3.4.75 \(\int \frac {1}{(d+e x)^{3/2} (b x+c x^2)^2} \, dx\) [375]
Optimal. Leaf size=206 \[ -\frac {e \left (2 c^2 d^2-2 b c d e+3 b^2 e^2\right )}{b^2 d^2 (c d-b e)^2 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )}+\frac {(4 c d+3 b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3 d^{5/2}}-\frac {c^{5/2} (4 c d-7 b e) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 (c d-b e)^{5/2}} \]
[Out]
(3*b*e+4*c*d)*arctanh((e*x+d)^(1/2)/d^(1/2))/b^3/d^(5/2)-c^(5/2)*(-7*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)^(5/2)-e*(3*b^2*e^2-2*b*c*d*e+2*c^2*d^2)/b^2/d^2/(-b*e+c*d)^2/(e*x+d)^(1/2)+(-
b*(-b*e+c*d)-c*(-b*e+2*c*d)*x)/b^2/d/(-b*e+c*d)/(c*x^2+b*x)/(e*x+d)^(1/2)
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Rubi [A]
time = 0.24, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps
used = 6, 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^{5/2} (4 c d-7 b e) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 (c d-b e)^{5/2}}+\frac {(3 b e+4 c d) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3 d^{5/2}}-\frac {e \left (3 b^2 e^2-2 b c d e+2 c^2 d^2\right )}{b^2 d^2 \sqrt {d+e x} (c d-b e)^2}-\frac {c x (2 c d-b e)+b (c d-b e)}{b^2 d \left (b x+c x^2\right ) \sqrt {d+e x} (c d-b e)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/((d + e*x)^(3/2)*(b*x + c*x^2)^2),x]
[Out]
-((e*(2*c^2*d^2 - 2*b*c*d*e + 3*b^2*e^2))/(b^2*d^2*(c*d - b*e)^2*Sqrt[d + e*x])) - (b*(c*d - b*e) + c*(2*c*d -
b*e)*x)/(b^2*d*(c*d - b*e)*Sqrt[d + e*x]*(b*x + c*x^2)) + ((4*c*d + 3*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b
^3*d^(5/2)) - (c^(5/2)*(4*c*d - 7*b*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*(c*d - b*e)^(5/2
))
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)^{3/2} \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) \sqrt {d+e x} \left (b x+c x^2\right )}-\frac {\int \frac {\frac {1}{2} (c d-b e) (4 c d+3 b e)+\frac {3}{2} c e (2 c d-b e) x}{(d+e x)^{3/2} \left (b x+c x^2\right )} \, dx}{b^2 d (c d-b e)}\\ &=-\frac {e \left (2 c^2 d^2-2 b c d e+3 b^2 e^2\right )}{b^2 d^2 (c d-b e)^2 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )}-\frac {\int \frac {\frac {1}{2} (c d-b e)^2 (4 c d+3 b e)+\frac {1}{2} c e \left (2 c^2 d^2-2 b c d e+3 b^2 e^2\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{b^2 d^2 (c d-b e)^2}\\ &=-\frac {e \left (2 c^2 d^2-2 b c d e+3 b^2 e^2\right )}{b^2 d^2 (c d-b e)^2 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )}-\frac {2 \text {Subst}\left (\int \frac {\frac {1}{2} e (c d-b e)^2 (4 c d+3 b e)-\frac {1}{2} c d e \left (2 c^2 d^2-2 b c d e+3 b^2 e^2\right )+\frac {1}{2} c e \left (2 c^2 d^2-2 b c d e+3 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^2 (c d-b e)^2}\\ &=-\frac {e \left (2 c^2 d^2-2 b c d e+3 b^2 e^2\right )}{b^2 d^2 (c d-b e)^2 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )}+\frac {\left (c^3 (4 c d-7 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)^2}-\frac {(c (4 c d+3 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^2}\\ &=-\frac {e \left (2 c^2 d^2-2 b c d e+3 b^2 e^2\right )}{b^2 d^2 (c d-b e)^2 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )}+\frac {(4 c d+3 b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3 d^{5/2}}-\frac {c^{5/2} (4 c d-7 b e) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 (c d-b e)^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 1.02, size = 202, normalized size = 0.98 \begin {gather*} \frac {-\frac {b \left (2 c^3 d^2 x (d+e x)+b^3 e^2 (d+3 e x)+b c^2 d \left (d^2-d e x-2 e^2 x^2\right )+b^2 c e \left (-2 d^2-d e x+3 e^2 x^2\right )\right )}{d^2 (c d-b e)^2 x (b+c x) \sqrt {d+e x}}+\frac {c^{5/2} (4 c d-7 b e) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{(-c d+b e)^{5/2}}+\frac {(4 c d+3 b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{5/2}}}{b^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/((d + e*x)^(3/2)*(b*x + c*x^2)^2),x]
[Out]
(-((b*(2*c^3*d^2*x*(d + e*x) + b^3*e^2*(d + 3*e*x) + b*c^2*d*(d^2 - d*e*x - 2*e^2*x^2) + b^2*c*e*(-2*d^2 - d*e
*x + 3*e^2*x^2)))/(d^2*(c*d - b*e)^2*x*(b + c*x)*Sqrt[d + e*x])) + (c^(5/2)*(4*c*d - 7*b*e)*ArcTan[(Sqrt[c]*Sq
rt[d + e*x])/Sqrt[-(c*d) + b*e]])/(-(c*d) + b*e)^(5/2) + ((4*c*d + 3*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/d^(5
/2))/b^3
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Maple [A]
time = 0.53, size = 174, normalized size = 0.84
| | |
| method |
result |
size |
| | |
| derivativedivides |
\(2 e^{3} \left (-\frac {c^{3} \left (\frac {b e \sqrt {e x +d}}{2 c \left (e x +d \right )+2 b e -2 c d}+\frac {\left (7 b e -4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}\right )}{b^{3} e^{3} \left (b e -c d \right )^{2}}-\frac {1}{d^{2} \left (b e -c d \right )^{2} \sqrt {e x +d}}+\frac {-\frac {b \sqrt {e x +d}}{2 x}+\frac {\left (3 b e +4 c d \right ) \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}}{b^{3} d^{2} e^{3}}\right )\) |
\(174\) |
| default |
\(2 e^{3} \left (-\frac {c^{3} \left (\frac {b e \sqrt {e x +d}}{2 c \left (e x +d \right )+2 b e -2 c d}+\frac {\left (7 b e -4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}\right )}{b^{3} e^{3} \left (b e -c d \right )^{2}}-\frac {1}{d^{2} \left (b e -c d \right )^{2} \sqrt {e x +d}}+\frac {-\frac {b \sqrt {e x +d}}{2 x}+\frac {\left (3 b e +4 c d \right ) \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}}{b^{3} d^{2} e^{3}}\right )\) |
\(174\) |
| risch |
\(-\frac {\sqrt {e x +d}}{d^{2} b^{2} x}-\frac {e \,c^{3} \sqrt {e x +d}}{b^{2} \left (b e -c d \right )^{2} \left (c e x +b e \right )}-\frac {7 e \,c^{3} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{b^{2} \left (b e -c d \right )^{2} \sqrt {\left (b e -c d \right ) c}}+\frac {4 d \,c^{4} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{b^{3} \left (b e -c d \right )^{2} \sqrt {\left (b e -c d \right ) c}}-\frac {2 e^{3}}{d^{2} \left (b e -c d \right )^{2} \sqrt {e x +d}}+\frac {3 e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{d^{\frac {5}{2}} b^{2}}+\frac {4 \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) c}{d^{\frac {3}{2}} b^{3}}\) |
\(229\) |
| | |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(e*x+d)^(3/2)/(c*x^2+b*x)^2,x,method=_RETURNVERBOSE)
[Out]
2*e^3*(-c^3/b^3/e^3/(b*e-c*d)^2*(1/2*b*e*(e*x+d)^(1/2)/(c*(e*x+d)+b*e-c*d)+1/2*(7*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/d^2/(b*e-c*d)^2/(e*x+d)^(1/2)+1/b^3/d^2/e^3*(-1/2*b*(e*x+d)^
(1/2)/x+1/2*(3*b*e+4*c*d)/d^(1/2)*arctanh((e*x+d)^(1/2)/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)^(3/2)/(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. 592 vs.
