*a^5*b)*e^7)*x^2 + (50*B*a^4*b^2*d^3*e^4 + 5*
(7*B*a^5*b - 13*A*a^4*b^2)*d^2*e^5 + 2*(3*B*a^6 - 13*A*a^5*b)*d*e^6)*x)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a*
e)*sqrt(e*x + d)*sqrt(-b/(b*d - a*e))/(b*e*x + b*d)) - (1280*A*a^6*e^6 - 96*(B*a*b^5 + 4*A*b^6)*d^6 + 16*(52*B
*a^2*b^4 + 183*A*a*b^5)*d^5*e - 28*(127*B*a^3*b^3 + 358*A*a^2*b^4)*d^4*e^2 + 70*(174*B*a^4*b^2 + 301*A*a^3*b^3
)*d^3*e^3 + 5*(6625*B*a^5*b - 7119*A*a^4*b^2)*d^2*e^4 + 1280*(2*B*a^6 - 19*A*a^5*b)*d*e^5 + 3465*(10*B*b^6*d*e
^5 + (3*B*a*b^5 - 13*A*b^6)*e^6)*x^6 + 2310*(20*B*b^6*d^2*e^4 + 2*(38*B*a*b^5 - 13*A*b^6)*d*e^5 + 7*(3*B*a^2*b
^4 - 13*A*a*b^5)*e^6)*x^5 + 231*(30*B*b^6*d^3*e^3 + 13*(73*B*a*b^5 - 3*A*b^6)*d^2*e^4 + 2*(781*B*a^2*b^4 - 611
*A*a*b^5)*d*e^5 + 128*(3*B*a^3*b^3 - 13*A*a^2*b^4)*e^6)*x^4 - 66*(30*B*b^6*d^4*e^2 - 3*(167*B*a*b^5 + 13*A*b^6
)*d^3*e^3 - (6223*B*a^2*b^4 - 663*A*a*b^5)*d^2*e^4 - (5771*B*a^3*b^3 - 7891*A*a^2*b^4)*d*e^5 - 395*(3*B*a^4*b^
2 - 13*A*a^3*b^3)*e^6)*x^3 + 11*(80*B*b^6*d^5*e - 4*(209*B*a*b^5 + 26*A*b^6)*d^4*e^2 + 2*(2811*B*a^2*b^4 + 559
*A*a*b^5)*d^3*e^3 + 4*(8566*B*a^3*b^3 - 1911*A*a^2*b^4)*d^2*e^4 + 50*(388*B*a^4*b^2 - 845*A*a^3*b^3)*d*e^5 + 9
65*(3*B*a^5*b - 13*A*a^4*b^2)*e^6)*x^2 - 2*(240*B*b^6*d^6 - 8*(251*B*a*b^5 + 39*A*b^6)*d^5*e + 2*(4133*B*a^2*b
^4 + 1352*A*a*b^5)*d^4*e^2 - 7*(3969*B*a^3*b^3 + 1651*A*a^2*b^4)*d^3*e^3 - 5*(16657*B*a^4*b^2 - 7917*A*a^3*b^3
)*d^2*e^4 - 5*(5729*B*a^5*b - 19279*A*a^4*b^2)*d*e^5 - 640*(3*B*a^6 - 13*A*a^5*b)*e^6)*x)*sqrt(e*x + d))/(a^5*
b^7*d^9 - 7*a^6*b^6*d^8*e + 21*a^7*b^5*d^7*e^2 - 35*a^8*b^4*d^6*e^3 + 35*a^9*b^3*d^5*e^4 - 21*a^10*b^2*d^4*e^5
+ 7*a^11*b*d^3*e^6 - a^12*d^2*e^7 + (b^12*d^7*e^2 - 7*a*b^11*d^6*e^3 + 21*a^2*b^10*d^5*e^4 - 35*a^3*b^9*d^4*e
^5 + 35*a^4*b^8*d^3*e^6 - 21*a^5*b^7*d^2*e^7 + 7*a^6*b^6*d*e^8 - a^7*b^5*e^9)*x^7 + (2*b^12*d^8*e - 9*a*b^11*d
^7*e^2 + 7*a^2*b^10*d^6*e^3 + 35*a^3*b^9*d^5*e^4 - 105*a^4*b^8*d^4*e^5 + 133*a^5*b^7*d^3*e^6 - 91*a^6*b^6*d^2*
e^7 + 33*a^7*b^5*d*e^8 - 5*a^8*b^4*e^9)*x^6 + (b^12*d^9 + 3*a*b^11*d^8*e - 39*a^2*b^10*d^7*e^2 + 105*a^3*b^9*d
^6*e^3 - 105*a^4*b^8*d^5*e^4 - 21*a^5*b^7*d^4*e^5 + 147*a^6*b^6*d^3*e^6 - 141*a^7*b^5*d^2*e^7 + 60*a^8*b^4*d*e
^8 - 10*a^9*b^3*e^9)*x^5 + 5*(a*b^11*d^9 - 3*a^2*b^10*d^8*e - 5*a^3*b^9*d^7*e^2 + 35*a^4*b^8*d^6*e^3 - 63*a^5*
b^7*d^5*e^4 + 49*a^6*b^6*d^4*e^5 - 7*a^7*b^5*d^3*e^6 - 15*a^8*b^4*d^2*e^7 + 10*a^9*b^3*d*e^8 - 2*a^10*b^2*e^9)
