3.4.85 \(\int \frac {1}{(d+e x)^{5/2} (b x+c x^2)^3} \, dx\) [385]
Optimal. Leaf size=470 \[ \frac {e \left (72 c^4 d^4-144 b c^3 d^3 e+27 b^2 c^2 d^2 e^2+45 b^3 c d e^3-35 b^4 e^4\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac {e (2 c d-b e) \left (12 c^4 d^4-24 b c^3 d^3 e+2 b^2 c^2 d^2 e^2+10 b^3 c d e^3-35 b^4 e^4\right )}{4 b^4 d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 c^2 d^2-3 b c d e-7 b^2 e^2\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-7 b^2 e^2\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}-\frac {\left (48 c^2 d^2+60 b c d e+35 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{9/2}}+\frac {c^{9/2} \left (48 c^2 d^2-156 b c d e+143 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{9/2}} \]
[Out]
1/12*e*(-35*b^4*e^4+45*b^3*c*d*e^3+27*b^2*c^2*d^2*e^2-144*b*c^3*d^3*e+72*c^4*d^4)/b^4/d^3/(-b*e+c*d)^3/(e*x+d)
^(3/2)+1/2*(-b*(-b*e+c*d)-c*(-b*e+2*c*d)*x)/b^2/d/(-b*e+c*d)/(e*x+d)^(3/2)/(c*x^2+b*x)^2+1/4*(b*(-b*e+c*d)*(-7
*b^2*e^2-3*b*c*d*e+12*c^2*d^2)+c*(-b*e+2*c*d)*(-7*b^2*e^2-12*b*c*d*e+12*c^2*d^2)*x)/b^4/d^2/(-b*e+c*d)^2/(e*x+
d)^(3/2)/(c*x^2+b*x)-1/4*(35*b^2*e^2+60*b*c*d*e+48*c^2*d^2)*arctanh((e*x+d)^(1/2)/d^(1/2))/b^5/d^(9/2)+1/4*c^(
9/2)*(143*b^2*e^2-156*b*c*d*e+48*c^2*d^2)*arctanh(c^(1/2)*(e*x+d)^(1/2)/(-b*e+c*d)^(1/2))/b^5/(-b*e+c*d)^(9/2)
+1/4*e*(-b*e+2*c*d)*(-35*b^4*e^4+10*b^3*c*d*e^3+2*b^2*c^2*d^2*e^2-24*b*c^3*d^3*e+12*c^4*d^4)/b^4/d^4/(-b*e+c*d
)^4/(e*x+d)^(1/2)
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Rubi [A]
time = 0.58, antiderivative size = 470, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {754, 836, 842,
840, 1180, 214} \begin {gather*} -\frac {c x (2 c d-b e)+b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 (d+e x)^{3/2} (c d-b e)}-\frac {\left (35 b^2 e^2+60 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{9/2}}+\frac {c^{9/2} \left (143 b^2 e^2-156 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{9/2}}+\frac {c x (2 c d-b e) \left (-7 b^2 e^2-12 b c d e+12 c^2 d^2\right )+b (c d-b e) \left (-7 b^2 e^2-3 b c d e+12 c^2 d^2\right )}{4 b^4 d^2 \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)^2}+\frac {e (2 c d-b e) \left (-35 b^4 e^4+10 b^3 c d e^3+2 b^2 c^2 d^2 e^2-24 b c^3 d^3 e+12 c^4 d^4\right )}{4 b^4 d^4 \sqrt {d+e x} (c d-b e)^4}+\frac {e \left (-35 b^4 e^4+45 b^3 c d e^3+27 b^2 c^2 d^2 e^2-144 b c^3 d^3 e+72 c^4 d^4\right )}{12 b^4 d^3 (d+e x)^{3/2} (c d-b e)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/((d + e*x)^(5/2)*(b*x + c*x^2)^3),x]
[Out]
(e*(72*c^4*d^4 - 144*b*c^3*d^3*e + 27*b^2*c^2*d^2*e^2 + 45*b^3*c*d*e^3 - 35*b^4*e^4))/(12*b^4*d^3*(c*d - b*e)^
3*(d + e*x)^(3/2)) + (e*(2*c*d - b*e)*(12*c^4*d^4 - 24*b*c^3*d^3*e + 2*b^2*c^2*d^2*e^2 + 10*b^3*c*d*e^3 - 35*b
^4*e^4))/(4*b^4*d^4*(c*d - b*e)^4*Sqrt[d + e*x]) - (b*(c*d - b*e) + c*(2*c*d - b*e)*x)/(2*b^2*d*(c*d - b*e)*(d
+ e*x)^(3/2)*(b*x + c*x^2)^2) + (b*(c*d - b*e)*(12*c^2*d^2 - 3*b*c*d*e - 7*b^2*e^2) + c*(2*c*d - b*e)*(12*c^2
*d^2 - 12*b*c*d*e - 7*b^2*e^2)*x)/(4*b^4*d^2*(c*d - b*e)^2*(d + e*x)^(3/2)*(b*x + c*x^2)) - ((48*c^2*d^2 + 60*
b*c*d*e + 35*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5*d^(9/2)) + (c^(9/2)*(48*c^2*d^2 - 156*b*c*d*e + 1
43*b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*(c*d - b*e)^(9/2))
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 754
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(b
*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e +
a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b
*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]
Rule 836
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)
*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])
Rule 840
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
Rule 842
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e
*f - d*g)*((d + e*x)^(m + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[(d +
e*x)^(m + 1)*(Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c,
d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && FractionQ[m] && LtQ[m, -1]
Rule 1180
