3.4.35 \(\int \frac {1}{(d+e x)^2 (b x+c x^2)^{5/2}} \, dx\) [335]

Optimal. Leaf size=348 \[ -\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) (d+e x) \left (b x+c x^2\right )^{3/2}}+\frac {2 \left (b (c d-b e) \left (8 c^2 d^2-2 b c d e-5 b^2 e^2\right )+c (2 c d-b e) \left (8 c^2 d^2-8 b c d e-5 b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 (d+e x) \sqrt {b x+c x^2}}+\frac {e \left (32 c^4 d^4-64 b c^3 d^3 e+12 b^2 c^2 d^2 e^2+20 b^3 c d e^3-15 b^4 e^4\right ) \sqrt {b x+c x^2}}{3 b^4 d^3 (c d-b e)^3 (d+e x)}+\frac {5 e^4 (2 c d-b e) \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{2 d^{7/2} (c d-b e)^{7/2}} \]

[Out]

-2/3*(b*(-b*e+c*d)+c*(-b*e+2*c*d)*x)/b^2/d/(-b*e+c*d)/(e*x+d)/(c*x^2+b*x)^(3/2)+5/2*e^4*(-b*e+2*c*d)*arctanh(1
/2*(b*d+(-b*e+2*c*d)*x)/d^(1/2)/(-b*e+c*d)^(1/2)/(c*x^2+b*x)^(1/2))/d^(7/2)/(-b*e+c*d)^(7/2)+2/3*(b*(-b*e+c*d)
*(-5*b^2*e^2-2*b*c*d*e+8*c^2*d^2)+c*(-b*e+2*c*d)*(-5*b^2*e^2-8*b*c*d*e+8*c^2*d^2)*x)/b^4/d^2/(-b*e+c*d)^2/(e*x
+d)/(c*x^2+b*x)^(1/2)+1/3*e*(-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)*(c*x^2+b
*x)^(1/2)/b^4/d^3/(-b*e+c*d)^3/(e*x+d)

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Rubi [A]
time = 0.25, 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 = {754, 836, 820, 738, 212} \begin {gather*} -\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}} \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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 738

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 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 820

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(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[S
implify[m + 2*p + 3], 0]

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])

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 ) \text {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 = 1.83, size = 379, normalized size = 1.09 \begin {gather*} \frac {x \left (\frac {\sqrt {d} (b+c x) \left (-32 c^6 d^4 x^3 (d+e x)+16 b c^5 d^3 x^2 \left (-3 d^2+d e x+4 e^2 x^2\right )+b^6 e^3 \left (-2 d^2+10 d e x+15 e^2 x^2\right )-12 b^2 c^4 d^2 x \left (d^3-7 d^2 e x-7 d e^2 x^2+e^3 x^3\right )+6 b^5 c e^2 \left (d^3-3 d^2 e x+5 e^3 x^3\right )+2 b^3 c^3 d \left (d^4+13 d^3 e x+3 d^2 e^2 x^2-19 d e^3 x^3-10 e^4 x^4\right )-3 b^4 c^2 e \left (2 d^4+2 d^3 e x+14 d^2 e^2 x^2+10 d e^3 x^3-5 e^4 x^4\right )\right )}{b^4 (-c d+b e)^3 (d+e x)}+\frac {15 e^4 (2 c d-b e) x^{3/2} (b+c x)^{5/2} \tan ^{-1}\left (\frac {-e \sqrt {x} \sqrt {b+c x}+\sqrt {c} (d+e x)}{\sqrt {d} \sqrt {-c d+b e}}\right )}{(-c d+b e)^{7/2}}\right )}{3 d^{7/2} (x (b+c x))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^2*(b*x + c*x^2)^(5/2)),x]

[Out]

