3.4.13 \(\int \frac {1}{(d+e x)^2 (b x+c x^2)^{5/2}} \, dx\)
Optimal. Leaf size=348 \[ -\frac {2 (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (d+e x) (c d-b e)}+\frac {2 \left (c x (2 c d-b e) \left (-5 b^2 e^2-8 b c d e+8 c^2 d^2\right )+b (c d-b e) \left (-5 b^2 e^2-2 b c d e+8 c^2 d^2\right )\right )}{3 b^4 d^2 \sqrt {b x+c x^2} (d+e x) (c d-b e)^2}+\frac {e \sqrt {b x+c x^2} \left (-15 b^4 e^4+20 b^3 c d e^3+12 b^2 c^2 d^2 e^2-64 b c^3 d^3 e+32 c^4 d^4\right )}{3 b^4 d^3 (d+e x) (c d-b e)^3}+\frac {5 e^4 (2 c d-b e) \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{2 d^{7/2} (c d-b e)^{7/2}} \]
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Rubi [A] time = 0.35, antiderivative size = 348, normalized size of antiderivative = 1.00,
number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used =
{740, 822, 806, 724, 206} \begin {gather*} \frac {e \sqrt {b x+c x^2} \left (12 b^2 c^2 d^2 e^2+20 b^3 c d e^3-15 b^4 e^4-64 b c^3 d^3 e+32 c^4 d^4\right )}{3 b^4 d^3 (d+e x) (c d-b e)^3}+\frac {2 \left (c x (2 c d-b e) \left (-5 b^2 e^2-8 b c d e+8 c^2 d^2\right )+b (c d-b e) \left (-5 b^2 e^2-2 b c d e+8 c^2 d^2\right )\right )}{3 b^4 d^2 \sqrt {b x+c x^2} (d+e x) (c d-b e)^2}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (d+e x) (c d-b e)}+\frac {5 e^4 (2 c d-b e) \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{2 d^{7/2} (c d-b e)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/((d + e*x)^2*(b*x + c*x^2)^(5/2)),x]
[Out]
(-2*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(3*b^2*d*(c*d - b*e)*(d + e*x)*(b*x + c*x^2)^(3/2)) + (2*(b*(c*d - b*
e)*(8*c^2*d^2 - 2*b*c*d*e - 5*b^2*e^2) + c*(2*c*d - b*e)*(8*c^2*d^2 - 8*b*c*d*e - 5*b^2*e^2)*x))/(3*b^4*d^2*(c
*d - b*e)^2*(d + e*x)*Sqrt[b*x + c*x^2]) + (e*(32*c^4*d^4 - 64*b*c^3*d^3*e + 12*b^2*c^2*d^2*e^2 + 20*b^3*c*d*e
^3 - 15*b^4*e^4)*Sqrt[b*x + c*x^2])/(3*b^4*d^3*(c*d - b*e)^3*(d + e*x)) + (5*e^4*(2*c*d - b*e)*ArcTanh[(b*d +
(2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(2*d^(7/2)*(c*d - b*e)^(7/2))
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 724
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]
Rule 740
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 806
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]
Rule 822
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])
