3.4.67 \(\int \frac {1}{(d+e x)^{5/2} (b x+c x^2)} \, dx\) [367]
Optimal. Leaf size=138 \[ -\frac {2 e}{3 d (c d-b e) (d+e x)^{3/2}}-\frac {2 e (2 c d-b e)}{d^2 (c d-b e)^2 \sqrt {d+e x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b d^{5/2}}+\frac {2 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b (c d-b e)^{5/2}} \]
[Out]
-2/3*e/d/(-b*e+c*d)/(e*x+d)^(3/2)-2*arctanh((e*x+d)^(1/2)/d^(1/2))/b/d^(5/2)+2*c^(5/2)*arctanh(c^(1/2)*(e*x+d)
^(1/2)/(-b*e+c*d)^(1/2))/b/(-b*e+c*d)^(5/2)-2*e*(-b*e+2*c*d)/d^2/(-b*e+c*d)^2/(e*x+d)^(1/2)
________________________________________________________________________________________
Rubi [A]
time = 0.17, antiderivative size = 138, 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 = {723, 842, 840,
1180, 214} \begin {gather*} \frac {2 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b (c d-b e)^{5/2}}-\frac {2 e (2 c d-b e)}{d^2 \sqrt {d+e x} (c d-b e)^2}-\frac {2 e}{3 d (d+e x)^{3/2} (c d-b e)}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b d^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/((d + e*x)^(5/2)*(b*x + c*x^2)),x]
[Out]
(-2*e)/(3*d*(c*d - b*e)*(d + e*x)^(3/2)) - (2*e*(2*c*d - b*e))/(d^2*(c*d - b*e)^2*Sqrt[d + e*x]) - (2*ArcTanh[
Sqrt[d + e*x]/Sqrt[d]])/(b*d^(5/2)) + (2*c^(5/2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*(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 723
Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[e*((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 - b*e - c
*e*x, x]/(a + b*x + c*x^2)), 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[m, -1]
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 )} \, dx &=-\frac {2 e}{3 d (c d-b e) (d+e x)^{3/2}}+\frac {\int \frac {c d-b e-c e x}{(d+e x)^{3/2} \left (b x+c x^2\right )} \, dx}{d (c d-b e)}\\ &=-\frac {2 e}{3 d (c d-b e) (d+e x)^{3/2}}-\frac {2 e (2 c d-b e)}{d^2 (c d-b e)^2 \sqrt {d+e x}}+\frac {\int \frac {(c d-b e)^2-c e (2 c d-b e) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{d^2 (c d-b e)^2}\\ &=-\frac {2 e}{3 d (c d-b e) (d+e x)^{3/2}}-\frac {2 e (2 c d-b e)}{d^2 (c d-b e)^2 \sqrt {d+e x}}+\frac {2 \text {Subst}\left (\int \frac {e (c d-b e)^2+c d e (2 c d-b e)-c e (2 c d-b e) 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 )}{d^2 (c d-b e)^2}\\ &=-\frac {2 e}{3 d (c d-b e) (d+e x)^{3/2}}-\frac {2 e (2 c d-b e)}{d^2 (c d-b e)^2 \sqrt {d+e x}}+\frac {(2 c) \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 d^2}-\frac {\left (2 c^3\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 (c d-b e)^2}\\ &=-\frac {2 e}{3 d (c d-b e) (d+e x)^{3/2}}-\frac {2 e (2 c d-b e)}{d^2 (c d-b e)^2 \sqrt {d+e x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b d^{5/2}}+\frac {2 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b (c d-b e)^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.61, size = 128, normalized size = 0.93 \begin {gather*} \frac {2 e (b e (4 d+3 e x)-c d (7 d+6 e x))}{3 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {2 c^{5/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{b (-c d+b e)^{5/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b d^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/((d + e*x)^(5/2)*(b*x + c*x^2)),x]