(x*((Sqrt[d]*(b + c*x)*(-32*c^6*d^4*x^3*(d + e*x) + 16*b*c^5*d^3*x^2*(-3*d^2 + d*e*x + 4*e^2*x^2) + b^6*e^3*(-
2*d^2 + 10*d*e*x + 15*e^2*x^2) - 12*b^2*c^4*d^2*x*(d^3 - 7*d^2*e*x - 7*d*e^2*x^2 + e^3*x^3) + 6*b^5*c*e^2*(d^3
- 3*d^2*e*x + 5*e^3*x^3) + 2*b^3*c^3*d*(d^4 + 13*d^3*e*x + 3*d^2*e^2*x^2 - 19*d*e^3*x^3 - 10*e^4*x^4) - 3*b^4
*c^2*e*(2*d^4 + 2*d^3*e*x + 14*d^2*e^2*x^2 + 10*d*e^3*x^3 - 5*e^4*x^4)))/(b^4*(-(c*d) + b*e)^3*(d + e*x)) + (1
5*e^4*(2*c*d - b*e)*x^(3/2)*(b + c*x)^(5/2)*ArcTan[(-(e*Sqrt[x]*Sqrt[b + c*x]) + Sqrt[c]*(d + e*x))/(Sqrt[d]*S
qrt[-(c*d) + b*e])])/(-(c*d) + b*e)^(7/2)))/(3*d^(7/2)*(x*(b + c*x))^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(988\) vs. \(2(322)=644\).
time = 0.61, size = 989, normalized size = 2.84

method	result	size
risch	\(-\frac {2 \left (c x +b \right ) \left (-6 b e x -8 c d x +b d \right )}{3 b^{4} d^{3} \sqrt {x \left (c x +b \right )}\, x}+\frac {2 c^{2} \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}}{3 b^{3} \left (b e -c d \right )^{2} \left (\frac {b}{c}+x \right )^{2}}+\frac {4 c^{3} \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}}{3 b^{4} \left (b e -c d \right )^{2} \left (\frac {b}{c}+x \right )}+\frac {e^{4} \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{d^{3} \left (b e -c d \right )^{3} \left (x +\frac {d}{e}\right )}-\frac {5 b \,e^{4} \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 d^{3} \left (b e -c d \right )^{3} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {5 e^{3} \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right ) c}{d^{2} \left (b e -c d \right )^{3} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {8 c^{3} \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}\, e}{b^{3} \left (b e -c d \right )^{3} \left (\frac {b}{c}+x \right )}-\frac {4 d \,c^{4} \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}}{b^{4} \left (b e -c d \right )^{3} \left (\frac {b}{c}+x \right )}\)	\(610\)
default	\(\frac {\frac {e^{2}}{d \left (b e -c d \right ) \left (x +\frac {d}{e}\right ) \left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}\right )^{\frac {3}{2}}}+\frac {5 e \left (b e -2 c d \right ) \left (-\frac {e^{2}}{3 d \left (b e -c d \right ) \left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}\right )^{\frac {3}{2}}}+\frac {e \left (b e -2 c d \right ) \left (\frac {\frac {4 c \left (x +\frac {d}{e}\right )}{3}+\frac {2 \left (b e -2 c d \right )}{3 e}}{\left (-\frac {4 c d \left (b e -c d \right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}\right )^{\frac {3}{2}}}+\frac {16 c \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{3 \left (-\frac {4 c d \left (b e -c d \right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right )^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{2 d \left (b e -c d \right )}-\frac {e^{2} \left (-\frac {e^{2}}{d \left (b e -c d \right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {e \left (b e -2 c d \right ) \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{d \left (b e -c d \right ) \left (-\frac {4 c d \left (b e -c d \right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {e^{2} \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{d \left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{d \left (b e -c d \right )}\right )}{2 d \left (b e -c d \right )}+\frac {4 c \,e^{2} \left (\frac {\frac {4 c \left (x +\frac {d}{e}\right )}{3}+\frac {2 \left (b e -2 c d \right )}{3 e}}{\left (-\frac {4 c d \left (b e -c d \right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}\right )^{\frac {3}{2}}}+\frac {16 c \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{3 \left (-\frac {4 c d \left (b e -c d \right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right )^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{d \left (b e -c d \right )}}{e^{2}}\)	\(989\)

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,method=_RETURNVERBOSE)

[Out]