\(2 (199) = 398\).
time = 3.39, size = 2408, normalized size = 11.69 \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)^(3/2)/(c*x^2+b*x)^2,x, algorithm="fricas")
[Out]
[-1/2*((4*c^4*d^5*x^2 + 4*b*c^3*d^5*x - 7*(b*c^3*d^3*x^3 + b^2*c^2*d^3*x^2)*e^2 + (4*c^4*d^4*x^3 - 3*b*c^3*d^4
*x^2 - 7*b^2*c^2*d^4*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)) - (4*c^4*d^4*x^2 + 4*b*c^3*d^4*x + 3*(b^3*c*x^3 + b^4*x^2)*e^4 - (2*b^2*c^2*d*x^3 - b^
3*c*d*x^2 - 3*b^4*d*x)*e^3 - (5*b*c^3*d^2*x^3 + 7*b^2*c^2*d^2*x^2 + 2*b^3*c*d^2*x)*e^2 + (4*c^4*d^3*x^3 - b*c^
3*d^3*x^2 - 5*b^2*c^2*d^3*x)*e)*sqrt(d)*log((x*e + 2*sqrt(x*e + d)*sqrt(d) + 2*d)/x) + 2*(2*b*c^3*d^4*x + b^2*
c^2*d^4 + 3*(b^3*c*d*x^2 + b^4*d*x)*e^3 - (2*b^2*c^2*d^2*x^2 + b^3*c*d^2*x - b^4*d^2)*e^2 + (2*b*c^3*d^3*x^2 -
b^2*c^2*d^3*x - 2*b^3*c*d^3)*e)*sqrt(x*e + d))/(b^3*c^3*d^6*x^2 + b^4*c^2*d^6*x + (b^5*c*d^3*x^3 + b^6*d^3*x^
2)*e^3 - (2*b^4*c^2*d^4*x^3 + b^5*c*d^4*x^2 - b^6*d^4*x)*e^2 + (b^3*c^3*d^5*x^3 - b^4*c^2*d^5*x^2 - 2*b^5*c*d^
5*x)*e), -1/2*(2*(4*c^4*d^5*x^2 + 4*b*c^3*d^5*x - 7*(b*c^3*d^3*x^3 + b^2*c^2*d^3*x^2)*e^2 + (4*c^4*d^4*x^3 - 3
*b*c^3*d^4*x^2 - 7*b^2*c^2*d^4*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)) - (4*c^4*d^4*x^2 + 4*b*c^3*d^4*x + 3*(b^3*c*x^3 + b^4*x^2)*e^4 - (2*b^2*c^2*d*x^3 - b^3*c*d*
x^2 - 3*b^4*d*x)*e^3 - (5*b*c^3*d^2*x^3 + 7*b^2*c^2*d^2*x^2 + 2*b^3*c*d^2*x)*e^2 + (4*c^4*d^3*x^3 - b*c^3*d^3*
x^2 - 5*b^2*c^2*d^3*x)*e)*sqrt(d)*log((x*e + 2*sqrt(x*e + d)*sqrt(d) + 2*d)/x) + 2*(2*b*c^3*d^4*x + b^2*c^2*d^
4 + 3*(b^3*c*d*x^2 + b^4*d*x)*e^3 - (2*b^2*c^2*d^2*x^2 + b^3*c*d^2*x - b^4*d^2)*e^2 + (2*b*c^3*d^3*x^2 - b^2*c
^2*d^3*x - 2*b^3*c*d^3)*e)*sqrt(x*e + d))/(b^3*c^3*d^6*x^2 + b^4*c^2*d^6*x + (b^5*c*d^3*x^3 + b^6*d^3*x^2)*e^3
- (2*b^4*c^2*d^4*x^3 + b^5*c*d^4*x^2 - b^6*d^4*x)*e^2 + (b^3*c^3*d^5*x^3 - b^4*c^2*d^5*x^2 - 2*b^5*c*d^5*x)*e
), -1/2*(2*(4*c^4*d^4*x^2 + 4*b*c^3*d^4*x + 3*(b^3*c*x^3 + b^4*x^2)*e^4 - (2*b^2*c^2*d*x^3 - b^3*c*d*x^2 - 3*b
^4*d*x)*e^3 - (5*b*c^3*d^2*x^3 + 7*b^2*c^2*d^2*x^2 + 2*b^3*c*d^2*x)*e^2 + (4*c^4*d^3*x^3 - b*c^3*d^3*x^2 - 5*b
^2*c^2*d^3*x)*e)*sqrt(-d)*arctan(sqrt(x*e + d)*sqrt(-d)/d) + (4*c^4*d^5*x^2 + 4*b*c^3*d^5*x - 7*(b*c^3*d^3*x^3
