3.4.76 \(\int \frac {1}{(d+e x)^{5/2} (b x+c x^2)^2} \, dx\) [376]

Optimal. Leaf size=267 \[ -\frac {e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right )}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right )}{b^2 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac {(4 c d+5 b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3 d^{7/2}}-\frac {c^{7/2} (4 c d-9 b e) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 (c d-b e)^{7/2}} \]

[Out]

-1/3*e*(5*b^2*e^2-6*b*c*d*e+6*c^2*d^2)/b^2/d^2/(-b*e+c*d)^2/(e*x+d)^(3/2)+(-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)+(5*b*e+4*c*d)*arctanh((e*x+d)^(1/2)/d^(1/2))/b^3/d^(7/2)-c^(7/2)*(-9*b
*e+4*c*d)*arctanh(c^(1/2)*(e*x+d)^(1/2)/(-b*e+c*d)^(1/2))/b^3/(-b*e+c*d)^(7/2)-e*(-b*e+2*c*d)*(5*b^2*e^2-b*c*d
*e+c^2*d^2)/b^2/d^3/(-b*e+c*d)^3/(e*x+d)^(1/2)

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Rubi [A]
time = 0.35, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {754, 842, 840, 1180, 214} \begin {gather*} -\frac {c^{7/2} (4 c d-9 b e) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 (c d-b e)^{7/2}}+\frac {(5 b e+4 c d) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3 d^{7/2}}-\frac {e \left (5 b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 b^2 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac {e (2 c d-b e) \left (5 b^2 e^2-b c d e+c^2 d^2\right )}{b^2 d^3 \sqrt {d+e x} (c d-b e)^3}-\frac {c x (2 c d-b e)+b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(e*(6*c^2*d^2 - 6*b*c*d*e + 5*b^2*e^2))/(b^2*d^2*(c*d - b*e)^2*(d + e*x)^(3/2)) - (e*(2*c*d - b*e)*(c^2*d
^2 - b*c*d*e + 5*b^2*e^2))/(b^2*d^3*(c*d - b*e)^3*Sqrt[d + e*x]) - (b*(c*d - b*e) + c*(2*c*d - b*e)*x)/(b^2*d*
(c*d - b*e)*(d + e*x)^(3/2)*(b*x + c*x^2)) + ((4*c*d + 5*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b^3*d^(7/2)) -
(c^(7/2)*(4*c*d - 9*b*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*(c*d - b*e)^(7/2))

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 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 )^2} \, dx &=-\frac {b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )}-\frac {\int \frac {\frac {1}{2} (c d-b e) (4 c d+5 b e)+\frac {5}{2} c e (2 c d-b e) x}{(d+e x)^{5/2} \left (b x+c x^2\right )} \, dx}{b^2 d (c d-b e)}\\ &=-\frac {e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right )}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )}-\frac {\int \frac {\frac {1}{2} (c d-b e)^2 (4 c d+5 b e)+\frac {1}{2} c e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right ) x}{(d+e x)^{3/2} \left (b x+c x^2\right )} \, dx}{b^2 d^2 (c d-b e)^2}\\ &=-\frac {e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right )}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right )}{b^2 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )}-\frac {\int \frac {\frac {1}{2} (c d-b e)^3 (4 c d+5 b e)+\frac {1}{2} c e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{b^2 d^3 (c d-b e)^3}\\ &=-\frac {e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right )}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right )}{b^2 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )}-\frac {2 \text {Subst}\left (\int \frac {\frac {1}{2} e (c d-b e)^3 (4 c d+5 b e)-\frac {1}{2} c d e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right )+\frac {1}{2} c e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\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^2 d^3 (c d-b e)^3}\\ &=-\frac {e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right )}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right )}{b^2 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac {\left (c^4 (4 c d-9 b e)\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 )}{b^3 (c d-b e)^3}-\frac {(c (4 c d+5 b e)) \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 )}{b^3 d^3}\\ &=-\frac {e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right )}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right )}{b^2 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac {(4 c d+5 b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3 d^{7/2}}-\frac {c^{7/2} (4 c d-9 b e) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 (c d-b e)^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 1.16, size = 272, normalized size = 1.02 \begin {gather*} \frac {\frac {b \left (-6 c^4 d^3 x (d+e x)^2-3 b c^3 d^2 (d-3 e x) (d+e x)^2+b^4 e^3 \left (3 d^2+20 d e x+15 e^2 x^2\right )+b^2 c^2 d e \left (9 d^3+9 d^2 e x-35 d e^2 x^2-33 e^3 x^3\right )+b^3 c e^2 \left (-9 d^3-41 d^2 e x-13 d e^2 x^2+15 e^3 x^3\right )\right )}{d^3 (c d-b e)^3 x (b+c x) (d+e x)^{3/2}}-\frac {3 c^{7/2} (4 c d-9 b e) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{(-c d+b e)^{7/2}}+\frac {3 (4 c d+5 b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{7/2}}}{3 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*(-6*c^4*d^3*x*(d + e*x)^2 - 3*b*c^3*d^2*(d - 3*e*x)*(d + e*x)^2 + b^4*e^3*(3*d^2 + 20*d*e*x + 15*e^2*x^2)
+ b^2*c^2*d*e*(9*d^3 + 9*d^2*e*x - 35*d*e^2*x^2 - 33*e^3*x^3) + b^3*c*e^2*(-9*d^3 - 41*d^2*e*x - 13*d*e^2*x^2
+ 15*e^3*x^3)))/(d^3*(c*d - b*e)^3*x*(b + c*x)*(d + e*x)^(3/2)) - (3*c^(7/2)*(4*c*d - 9*b*e)*ArcTan[(Sqrt[c]*S
qrt[d + e*x])/Sqrt[-(c*d) + b*e]])/(-(c*d) + b*e)^(7/2) + (3*(4*c*d + 5*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/d
^(7/2))/(3*b^3)