1/e^2*(1/d/(b*e-c*d)*e^2/(x+d/e)/(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)+5/2*e*(b*e-2*c*d)
/d/(b*e-c*d)*(-1/3/d/(b*e-c*d)*e^2/(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)+1/2*e*(b*e-2*c*
d)/d/(b*e-c*d)*(2/3*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/(-4*c*d*(b*e-c*d)/e^2-1/e^2*(b*e-2*c*d)^2)/(c*(x+d/e)^2+1/e*
(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)+16/3*c/(-4*c*d*(b*e-c*d)/e^2-1/e^2*(b*e-2*c*d)^2)^2*(2*c*(x+d/e)+1/
e*(b*e-2*c*d))/(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))-1/d/(b*e-c*d)*e^2*(-1/d/(b*e-c*d)*
e^2/(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)+e*(b*e-2*c*d)/d/(b*e-c*d)*(2*c*(x+d/e)+1/e*(b*
e-2*c*d))/(-4*c*d*(b*e-c*d)/e^2-1/e^2*(b*e-2*c*d)^2)/(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/
2)+1/d/(b*e-c*d)*e^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+1/e*(b*e-2*c*d)*(x+d/e)+2*(-d*(b*e-c*d)/e
^2)^(1/2)*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))))+4*c/d/(b*e-c*d)*e^2*(2/3*(2*
c*(x+d/e)+1/e*(b*e-2*c*d))/(-4*c*d*(b*e-c*d)/e^2-1/e^2*(b*e-2*c*d)^2)/(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(
b*e-c*d)/e^2)^(3/2)+16/3*c/(-4*c*d*(b*e-c*d)/e^2-1/e^2*(b*e-2*c*d)^2)^2*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/(c*(x+d/
e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)))