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) (d+e x) \left (b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {\frac {1}{2} \left (8 c^2 d^2-2 b c d e-5 b^2 e^2\right )+3 c e (2 c d-b e) x}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2 d (c d-b e)}\\ &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) (d+e x) \left (b x+c x^2\right )^{3/2}}+\frac {2 \left (b (c d-b e) \left (8 c^2 d^2-2 b c d e-5 b^2 e^2\right )+c (2 c d-b e) \left (8 c^2 d^2-8 b c d e-5 b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 (d+e x) \sqrt {b x+c x^2}}+\frac {4 \int \frac {\frac {1}{4} b e \left (16 c^3 d^3-16 b c^2 d^2 e-10 b^2 c d e^2+15 b^3 e^3\right )+\frac {1}{2} c e (2 c d-b e) \left (8 c^2 d^2-8 b c d e-5 b^2 e^2\right ) x}{(d+e x)^2 \sqrt {b x+c x^2}} \, dx}{3 b^4 d^2 (c d-b e)^2}\\ &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) (d+e x) \left (b x+c x^2\right )^{3/2}}+\frac {2 \left (b (c d-b e) \left (8 c^2 d^2-2 b c d e-5 b^2 e^2\right )+c (2 c d-b e) \left (8 c^2 d^2-8 b c d e-5 b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 (d+e x) \sqrt {b x+c x^2}}+\frac {e \left (32 c^4 d^4-64 b c^3 d^3 e+12 b^2 c^2 d^2 e^2+20 b^3 c d e^3-15 b^4 e^4\right ) \sqrt {b x+c x^2}}{3 b^4 d^3 (c d-b e)^3 (d+e x)}+\frac {\left (5 e^4 (2 c d-b e)\right ) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{2 d^3 (c d-b e)^3}\\ &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) (d+e x) \left (b x+c x^2\right )^{3/2}}+\frac {2 \left (b (c d-b e) \left (8 c^2 d^2-2 b c d e-5 b^2 e^2\right )+c (2 c d-b e) \left (8 c^2 d^2-8 b c d e-5 b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 (d+e x) \sqrt {b x+c x^2}}+\frac {e \left (32 c^4 d^4-64 b c^3 d^3 e+12 b^2 c^2 d^2 e^2+20 b^3 c d e^3-15 b^4 e^4\right ) \sqrt {b x+c x^2}}{3 b^4 d^3 (c d-b e)^3 (d+e x)}-\frac {\left (5 e^4 (2 c d-b e)\right ) \operatorname {Subst}\left (\int \frac {1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac {-b d-(2 c d-b e) x}{\sqrt {b x+c x^2}}\right )}{d^3 (c d-b e)^3}\\ &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) (d+e x) \left (b x+c x^2\right )^{3/2}}+\frac {2 \left (b (c d-b e) \left (8 c^2 d^2-2 b c d e-5 b^2 e^2\right )+c (2 c d-b e) \left (8 c^2 d^2-8 b c d e-5 b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 (d+e x) \sqrt {b x+c x^2}}+\frac {e \left (32 c^4 d^4-64 b c^3 d^3 e+12 b^2 c^2 d^2 e^2+20 b^3 c d e^3-15 b^4 e^4\right ) \sqrt {b x+c x^2}}{3 b^4 d^3 (c d-b e)^3 (d+e x)}+\frac {5 e^4 (2 c d-b e) \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{2 d^{7/2} (c d-b e)^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.93, size = 351, normalized size = 1.01 \begin {gather*} -\frac {9 b^4 d^{5/2} e (c d-b e)^{5/2}+(d+e x) \left (3 b^3 d^{3/2} (2 c d-5 b e) (c d-b e)^{5/2}-9 b^2 \sqrt {d} x (c d-b e)^{5/2} \left (4 c^2 d^2-5 b^2 e^2\right )-3 b c \sqrt {d} x^2 (c d-b e)^{3/2} \left (15 b^3 e^3-10 b^2 c d e^2-16 b c^2 d^2 e+16 c^3 d^3\right )+x^{3/2} (b+c x) \left (45 b^4 e^4 \sqrt {b+c x} (b e-2 c d) \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt {c d-b e}}{\sqrt {d} \sqrt {b+c x}}\right )-3 c \sqrt {d} \sqrt {x} \sqrt {c d-b e} \left (-15 b^4 e^4+20 b^3 c d e^3+12 b^2 c^2 d^2 e^2-64 b c^3 d^3 e+32 c^4 d^4\right )\right )\right )}{9 b^4 d^{7/2} (x (b+c x))^{3/2} (d+e x) (c d-b e)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/((d + e*x)^2*(b*x + c*x^2)^(5/2)),x]
[Out]
-1/9*(9*b^4*d^(5/2)*e*(c*d - b*e)^(5/2) + (d + e*x)*(3*b^3*d^(3/2)*(2*c*d - 5*b*e)*(c*d - b*e)^(5/2) - 9*b^2*S
qrt[d]*(c*d - b*e)^(5/2)*(4*c^2*d^2 - 5*b^2*e^2)*x - 3*b*c*Sqrt[d]*(c*d - b*e)^(3/2)*(16*c^3*d^3 - 16*b*c^2*d^
2*e - 10*b^2*c*d*e^2 + 15*b^3*e^3)*x^2 + x^(3/2)*(b + c*x)*(-3*c*Sqrt[d]*Sqrt[c*d - b*e]*(32*c^4*d^4 - 64*b*c^
3*d^3*e + 12*b^2*c^2*d^2*e^2 + 20*b^3*c*d*e^3 - 15*b^4*e^4)*Sqrt[x] + 45*b^4*e^4*(-2*c*d + b*e)*Sqrt[b + c*x]*
ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])))/(b^4*d^(7/2)*(c*d - b*e)^(7/2)*(x*(b + c*x))^(3/
2)*(d + e*x))
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IntegrateAlgebraic [A] time = 2.54, size = 475, normalized size = 1.36 \begin {gather*} \frac {\sqrt {b x+c x^2} \left (-2 b^6 d^2 e^3+10 b^6 d e^4 x+15 b^6 e^5 x^2+6 b^5 c d^3 e^2-18 b^5 c d^2 e^3 x+30 b^5 c e^5 x^3-6 b^4 c^2 d^4 e-6 b^4 c^2 d^3 e^2 x-42 b^4 c^2 d^2 e^3 x^2-30 b^4 c^2 d e^4 x^3+15 b^4 c^2 e^5 x^4+2 b^3 c^3 d^5+26 b^3 c^3 d^4 e x+6 b^3 c^3 d^3 e^2 x^2-38 b^3 c^3 d^2 e^3 x^3-20 b^3 c^3 d e^4 x^4-12 b^2 c^4 d^5 x+84 b^2 c^4 d^4 e x^2+84 b^2 c^4 d^3 e^2 x^3-12 b^2 c^4 d^2 e^3 x^4-48 b c^5 d^5 x^2+16 b c^5 d^4 e x^3+64 b c^5 d^3 e^2 x^4-32 c^6 d^5 x^3-32 c^6 d^4 e x^4\right )}{3 b^4 d^3 x^2 (b+c x)^2 (d+e x) (b e-c d)^3}+\frac {5 \left (2 c d e^4-b e^5\right ) \tanh ^{-1}\left (\frac {-e \sqrt {b x+c x^2}+\sqrt {c} d+\sqrt {c} e x}{\sqrt {d} \sqrt {c d-b e}}\right )}{d^{7/2} (c d-b e)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[1/((d + e*x)^2*(b*x + c*x^2)^(5/2)),x]