*x^4 + 5*(2*a^2*b^10*d^9 - 10*a^3*b^9*d^8*e + 15*a^4*b^8*d^7*e^2 + 7*a^5*b^7*d^6*e^3 - 49*a^6*b^6*d^5*e^4 + 63
*a^7*b^5*d^4*e^5 - 35*a^8*b^4*d^3*e^6 + 5*a^9*b^3*d^2*e^7 + 3*a^10*b^2*d*e^8 - a^11*b*e^9)*x^3 + (10*a^3*b^9*d
^9 - 60*a^4*b^8*d^8*e + 141*a^5*b^7*d^7*e^2 - 147*a^6*b^6*d^6*e^3 + 21*a^7*b^5*d^5*e^4 + 105*a^8*b^4*d^4*e^5 -
105*a^9*b^3*d^3*e^6 + 39*a^10*b^2*d^2*e^7 - 3*a^11*b*d*e^8 - a^12*e^9)*x^2 + (5*a^4*b^8*d^9 - 33*a^5*b^7*d^8*
e + 91*a^6*b^6*d^7*e^2 - 133*a^7*b^5*d^6*e^3 + 105*a^8*b^4*d^5*e^4 - 35*a^9*b^3*d^4*e^5 - 7*a^10*b^2*d^3*e^6 +
9*a^11*b*d^2*e^7 - 2*a^12*d*e^8)*x)]
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giac [B] time = 0.42, size = 1103, normalized size = 2.76
result too large to display
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)^3,x, algorithm="giac")
[Out]
231/128*(10*B*b^2*d*e^4 + 3*B*a*b*e^5 - 13*A*b^2*e^5)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^7*d^7 -
7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d
*e^6 - a^7*e^7)*sqrt(-b^2*d + a*b*e)) + 2/3*(15*(x*e + d)*B*b*d*e^4 + B*b*d^2*e^4 + 3*(x*e + d)*B*a*e^5 - 18*(
x*e + d)*A*b*e^5 - B*a*d*e^5 - A*b*d*e^5 + A*a*e^6)/((b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^3*b^
4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^7)*(x*e + d)^(3/2)) + 1/1920*(1545
0*(x*e + d)^(9/2)*B*b^6*d*e^4 - 66980*(x*e + d)^(7/2)*B*b^6*d^2*e^4 + 110080*(x*e + d)^(5/2)*B*b^6*d^3*e^4 - 8
1500*(x*e + d)^(3/2)*B*b^6*d^4*e^4 + 22950*sqrt(x*e + d)*B*b^6*d^5*e^4 + 6555*(x*e + d)^(9/2)*B*a*b^5*e^5 - 22
005*(x*e + d)^(9/2)*A*b^6*e^5 + 37670*(x*e + d)^(7/2)*B*a*b^5*d*e^5 + 96290*(x*e + d)^(7/2)*A*b^6*d*e^5 - 1698
56*(x*e + d)^(5/2)*B*a*b^5*d^2*e^5 - 160384*(x*e + d)^(5/2)*A*b^6*d^2*e^5 + 204690*(x*e + d)^(3/2)*B*a*b^5*d^3
*e^5 + 121310*(x*e + d)^(3/2)*A*b^6*d^3*e^5 - 79155*sqrt(x*e + d)*B*a*b^5*d^4*e^5 - 35595*sqrt(x*e + d)*A*b^6*
d^4*e^5 + 29310*(x*e + d)^(7/2)*B*a^2*b^4*e^6 - 96290*(x*e + d)^(7/2)*A*a*b^5*e^6 + 9472*(x*e + d)^(5/2)*B*a^2
*b^4*d*e^6 + 320768*(x*e + d)^(5/2)*A*a*b^5*d*e^6 - 125070*(x*e + d)^(3/2)*B*a^2*b^4*d^2*e^6 - 363930*(x*e + d
)^(3/2)*A*a*b^5*d^2*e^6 + 87120*sqrt(x*e + d)*B*a^2*b^4*d^3*e^6 + 142380*sqrt(x*e + d)*A*a*b^5*d^3*e^6 + 50304
*(x*e + d)^(5/2)*B*a^3*b^3*e^7 - 160384*(x*e + d)^(5/2)*A*a^2*b^4*e^7 - 37930*(x*e + d)^(3/2)*B*a^3*b^3*d*e^7
+ 363930*(x*e + d)^(3/2)*A*a^2*b^4*d*e^7 - 15930*sqrt(x*e + d)*B*a^3*b^3*d^2*e^7 - 213570*sqrt(x*e + d)*A*a^2*