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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(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. 902 vs. \(2 (334) = 668\).
time = 1.98, size = 1816, normalized size = 5.22 \begin {gather*} \left [-\frac {15 \, \sqrt {c d^{2} - b d e} {\left ({\left (b^{5} c^{2} x^{5} + 2 \, b^{6} c x^{4} + b^{7} x^{3}\right )} e^{6} - {\left (2 \, b^{4} c^{3} d x^{5} + 3 \, b^{5} c^{2} d x^{4} - b^{7} d x^{2}\right )} e^{5} - 2 \, {\left (b^{4} c^{3} d^{2} x^{4} + 2 \, b^{5} c^{2} d^{2} x^{3} + b^{6} c d^{2} x^{2}\right )} e^{4}\right )} \log \left (\frac {2 \, c d x - b x e + b d + 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{x e + d}\right ) - 2 \, {\left (32 \, c^{7} d^{7} x^{3} + 48 \, b c^{6} d^{7} x^{2} + 12 \, b^{2} c^{5} d^{7} x - 2 \, b^{3} c^{4} d^{7} + 15 \, {\left (b^{5} c^{2} d x^{4} + 2 \, b^{6} c d x^{3} + b^{7} d x^{2}\right )} e^{6} - 5 \, {\left (7 \, b^{4} c^{3} d^{2} x^{4} + 12 \, b^{5} c^{2} d^{2} x^{3} + 3 \, b^{6} c d^{2} x^{2} - 2 \, b^{7} d^{2} x\right )} e^{5} + 2 \, {\left (4 \, b^{3} c^{4} d^{3} x^{4} - 4 \, b^{4} c^{3} d^{3} x^{3} - 21 \, b^{5} c^{2} d^{3} x^{2} - 14 \, b^{6} c d^{3} x - b^{7} d^{3}\right )} e^{4} + 2 \, {\left (38 \, b^{2} c^{5} d^{4} x^{4} + 61 \, b^{3} c^{4} d^{4} x^{3} + 24 \, b^{4} c^{3} d^{4} x^{2} + 6 \, b^{5} c^{2} d^{4} x + 4 \, b^{6} c d^{4}\right )} e^{3} - 2 \, {\left (48 \, b c^{6} d^{5} x^{4} + 34 \, b^{2} c^{5} d^{5} x^{3} - 39 \, b^{3} c^{4} d^{5} x^{2} - 16 \, b^{4} c^{3} d^{5} x + 6 \, b^{5} c^{2} d^{5}\right )} e^{2} + 2 \, {\left (16 \, c^{7} d^{6} x^{4} - 24 \, b c^{6} d^{6} x^{3} - 66 \, b^{2} c^{5} d^{6} x^{2} - 19 \, b^{3} c^{4} d^{6} x + 4 \, b^{4} c^{3} d^{6}\right )} e\right )} \sqrt {c x^{2} + b x}}{6 \, {\left (b^{4} c^{6} d^{9} x^{4} + 2 \, b^{5} c^{5} d^{9} x^{3} + b^{6} c^{4} d^{9} x^{2} + {\left (b^{8} c^{2} d^{4} x^{5} + 2 \, b^{9} c d^{4} x^{4} + b^{10} d^{4} x^{3}\right )} e^{5} - {\left (4 \, b^{7} c^{3} d^{5} x^{5} + 7 \, b^{8} c^{2} d^{5} x^{4} + 2 \, b^{9} c d^{5} x^{3} - b^{10} d^{5} x^{2}\right )} e^{4} + 2 \, {\left (3 \, b^{6} c^{4} d^{6} x^{5} + 4 \, b^{7} c^{3} d^{6} x^{4} - b^{8} c^{2} d^{6} x^{3} - 2 \, b^{9} c d^{6} x^{2}\right )} e^{3} - 2 \, {\left (2 \, b^{5} c^{5} d^{7} x^{5} + b^{6} c^{4} d^{7} x^{4} - 4 \, b^{7} c^{3} d^{7} x^{3} - 3 \, b^{8} c^{2} d^{7} x^{2}\right )} e^{2} + {\left (b^{4} c^{6} d^{8} x^{5} - 