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^{5/2} \left (b x+c x^2\right )^3} \, dx &=-\frac {b (c d-b e)+c (2 c d-b e) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}-\frac {\int \frac {\frac {1}{2} \left (12 c^2 d^2-3 b c d e-7 b^2 e^2\right )+\frac {9}{2} c e (2 c d-b e) x}{(d+e x)^{5/2} \left (b x+c x^2\right )^2} \, dx}{2 b^2 d (c d-b e)}\\ &=-\frac {b (c d-b e)+c (2 c d-b e) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 c^2 d^2-3 b c d e-7 b^2 e^2\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-7 b^2 e^2\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac {\int \frac {\frac {1}{4} (c d-b e)^2 \left (48 c^2 d^2+60 b c d e+35 b^2 e^2\right )+\frac {5}{4} c e (2 c d-b e) \left (12 c^2 d^2-12 b c d e-7 b^2 e^2\right ) x}{(d+e x)^{5/2} \left (b x+c x^2\right )} \, dx}{2 b^4 d^2 (c d-b e)^2}\\ &=\frac {e \left (72 c^4 d^4-144 b c^3 d^3 e+27 b^2 c^2 d^2 e^2+45 b^3 c d e^3-35 b^4 e^4\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}-\frac {b (c d-b e)+c (2 c d-b e) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 c^2 d^2-3 b c d e-7 b^2 e^2\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-7 b^2 e^2\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac {\int \frac {\frac {1}{4} (c d-b e)^3 \left (48 c^2 d^2+60 b c d e+35 b^2 e^2\right )+\frac {1}{4} c e \left (72 c^4 d^4-144 b c^3 d^3 e+27 b^2 c^2 d^2 e^2+45 b^3 c d e^3-35 b^4 e^4\right ) x}{(d+e x)^{3/2} \left (b x+c x^2\right )} \, dx}{2 b^4 d^3 (c d-b e)^3}\\ &=\frac {e \left (72 c^4 d^4-144 b c^3 d^3 e+27 b^2 c^2 d^2 e^2+45 b^3 c d e^3-35 b^4 e^4\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac {e (2 c d-b e) \left (12 c^4 d^4-24 b c^3 d^3 e+2 b^2 c^2 d^2 e^2+10 b^3 c d e^3-35 b^4 e^4\right )}{4 b^4 d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 c^2 d^2-3 b c d e-7 b^2 e^2\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-7 b^2 e^2\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac {\int \frac {\frac {1}{4} (c d-b e)^4 \left (48 c^2 d^2+60 b c d e+35 b^2 e^2\right )+\frac {1}{4} c e (2 c d-b e) \left (12 c^4 d^4-24 b c^3 d^3 e+2 b^2 c^2 d^2 e^2+10 b^3 c d e^3-35 b^4 e^4\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 d^4 (c d-b e)^4}\\ &=\frac {e \left (72 c^4 d^4-144 b c^3 d^3 e+27 b^2 c^2 d^2 e^2+45 b^3 c d e^3-35 b^4 e^4\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac {e (2 c d-b e) \left (12 c^4 d^4-24 b c^3 d^3 e+2 b^2 c^2 d^2 e^2+10 b^3 c d e^3-35 b^4 e^4\right )}{4 b^4 d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 c^2 d^2-3 b c d e-7 b^2 e^2\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-7 b^2 e^2\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac {\text {Subst}\left (\int \frac {\frac {1}{4} e (c d-b e)^4 \left (48 c^2 d^2+60 b c d e+35 b^2 e^2\right )-\frac {1}{4} c d e (2 c d-b e) \left (12 c^4 d^4-24 b c^3 d^3 e+2 b^2 c^2 d^2 e^2+10 b^3 c d e^3-35 b^4 e^4\right )+\frac {1}{4} c e (2 c d-b e) \left (12 c^4 d^4-24 b c^3 d^3 e+2 b^2 c^2 d^2 e^2+10 b^3 c d e^3-35 b^4 e^4\right ) x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{b^4 d^4 (c d-b e)^4}\\ &=\frac {e \left (72 c^4 d^4-144 b c^3 d^3 e+27 b^2 c^2 d^2 e^2+45 b^3 c d e^3-35 b^4 e^4\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac {e (2 c d-b e) \left (12 c^4 d^4-24 b c^3 d^3 e+2 b^2 c^2 d^2 e^2+10 b^3 c d e^3-35 b^4 e^4\right )}{4 b^4 d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 c^2 d^2-3 b c d e-7 b^2 e^2\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-7 b^2 e^2\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac {\left (c \left (48 c^2 d^2+60 b c d e+35 b^2 e^2\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 b^5 d^4}-\frac {\left (c^5 \left (48 c^2 d^2-156 b c d e+143 b^2 e^2\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 b^5 (c d-b e)^4}\\ &=\frac {e \left (72 c^4 d^4-144 b c^3 d^3 e+27 b^2 c^2 d^2 e^2+45 b^3 c d e^3-35 b^4 e^4\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac {e (2 c d-b e) \left (12 c^4 d^4-24 b c^3 d^3 e+2 b^2 c^2 d^2 e^2+10 b^3 c d e^3-35 b^4 e^4\right )}{4 b^4 d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 c^2 d^2-3 b c d e-7 b^2 e^2\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-7 b^2 