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Maple [A]
time = 0.56, size = 204, normalized size = 0.76

method	result	size
derivativedivides	\(2 e^{3} \left (\frac {-\frac {b \sqrt {e x +d}}{2 x}+\frac {\left (5 b e +4 c d \right ) \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}}{b^{3} d^{3} e^{3}}+\frac {c^{4} \left (\frac {b e \sqrt {e x +d}}{2 c \left (e x +d \right )+2 b e -2 c d}+\frac {\left (9 b e -4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}\right )}{b^{3} e^{3} \left (b e -c d \right )^{3}}-\frac {1}{3 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 b e -4 c d}{d^{3} \left (b e -c d \right )^{3} \sqrt {e x +d}}\right )\)	\(204\)
default	\(2 e^{3} \left (\frac {-\frac {b \sqrt {e x +d}}{2 x}+\frac {\left (5 b e +4 c d \right ) \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}}{b^{3} d^{3} e^{3}}+\frac {c^{4} \left (\frac {b e \sqrt {e x +d}}{2 c \left (e x +d \right )+2 b e -2 c d}+\frac {\left (9 b e -4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}\right )}{b^{3} e^{3} \left (b e -c d \right )^{3}}-\frac {1}{3 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 b e -4 c d}{d^{3} \left (b e -c d \right )^{3} \sqrt {e x +d}}\right )\)	\(204\)
risch	\(-\frac {\sqrt {e x +d}}{d^{3} b^{2} x}+\frac {e \,c^{4} \sqrt {e x +d}}{b^{2} \left (b e -c d \right )^{3} \left (c e x +b e \right )}+\frac {9 e \,c^{4} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{b^{2} \left (b e -c d \right )^{3} \sqrt {\left (b e -c d \right ) c}}-\frac {4 d \,c^{5} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{b^{3} \left (b e -c d \right )^{3} \sqrt {\left (b e -c d \right ) c}}-\frac {2 e^{3}}{3 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {4 e^{4} b}{d^{3} \left (b e -c d \right )^{3} \sqrt {e x +d}}+\frac {8 e^{3} c}{d^{2} \left (b e -c d \right )^{3} \sqrt {e x +d}}+\frac {5 e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{d^{\frac {7}{2}} b^{2}}+\frac {4 \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) c}{d^{\frac {5}{2}} b^{3}}\)	\(280\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(5/2)/(c*x^2+b*x)^2,x,method=_RETURNVERBOSE)

[Out]