+ b^2*c^2*d^3*x^2)*e^2 + (4*c^4*d^4*x^3 - 3*b*c^3*d^4*x^2 - 7*b^2*c^2*d^4*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*(2*b*c^3*d^4*x + b^2*c^2*d^4
+ 3*(b^3*c*d*x^2 + b^4*d*x)*e^3 - (2*b^2*c^2*d^2*x^2 + b^3*c*d^2*x - b^4*d^2)*e^2 + (2*b*c^3*d^3*x^2 - b^2*c^
2*d^3*x - 2*b^3*c*d^3)*e)*sqrt(x*e + d))/(b^3*c^3*d^6*x^2 + b^4*c^2*d^6*x + (b^5*c*d^3*x^3 + b^6*d^3*x^2)*e^3
- (2*b^4*c^2*d^4*x^3 + b^5*c*d^4*x^2 - b^6*d^4*x)*e^2 + (b^3*c^3*d^5*x^3 - b^4*c^2*d^5*x^2 - 2*b^5*c*d^5*x)*e)
, -((4*c^4*d^5*x^2 + 4*b*c^3*d^5*x - 7*(b*c^3*d^3*x^3 + b^2*c^2*d^3*x^2)*e^2 + (4*c^4*d^4*x^3 - 3*b*c^3*d^4*x^
2 - 7*b^2*c^2*d^4*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)) + (4*c^4*d^4*x^2 + 4*b*c^3*d^4*x + 3*(b^3*c*x^3 + b^4*x^2)*e^4 - (2*b^2*c^2*d*x^3 - b^3*c*d*x^2 - 3*b^4*d
*x)*e^3 - (5*b*c^3*d^2*x^3 + 7*b^2*c^2*d^2*x^2 + 2*b^3*c*d^2*x)*e^2 + (4*c^4*d^3*x^3 - b*c^3*d^3*x^2 - 5*b^2*c
^2*d^3*x)*e)*sqrt(-d)*arctan(sqrt(x*e + d)*sqrt(-d)/d) + (2*b*c^3*d^4*x + b^2*c^2*d^4 + 3*(b^3*c*d*x^2 + b^4*d
*x)*e^3 - (2*b^2*c^2*d^2*x^2 + b^3*c*d^2*x - b^4*d^2)*e^2 + (2*b*c^3*d^3*x^2 - b^2*c^2*d^3*x - 2*b^3*c*d^3)*e)
*sqrt(x*e + d))/(b^3*c^3*d^6*x^2 + b^4*c^2*d^6*x + (b^5*c*d^3*x^3 + b^6*d^3*x^2)*e^3 - (2*b^4*c^2*d^4*x^3 + b^
5*c*d^4*x^2 - b^6*d^4*x)*e^2 + (b^3*c^3*d^5*x^3 - b^4*c^2*d^5*x^2 - 2*b^5*c*d^5*x)*e)]
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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 {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)**(3/2)/(c*x**2+b*x)**2,x)
[Out]
Integral(1/(x**2*(b + c*x)**2*(d + e*x)**(3/2)), x)
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Giac [A]
time = 1.60, size = 353, normalized size = 1.71 \begin {gather*} \frac {{\left (4 \, c^{4} d - 7 \, b c^{3} e\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{{\left (b^{3} c^{2} d^{2} - 2 \, b^{4} c d e + b^{5} e^{2}\right )} \sqrt {-c^{2} d + b c e}} - \frac {2 \, {\left (x e + d\right )}^{2} c^{3} d^{2} e - 2 \, {\left (x e + d\right )} c^{3} d^{3} e - 2 \, {\left (x e + d\right )}^{2} b c^{2} d e^{2} + 3 \, {\left (x e + d\right )} b c^{2} d^{2} e^{2} + 3 \, {\left (x e + d\right )}^{2} b^{2} c e^{3} - 7 \, {\left (x e + d\right )} b^{2} c d e^{3} + 2 \, b^{2} c d^{2} e^{3} + 3 \, {\left (x e + d\right )} b^{3} e^{4} - 2 \, b^{3} d e^{4}}{{\left (b^{2} c^{2} d^{4} - 2 \, b^{3} c d^{3} e + b^{4} d^{2} e^{2}\right )} {\left ({\left (x e + d\right )}^{\frac {5}{2}} c - 2 \, {\left (x e + d\right )}^{\frac {3}{2}} c d + \sqrt {x e + d} c d^{2} + {\left (x e + d\right )}^{\frac {3}{2}} b e - \sqrt {x e + d} b d e\right )}} - \frac {{\left (4 \, c d + 3 \, b e\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^(3/2)/(c*x^2+b*x)^2,x, algorithm="giac")
[Out]
(4*c^4*d - 7*b*c^3*e)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b^3*c^2*d^2 - 2*b^4*c*d*e + b^5*e^2)*sqrt
(-c^2*d + b*c*e)) - (2*(x*e + d)^2*c^3*d^2*e - 2*(x*e + d)*c^3*d^3*e - 2*(x*e + d)^2*b*c^2*d*e^2 + 3*(x*e + d)
*b*c^2*d^2*e^2 + 3*(x*e + d)^2*b^2*c*e^3 - 7*(x*e + d)*b^2*c*d*e^3 + 2*b^2*c*d^2*e^3 + 3*(x*e + d)*b^3*e^4 - 2
*b^3*d*e^4)/((b^2*c^2*d^4 - 2*b^3*c*d^3*e + b^4*d^2*e^2)*((x*e + d)^(5/2)*c - 2*(x*e + d)^(3/2)*c*d + sqrt(x*e
+ d)*c*d^2 + (x*e + d)^(3/2)*b*e - sqrt(x*e + d)*b*d*e)) - (4*c*d + 3*b*e)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^
3*sqrt(-d)*d^2)
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Mupad [B]
time = 2.43, size = 2500, normalized size = 12.14 \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)^(3/2)),x)
[Out]
(atan((((-c^5*(b*e - c*d)^5)^(1/2)*(7*b*e - 4*c*d)*((d + e*x)^(1/2)*(64*b^6*c^15*d^18*e^2 - 576*b^7*c^14*d^17*
e^3 + 2228*b^8*c^13*d^16*e^4 - 4768*b^9*c^12*d^15*e^5 + 5960*b^10*c^11*d^14*e^6 - 3976*b^11*c^10*d^13*e^7 + 57
8*b^12*c^9*d^12*e^8 + 1004*b^13*c^8*d^11*e^9 - 442*b^14*c^7*d^10*e^10 - 320*b^15*c^6*d^9*e^11 + 362*b^16*c^5*d
^8*e^12 - 132*b^17*c^4*d^7*e^13 + 18*b^18*c^3*d^6*e^14) + ((-c^5*(b*e - c*d)^5)^(1/2)*(7*b*e - 4*c*d)*(8*b^10*
c^13*d^19*e^3 - 76*b^11*c^12*d^18*e^4 + 300*b^12*c^11*d^17*e^5 - 612*b^13*c^10*d^16*e^6 + 576*b^14*c^9*d^15*e^
7 + 168*b^15*c^8*d^14*e^8 - 1176*b^16*c^7*d^13*e^9 + 1560*b^17*c^6*d^12*e^10 - 1128*b^18*c^5*d^11*e^11 + 484*b
^19*c^4*d^10*e^12 - 116*b^20*c^3*d^9*e^13 + 12*b^21*c^2*d^8*e^14 - ((-c^5*(b*e - c*d)^5)^(1/2)*(7*b*e - 4*c*d)