[Out]
(Sqrt[b*x + c*x^2]*(2*b^3*c^3*d^5 - 6*b^4*c^2*d^4*e + 6*b^5*c*d^3*e^2 - 2*b^6*d^2*e^3 - 12*b^2*c^4*d^5*x + 26*
b^3*c^3*d^4*e*x - 6*b^4*c^2*d^3*e^2*x - 18*b^5*c*d^2*e^3*x + 10*b^6*d*e^4*x - 48*b*c^5*d^5*x^2 + 84*b^2*c^4*d^
4*e*x^2 + 6*b^3*c^3*d^3*e^2*x^2 - 42*b^4*c^2*d^2*e^3*x^2 + 15*b^6*e^5*x^2 - 32*c^6*d^5*x^3 + 16*b*c^5*d^4*e*x^
3 + 84*b^2*c^4*d^3*e^2*x^3 - 38*b^3*c^3*d^2*e^3*x^3 - 30*b^4*c^2*d*e^4*x^3 + 30*b^5*c*e^5*x^3 - 32*c^6*d^4*e*x
^4 + 64*b*c^5*d^3*e^2*x^4 - 12*b^2*c^4*d^2*e^3*x^4 - 20*b^3*c^3*d*e^4*x^4 + 15*b^4*c^2*e^5*x^4))/(3*b^4*d^3*(-
(c*d) + b*e)^3*x^2*(b + c*x)^2*(d + e*x)) + (5*(2*c*d*e^4 - b*e^5)*ArcTanh[(Sqrt[c]*d + Sqrt[c]*e*x - e*Sqrt[b
*x + c*x^2])/(Sqrt[d]*Sqrt[c*d - b*e])])/(d^(7/2)*(c*d - b*e)^(7/2))
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fricas [B] time = 0.54, size = 1789, normalized size = 5.14
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^2/(c*x^2+b*x)^(5/2),x, algorithm="fricas")
[Out]
[1/6*(15*((2*b^4*c^3*d*e^5 - b^5*c^2*e^6)*x^5 + (2*b^4*c^3*d^2*e^4 + 3*b^5*c^2*d*e^5 - 2*b^6*c*e^6)*x^4 + (4*b
^5*c^2*d^2*e^4 - b^7*e^6)*x^3 + (2*b^6*c*d^2*e^4 - b^7*d*e^5)*x^2)*sqrt(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e
)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d)) - 2*(2*b^3*c^4*d^7 - 8*b^4*c^3*d^6*e + 12*b^5*c^2*d^
5*e^2 - 8*b^6*c*d^4*e^3 + 2*b^7*d^3*e^4 - (32*c^7*d^6*e - 96*b*c^6*d^5*e^2 + 76*b^2*c^5*d^4*e^3 + 8*b^3*c^4*d^
3*e^4 - 35*b^4*c^3*d^2*e^5 + 15*b^5*c^2*d*e^6)*x^4 - 2*(16*c^7*d^7 - 24*b*c^6*d^6*e - 34*b^2*c^5*d^5*e^2 + 61*
b^3*c^4*d^4*e^3 - 4*b^4*c^3*d^3*e^4 - 30*b^5*c^2*d^2*e^5 + 15*b^6*c*d*e^6)*x^3 - 3*(16*b*c^6*d^7 - 44*b^2*c^5*
d^6*e + 26*b^3*c^4*d^5*e^2 + 16*b^4*c^3*d^4*e^3 - 14*b^5*c^2*d^3*e^4 - 5*b^6*c*d^2*e^5 + 5*b^7*d*e^6)*x^2 - 2*
(6*b^2*c^5*d^7 - 19*b^3*c^4*d^6*e + 16*b^4*c^3*d^5*e^2 + 6*b^5*c^2*d^4*e^3 - 14*b^6*c*d^3*e^4 + 5*b^7*d^2*e^5)
*x)*sqrt(c*x^2 + b*x))/((b^4*c^6*d^8*e - 4*b^5*c^5*d^7*e^2 + 6*b^6*c^4*d^6*e^3 - 4*b^7*c^3*d^5*e^4 + b^8*c^2*d
^4*e^5)*x^5 + (b^4*c^6*d^9 - 2*b^5*c^5*d^8*e - 2*b^6*c^4*d^7*e^2 + 8*b^7*c^3*d^6*e^3 - 7*b^8*c^2*d^5*e^4 + 2*b
^9*c*d^4*e^5)*x^4 + (2*b^5*c^5*d^9 - 7*b^6*c^4*d^8*e + 8*b^7*c^3*d^7*e^2 - 2*b^8*c^2*d^6*e^3 - 2*b^9*c*d^5*e^4
+ b^10*d^4*e^5)*x^3 + (b^6*c^4*d^9 - 4*b^7*c^3*d^8*e + 6*b^8*c^2*d^7*e^2 - 4*b^9*c*d^6*e^3 + b^10*d^5*e^4)*x^