2*e^3*(1/b^3/d^3/e^3*(-1/2*b*(e*x+d)^(1/2)/x+1/2*(5*b*e+4*c*d)/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2)))+c^4/b^3
/e^3/(b*e-c*d)^3*(1/2*b*e*(e*x+d)^(1/2)/(c*(e*x+d)+b*e-c*d)+1/2*(9*b*e-4*c*d)/((b*e-c*d)*c)^(1/2)*arctan(c*(e*
x+d)^(1/2)/((b*e-c*d)*c)^(1/2)))-1/3/d^2/(b*e-c*d)^2/(e*x+d)^(3/2)-(2*b*e-4*c*d)/d^3/(b*e-c*d)^3/(e*x+d)^(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)^(5/2)/(c*x^2+b*x)^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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 981 vs. \(2 (260) = 520\).
time = 4.42, size = 3960, normalized size = 14.83 \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)^2,x, algorithm="fricas")

[Out]

[1/6*(3*(4*c^5*d^7*x^2 + 4*b*c^4*d^7*x - 9*(b*c^4*d^4*x^4 + b^2*c^3*d^4*x^3)*e^3 + 2*(2*c^5*d^5*x^4 - 7*b*c^4*
d^5*x^3 - 9*b^2*c^3*d^5*x^2)*e^2 + (8*c^5*d^6*x^3 - b*c^4*d^6*x^2 - 9*b^2*c^3*d^6*x)*e)*sqrt(c/(c*d - b*e))*lo
g((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*(4*c^5*d^6*x^2 + 4*b*
c^4*d^6*x - 5*(b^4*c*x^4 + b^5*x^3)*e^6 + (11*b^3*c^2*d*x^4 + b^4*c*d*x^3 - 10*b^5*d*x^2)*e^5 - (3*b^2*c^3*d^2
*x^4 - 19*b^3*c^2*d^2*x^3 - 17*b^4*c*d^2*x^2 + 5*b^5*d^2*x)*e^4 - (7*b*c^4*d^3*x^4 + 13*b^2*c^3*d^3*x^3 - 5*b^
3*c^2*d^3*x^2 - 11*b^4*c*d^3*x)*e^3 + (4*c^5*d^4*x^4 - 10*b*c^4*d^4*x^3 - 17*b^2*c^3*d^4*x^2 - 3*b^3*c^2*d^4*x
)*e^2 + (8*c^5*d^5*x^3 + b*c^4*d^5*x^2 - 7*b^2*c^3*d^5*x)*e)*sqrt(d)*log((x*e + 2*sqrt(x*e + d)*sqrt(d) + 2*d)
/x) - 2*(6*b*c^4*d^6*x + 3*b^2*c^3*d^6 - 15*(b^4*c*d*x^3 + b^5*d*x^2)*e^5 + (33*b^3*c^2*d^2*x^3 + 13*b^4*c*d^2
*x^2 - 20*b^5*d^2*x)*e^4 - (9*b^2*c^3*d^3*x^3 - 35*b^3*c^2*d^3*x^2 - 41*b^4*c*d^3*x + 3*b^5*d^3)*e^3 + 3*(2*b*
c^4*d^4*x^3 - 5*b^2*c^3*d^4*x^2 - 3*b^3*c^2*d^4*x + 3*b^4*c*d^4)*e^2 + 3*(4*b*c^4*d^5*x^2 - b^2*c^3*d^5*x - 3*
b^3*c^2*d^5)*e)*sqrt(x*e + d))/(b^3*c^4*d^9*x^2 + b^4*c^3*d^9*x - (b^6*c*d^4*x^4 + b^7*d^4*x^3)*e^5 + (3*b^5*c