2 \, b^{5} c^{5} d^{8} x^{4} - 7 \, b^{6} c^{4} d^{8} x^{3} - 4 \, b^{7} c^{3} d^{8} x^{2}\right )} e\right )}}, -\frac {15 \, \sqrt {-c d^{2} + b d e} {\left ({\left (b^{5} c^{2} x^{5} + 2 \, b^{6} c x^{4} + b^{7} x^{3}\right )} e^{6} - {\left (2 \, b^{4} c^{3} d x^{5} + 3 \, b^{5} c^{2} d x^{4} - b^{7} d x^{2}\right )} e^{5} - 2 \, {\left (b^{4} c^{3} d^{2} x^{4} + 2 \, b^{5} c^{2} d^{2} x^{3} + b^{6} c d^{2} x^{2}\right )} e^{4}\right )} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{c d x - b x e}\right ) - {\left (32 \, c^{7} d^{7} x^{3} + 48 \, b c^{6} d^{7} x^{2} + 12 \, b^{2} c^{5} d^{7} x - 2 \, b^{3} c^{4} d^{7} + 15 \, {\left (b^{5} c^{2} d x^{4} + 2 \, b^{6} c d x^{3} + b^{7} d x^{2}\right )} e^{6} - 5 \, {\left (7 \, b^{4} c^{3} d^{2} x^{4} + 12 \, b^{5} c^{2} d^{2} x^{3} + 3 \, b^{6} c d^{2} x^{2} - 2 \, b^{7} d^{2} x\right )} e^{5} + 2 \, {\left (4 \, b^{3} c^{4} d^{3} x^{4} - 4 \, b^{4} c^{3} d^{3} x^{3} - 21 \, b^{5} c^{2} d^{3} x^{2} - 14 \, b^{6} c d^{3} x - b^{7} d^{3}\right )} e^{4} + 2 \, {\left (38 \, b^{2} c^{5} d^{4} x^{4} + 61 \, b^{3} c^{4} d^{4} x^{3} + 24 \, b^{4} c^{3} d^{4} x^{2} + 6 \, b^{5} c^{2} d^{4} x + 4 \, b^{6} c d^{4}\right )} e^{3} - 2 \, {\left (48 \, b c^{6} d^{5} x^{4} + 34 \, b^{2} c^{5} d^{5} x^{3} - 39 \, b^{3} c^{4} d^{5} x^{2} - 16 \, b^{4} c^{3} d^{5} x + 6 \, b^{5} c^{2} d^{5}\right )} e^{2} + 2 \, {\left (16 \, c^{7} d^{6} x^{4} - 24 \, b c^{6} d^{6} x^{3} - 66 \, b^{2} c^{5} d^{6} x^{2} - 19 \, b^{3} c^{4} d^{6} x + 4 \, b^{4} c^{3} d^{6}\right )} e\right )} \sqrt {c x^{2} + b x}}{3 \, {\left (b^{4} c^{6} d^{9} x^{4} + 2 \, b^{5} c^{5} d^{9} x^{3} + b^{6} c^{4} d^{9} x^{2} + {\left (b^{8} c^{2} d^{4} x^{5} + 2 \, b^{9} c d^{4} x^{4} + b^{10} d^{4} x^{3}\right )} e^{5} - {\left (4 \, b^{7} c^{3} d^{5} x^{5} + 7 \, b^{8} c^{2} d^{5} x^{4} + 2 \, b^{9} c d^{5} x^{3} - b^{10} d^{5} x^{2}\right )} e^{4} + 2 \, {\left (3 \, b^{6} c^{4} d^{6} x^{5} + 4 \, b^{7} c^{3} d^{6} x^{4} - b^{8} c^{2} d^{6} x^{3} - 2 \, b^{9} c d^{6} x^{2}\right )} e^{3} - 2 \, {\left (2 \, b^{5} c^{5} d^{7} x^{5} + b^{6} c^{4} d^{7} x^{4} - 4 \, b^{7} c^{3} d^{7} x^{3} - 3 \, b^{8} c^{2} d^{7} x^{2}\right )} e^{2} + {\left (b^{4} c^{6} d^{8} x^{5} - 2 \, b^{5} c^{5} d^{8} x^{4} - 7 \, b^{6} c^{4} d^{8} x^{3} - 4 \, b^{7} c^{3} d^{8} x^{2}\right )} e\right )}}\right ] \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="fricas")