b^4*d^2*e^7 + 39810*(x*e + d)^(3/2)*B*a^4*b^2*e^8 - 121310*(x*e + d)^(3/2)*A*a^3*b^3*e^8 - 27630*sqrt(x*e + d)
*B*a^4*b^2*d*e^8 + 142380*sqrt(x*e + d)*A*a^3*b^3*d*e^8 + 12645*sqrt(x*e + d)*B*a^5*b*e^9 - 35595*sqrt(x*e + d
)*A*a^4*b^2*e^9)/((b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21
*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^7)*((x*e + d)*b - b*d + a*e)^5)
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maple [B] time = 0.09, size = 1653, normalized size = 4.13
result too large to display
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)^3,x)
[Out]
3793/192*e^7/(a*e-b*d)^7*b^3/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a^3*d+531/64*e^7/(a*e-b*d)^7*b^3/(b*e*x+a*e)^5*(e*x
+d)^(1/2)*B*a^3*d^2+4169/64*e^6/(a*e-b*d)^7*b^4/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a^2*d^2-6823/64*e^5/(a*e-b*d)^7*
b^5/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a*d^3-363/8*e^6/(a*e-b*d)^7*b^4/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^2*d^3+5277/1
28*e^5/(a*e-b*d)^7*b^5/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a*d^4+7119/64*e^7/(a*e-b*d)^7*b^4/(b*e*x+a*e)^5*(e*x+d)^(
1/2)*A*a^2*d^2-2373/32*e^6/(a*e-b*d)^7*b^5/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a*d^3-2506/15*e^6/(a*e-b*d)^7*b^5/(b*
e*x+a*e)^5*(e*x+d)^(5/2)*A*a*d-74/15*e^6/(a*e-b*d)^7*b^4/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*a^2*d+921/64*e^8/(a*e-b
*d)^7*b^2/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^4*d-12131/64*e^7/(a*e-b*d)^7*b^4/(b*e*x+a*e)^5*(e*x+d)^(3/2)*A*a^2*d
+12131/64*e^6/(a*e-b*d)^7*b^5/(b*e*x+a*e)^5*(e*x+d)^(3/2)*A*a*d^2-2373/32*e^8/(a*e-b*d)^7*b^3/(b*e*x+a*e)^5*(e
*x+d)^(1/2)*A*a^3*d-3767/192*e^5/(a*e-b*d)^7*b^5/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*a*d+1327/15*e^5/(a*e-b*d)^7*b^5
/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*a*d^2-2/3*e^5/(a*e-b*d)^6/(e*x+d)^(3/2)*A+2/3*e^4/(a*e-b*d)^6/(e*x+d)^(3/2)*B*d
+12*e^5/(a*e-b*d)^7/(e*x+d)^(1/2)*A*b-2*e^5/(a*e-b*d)^7/(e*x+d)^(1/2)*a*B-10*e^4/(a*e-b*d)^7/(e*x+d)^(1/2)*B*b
*d+3003/128*e^5/(a*e-b*d)^7*b^2/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*A+1467/128*e^5
/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*(e*x+d)^(9/2)*A+12131/192*e^8/(a*e-b*d)^7*b^3/(b*e*x+a*e)^5*(e*x+d)^(3/2)*A*a^3
-12131/192*e^5/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*A*d^3-1155/64*e^4/(a*e-b*d)^7*b^2/((a*e-b*d)*b)^(1/