^2*d^5*x^4 + b^6*c*d^5*x^3 - 2*b^7*d^5*x^2)*e^4 - (3*b^4*c^3*d^6*x^4 - 3*b^5*c^2*d^6*x^3 - 5*b^6*c*d^6*x^2 + b
^7*d^6*x)*e^3 + (b^3*c^4*d^7*x^4 - 5*b^4*c^3*d^7*x^3 - 3*b^5*c^2*d^7*x^2 + 3*b^6*c*d^7*x)*e^2 + (2*b^3*c^4*d^8
*x^3 - b^4*c^3*d^8*x^2 - 3*b^5*c^2*d^8*x)*e), -1/6*(6*(4*c^5*d^7*x^2 + 4*b*c^4*d^7*x - 9*(b*c^4*d^4*x^4 + b^2*
c^3*d^4*x^3)*e^3 + 2*(2*c^5*d^5*x^4 - 7*b*c^4*d^5*x^3 - 9*b^2*c^3*d^5*x^2)*e^2 + (8*c^5*d^6*x^3 - b*c^4*d^6*x^
2 - 9*b^2*c^3*d^6*x)*e)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(x*e + d)*sqrt(-c/(c*d - b*e))/(c*x*e + c
*d)) - 3*(4*c^5*d^6*x^2 + 4*b*c^4*d^6*x - 5*(b^4*c*x^4 + b^5*x^3)*e^6 + (11*b^3*c^2*d*x^4 + b^4*c*d*x^3 - 10*b
^5*d*x^2)*e^5 - (3*b^2*c^3*d^2*x^4 - 19*b^3*c^2*d^2*x^3 - 17*b^4*c*d^2*x^2 + 5*b^5*d^2*x)*e^4 - (7*b*c^4*d^3*x
^4 + 13*b^2*c^3*d^3*x^3 - 5*b^3*c^2*d^3*x^2 - 11*b^4*c*d^3*x)*e^3 + (4*c^5*d^4*x^4 - 10*b*c^4*d^4*x^3 - 17*b^2
*c^3*d^4*x^2 - 3*b^3*c^2*d^4*x)*e^2 + (8*c^5*d^5*x^3 + b*c^4*d^5*x^2 - 7*b^2*c^3*d^5*x)*e)*sqrt(d)*log((x*e +
2*sqrt(x*e + d)*sqrt(d) + 2*d)/x) + 2*(6*b*c^4*d^6*x + 3*b^2*c^3*d^6 - 15*(b^4*c*d*x^3 + b^5*d*x^2)*e^5 + (33*
b^3*c^2*d^2*x^3 + 13*b^4*c*d^2*x^2 - 20*b^5*d^2*x)*e^4 - (9*b^2*c^3*d^3*x^3 - 35*b^3*c^2*d^3*x^2 - 41*b^4*c*d^
3*x + 3*b^5*d^3)*e^3 + 3*(2*b*c^4*d^4*x^3 - 5*b^2*c^3*d^4*x^2 - 3*b^3*c^2*d^4*x + 3*b^4*c*d^4)*e^2 + 3*(4*b*c^
4*d^5*x^2 - b^2*c^3*d^5*x - 3*b^3*c^2*d^5)*e)*sqrt(x*e + d))/(b^3*c^4*d^9*x^2 + b^4*c^3*d^9*x - (b^6*c*d^4*x^4
+ b^7*d^4*x^3)*e^5 + (3*b^5*c^2*d^5*x^4 + b^6*c*d^5*x^3 - 2*b^7*d^5*x^2)*e^4 - (3*b^4*c^3*d^6*x^4 - 3*b^5*c^2
*d^6*x^3 - 5*b^6*c*d^6*x^2 + b^7*d^6*x)*e^3 + (b^3*c^4*d^7*x^4 - 5*b^4*c^3*d^7*x^3 - 3*b^5*c^2*d^7*x^2 + 3*b^6
*c*d^7*x)*e^2 + (2*b^3*c^4*d^8*x^3 - b^4*c^3*d^8*x^2 - 3*b^5*c^2*d^8*x)*e), -1/6*(6*(4*c^5*d^6*x^2 + 4*b*c^4*d
^6*x - 5*(b^4*c*x^4 + b^5*x^3)*e^6 + (11*b^3*c^2*d*x^4 + b^4*c*d*x^3 - 10*b^5*d*x^2)*e^5 - (3*b^2*c^3*d^2*x^4
- 19*b^3*c^2*d^2*x^3 - 17*b^4*c*d^2*x^2 + 5*b^5*d^2*x)*e^4 - (7*b*c^4*d^3*x^4 + 13*b^2*c^3*d^3*x^3 - 5*b^3*c^2