2), 1/3*(15*((2*b^4*c^3*d*e^5 - b^5*c^2*e^6)*x^5 + (2*b^4*c^3*d^2*e^4 + 3*b^5*c^2*d*e^5 - 2*b^6*c*e^6)*x^4 + (
4*b^5*c^2*d^2*e^4 - b^7*e^6)*x^3 + (2*b^6*c*d^2*e^4 - b^7*d*e^5)*x^2)*sqrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^2
+ b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) - (2*b^3*c^4*d^7 - 8*b^4*c^3*d^6*e + 12*b^5*c^2*d^5*e^2 - 8*b^6*c
*d^4*e^3 + 2*b^7*d^3*e^4 - (32*c^7*d^6*e - 96*b*c^6*d^5*e^2 + 76*b^2*c^5*d^4*e^3 + 8*b^3*c^4*d^3*e^4 - 35*b^4*
c^3*d^2*e^5 + 15*b^5*c^2*d*e^6)*x^4 - 2*(16*c^7*d^7 - 24*b*c^6*d^6*e - 34*b^2*c^5*d^5*e^2 + 61*b^3*c^4*d^4*e^3
- 4*b^4*c^3*d^3*e^4 - 30*b^5*c^2*d^2*e^5 + 15*b^6*c*d*e^6)*x^3 - 3*(16*b*c^6*d^7 - 44*b^2*c^5*d^6*e + 26*b^3*
c^4*d^5*e^2 + 16*b^4*c^3*d^4*e^3 - 14*b^5*c^2*d^3*e^4 - 5*b^6*c*d^2*e^5 + 5*b^7*d*e^6)*x^2 - 2*(6*b^2*c^5*d^7
- 19*b^3*c^4*d^6*e + 16*b^4*c^3*d^5*e^2 + 6*b^5*c^2*d^4*e^3 - 14*b^6*c*d^3*e^4 + 5*b^7*d^2*e^5)*x)*sqrt(c*x^2
+ b*x))/((b^4*c^6*d^8*e - 4*b^5*c^5*d^7*e^2 + 6*b^6*c^4*d^6*e^3 - 4*b^7*c^3*d^5*e^4 + b^8*c^2*d^4*e^5)*x^5 + (
b^4*c^6*d^9 - 2*b^5*c^5*d^8*e - 2*b^6*c^4*d^7*e^2 + 8*b^7*c^3*d^6*e^3 - 7*b^8*c^2*d^5*e^4 + 2*b^9*c*d^4*e^5)*x
^4 + (2*b^5*c^5*d^9 - 7*b^6*c^4*d^8*e + 8*b^7*c^3*d^7*e^2 - 2*b^8*c^2*d^6*e^3 - 2*b^9*c*d^5*e^4 + b^10*d^4*e^5
)*x^3 + (b^6*c^4*d^9 - 4*b^7*c^3*d^8*e + 6*b^8*c^2*d^7*e^2 - 4*b^9*c*d^6*e^3 + b^10*d^5*e^4)*x^2)]
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giac [B] time = 1.58, size = 1363, normalized size = 3.92
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^2/(c*x^2+b*x)^(5/2),x, algorithm="giac")
[Out]
-1/6*((64*sqrt(c*d^2 - b*d*e)*c^5*d^4*e^2 - 128*sqrt(c*d^2 - b*d*e)*b*c^4*d^3*e^3 + 24*sqrt(c*d^2 - b*d*e)*b^2
*c^3*d^2*e^4 - 30*b^4*c^(3/2)*d*e^6*log(abs(2*c*d - b*e - 2*sqrt(c*d^2 - b*d*e)*sqrt(c))) + 40*sqrt(c*d^2 - b*
d*e)*b^3*c^2*d*e^5 + 15*b^5*sqrt(c)*e^7*log(abs(2*c*d - b*e - 2*sqrt(c*d^2 - b*d*e)*sqrt(c))) - 30*sqrt(c*d^2
- b*d*e)*b^4*c*e^6)*sgn(1/(x*e + d))/(sqrt(c*d^2 - b*d*e)*b^4*c^(7/2)*d^6 - 3*sqrt(c*d^2 - b*d*e)*b^5*c^(5/2)*
d^5*e + 3*sqrt(c*d^2 - b*d*e)*b^6*c^(3/2)*d^4*e^2 - sqrt(c*d^2 - b*d*e)*b^7*sqrt(c)*d^3*e^3) + 2*((((4*(8*c^6*
d^7*e^16 - 28*b*c^5*d^6*e^17 + 30*b^2*c^4*d^5*e^18 - 5*b^3*c^3*d^4*e^19 - 18*b^4*c^2*d^3*e^20 + 18*b^5*c*d^2*e
^21 - 5*b^6*d*e^22)/(b^4*c^3*d^6*e^11*sgn(1/(x*e + d)) - 3*b^5*c^2*d^5*e^12*sgn(1/(x*e + d)) + 3*b^6*c*d^4*e^1