[Out]

[-1/6*(15*sqrt(c*d^2 - b*d*e)*((b^5*c^2*x^5 + 2*b^6*c*x^4 + b^7*x^3)*e^6 - (2*b^4*c^3*d*x^5 + 3*b^5*c^2*d*x^4
- b^7*d*x^2)*e^5 - 2*(b^4*c^3*d^2*x^4 + 2*b^5*c^2*d^2*x^3 + b^6*c*d^2*x^2)*e^4)*log((2*c*d*x - b*x*e + b*d + 2
*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(x*e + d)) - 2*(32*c^7*d^7*x^3 + 48*b*c^6*d^7*x^2 + 12*b^2*c^5*d^7*x -
2*b^3*c^4*d^7 + 15*(b^5*c^2*d*x^4 + 2*b^6*c*d*x^3 + b^7*d*x^2)*e^6 - 5*(7*b^4*c^3*d^2*x^4 + 12*b^5*c^2*d^2*x^
3 + 3*b^6*c*d^2*x^2 - 2*b^7*d^2*x)*e^5 + 2*(4*b^3*c^4*d^3*x^4 - 4*b^4*c^3*d^3*x^3 - 21*b^5*c^2*d^3*x^2 - 14*b^
6*c*d^3*x - b^7*d^3)*e^4 + 2*(38*b^2*c^5*d^4*x^4 + 61*b^3*c^4*d^4*x^3 + 24*b^4*c^3*d^4*x^2 + 6*b^5*c^2*d^4*x +
4*b^6*c*d^4)*e^3 - 2*(48*b*c^6*d^5*x^4 + 34*b^2*c^5*d^5*x^3 - 39*b^3*c^4*d^5*x^2 - 16*b^4*c^3*d^5*x + 6*b^5*c
^2*d^5)*e^2 + 2*(16*c^7*d^6*x^4 - 24*b*c^6*d^6*x^3 - 66*b^2*c^5*d^6*x^2 - 19*b^3*c^4*d^6*x + 4*b^4*c^3*d^6)*e)
*sqrt(c*x^2 + b*x))/(b^4*c^6*d^9*x^4 + 2*b^5*c^5*d^9*x^3 + b^6*c^4*d^9*x^2 + (b^8*c^2*d^4*x^5 + 2*b^9*c*d^4*x^
4 + b^10*d^4*x^3)*e^5 - (4*b^7*c^3*d^5*x^5 + 7*b^8*c^2*d^5*x^4 + 2*b^9*c*d^5*x^3 - b^10*d^5*x^2)*e^4 + 2*(3*b^
6*c^4*d^6*x^5 + 4*b^7*c^3*d^6*x^4 - b^8*c^2*d^6*x^3 - 2*b^9*c*d^6*x^2)*e^3 - 2*(2*b^5*c^5*d^7*x^5 + b^6*c^4*d^
7*x^4 - 4*b^7*c^3*d^7*x^3 - 3*b^8*c^2*d^7*x^2)*e^2 + (b^4*c^6*d^8*x^5 - 2*b^5*c^5*d^8*x^4 - 7*b^6*c^4*d^8*x^3
- 4*b^7*c^3*d^8*x^2)*e), -1/3*(15*sqrt(-c*d^2 + b*d*e)*((b^5*c^2*x^5 + 2*b^6*c*x^4 + b^7*x^3)*e^6 - (2*b^4*c^3
*d*x^5 + 3*b^5*c^2*d*x^4 - b^7*d*x^2)*e^5 - 2*(b^4*c^3*d^2*x^4 + 2*b^5*c^2*d^2*x^3 + b^6*c*d^2*x^2)*e^4)*arcta
n(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/(c*d*x - b*x*e)) - (32*c^7*d^7*x^3 + 48*b*c^6*d^7*x^2 + 12*b^2*c^5*d
^7*x - 2*b^3*c^4*d^7 + 15*(b^5*c^2*d*x^4 + 2*b^6*c*d*x^3 + b^7*d*x^2)*e^6 - 5*(7*b^4*c^3*d^2*x^4 + 12*b^5*c^2*
d^2*x^3 + 3*b^6*c*d^2*x^2 - 2*b^7*d^2*x)*e^5 + 2*(4*b^3*c^4*d^3*x^4 - 4*b^4*c^3*d^3*x^3 - 21*b^5*c^2*d^3*x^2 -
14*b^6*c*d^3*x - b^7*d^3)*e^4 + 2*(38*b^2*c^5*d^4*x^4 + 61*b^3*c^4*d^4*x^3 + 24*b^4*c^3*d^4*x^2 + 6*b^5*c^2*d
^4*x + 4*b^6*c*d^4)*e^3 - 2*(48*b*c^6*d^5*x^4 + 34*b^2*c^5*d^5*x^3 - 39*b^3*c^4*d^5*x^2 - 16*b^4*c^3*d^5*x + 6
*b^5*c^2*d^5)*e^2 + 2*(16*c^7*d^6*x^4 - 24*b*c^6*d^6*x^3 - 66*b^2*c^5*d^6*x^2 - 19*b^3*c^4*d^6*x + 4*b^4*c^3*d
^6)*e)*sqrt(c*x^2 + b*x))/(b^4*c^6*d^9*x^4 + 2*b^5*c^5*d^9*x^3 + b^6*c^4*d^9*x^2 + (b^8*c^2*d^4*x^5 + 2*b^9*c*
d^4*x^4 + b^10*d^4*x^3)*e^5 - (4*b^7*c^3*d^5*x^5 + 7*b^8*c^2*d^5*x^4 + 2*b^9*c*d^5*x^3 - b^10*d^5*x^2)*e^4 + 2
*(3*b^6*c^4*d^6*x^5 + 4*b^7*c^3*d^6*x^4 - b^8*c^2*d^6*x^3 - 2*b^9*c*d^6*x^2)*e^3 - 2*(2*b^5*c^5*d^7*x^5 + b^6*
c^4*d^7*x^4 - 4*b^7*c^3*d^7*x^3 - 3*b^8*c^2*d^7*x^2)*e^2 + (b^4*c^6*d^8*x^5 - 2*b^5*c^5*d^8*x^4 - 7*b^6*c^4*d^
8*x^3 - 4*b^7*c^3*d^8*x^2)*e)]