*d^3*x^2 - 11*b^4*c*d^3*x)*e^3 + (4*c^5*d^4*x^4 - 10*b*c^4*d^4*x^3 - 17*b^2*c^3*d^4*x^2 - 3*b^3*c^2*d^4*x)*e^2
+ (8*c^5*d^5*x^3 + b*c^4*d^5*x^2 - 7*b^2*c^3*d^5*x)*e)*sqrt(-d)*arctan(sqrt(x*e + d)*sqrt(-d)/d) - 3*(4*c^5*d
^7*x^2 + 4*b*c^4*d^7*x - 9*(b*c^4*d^4*x^4 + b^2*c^3*d^4*x^3)*e^3 + 2*(2*c^5*d^5*x^4 - 7*b*c^4*d^5*x^3 - 9*b^2*
c^3*d^5*x^2)*e^2 + (8*c^5*d^6*x^3 - b*c^4*d^6*x^2 - 9*b^2*c^3*d^6*x)*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)) + 2*(6*b*c^4*d^6*x + 3*b^2*c^3*d^6 - 15*(
b^4*c*d*x^3 + b^5*d*x^2)*e^5 + (33*b^3*c^2*d^2*x^3 + 13*b^4*c*d^2*x^2 - 20*b^5*d^2*x)*e^4 - (9*b^2*c^3*d^3*x^3
- 35*b^3*c^2*d^3*x^2 - 41*b^4*c*d^3*x + 3*b^5*d^3)*e^3 + 3*(2*b*c^4*d^4*x^3 - 5*b^2*c^3*d^4*x^2 - 3*b^3*c^2*d
^4*x + 3*b^4*c*d^4)*e^2 + 3*(4*b*c^4*d^5*x^2 - b^2*c^3*d^5*x - 3*b^3*c^2*d^5)*e)*sqrt(x*e + d))/(b^3*c^4*d^9*x
^2 + b^4*c^3*d^9*x - (b^6*c*d^4*x^4 + b^7*d^4*x^3)*e^5 + (3*b^5*c^2*d^5*x^4 + b^6*c*d^5*x^3 - 2*b^7*d^5*x^2)*e
^4 - (3*b^4*c^3*d^6*x^4 - 3*b^5*c^2*d^6*x^3 - 5*b^6*c*d^6*x^2 + b^7*d^6*x)*e^3 + (b^3*c^4*d^7*x^4 - 5*b^4*c^3*
d^7*x^3 - 3*b^5*c^2*d^7*x^2 + 3*b^6*c*d^7*x)*e^2 + (2*b^3*c^4*d^8*x^3 - b^4*c^3*d^8*x^2 - 3*b^5*c^2*d^8*x)*e),
-1/3*(3*(4*c^5*d^7*x^2 + 4*b*c^4*d^7*x - 9*(b*c^4*d^4*x^4 + b^2*c^3*d^4*x^3)*e^3 + 2*(2*c^5*d^5*x^4 - 7*b*c^4
*d^5*x^3 - 9*b^2*c^3*d^5*x^2)*e^2 + (8*c^5*d^6*x^3 - b*c^4*d^6*x^2 - 9*b^2*c^3*d^6*x)*e)*sqrt(-c/(c*d - b*e))*
arctan(-(c*d - b*e)*sqrt(x*e + d)*sqrt(-c/(c*d - b*e))/(c*x*e + c*d)) + 3*(4*c^5*d^6*x^2 + 4*b*c^4*d^6*x - 5*(
b^4*c*x^4 + b^5*x^3)*e^6 + (11*b^3*c^2*d*x^4 + b^4*c*d*x^3 - 10*b^5*d*x^2)*e^5 - (3*b^2*c^3*d^2*x^4 - 19*b^3*c
^2*d^2*x^3 - 17*b^4*c*d^2*x^2 + 5*b^5*d^2*x)*e^4 - (7*b*c^4*d^3*x^4 + 13*b^2*c^3*d^3*x^3 - 5*b^3*c^2*d^3*x^2 -
11*b^4*c*d^3*x)*e^3 + (4*c^5*d^4*x^4 - 10*b*c^...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{2} \left (b + c x\right )^{2} \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)**2,x)