*(d + e*x)^(1/2)*(16*b^12*c^13*d^21*e^2 - 168*b^13*c^12*d^20*e^3 + 800*b^14*c^11*d^19*e^4 - 2280*b^15*c^10*d^1
8*e^5 + 4320*b^16*c^9*d^17*e^6 - 5712*b^17*c^8*d^16*e^7 + 5376*b^18*c^7*d^15*e^8 - 3600*b^19*c^6*d^14*e^9 + 16
80*b^20*c^5*d^13*e^10 - 520*b^21*c^4*d^12*e^11 + 96*b^22*c^3*d^11*e^12 - 8*b^23*c^2*d^10*e^13))/(2*(b^8*e^5 -
b^3*c^5*d^5 + 5*b^4*c^4*d^4*e - 10*b^5*c^3*d^3*e^2 + 10*b^6*c^2*d^2*e^3 - 5*b^7*c*d*e^4))))/(2*(b^8*e^5 - b^3*
c^5*d^5 + 5*b^4*c^4*d^4*e - 10*b^5*c^3*d^3*e^2 + 10*b^6*c^2*d^2*e^3 - 5*b^7*c*d*e^4)))*1i)/(2*(b^8*e^5 - b^3*c
^5*d^5 + 5*b^4*c^4*d^4*e - 10*b^5*c^3*d^3*e^2 + 10*b^6*c^2*d^2*e^3 - 5*b^7*c*d*e^4)) + ((-c^5*(b*e - c*d)^5)^(
1/2)*(7*b*e - 4*c*d)*((d + e*x)^(1/2)*(64*b^6*c^15*d^18*e^2 - 576*b^7*c^14*d^17*e^3 + 2228*b^8*c^13*d^16*e^4 -
4768*b^9*c^12*d^15*e^5 + 5960*b^10*c^11*d^14*e^6 - 3976*b^11*c^10*d^13*e^7 + 578*b^12*c^9*d^12*e^8 + 1004*b^1
3*c^8*d^11*e^9 - 442*b^14*c^7*d^10*e^10 - 320*b^15*c^6*d^9*e^11 + 362*b^16*c^5*d^8*e^12 - 132*b^17*c^4*d^7*e^1
3 + 18*b^18*c^3*d^6*e^14) - ((-c^5*(b*e - c*d)^5)^(1/2)*(7*b*e - 4*c*d)*(8*b^10*c^13*d^19*e^3 - 76*b^11*c^12*d
^18*e^4 + 300*b^12*c^11*d^17*e^5 - 612*b^13*c^10*d^16*e^6 + 576*b^14*c^9*d^15*e^7 + 168*b^15*c^8*d^14*e^8 - 11
76*b^16*c^7*d^13*e^9 + 1560*b^17*c^6*d^12*e^10 - 1128*b^18*c^5*d^11*e^11 + 484*b^19*c^4*d^10*e^12 - 116*b^20*c
^3*d^9*e^13 + 12*b^21*c^2*d^8*e^14 + ((-c^5*(b*e - c*d)^5)^(1/2)*(7*b*e - 4*c*d)*(d + e*x)^(1/2)*(16*b^12*c^13
*d^21*e^2 - 168*b^13*c^12*d^20*e^3 + 800*b^14*c^11*d^19*e^4 - 2280*b^15*c^10*d^18*e^5 + 4320*b^16*c^9*d^17*e^6
- 5712*b^17*c^8*d^16*e^7 + 5376*b^18*c^7*d^15*e^8 - 3600*b^19*c^6*d^14*e^9 + 1680*b^20*c^5*d^13*e^10 - 520*b^
21*c^4*d^12*e^11 + 96*b^22*c^3*d^11*e^12 - 8*b^23*c^2*d^10*e^13))/(2*(b^8*e^5 - b^3*c^5*d^5 + 5*b^4*c^4*d^4*e
- 10*b^5*c^3*d^3*e^2 + 10*b^6*c^2*d^2*e^3 - 5*b^7*c*d*e^4))))/(2*(b^8*e^5 - b^3*c^5*d^5 + 5*b^4*c^4*d^4*e - 10
*b^5*c^3*d^3*e^2 + 10*b^6*c^2*d^2*e^3 - 5*b^7*c*d*e^4)))*1i)/(2*(b^8*e^5 - b^3*c^5*d^5 + 5*b^4*c^4*d^4*e - 10*
b^5*c^3*d^3*e^2 + 10*b^6*c^2*d^2*e^3 - 5*b^7*c*d*e^4)))/(64*b^4*c^15*d^16*e^3 - 512*b^5*c^14*d^15*e^4 + 1804*b
^6*c^13*d^14*e^5 - 3668*b^7*c^12*d^13*e^6 + 4606*b^8*c^11*d^12*e^7 - 3248*b^9*c^10*d^11*e^8 + 322*b^10*c^9*d^1