e^2\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}-\frac {\left (48 c^2 d^2+60 b c d e+35 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{9/2}}+\frac {c^{9/2} \left (48 c^2 d^2-156 b c d e+143 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 2.04, size = 501, normalized size = 1.07 \begin {gather*} \frac {\frac {b \left (72 c^7 d^5 x^3 (d+e x)^2+36 b c^6 d^4 x^2 (3 d-5 e x) (d+e x)^2+3 b^2 c^5 d^3 x (d+e x)^2 \left (8 d^2-91 d e x+28 e^2 x^2\right )-3 b^3 c^4 d^2 (d+e x)^2 \left (2 d^3+21 d^2 e x-44 d e^2 x^2-18 e^3 x^3\right )+b^7 e^4 \left (-6 d^3+21 d^2 e x+140 d e^2 x^2+105 e^3 x^3\right )+2 b^6 c e^3 \left (12 d^4-30 d^3 e x-139 d^2 e^2 x^2+20 d e^3 x^3+105 e^4 x^4\right )+4 b^4 c^3 d e \left (6 d^5+15 d^4 e x+45 d^3 e^2 x^2+45 d^2 e^3 x^3-53 d e^4 x^4-60 e^5 x^5\right )+b^5 c^2 e^2 \left (-36 d^5+30 d^4 e x-30 d^3 e^2 x^2-565 d^2 e^3 x^3-340 d e^4 x^4+105 e^5 x^5\right )\right )}{d^4 (c d-b e)^4 x^2 (b+c x)^2 (d+e x)^{3/2}}-\frac {3 c^{9/2} \left (48 c^2 d^2-156 b c d e+143 b^2 e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{(-c d+b e)^{9/2}}-\frac {3 \left (48 c^2 d^2+60 b c d e+35 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{9/2}}}{12 b^5} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/((d + e*x)^(5/2)*(b*x + c*x^2)^3),x]
[Out]
((b*(72*c^7*d^5*x^3*(d + e*x)^2 + 36*b*c^6*d^4*x^2*(3*d - 5*e*x)*(d + e*x)^2 + 3*b^2*c^5*d^3*x*(d + e*x)^2*(8*
d^2 - 91*d*e*x + 28*e^2*x^2) - 3*b^3*c^4*d^2*(d + e*x)^2*(2*d^3 + 21*d^2*e*x - 44*d*e^2*x^2 - 18*e^3*x^3) + b^
7*e^4*(-6*d^3 + 21*d^2*e*x + 140*d*e^2*x^2 + 105*e^3*x^3) + 2*b^6*c*e^3*(12*d^4 - 30*d^3*e*x - 139*d^2*e^2*x^2
+ 20*d*e^3*x^3 + 105*e^4*x^4) + 4*b^4*c^3*d*e*(6*d^5 + 15*d^4*e*x + 45*d^3*e^2*x^2 + 45*d^2*e^3*x^3 - 53*d*e^
4*x^4 - 60*e^5*x^5) + b^5*c^2*e^2*(-36*d^5 + 30*d^4*e*x - 30*d^3*e^2*x^2 - 565*d^2*e^3*x^3 - 340*d*e^4*x^4 + 1
05*e^5*x^5)))/(d^4*(c*d - b*e)^4*x^2*(b + c*x)^2*(d + e*x)^(3/2)) - (3*c^(9/2)*(48*c^2*d^2 - 156*b*c*d*e + 143
*b^2*e^2)*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/(-(c*d) + b*e)^(9/2) - (3*(48*c^2*d^2 + 60*b*c*d
*e + 35*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/d^(9/2))/(12*b^5)
________________________________________________________________________________________
Maple [A]
time = 0.66, size = 326, normalized size = 0.69
| | |
| method |
result |
size |
| | |
| derivativedivides |
\(2 e^{5} \left (-\frac {\frac {-\frac {e b \left (11 b e +12 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8}+\left (\frac {13}{8} b^{2} d \,e^{2}+\frac {3}{2} b c \,d^{2} e \right ) \sqrt {e x +d}}{e^{2} x^{2}}+\frac {\left (35 b^{2} e^{2}+60 b c d e +48 d^{2} c^{2}\right ) \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 \sqrt {d}}}{d^{4} b^{5} e^{5}}-\frac {c^{5} \left (\frac {\left (\frac {23}{8} b^{2} e^{2} c -\frac {3}{2} d b e \,c^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}+\frac {b e \left (25 b^{2} e^{2}-37 b c d e +12 d^{2} c^{2}\right ) \sqrt {e x +d}}{8}}{\left (c \left (e x +d \right )+b e -c d \right )^{2}}+\frac {\left (143 b^{2} e^{2}-156 b c d e +48 d^{2} c^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{8 \sqrt {\left (b e -c d \right ) c}}\right )}{b^{5} e^{5} \left (b e -c d \right )^{4}}-\frac {-3 b e +6 c d}{d^{4} \left (b e -c d \right )^{4} \sqrt {e x +d}}+\frac {1}{3 d^{3} \left (b e -c d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}\right )\) |
\(326\) |
| default |
\(2 e^{5} \left (-\frac {\frac {-\frac {e b \left (11 b e +12 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8}+\left (\frac {13}{8} b^{2} d \,e^{2}+\frac {3}{2} b c \,d^{2} e \right ) \sqrt {e x +d}}{e^{2} x^{2}}+\frac {\left (35 b^{2} e^{2}+60 b c d e +48 d^{2} c^{2}\right ) \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 \sqrt {d}}}{d^{4} b^{5} e^{5}}-\frac {c^{5} \left (\frac {\left (\frac {23}{8} b^{2} e^{2} c -\frac {3}{2} d b e \,c^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}+\frac {b e \left (25 b^{2} e^{2}-37 b c d e +12 d^{2} c^{2}\right ) \sqrt {e x +d}}{8}}{\left (c \left (e x +d \right )+b e -c d \right )^{2}}+\frac {\left (143 b^{2} e^{2}-156 b c d e +48 d^{2} c^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{8 \sqrt {\left (b e -c d \right ) c}}\right )}{b^{5} e^{5} \left (b e -c d \right )^{4}}-\frac {-3 b e +6 c d}{d^{4} \left (b e -c d \right )^{4} \sqrt {e x +d}}+\frac {1}{3 d^{3} \left (b e -c d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}\right )\) |