[Out]

Integral(1/(x**2*(b + c*x)**2*(d + e*x)**(5/2)), x)

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Giac [A]
time = 1.40, size = 481, normalized size = 1.80 \begin {gather*} \frac {{\left (4 \, c^{5} d - 9 \, b c^{4} e\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{{\left (b^{3} c^{3} d^{3} - 3 \, b^{4} c^{2} d^{2} e + 3 \, b^{5} c d e^{2} - b^{6} e^{3}\right )} \sqrt {-c^{2} d + b c e}} - \frac {2 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{4} d^{3} e - 2 \, \sqrt {x e + d} c^{4} d^{4} e - 3 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{3} d^{2} e^{2} + 4 \, \sqrt {x e + d} b c^{3} d^{3} e^{2} + 3 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c^{2} d e^{3} - 6 \, \sqrt {x e + d} b^{2} c^{2} d^{2} e^{3} - {\left (x e + d\right )}^{\frac {3}{2}} b^{3} c e^{4} + 4 \, \sqrt {x e + d} b^{3} c d e^{4} - \sqrt {x e + d} b^{4} e^{5}}{{\left (b^{2} c^{3} d^{6} - 3 \, b^{3} c^{2} d^{5} e + 3 \, b^{4} c d^{4} e^{2} - b^{5} d^{3} e^{3}\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 )}} - \frac {2 \, {\left (12 \, {\left (x e + d\right )} c d e^{3} + c d^{2} e^{3} - 6 \, {\left (x e + d\right )} b e^{4} - b d e^{4}\right )}}{3 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} {\left (x e + d\right )}^{\frac {3}{2}}} - \frac {{\left (4 \, c d + 5 \, b e\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d} d^{3}} \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)^2,x, algorithm="giac")

[Out]

(4*c^5*d - 9*b*c^4*e)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b^3*c^3*d^3 - 3*b^4*c^2*d^2*e + 3*b^5*c*d
*e^2 - b^6*e^3)*sqrt(-c^2*d + b*c*e)) - (2*(x*e + d)^(3/2)*c^4*d^3*e - 2*sqrt(x*e + d)*c^4*d^4*e - 3*(x*e + d)
^(3/2)*b*c^3*d^2*e^2 + 4*sqrt(x*e + d)*b*c^3*d^3*e^2 + 3*(x*e + d)^(3/2)*b^2*c^2*d*e^3 - 6*sqrt(x*e + d)*b^2*c
^2*d^2*e^3 - (x*e + d)^(3/2)*b^3*c*e^4 + 4*sqrt(x*e + d)*b^3*c*d*e^4 - sqrt(x*e + d)*b^4*e^5)/((b^2*c^3*d^6 -
3*b^3*c^2*d^5*e + 3*b^4*c*d^4*e^2 - b^5*d^3*e^3)*((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*
d*e)) - 2/3*(12*(x*e + d)*c*d*e^3 + c*d^2*e^3 - 6*(x*e + d)*b*e^4 - b*d*e^4)/((c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2
*c*d^4*e^2 - b^3*d^3*e^3)*(x*e + d)^(3/2)) - (4*c*d + 5*b*e)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^3*sqrt(-d)*d^3)

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Mupad [B]
time = 3.15, size = 2500, normalized size = 9.36 \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)^2*(d + e*x)^(5/2)),x)

[Out]