0*e^9 + 1756*b^11*c^8*d^9*e^10 - 1742*b^12*c^7*d^8*e^11 + 744*b^13*c^6*d^7*e^12 - 126*b^14*c^5*d^6*e^13 - ((-c
^5*(b*e - c*d)^5)^(1/2)*(7*b*e - 4*c*d)*((d + e*x)^(1/2)*(64*b^6*c^15*d^18*e^2 - 576*b^7*c^14*d^17*e^3 + 2228*
b^8*c^13*d^16*e^4 - 4768*b^9*c^12*d^15*e^5 + 5960*b^10*c^11*d^14*e^6 - 3976*b^11*c^10*d^13*e^7 + 578*b^12*c^9*
d^12*e^8 + 1004*b^13*c^8*d^11*e^9 - 442*b^14*c^7*d^10*e^10 - 320*b^15*c^6*d^9*e^11 + 362*b^16*c^5*d^8*e^12 - 1
32*b^17*c^4*d^7*e^13 + 18*b^18*c^3*d^6*e^14) + ((-c^5*(b*e - c*d)^5)^(1/2)*(7*b*e - 4*c*d)*(8*b^10*c^13*d^19*e
^3 - 76*b^11*c^12*d^18*e^4 + 300*b^12*c^11*d^17*e^5 - 612*b^13*c^10*d^16*e^6 + 576*b^14*c^9*d^15*e^7 + 168*b^1
5*c^8*d^14*e^8 - 1176*b^16*c^7*d^13*e^9 + 1560*b^17*c^6*d^12*e^10 - 1128*b^18*c^5*d^11*e^11 + 484*b^19*c^4*d^1
0*e^12 - 116*b^20*c^3*d^9*e^13 + 12*b^21*c^2*d^8*e^14 - ((-c^5*(b*e - c*d)^5)^(1/2)*(7*b*e - 4*c*d)*(d + e*x)^
(1/2)*(16*b^12*c^13*d^21*e^2 - 168*b^13*c^12*d^20*e^3 + 800*b^14*c^11*d^19*e^4 - 2280*b^15*c^10*d^18*e^5 + 432
0*b^16*c^9*d^17*e^6 - 5712*b^17*c^8*d^16*e^7 + 5376*b^18*c^7*d^15*e^8 - 3600*b^19*c^6*d^14*e^9 + 1680*b^20*c^5
*d^13*e^10 - 520*b^21*c^4*d^12*e^11 + 96*b^22*c^3*d^11*e^12 - 8*b^23*c^2*d^10*e^13))/(2*(b^8*e^5 - b^3*c^5*d^5
+ 5*b^4*c^4*d^4*e - 10*b^5*c^3*d^3*e^2 + 10*b^6*c^2*d^2*e^3 - 5*b^7*c*d*e^4))))/(2*(b^8*e^5 - b^3*c^5*d^5 + 5
*b^4*c^4*d^4*e - 10*b^5*c^3*d^3*e^2 + 10*b^6*c^2*d^2*e^3 - 5*b^7*c*d*e^4))))/(2*(b^8*e^5 - b^3*c^5*d^5 + 5*b^4
*c^4*d^4*e - 10*b^5*c^3*d^3*e^2 + 10*b^6*c^2*d^2*e^3 - 5*b^7*c*d*e^4)) + ((-c^5*(b*e - c*d)^5)^(1/2)*(7*b*e -
4*c*d)*((d + e*x)^(1/2)*(64*b^6*c^15*d^18*e^2 - 576*b^7*c^14*d^17*e^3 + 2228*b^8*c^13*d^16*e^4 - 4768*b^9*c^12
*d^15*e^5 + 5960*b^10*c^11*d^14*e^6 - 3976*b^11*c^10*d^13*e^7 + 578*b^12*c^9*d^12*e^8 + 1004*b^13*c^8*d^11*e^9
- 442*b^14*c^7*d^10*e^10 - 320*b^15*c^6*d^9*e^11 + 362*b^16*c^5*d^8*e^12 - 132*b^17*c^4*d^7*e^13 + 18*b^18*c^
3*d^6*e^14) - ((-c^5*(b*e - c*d)^5)^(1/2)*(7*b*e - 4*c*d)*(8*b^10*c^13*d^19*e^3 - 76*b^11*c^12*d^18*e^4 + 300*
b^12*c^11*d^17*e^5 - 612*b^13*c^10*d^16*e^6 + 5...
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