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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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1363 vs. \(2 (334) = 668\).
time = 1.99, size = 1363, normalized size = 3.92 \begin {gather*} -\frac {1}{6} \, {\left (\frac {{\left (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^{\frac {3}{2}} d e^{6} \log \left ({\left | 2 \, c d - b e - 2 \, \sqrt {c d^{2} - b d e} \sqrt {c} \right |}\right ) + 40 \, \sqrt {c d^{2} - b d e} b^{3} c^{2} d e^{5} + 15 \, b^{5} \sqrt {c} e^{7} \log \left ({\left | 2 \, c d - b e - 2 \, \sqrt {c d^{2} - b d e} \sqrt {c} \right |}\right ) - 30 \, \sqrt {c d^{2} - b d e} b^{4} c e^{6}\right )} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{\sqrt {c d^{2} - b d e} b^{4} c^{\frac {7}{2}} d^{6} - 3 \, \sqrt {c d^{2} - b d e} b^{5} c^{\frac {5}{2}} d^{5} e + 3 \, \sqrt {c d^{2} - b d e} b^{6} c^{\frac {3}{2}} d^{4} e^{2} - \sqrt {c d^{2} - b d e} b^{7} \sqrt {c} d^{3} e^{3}} + \frac {2 \, {\left (\frac {{\left (\frac {{\left (\frac {{\left (\frac {4 \, {\left (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}\right )}}{b^{4} c^{3} d^{6} e^{11} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 3 \, b^{5} c^{2} d^{5} e^{12} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 3 \, b^{6} c d^{4} e^{13} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - b^{7} d^{3} e^{14} \mathrm {sgn}\left (\frac {1}{x e + d}\right )} + \frac {3 \, {\left (b^{4} c^{2} d^{4} e^{21} - 2 \, b^{5} c d^{3} e^{22} + b^{6} d^{2} e^{23}\right )} e^{\left (-1\right )}}{{\left (b^{4} c^{3} d^{6} e^{11} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 3 \, b^{5} c^{2} d^{5} e^{12} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 3 \, b^{6} c d^{4} e^{13} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - b^{7} d^{3} e^{14} \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )} {\left (x e + d\right )}}\right )} e^{\left (-1\right )}}{x e + d} - \frac {3 \, {\left (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}\right )}}{b^{4} c^{3} d^{6} e^{11} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 3 \, b^{5} c^{2} d^{5} e^{12} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 3 \, b^{6} c d^{4} e^{13} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - b^{7} d^{3} e^{14} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}\right )} e^{\left (-1\right )}}{x e + d} + \frac {6 \, {\left (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}\right )}}{b^{4} c^{3} d^{6} e^{11} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 3 \, b^{5} c^{2} d^{5} e^{12} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 3 \, b^{6} c d^{4} e^{13} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - b^{7} d^{3} e^{14} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}\right )} e^{\left (-1\right )}}{x e + d} - \frac {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} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 3 \, b^{5} c^{2} d^{5} e^{12} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 3 \, b^{6} c d^{4} e^{13} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - b^{7} d^{3} e^{14} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}\right )}}{{\left (c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {b e}{x e + d} - \frac {b d e}{{\left (x e + d\right )}^{2}}\right )}^{\frac {3}{2}}} + \frac {15 \, {\left (2 \, c d e^{7} - b e^{8}\right )} \log \left ({\left | 2 \, c d - b e - 2 \, \sqrt {c d^{2} - b d e} {\left (\sqrt {c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {b e}{x e + d} - \frac {b d e}{{\left (x e + d\right )}^{2}}} + \frac {\sqrt {c d^{2} e^{2} - b d e^{3}} e^{\left (-1\right )}}{x e + d}\right )} \right |}\right )}{{\left (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}\right )} \sqrt {c d^{2} - b d e} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}\right )} e^{\left (-2\right )} \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="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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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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