((10*e^3*(b*e - 2*c*d)*(d + e*x))/(3*(c*d^2 - b*d*e)^2) - (2*e^3)/(3*(c*d^2 - b*d*e)) + (e*(d + e*x)^2*(15*b^4
*e^4 + 6*c^4*d^4 + 64*b^2*c^2*d^2*e^2 - 12*b*c^3*d^3*e - 58*b^3*c*d*e^3))/(3*b^2*(c*d^2 - b*d*e)^3) + (e*(b*e
- 2*c*d)*(d + e*x)^3*(c^3*d^2 + 5*b^2*c*e^2 - b*c^2*d*e))/(b^2*(c*d^2 - b*d*e)^3))/(c*(d + e*x)^(7/2) + (c*d^2
- b*d*e)*(d + e*x)^(3/2) + (b*e - 2*c*d)*(d + e*x)^(5/2)) + (atan((((-c^7*(b*e - c*d)^7)^(1/2)*(9*b*e - 4*c*d
)*((d + e*x)^(1/2)*(64*b^6*c^20*d^26*e^2 - 832*b^7*c^19*d^25*e^3 + 4820*b^8*c^18*d^24*e^4 - 16240*b^9*c^17*d^2
3*e^5 + 34490*b^10*c^16*d^22*e^6 - 45430*b^11*c^15*d^21*e^7 + 29414*b^12*c^14*d^20*e^8 + 10670*b^13*c^13*d^19*
e^9 - 39550*b^14*c^12*d^18*e^10 + 25730*b^15*c^11*d^17*e^11 + 19048*b^16*c^10*d^16*e^12 - 53852*b^17*c^9*d^15*
e^13 + 55510*b^18*c^8*d^14*e^14 - 35210*b^19*c^7*d^13*e^15 + 14830*b^20*c^6*d^12*e^16 - 4082*b^21*c^5*d^11*e^1
7 + 670*b^22*c^4*d^10*e^18 - 50*b^23*c^3*d^9*e^19) + ((-c^7*(b*e - c*d)^7)^(1/2)*(9*b*e - 4*c*d)*(8*b^10*c^18*
d^28*e^3 - 112*b^11*c^17*d^27*e^4 + 664*b^12*c^16*d^26*e^5 - 2080*b^13*c^15*d^25*e^6 + 2996*b^14*c^14*d^24*e^7
+ 2528*b^15*c^13*d^23*e^8 - 23056*b^16*c^12*d^22*e^9 + 59312*b^17*c^11*d^21*e^10 - 95700*b^18*c^10*d^20*e^11
+ 109648*b^19*c^9*d^19*e^12 - 92840*b^20*c^8*d^18*e^13 + 58688*b^21*c^7*d^17*e^14 - 27476*b^22*c^6*d^16*e^15 +
9280*b^23*c^5*d^15*e^16 - 2144*b^24*c^4*d^14*e^17 + 304*b^25*c^3*d^13*e^18 - 20*b^26*c^2*d^12*e^19 - ((-c^7*(
b*e - c*d)^7)^(1/2)*(9*b*e - 4*c*d)*(d + e*x)^(1/2)*(16*b^12*c^18*d^31*e^2 - 248*b^13*c^17*d^30*e^3 + 1800*b^1
4*c^16*d^29*e^4 - 8120*b^15*c^15*d^28*e^5 + 25480*b^16*c^14*d^27*e^6 - 58968*b^17*c^13*d^26*e^7 + 104104*b^18*
c^12*d^25*e^8 - 143000*b^19*c^11*d^24*e^9 + 154440*b^20*c^10*d^23*e^10 - 131560*b^21*c^9*d^22*e^11 + 88088*b^2
2*c^8*d^21*e^12 - 45864*b^23*c^7*d^20*e^13 + 18200*b^24*c^6*d^19*e^14 - 5320*b^25*c^5*d^18*e^15 + 1080*b^26*c^
4*d^17*e^16 - 136*b^27*c^3*d^16*e^17 + 8*b^28*c^2*d^15*e^18))/(2*(b^10*e^7 - b^3*c^7*d^7 + 7*b^4*c^6*d^6*e - 2
1*b^5*c^5*d^5*e^2 + 35*b^6*c^4*d^4*e^3 - 35*b^7*c^3*d^3*e^4 + 21*b^8*c^2*d^2*e^5 - 7*b^9*c*d*e^6))))/(2*(b^10*
e^7 - b^3*c^7*d^7 + 7*b^4*c^6*d^6*e - 21*b^5*c^5*d^5*e^2 + 35*b^6*c^4*d^4*e^3 - 35*b^7*c^3*d^3*e^4 + 21*b^8*c^
2*d^2*e^5 - 7*b^9*c*d*e^6)))*1i)/(2*(b^10*e^7 - b^3*c^7*d^7 + 7*b^4*c^6*d^6*e - 21*b^5*c^5*d^5*e^2 + 35*b^6*c^
4*d^4*e^3 - 35*b^7*c^3*d^3*e^4 + 21*b^8*c^2*d^2*e^5 - 7*b^9*c*d*e^6)) + ((-c^7*(b*e - c*d)^7)^(1/2)*(9*b*e - 4
*c*d)*((d + e*x)^(1/2)*(64*b^6*c^20*d^26*e^2 - 832*b^7*c^19*d^25*e^3 + 4820*b^8*c^18*d^24*e^4 - 16240*b^9*c^17