\(326\) |
| risch |
\(-\frac {\sqrt {e x +d}\, \left (-11 b e x -12 c d x +2 b d \right )}{4 d^{4} b^{4} x^{2}}-\frac {23 e^{2} c^{6} \left (e x +d \right )^{\frac {3}{2}}}{4 b^{3} \left (b e -c d \right )^{4} \left (c e x +b e \right )^{2}}+\frac {3 d e \,c^{7} \left (e x +d \right )^{\frac {3}{2}}}{b^{4} \left (b e -c d \right )^{4} \left (c e x +b e \right )^{2}}-\frac {25 e^{3} c^{5} \sqrt {e x +d}}{4 b^{2} \left (b e -c d \right )^{4} \left (c e x +b e \right )^{2}}+\frac {37 d \,e^{2} c^{6} \sqrt {e x +d}}{4 b^{3} \left (b e -c d \right )^{4} \left (c e x +b e \right )^{2}}-\frac {3 d^{2} e \,c^{7} \sqrt {e x +d}}{b^{4} \left (b e -c d \right )^{4} \left (c e x +b e \right )^{2}}-\frac {143 e^{2} c^{5} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{4 b^{3} \left (b e -c d \right )^{4} \sqrt {\left (b e -c d \right ) c}}+\frac {39 d e \,c^{6} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{b^{4} \left (b e -c d \right )^{4} \sqrt {\left (b e -c d \right ) c}}-\frac {12 d^{2} c^{7} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{b^{5} \left (b e -c d \right )^{4} \sqrt {\left (b e -c d \right ) c}}+\frac {2 e^{5}}{3 d^{3} \left (b e -c d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}+\frac {6 b \,e^{6}}{d^{4} \left (b e -c d \right )^{4} \sqrt {e x +d}}-\frac {12 e^{5} c}{d^{3} \left (b e -c d \right )^{4} \sqrt {e x +d}}-\frac {35 e^{2} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{4 b^{3} d^{\frac {9}{2}}}-\frac {15 e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) c}{b^{4} d^{\frac {7}{2}}}-\frac {12 \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) c^{2}}{b^{5} d^{\frac {5}{2}}}\) |
\(535\) |
| | |
|
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(e*x+d)^(5/2)/(c*x^2+b*x)^3,x,method=_RETURNVERBOSE)
[Out]
2*e^5*(-1/d^4/b^5/e^5*((-1/8*e*b*(11*b*e+12*c*d)*(e*x+d)^(3/2)+(13/8*b^2*d*e^2+3/2*b*c*d^2*e)*(e*x+d)^(1/2))/e
^2/x^2+1/8*(35*b^2*e^2+60*b*c*d*e+48*c^2*d^2)/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2)))-c^5/b^5/e^5/(b*e-c*d)^4*
(((23/8*b^2*e^2*c-3/2*d*b*e*c^2)*(e*x+d)^(3/2)+1/8*b*e*(25*b^2*e^2-37*b*c*d*e+12*c^2*d^2)*(e*x+d)^(1/2))/(c*(e
*x+d)+b*e-c*d)^2+1/8*(143*b^2*e^2-156*b*c*d*e+48*c^2*d^2)/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d
)*c)^(1/2)))-1/d^4/(b*e-c*d)^4*(-3*b*e+6*c*d)/(e*x+d)^(1/2)+1/3/d^3/(b*e-c*d)^3/(e*x+d)^(3/2))
________________________________________________________________________________________
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(1/(e*x+d)^(5/2)/(c*x^2+b*x)^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(c*d-%e*b>0)', see `assume?` fo
r more detai
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1787 vs.
\(2 (449) = 898\).
time = 14.63, size = 7184, normalized size = 15.29 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^(5/2)/(c*x^2+b*x)^3,x, algorithm="fricas")
[Out]
[1/24*(3*(48*c^8*d^9*x^4 + 96*b*c^7*d^9*x^3 + 48*b^2*c^6*d^9*x^2 + 143*(b^2*c^6*d^5*x^6 + 2*b^3*c^5*d^5*x^5 +
b^4*c^4*d^5*x^4)*e^4 - 26*(6*b*c^7*d^6*x^6 + b^2*c^6*d^6*x^5 - 16*b^3*c^5*d^6*x^4 - 11*b^4*c^4*d^6*x^3)*e^3 +
(48*c^8*d^7*x^6 - 216*b*c^7*d^7*x^5 - 433*b^2*c^6*d^7*x^4 - 26*b^3*c^5*d^7*x^3 + 143*b^4*c^4*d^7*x^2)*e^2 + 12
*(8*c^8*d^8*x^5 + 3*b*c^7*d^8*x^4 - 18*b^2*c^6*d^8*x^3 - 13*b^3*c^5*d^8*x^2)*e)*sqrt(c/(c*d - b*e))*log((2*c*d
+ 2*(c*d - b*e)*sqrt(x*e + d)*sqrt(c/(c*d - b*e)) + (c*x - b)*e)/(c*x + b)) + 3*(48*c^8*d^8*x^4 + 96*b*c^7*d^
8*x^3 + 48*b^2*c^6*d^8*x^2 + 35*(b^6*c^2*x^6 + 2*b^7*c*x^5 + b^8*x^4)*e^8 - 10*(8*b^5*c^3*d*x^6 + 9*b^6*c^2*d*
x^5 - 6*b^7*c*d*x^4 - 7*b^8*d*x^3)*e^7 + (18*b^4*c^4*d^2*x^6 - 124*b^5*c^3*d^2*x^5 - 267*b^6*c^2*d^2*x^4 - 90*
b^7*c*d^2*x^3 + 35*b^8*d^2*x^2)*e^6 + 4*(7*b^3*c^5*d^3*x^6 + 23*b^4*c^4*d^3*x^5 + 5*b^5*c^3*d^3*x^4 - 31*b^6*c