3*sgn(1/(x*e + d)) - b^7*d^3*e^14*sgn(1/(x*e + d))) + 3*(b^4*c^2*d^4*e^21 - 2*b^5*c*d^3*e^22 + b^6*d^2*e^23)*e
^(-1)/((b^4*c^3*d^6*e^11*sgn(1/(x*e + d)) - 3*b^5*c^2*d^5*e^12*sgn(1/(x*e + d)) + 3*b^6*c*d^4*e^13*sgn(1/(x*e
+ d)) - b^7*d^3*e^14*sgn(1/(x*e + d)))*(x*e + d)))*e^(-1)/(x*e + d) - 3*(32*c^6*d^6*e^15 - 96*b*c^5*d^5*e^16 +
80*b^2*c^4*d^4*e^17 - 46*b^4*c^2*d^2*e^19 + 30*b^5*c*d*e^20 - 5*b^6*e^21)/(b^4*c^3*d^6*e^11*sgn(1/(x*e + d))
- 3*b^5*c^2*d^5*e^12*sgn(1/(x*e + d)) + 3*b^6*c*d^4*e^13*sgn(1/(x*e + d)) - b^7*d^3*e^14*sgn(1/(x*e + d))))*e^
(-1)/(x*e + d) + 6*(16*c^6*d^5*e^14 - 40*b*c^5*d^4*e^15 + 22*b^2*c^4*d^3*e^16 + 7*b^3*c^3*d^2*e^17 - 15*b^4*c^
2*d*e^18 + 5*b^5*c*e^19)/(b^4*c^3*d^6*e^11*sgn(1/(x*e + d)) - 3*b^5*c^2*d^5*e^12*sgn(1/(x*e + d)) + 3*b^6*c*d^
4*e^13*sgn(1/(x*e + d)) - b^7*d^3*e^14*sgn(1/(x*e + d))))*e^(-1)/(x*e + d) - (32*c^6*d^4*e^13 - 64*b*c^5*d^3*e
^14 + 12*b^2*c^4*d^2*e^15 + 20*b^3*c^3*d*e^16 - 15*b^4*c^2*e^17)/(b^4*c^3*d^6*e^11*sgn(1/(x*e + d)) - 3*b^5*c^
2*d^5*e^12*sgn(1/(x*e + d)) + 3*b^6*c*d^4*e^13*sgn(1/(x*e + d)) - b^7*d^3*e^14*sgn(1/(x*e + d))))/(c - 2*c*d/(
x*e + d) + c*d^2/(x*e + d)^2 + b*e/(x*e + d) - b*d*e/(x*e + d)^2)^(3/2) + 15*(2*c*d*e^7 - b*e^8)*log(abs(2*c*d
- b*e - 2*sqrt(c*d^2 - b*d*e)*(sqrt(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + b*e/(x*e + d) - b*d*e/(x*e + d)
^2) + sqrt(c*d^2*e^2 - b*d*e^3)*e^(-1)/(x*e + d))))/((c^3*d^6*e - 3*b*c^2*d^5*e^2 + 3*b^2*c*d^4*e^3 - b^3*d^3*
e^4)*sqrt(c*d^2 - b*d*e)*sgn(1/(x*e + d))))*e^(-2)
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maple [B] time = 0.06, size = 1857, normalized size = 5.34
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(e*x+d)^2/(c*x^2+b*x)^(5/2),x)
[Out]
80/3/(b*e-c*d)^2*c^3/b^3/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)+1/(b*e-c*d)/d/(x+d/e)/((x+d
/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)-10/3/(b*e-c*d)^2/b/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*
d)*(x+d/e)/e)^(3/2)*c^2+5*e^3/(b*e-c*d)^3/d^2/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d
/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*c+20/3*e^
2/(b*e-c*d)^2/d^2*c/b/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)-5/3*e^2/(b*e-c*d)^2/d^2/((x+d/
e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)*c*x-16/3*c^2/(b*e-c*d)/d/b^2/((x+d/e)^2*c-(b*e-c*d)*d/e^2+
(b*e-2*c*d)*(x+d/e)/e)^(3/2)*x+128/3*c^3/(b*e-c*d)/d/b^4/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(