2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*B*d-515/64*e^4/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*(e*x+d)^(9/2)*B*d-
765/64*e^4/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*d^5+3349/96*e^4/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*B*(e*x+
d)^(7/2)*d^2-1327/64*e^8/(a*e-b*d)^7*b^2/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a^4+2373/128*e^9/(a*e-b*d)^7*b^2/(b*e*x
+a*e)^5*(e*x+d)^(1/2)*A*a^4+2373/128*e^5/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*d^4+9629/192*e^6/(a*e-b
*d)^7*b^5/(b*e*x+a*e)^5*A*(e*x+d)^(7/2)*a-9629/192*e^5/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*A*(e*x+d)^(7/2)*d-977/64*
e^6/(a*e-b*d)^7*b^4/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*a^2-172/3*e^4/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*
d^3+4075/96*e^4/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*d^4-693/128*e^5/(a*e-b*d)^7*b/((a*e-b*d)*b)^(1/2
)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a*B-843/128*e^9/(a*e-b*d)^7*b/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^5-
437/128*e^5/(a*e-b*d)^7*b^5/(b*e*x+a*e)^5*(e*x+d)^(9/2)*B*a+1253/15*e^7/(a*e-b*d)^7*b^4/(b*e*x+a*e)^5*(e*x+d)^
(5/2)*A*a^2+1253/15*e^5/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*(e*x+d)^(5/2)*A*d^2-131/5*e^7/(a*e-b*d)^7*b^3/(b*e*x+a*e
)^5*(e*x+d)^(5/2)*B*a^3
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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)^(5/2)/(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 = 3.15, size = 755, normalized size = 1.89 \begin {gather*} -\frac {\frac {2\,\left (A\,e^5-B\,d\,e^4\right )}{3\,\left (a\,e-b\,d\right )}+\frac {2123\,{\left (d+e\,x\right )}^2\,\left (-13\,A\,b^2\,e^5+10\,B\,d\,b^2\,e^4+3\,B\,a\,b\,e^5\right )}{384\,{\left (a\,e-b\,d\right )}^3}+\frac {869\,{\left (d+e\,x\right )}^3\,\left (-13\,A\,b^3\,e^5+10\,B\,d\,b^3\,e^4+3\,B\,a\,b^2\,e^5\right )}{64\,{\left (a\,e-b\,d\right )}^4}+\frac {539\,{\left (d+e\,x\right )}^5\,\left (-13\,A\,b^5\,e^5+10\,B\,d\,b^5\,e^4+3\,B\,a\,b^4\,e^5\right )}{64\,{\left (a\,e-b\,d\right )}^6}+\frac {2\,\left (d+e\,x\right )\,\left (3\,B\,a\,e^5-13\,A\,b\,e^5+10\,B\,b\,d\,e^4\right )}{3\,{\left (a\,e-b\,d\right )}^2}+\frac {77\,b^3\,{\left (d+e\,x\right )}^4\,\left (3\,B\,a\,e^5-13\,A\,b\,e^5+10\,B\,b\,d\,e^4\right )}{5\,{\left (a\,e-b\,d\right )}^5}+\frac {231\,b^5\,{\left (d+e\,x\right )}^6\,\left (3\,B\,a\,e^5-13\,A\,b\,e^5+10\,B\,b\,d\,e^4\right )}{128\,{\left (a\,e-b\,d\right )}^7}}{{\left (d+e\,x\right )}^{3/2}\,\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 )}^{7/2}\,\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 )}^{5/2}\,\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 )}^{13/2}-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^{11/2}+{\left (d+e\,x\right )}^{9/2}\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )}-\frac {231\,\sqrt {b}\,e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^4\,\sqrt {d+e\,x}\,\left (3\,B\,a\,e-13\,A\,b\,e+10\,B\,b\,d\right )\,\left (a^7\,e^7-7\,a^6\,b\,d\,e^6+21\,a^5\,b^2\,d^2\,e^5-35\,a^4\,b^3\,d^3\,e^4+35\,a^3\,b^4\,d^4\,e^3-21\,a^2\,b^5\,d^5\,e^2+7\,a\,b^6\,d^6\,e-b^7\,d^7\right )}{{\left (a\,e-b\,d\right )}^{15/2}\,\left (3\,B\,a\,e^5-13\,A\,b\,e^5+10\,B\,b\,d\,e^4\right )}\right )\,\left (3\,B\,a\,e-13\,A\,b\,e+10\,B\,b\,d\right )}{128\,{\left (a\,e-b\,d\right )}^{15/2}} \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)^3),x)
[Out]
- ((2*(A*e^5 - B*d*e^4))/(3*(a*e - b*d)) + (2123*(d + e*x)^2*(3*B*a*b*e^5 - 13*A*b^2*e^5 + 10*B*b^2*d*e^4))/(3
84*(a*e - b*d)^3) + (869*(d + e*x)^3*(3*B*a*b^2*e^5 - 13*A*b^3*e^5 + 10*B*b^3*d*e^4))/(64*(a*e - b*d)^4) + (53
9*(d + e*x)^5*(3*B*a*b^4*e^5 - 13*A*b^5*e^5 + 10*B*b^5*d*e^4))/(64*(a*e - b*d)^6) + (2*(d + e*x)*(3*B*a*e^5 -
13*A*b*e^5 + 10*B*b*d*e^4))/(3*(a*e - b*d)^2) + (77*b^3*(d + e*x)^4*(3*B*a*e^5 - 13*A*b*e^5 + 10*B*b*d*e^4))/(
5*(a*e - b*d)^5) + (231*b^5*(d + e*x)^6*(3*B*a*e^5 - 13*A*b*e^5 + 10*B*b*d*e^4))/(128*(a*e - b*d)^7))/((d + e*
x)^(3/2)*(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)^(7/2)*(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)^(5/2)*(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)^(13/2) - (5*b^5*d - 5*a*b^
4*e)*(d + e*x)^(11/2) + (d + e*x)^(9/2)*(10*b^5*d^2 + 10*a^2*b^3*e^2 - 20*a*b^4*d*e)) - (231*b^(1/2)*e^4*atan(
(b^(1/2)*e^4*(d + e*x)^(1/2)*(3*B*a*e - 13*A*b*e + 10*B*b*d)*(a^7*e^7 - b^7*d^7 - 21*a^2*b^5*d^5*e^2 + 35*a^3*
b^4*d^4*e^3 - 35*a^4*b^3*d^3*e^4 + 21*a^5*b^2*d^2*e^5 + 7*a*b^6*d^6*e - 7*a^6*b*d*e^6))/((a*e - b*d)^(15/2)*(3
*B*a*e^5 - 13*A*b*e^5 + 10*B*b*d*e^4)))*(3*B*a*e - 13*A*b*e + 10*B*b*d))/(128*(a*e - b*d)^(15/2))
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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)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
Timed out
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