^2*d^3*x^3 - 20*b^7*c*d^3*x^2)*e^5 + (83*b^2*c^6*d^4*x^6 + 222*b^3*c^5*d^4*x^5 + 213*b^4*c^4*d^4*x^4 + 92*b^5*
c^3*d^4*x^3 + 18*b^6*c^2*d^4*x^2)*e^4 - 2*(66*b*c^7*d^5*x^6 + 49*b^2*c^6*d^5*x^5 - 114*b^3*c^5*d^5*x^4 - 111*b
^4*c^4*d^5*x^3 - 14*b^5*c^3*d^5*x^2)*e^3 + (48*c^8*d^6*x^6 - 168*b*c^7*d^6*x^5 - 397*b^2*c^6*d^6*x^4 - 98*b^3*
c^5*d^6*x^3 + 83*b^4*c^4*d^6*x^2)*e^2 + 12*(8*c^8*d^7*x^5 + 5*b*c^7*d^7*x^4 - 14*b^2*c^6*d^7*x^3 - 11*b^3*c^5*
d^7*x^2)*e)*sqrt(d)*log((x*e - 2*sqrt(x*e + d)*sqrt(d) + 2*d)/x) + 2*(72*b*c^7*d^8*x^3 + 108*b^2*c^6*d^8*x^2 +
24*b^3*c^5*d^8*x - 6*b^4*c^4*d^8 + 105*(b^6*c^2*d*x^5 + 2*b^7*c*d*x^4 + b^8*d*x^3)*e^7 - 20*(12*b^5*c^3*d^2*x
^5 + 17*b^6*c^2*d^2*x^4 - 2*b^7*c*d^2*x^3 - 7*b^8*d^2*x^2)*e^6 + (54*b^4*c^4*d^3*x^5 - 212*b^5*c^3*d^3*x^4 - 5
65*b^6*c^2*d^3*x^3 - 278*b^7*c*d^3*x^2 + 21*b^8*d^3*x)*e^5 + 6*(14*b^3*c^5*d^4*x^5 + 40*b^4*c^4*d^4*x^4 + 30*b
^5*c^3*d^4*x^3 - 5*b^6*c^2*d^4*x^2 - 10*b^7*c*d^4*x - b^8*d^4)*e^4 - 3*(60*b^2*c^6*d^5*x^5 + 35*b^3*c^5*d^5*x^
4 - 85*b^4*c^4*d^5*x^3 - 60*b^5*c^3*d^5*x^2 - 10*b^6*c^2*d^5*x - 8*b^7*c*d^5)*e^3 + 6*(12*b*c^7*d^6*x^5 - 42*b
^2*c^6*d^6*x^4 - 73*b^3*c^5*d^6*x^3 + 10*b^5*c^3*d^6*x - 6*b^6*c^2*d^6)*e^2 + 3*(48*b*c^7*d^7*x^4 + 12*b^2*c^6
*d^7*x^3 - 75*b^3*c^5*d^7*x^2 - 25*b^4*c^4*d^7*x + 8*b^5*c^3*d^7)*e)*sqrt(x*e + d))/(b^5*c^6*d^11*x^4 + 2*b^6*
c^5*d^11*x^3 + b^7*c^4*d^11*x^2 + (b^9*c^2*d^5*x^6 + 2*b^10*c*d^5*x^5 + b^11*d^5*x^4)*e^6 - 2*(2*b^8*c^3*d^6*x
^6 + 3*b^9*c^2*d^6*x^5 - b^11*d^6*x^3)*e^5 + (6*b^7*c^4*d^7*x^6 + 4*b^8*c^3*d^7*x^5 - 9*b^9*c^2*d^7*x^4 - 6*b^
10*c*d^7*x^3 + b^11*d^7*x^2)*e^4 - 4*(b^6*c^5*d^8*x^6 - b^7*c^4*d^8*x^5 - 4*b^8*c^3*d^8*x^4 - b^9*c^2*d^8*x^3
+ b^10*c*d^8*x^2)*e^3 + (b^5*c^6*d^9*x^6 - 6*b^6*c^5*d^9*x^5 - 9*b^7*c^4*d^9*x^4 + 4*b^8*c^3*d^9*x^3 + 6*b^9*c
^2*d^9*x^2)*e^2 + 2*(b^5*c^6*d^10*x^5 - 3*b^7*c^4*d^10*x^3 - 2*b^8*c^3*d^10*x^2)*e), 1/24*(6*(48*c^8*d^9*x^4 +
96*b*c^7*d^9*x^3 + 48*b^2*c^6*d^9*x^2 + 143*(b^2*c^6*d^5*x^6 + 2*b^3*c^5*d^5*x^5 + b^4*c^4*d^5*x^4)*e^4 - 26*
(6*b*c^7*d^6*x^6 + b^2*c^6*d^6*x^5 - 16*b^3*c^5*d^6*x^4 - 11*b^4*c^4*d^6*x^3)*e^3 + (48*c^8*d^7*x^6 - 216*b*c^
7*d^7*x^5 - 433*b^2*c^6*d^7*x^4 - 26*b^3*c^5*d^7*x^3 + 143*b^4*c^4*d^7*x^2)*e^2 + 12*(8*c^8*d^8*x^5 + 3*b*c^7*
d^8*x^4 - 18*b^2*c^6*d^8*x^3 - 13*b^3*c^5*d^8*x^2)*e)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(x*e + d)*s
qrt(-c/(c*d - b*e))/(c*x*e + c*d)) + 3*(48*c^8*d^8*x^4 + 96*b*c^7*d^8*x^3 + 48*b^2*c^6*d^8*x^2 + 35*(b^6*c^2*x
^6 + 2*b^7*c*x^5 + b^8*x^4)*e^8 - 10*(8*b^5*c^3*d*x^6 + 9*b^6*c^2*d*x^5 - 6*b^7*c*d*x^4 - 7*b^8*d*x^3)*e^7 + (
18*b^4*c^4*d^2*x^6 - 124*b^5*c^3*d^2*x^5 - 267*b^6*c^2*d^2*x^4 - 90*b^7*c*d^2*x^3 + 35*b^8*d^2*x^2)*e^6 + 4*(7
*b^3*c^5*d^3*x^6 + 23*b^4*c^4*d^3*x^5 + 5*b^5*c^3*d^3*x^4 - 31*b^6*c^2*d^3*x^3 - 20*b^7*c*d^3*x^2)*e^5 + (83*b
^2*c^6*d^4*x^6 + 222*b^3*c^5*d^4*x^5 + 213*b^4*c^4*d^4*x^4 + 92*b^5*c^3*d^4*x^3 + 18*b^6*c^2*d^4*x^2)*e^4 - 2*
(66*b*c^7*d^5*x^6 + 49*b^2*c^6*d^5*x^5 - 114*b^3*c^5*d^5*x^4 - 111*b^4*c^4*d^5*x^3 - 14*b^5*c^3*d^5*x^2)*e^3 +
(48*c^8*d^6*x^6 - 168*b*c^7*d^6*x^5 - 397*b^2*c^6*d^6*x^4 - 98*b^3*c^5*d^6*x^3 + 83*b^4*c^4*d^6*x^2)*e^2 + 12
*(8*c^8*d^7*x^5 + 5*b*c^7*d^7*x^4 - 14*b^2*c^6*d^7*x^3 - 11*b^3*c^5*d^7*x^2)*e)*sqrt(d)*log((x*e - 2*sqrt(x*e
+ d)*sqrt(d) + 2*d)/x) + 2*(72*b*c^7*d^8*x^3 + 108*b^2*c^6*d^8*x^2 + 24*b^3*c^5*d^8*x - 6*b^4*c^4*d^8 + 105*(b
^6*c^2*d*x^5 + 2*b^7*c*d*x^4 + b^8*d*x^3)*e^7 - 20*(12*b^5*c^3*d^2*x^5 + 17*b^6*c^2*d^2*x^4 - 2*b^7*c*d^2*x^3
- 7*b^8*d^2*x^2)*e^6 + (54*b^4*c^4*d^3*x^5 - 212*b^5*c^3*d^3*x^4 - 565*b^6*c^2*d^3*x^3 - 278*b^7*c*d^3*x^2 + 2
1*b^8*d^3*x)*e^5 + 6*(14*b^3*c^5*d^4*x^5 + 40*b^4*c^4*d^4*x^4 + 30*b^5*c^3*d^4*x^3 - 5*b^6*c^2*d^4*x^2 - 10*b^
7*c*d^4*x - b^8*d^4)*e^4 - 3*(60*b^2*c^6*d^5*x^5 + 35*b^3*c^5*d^5*x^4 - 85*b^4*c^4*d^5*x^3 - 60*b^5*c^3*d^5*x^
2 - 10*b^6*c^2*d^5*x - 8*b^7*c*d^5)*e^3 + 6*(12*b*c^7*d^6*x^5 - 42*b^2*c^6*d^6*x^4 - 73*b^3*c^5*d^6*x^3 + 10*b
^5*c^3*d^6*x - 6*b^6*c^2*d^6)*e^2 + 3*(48*b*c^7*d^7*x^4 + 12*b^2*c^6*d^7*x^3 - 75*b^3*c^5*d^7*x^2 - 25*b^4*c^4
*d^7*x + 8*b^5*c^3*d^7)*e)*sqrt(x*e + d))/(b^5*c^6*d^11*x^4 + 2*b^6*c^5*d^11*x^3 + b^7*c^4*d^11*x^2 + (b^9*c^2
*d^5*x^6 + 2*b^10*c*d^5*x^5 + b^11*d^5*x^4)*e^6...