1/2)*x-80/3*e/(b*e-c*d)^2/d*c^2/b^2/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)+10*e^2/(b*e-c*d)
^3/d/b/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*c^2+5*e^4/(b*e-c*d)^3/d^3/((x+d/e)^2*c-(b*e-c
*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*c-5/2*e^4/(b*e-c*d)^3/d^3/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d
/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2
))/(x+d/e))*b-20/3/(b*e-c*d)^2/b^2/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)*c^3*x-8/3*c/(b*e-
c*d)/d/b/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)+64/3*c^2/(b*e-c*d)/d/b^3/((x+d/e)^2*c-(b*e-
c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)+5*e^4/(b*e-c*d)^3/d^3/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)
/e)^(1/2)*b-15*e^3/(b*e-c*d)^3/d^2/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*c-5/3*e^2/(b*e-c*
d)^2/d^2/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)*b+5*e/(b*e-c*d)^2/d/((x+d/e)^2*c-(b*e-c*d)*
d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)*c+160/3/(b*e-c*d)^2*c^4/b^4/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e
)/e)^(1/2)*x+20*e^2/(b*e-c*d)^3/d/b^2/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*c^3+20/3*e/(
b*e-c*d)^2/d/b/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)*c^2*x-20*e^3/(b*e-c*d)^3/d^2/b/((x+d/
e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*c^2+40/3*e^2/(b*e-c*d)^2/d^2*c^2/b^2/((x+d/e)^2*c-(b*e-c
*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x-160/3*e/(b*e-c*d)^2/d*c^3/b^3/((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d
)*(x+d/e)/e)^(1/2)*x
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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(1/(e*x+d)^2/(c*x^2+b*x)^(5/2),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(b*e-c*d>0)', see `assume?` for
more details)Is b*e-c*d positive, negative or zero?
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (c\,x^2+b\,x\right )}^{5/2}\,{\left (d+e\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((b*x + c*x^2)^(5/2)*(d + e*x)^2),x)
[Out]
int(1/((b*x + c*x^2)^(5/2)*(d + e*x)^2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (d + e x\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)**2/(c*x**2+b*x)**(5/2),x)
[Out]
Integral(1/((x*(b + c*x))**(5/2)*(d + e*x)**2), x)
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