*d^23*e^5 + 34490*b^10*c^16*d^22*e^6 - 45430*b^11*c^15*d^21*e^7 + 29414*b^12*c^14*d^20*e^8 + 10670*b^13*c^13*d
^19*e^9 - 39550*b^14*c^12*d^18*e^10 + 25730*b^15*c^11*d^17*e^11 + 19048*b^16*c^10*d^16*e^12 - 53852*b^17*c^9*d
^15*e^13 + 55510*b^18*c^8*d^14*e^14 - 35210*b^19*c^7*d^13*e^15 + 14830*b^20*c^6*d^12*e^16 - 4082*b^21*c^5*d^11
*e^17 + 670*b^22*c^4*d^10*e^18 - 50*b^23*c^3*d^9*e^19) - ((-c^7*(b*e - c*d)^7)^(1/2)*(9*b*e - 4*c*d)*(8*b^10*c
^18*d^28*e^3 - 112*b^11*c^17*d^27*e^4 + 664*b^12*c^16*d^26*e^5 - 2080*b^13*c^15*d^25*e^6 + 2996*b^14*c^14*d^24
*e^7 + 2528*b^15*c^13*d^23*e^8 - 23056*b^16*c^12*d^22*e^9 + 59312*b^17*c^11*d^21*e^10 - 95700*b^18*c^10*d^20*e
^11 + 109648*b^19*c^9*d^19*e^12 - 92840*b^20*c^8*d^18*e^13 + 58688*b^21*c^7*d^17*e^14 - 27476*b^22*c^6*d^16*e^
15 + 9280*b^23*c^5*d^15*e^16 - 2144*b^24*c^4*d^14*e^17 + 304*b^25*c^3*d^13*e^18 - 20*b^26*c^2*d^12*e^19 + ((-c
^7*(b*e - c*d)^7)^(1/2)*(9*b*e - 4*c*d)*(d + e*x)^(1/2)*(16*b^12*c^18*d^31*e^2 - 248*b^13*c^17*d^30*e^3 + 1800
*b^14*c^16*d^29*e^4 - 8120*b^15*c^15*d^28*e^5 + 25480*b^16*c^14*d^27*e^6 - 58968*b^17*c^13*d^26*e^7 + 104104*b
^18*c^12*d^25*e^8 - 143000*b^19*c^11*d^24*e^9 + 154440*b^20*c^10*d^23*e^10 - 131560*b^21*c^9*d^22*e^11 + 88088
*b^22*c^8*d^21*e^12 - 45864*b^23*c^7*d^20*e^13 + 18200*b^24*c^6*d^19*e^14 - 5320*b^25*c^5*d^18*e^15 + 1080*b^2
6*c^4*d^17*e^16 - 136*b^27*c^3*d^16*e^17 + 8*b^28*c^2*d^15*e^18))/(2*(b^10*e^7 - b^3*c^7*d^7 + 7*b^4*c^6*d^6*e
- 21*b^5*c^5*d^5*e^2 + 35*b^6*c^4*d^4*e^3 - 35*b^7*c^3*d^3*e^4 + 21*b^8*c^2*d^2*e^5 - 7*b^9*c*d*e^6))))/(2*(b
^10*e^7 - b^3*c^7*d^7 + 7*b^4*c^6*d^6*e - 21*b^5*c^5*d^5*e^2 + 35*b^6*c^4*d^4*e^3 - 35*b^7*c^3*d^3*e^4 + 21*b^
8*c^2*d^2*e^5 - 7*b^9*c*d*e^6)))*1i)/(2*(b^10*e^7 - b^3*c^7*d^7 + 7*b^4*c^6*d^6*e - 21*b^5*c^5*d^5*e^2 + 35*b^
6*c^4*d^4*e^3 - 35*b^7*c^3*d^3*e^4 + 21*b^8*c^2*d^2*e^5 - 7*b^9*c*d*e^6)))/(((-c^7*(b*e - c*d)^7)^(1/2)*(9*b*e
- 4*c*d)*((d + e*x)^(1/2)*(64*b^6*c^20*d^26*e^2 - 832*b^7*c^19*d^25*e^3 + 4820*b^8*c^18*d^24*e^4 - 16240*b^9*
c^17*d^23*e^5 + 34490*b^10*c^16*d^22*e^6 - 45430*b^11*c^15*d^21*e^7 + 29414*b^12*c^14*d^20*e^8 + 10670*b^13*c^
13*d^19*e^9 - 39550*b^14*c^12*d^18*e^10 + 25730*b^15*c^11*d^17*e^11 + 19048*b^16*c^10*d^16*e^12 - 53852*b^17*c
^9*d^15*e^13 + 55510*b^18*c^8*d^14*e^14 - 35210*b^19*c^7*d^13*e^15 + 14830*b^20*c^6*d^12*e^16 - 4082*b^21*c^5*
d^11*e^17 + 670*b^22*c^4*d^10*e^18 - 50*b^23*c^3*d^9*e^19) - ((-c^7*(b*e - c*d)^7)^(1/2)*(9*b*e - 4*c*d)*(8*b^
10*c^18*d^28*e^3 - 112*b^11*c^17*d^27*e^4 + 664...

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