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{3} \left (b + c x\right )^{3} \left (d + e x\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)**(5/2)/(c*x**2+b*x)**3,x)
[Out]
Integral(1/(x**3*(b + c*x)**3*(d + e*x)**(5/2)), x)
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 942 vs.
\(2 (449) = 898\).
time = 1.48, size = 942, normalized size = 2.00 \begin {gather*} -\frac {{\left (48 \, c^{7} d^{2} - 156 \, b c^{6} d e + 143 \, b^{2} c^{5} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{4 \, {\left (b^{5} c^{4} d^{4} - 4 \, b^{6} c^{3} d^{3} e + 6 \, b^{7} c^{2} d^{2} e^{2} - 4 \, b^{8} c d e^{3} + b^{9} e^{4}\right )} \sqrt {-c^{2} d + b c e}} - \frac {2 \, {\left (18 \, {\left (x e + d\right )} c d e^{5} + c d^{2} e^{5} - 9 \, {\left (x e + d\right )} b e^{6} - b d e^{6}\right )}}{3 \, {\left (c^{4} d^{8} - 4 \, b c^{3} d^{7} e + 6 \, b^{2} c^{2} d^{6} e^{2} - 4 \, b^{3} c d^{5} e^{3} + b^{4} d^{4} e^{4}\right )} {\left (x e + d\right )}^{\frac {3}{2}}} + \frac {24 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{7} d^{5} e - 72 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{7} d^{6} e + 72 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{7} d^{7} e - 24 \, \sqrt {x e + d} c^{7} d^{8} e - 60 \, {\left (x e + d\right )}^{\frac {7}{2}} b c^{6} d^{4} e^{2} + 216 \, {\left (x e + d\right )}^{\frac {5}{2}} b c^{6} d^{5} e^{2} - 252 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{6} d^{6} e^{2} + 96 \, \sqrt {x e + d} b c^{6} d^{7} e^{2} + 28 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{2} c^{5} d^{3} e^{3} - 175 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{2} c^{5} d^{4} e^{3} + 274 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c^{5} d^{5} e^{3} - 127 \, \sqrt {x e + d} b^{2} c^{5} d^{6} e^{3} + 18 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{3} c^{4} d^{2} e^{4} - 10 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{3} c^{4} d^{3} e^{4} - 55 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} c^{4} d^{4} e^{4} + 45 \, \sqrt {x e + d} b^{3} c^{4} d^{5} e^{4} - 32 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{4} c^{3} d e^{5} + 140 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{4} c^{3} d^{2} e^{5} - 180 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{4} c^{3} d^{3} e^{5} + 80 \, \sqrt {x e + d} b^{4} c^{3} d^{4} e^{5} + 11 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{5} c^{2} e^{6} - 99 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{5} c^{2} d e^{6} + 199 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{5} c^{2} d^{2} e^{6} - 123 \, \sqrt {x e + d} b^{5} c^{2} d^{3} e^{6} + 22 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{6} c e^{7} - 80 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{6} c d e^{7} + 66 \, \sqrt {x e + d} b^{6} c d^{2} e^{7} + 11 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{7} e^{8} - 13 \, \sqrt {x e + d} b^{7} d e^{8}}{4 \, {\left (b^{4} c^{4} d^{8} - 4 \, b^{5} c^{3} d^{7} e + 6 \, b^{6} c^{2} d^{6} e^{2} - 4 \, b^{7} c d^{5} e^{3} + b^{8} d^{4} e^{4}\right )} {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + {\left (x e + d\right )} b e - b d e\right )}^{2}} + \frac {{\left (48 \, c^{2} d^{2} + 60 \, b c d e + 35 \, b^{2} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{4 \, b^{5} \sqrt {-d} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^(5/2)/(c*x^2+b*x)^3,x, algorithm="giac")
[Out]
-1/4*(48*c^7*d^2 - 156*b*c^6*d*e + 143*b^2*c^5*e^2)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b^5*c^4*d^4
- 4*b^6*c^3*d^3*e + 6*b^7*c^2*d^2*e^2 - 4*b^8*c*d*e^3 + b^9*e^4)*sqrt(-c^2*d + b*c*e)) - 2/3*(18*(x*e + d)*c*
d*e^5 + c*d^2*e^5 - 9*(x*e + d)*b*e^6 - b*d*e^6)/((c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 - 4*b^3*c*d^5*e
^3 + b^4*d^4*e^4)*(x*e + d)^(3/2)) + 1/4*(24*(x*e + d)^(7/2)*c^7*d^5*e - 72*(x*e + d)^(5/2)*c^7*d^6*e + 72*(x*
e + d)^(3/2)*c^7*d^7*e - 24*sqrt(x*e + d)*c^7*d^8*e - 60*(x*e + d)^(7/2)*b*c^6*d^4*e^2 + 216*(x*e + d)^(5/2)*b
*c^6*d^5*e^2 - 252*(x*e + d)^(3/2)*b*c^6*d^6*e^2 + 96*sqrt(x*e + d)*b*c^6*d^7*e^2 + 28*(x*e + d)^(7/2)*b^2*c^5
*d^3*e^3 - 175*(x*e + d)^(5/2)*b^2*c^5*d^4*e^3 + 274*(x*e + d)^(3/2)*b^2*c^5*d^5*e^3 - 127*sqrt(x*e + d)*b^2*c
^5*d^6*e^3 + 18*(x*e + d)^(7/2)*b^3*c^4*d^2*e^4 - 10*(x*e + d)^(5/2)*b^3*c^4*d^3*e^4 - 55*(x*e + d)^(3/2)*b^3*
c^4*d^4*e^4 + 45*sqrt(x*e + d)*b^3*c^4*d^5*e^4 - 32*(x*e + d)^(7/2)*b^4*c^3*d*e^5 + 140*(x*e + d)^(5/2)*b^4*c^
3*d^2*e^5 - 180*(x*e + d)^(3/2)*b^4*c^3*d^3*e^5 + 80*sqrt(x*e + d)*b^4*c^3*d^4*e^5 + 11*(x*e + d)^(7/2)*b^5*c^
2*e^6 - 99*(x*e + d)^(5/2)*b^5*c^2*d*e^6 + 199*(x*e + d)^(3/2)*b^5*c^2*d^2*e^6 - 123*sqrt(x*e + d)*b^5*c^2*d^3
*e^6 + 22*(x*e + d)^(5/2)*b^6*c*e^7 - 80*(x*e + d)^(3/2)*b^6*c*d*e^7 + 66*sqrt(x*e + d)*b^6*c*d^2*e^7 + 11*(x*
e + d)^(3/2)*b^7*e^8 - 13*sqrt(x*e + d)*b^7*d*e^8)/((b^4*c^4*d^8 - 4*b^5*c^3*d^7*e + 6*b^6*c^2*d^6*e^2 - 4*b^7
*c*d^5*e^3 + b^8*d^4*e^4)*((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e)^2) + 1/4*(48*c^2*d
^2 + 60*b*c*d*e + 35*b^2*e^2)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^5*sqrt(-d)*d^4)
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Mupad [B]
time = 4.82, size = 2500, normalized size = 5.32 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((b*x + c*x^2)^3*(d + e*x)^(5/2)),x)
[Out]
- ((2*e^5)/(3*(c*d^2 - b*d*e)) - (14*e^5*(b*e - 2*c*d)*(d + e*x))/(3*(c*d^2 - b*d*e)^2) - (e*(d + e*x)^5*(24*c
^7*d^5 + 35*b^5*c^2*e^5 - 80*b^4*c^3*d*e^4 + 28*b^2*c^5*d^3*e^2 + 18*b^3*c^4*d^2*e^3 - 60*b*c^6*d^4*e))/(4*b^4
*(c*d^2 - b*d*e)^4) + (e*(d + e*x)^4*(216*c^7*d^6 - 210*b^6*c*e^6 + 865*b^5*c^2*d*e^5 + 525*b^2*c^5*d^4*e^2 +
30*b^3*c^4*d^3*e^3 - 988*b^4*c^3*d^2*e^4 - 648*b*c^6*d^5*e))/(12*b^4*(c*d^2 - b*d*e)^4) + (e*(d + e*x)^2*(72*c
^6*d^6 - 175*b^6*e^6 + 165*b^2*c^4*d^4*e^2 + 30*b^3*c^3*d^3*e^3 - 738*b^4*c^2*d^2*e^4 - 216*b*c^5*d^5*e + 687*
b^5*c*d*e^5))/(12*b^4*(c*d^2 - b*d*e)^3) - (e*(d + e*x)^3*(105*b^7*e^7 + 216*c^7*d^7 + 822*b^2*c^5*d^5*e^2 - 1
65*b^3*c^4*d^4*e^3 - 1372*b^4*c^3*d^3*e^4 + 1845*b^5*c^2*d^2*e^5 - 756*b*c^6*d^6*e - 800*b^6*c*d*e^6))/(12*b^4
*(c*d^2 - b*d*e)^4))/(c^2*(d + e*x)^(11/2) - (4*c^2*d - 2*b*c*e)*(d + e*x)^(9/2) - (d + e*x)^(5/2)*(4*c^2*d^3
+ 2*b^2*d*e^2 - 6*b*c*d^2*e) + (d + e*x)^(7/2)*(b^2*e^2 + 6*c^2*d^2 - 6*b*c*d*e) + (d + e*x)^(3/2)*(c^2*d^4 +
b^2*d^2*e^2 - 2*b*c*d^3*e)) - (atan(((((d + e*x)^(1/2)*(589824*b^12*c^27*d^36*e^2 - 10616832*b^13*c^26*d^35*e^
3 + 89518080*b^14*c^25*d^34*e^4 - 468971520*b^15*c^24*d^33*e^5 + 1707439360*b^16*c^23*d^32*e^6 - 4579446784*b^
17*c^22*d^31*e^7 + 9364822016*b^18*c^21*d^30*e^8 - 14937190400*b^19*c^20*d^29*e^9 + 18936107520*b^20*c^19*d^28
*e^10 - 19535324160*b^21*c^18*d^27*e^11 + 17074641408*b^22*c^17*d^26*e^12 - 13484230656*b^23*c^16*d^25*e^13 +
10265639040*b^24*c^15*d^24*e^14 - 7643066880*b^25*c^14*d^23*e^15 + 5421597440*b^26*c^13*d^22*e^16 - 3708136960
*b^27*c^12*d^21*e^17 + 2608529792*b^28*c^11*d^20*e^18 - 1894041600*b^29*c^10*d^19*e^19 + 1274465280*b^30*c^9*d
^18*e^20 - 707773440*b^31*c^8*d^17*e^21 + 301648512*b^32*c^7*d^16*e^22 - 93688320*b^33*c^6*d^15*e^23 + 1993088
0*b^34*c^5*d^14*e^24 - 2598400*b^35*c^4*d^13*e^25 + 156800*b^36*c^3*d^12*e^26) + ((35*b^2*e^2 + 48*c^2*d^2 + 6
0*b*c*d*e)*(24576*b^18*c^24*d^38*e^3 - 466944*b^19*c^23*d^37*e^4 + 4185088*b^20*c^22*d^36*e^5 - 23500800*b^21*
c^21*d^35*e^6 + 92710912*b^22*c^20*d^34*e^7 - 273566720*b^23*c^19*d^33*e^8 + 629578752*b^24*c^18*d^32*e^9 - 11
69833984*b^25*c^17*d^31*e^10 + 1818910720*b^26*c^16*d^30*e^11 - 2465058816*b^27*c^15*d^29*e^12 + 3031169024*b^
28*c^14*d^28*e^13 - 3457871872*b^29*c^13*d^27*e^14 + 3626348544*b^30*c^12*d^26*e^15 - 3385559040*b^31*c^11*d^2
5*e^16 + 2714064896*b^32*c^10*d^24*e^17 - 1813512192*b^33*c^9*d^23*e^18 + 986251264*b^34*c^8*d^22*e^19 - 42681
5488*b^35*c^7*d^21*e^20 + 143109120*b^36*c^6*d^20*e^21 - 35796992*b^37*c^5*d^19*e^22 + 6285312*b^38*c^4*d^18*e
^23 - 691200*b^39*c^3*d^17*e^24 + 35840*b^40*c^2*d^16*e^25 - ((d + e*x)^(1/2)*(35*b^2*e^2 + 48*c^2*d^2 + 60*b*
c*d*e)*(16384*b^22*c^23*d^41*e^2 - 335872*b^23*c^22*d^40*e^3 + 3276800*b^24*c^21*d^39*e^4 - 20234240*b^25*c^20
*d^38*e^5 + 88719360*b^26*c^19*d^37*e^6 - 293707776*b^27*c^18*d^36*e^7 + 762052608*b^28*c^17*d^35*e^8 - 158760
9600*b^29*c^16*d^34*e^9 + 2698936320*b^30*c^15*d^33*e^10 - 3783802880*b^31*c^14*d^32*e^11 + 4402970624*b^32*c^
13*d^31*e^12 - 4265377792*b^33*c^12*d^30*e^13 + 3439820800*b^34*c^11*d^29*e^14 - 2302033920*b^35*c^10*d^28*e^1
5 + 1270087680*b^36*c^9*d^27*e^16 - 571539456*b^37*c^8*d^26*e^17 + 206389248*b^38*c^7*d^25*e^18 - 58368000*b^3
9*c^6*d^24*e^19 + 12451840*b^40*c^5*d^23*e^20 - 1884160*b^41*c^4*d^22*e^21 + 180224*b^42*c^3*d^21*e^22 - 8192*
b^43*c^2*d^20*e^23))/(8*b^5*(d^9)^(1/2))))/(8*b^5*(d^9)^(1/2)))*(35*b^2*e^2 + 48*c^2*d^2 + 60*b*c*d*e)*1i)/(8*
b^5*(d^9)^(1/2)) + (((d + e*x)^(1/2)*(589824*b^12*c^27*d^36*e^2 - 10616832*b^13*c^26*d^35*e^3 + 89518080*b^14*
c^25*d^34*e^4 - 468971520*b^15*c^24*d^33*e^5 + 1707439360*b^16*c^23*d^32*e^6 - 4579446784*b^17*c^22*d^31*e^7 +
9364822016*b^18*c^21*d^30*e^8 - 14937190400*b^19*c^20*d^29*e^9 + 18936107520*b^20*c^19*d^28*e^10 - 1953532416
0*b^21*c^18*d^27*e^11 + 17074641408*b^22*c^17*d^26*e^12 - 13484230656*b^23*c^16*d^25*e^13 + 10265639040*b^24*c
^15*d^24*e^14 - 7643066880*b^25*c^14*d^23*e^15 + 5421597440*b^26*c^13*d^22*e^16 - 3708136960*b^27*c^12*d^21*e^
17 + 2608529792*b^28*c^11*d^20*e^18 - 1894041600*b^29*c^10*d^19*e^19 + 1274465280*b^30*c^9*d^18*e^20 - 7077734
40*b^31*c^8*d^17*e^21 + 301648512*b^32*c^7*d^16*e^22 - 93688320*b^33*c^6*d^15*e^23 + 19930880*b^34*c^5*d^14*e^
24 - 2598400*b^35*c^4*d^13*e^25 + 156800*b^36*c^3*d^12*e^26) - ((35*b^2*e^2 + 48*c^2*d^2 + 60*b*c*d*e)*(24576*
b^18*c^24*d^38*e^3 - 466944*b^19*c^23*d^37*e^4 + 4185088*b^20*c^22*d^36*e^5 - 23500800*b^21*c^21*d^35*e^6 + 92
710912*b^22*c^20*d^34*e^7 - 273566720*b^23*c^19*d^33*e^8 + 629578752*b^24*c^18*d^32*e^9 - 1169833984*b^25*c^17
*d^31*e^10 + 1818910720*b^26*c^16*d^30*e^11 - 2465058816*b^27*c^15*d^29*e^12 + 3031169024*b^28*c^14*d^28*e^13
- 3457871872*b^29*c^13*d^27*e^14 + 3626348544*b^30*c^12*d^26*e^15 - 3385559040*b^31*c^11*d^25*e^16 + 271406489
6*b^32*c^10*d^24*e^17 - 1813512192*b^33*c^9*d^23*e^18 + 986251264*b^34*c^8*d^22*e^19 - 426815488*b^35*c^7*d^21
*e^20 + 143109120*b^36*c^6*d^20*e^21 - 35796992*b^37*c^5*d^19*e^22 + 6285312*b^38*c^4*d^18*e^23 - 691200*b^39*
c^3*d^17*e^24 + 35840*b^40*c^2*d^16*e^25 + ((d ...
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