[Out]
(2*e*(b*e*(4*d + 3*e*x) - c*d*(7*d + 6*e*x)))/(3*d^2*(c*d - b*e)^2*(d + e*x)^(3/2)) - (2*c^(5/2)*ArcTan[(Sqrt[
c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/(b*(-(c*d) + b*e)^(5/2)) - (2*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*d^(5/2
))
________________________________________________________________________________________
Maple [A]
time = 0.43, size = 136, normalized size = 0.99
| | |
| method |
result |
size |
| | |
| derivativedivides |
\(2 e \left (-\frac {c^{3} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{2} b e \sqrt {\left (b e -c d \right ) c}}-\frac {\arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{e b \,d^{\frac {5}{2}}}-\frac {-b e +2 c d}{d^{2} \left (b e -c d \right )^{2} \sqrt {e x +d}}+\frac {1}{3 d \left (b e -c d \right ) \left (e x +d \right )^{\frac {3}{2}}}\right )\) |
\(136\) |
| default |
\(2 e \left (-\frac {c^{3} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{2} b e \sqrt {\left (b e -c d \right ) c}}-\frac {\arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{e b \,d^{\frac {5}{2}}}-\frac {-b e +2 c d}{d^{2} \left (b e -c d \right )^{2} \sqrt {e x +d}}+\frac {1}{3 d \left (b e -c d \right ) \left (e x +d \right )^{\frac {3}{2}}}\right )\) |
\(136\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(e*x+d)^(5/2)/(c*x^2+b*x),x,method=_RETURNVERBOSE)
[Out]
2*e*(-1/(b*e-c*d)^2*c^3/b/e/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))-1/e/b/d^(5/2)*arct
anh((e*x+d)^(1/2)/d^(1/2))-1/d^2/(b*e-c*d)^2*(-b*e+2*c*d)/(e*x+d)^(1/2)+1/3/d/(b*e-c*d)/(e*x+d)^(3/2))
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^(5/2)/(c*x^2+b*x),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. 362 vs.
\(2 (127) = 254\).
time = 2.65, size = 1481, normalized size = 10.73 \begin {gather*} \left [\frac {3 \, {\left (c^{2} d^{3} x^{2} e^{2} + 2 \, c^{2} d^{4} x e + c^{2} d^{5}\right )} \sqrt {\frac {c}{c d - b e}} \log \left (\frac {2 \, c d + 2 \, {\left (c d - b e\right )} \sqrt {x e + d} \sqrt {\frac {c}{c d - b e}} + {\left (c x - b\right )} e}{c x + b}\right ) + 3 \, {\left (c^{2} d^{4} + b^{2} x^{2} e^{4} - 2 \, {\left (b c d x^{2} - b^{2} d x\right )} e^{3} + {\left (c^{2} d^{2} x^{2} - 4 \, b c d^{2} x + b^{2} d^{2}\right )} e^{2} + 2 \, {\left (c^{2} d^{3} x - b c d^{3}\right )} e\right )} \sqrt {d} \log \left (\frac {x e - 2 \, \sqrt {x e + d} \sqrt {d} + 2 \, d}{x}\right ) - 2 \, {\left (7 \, b c d^{3} e - 3 \, b^{2} d x e^{3} + 2 \, {\left (3 \, b c d^{2} x - 2 \, b^{2} d^{2}\right )} e^{2}\right )} \sqrt {x e + d}}{3 \, {\left (b c^{2} d^{7} + b^{3} d^{3} x^{2} e^{4} - 2 \, {\left (b^{2} c d^{4} x^{2} - b^{3} d^{4} x\right )} e^{3} + {\left (b c^{2} d^{5} x^{2} - 4 \, b^{2} c d^{5} x + b^{3} d^{5}\right )} e^{2} + 2 \, {\left (b c^{2} d^{6} x - b^{2} c d^{6}\right )} e\right )}}, \frac {6 \, {\left (c^{2} d^{3} x^{2} e^{2} + 2 \, c^{2} d^{4} x e + c^{2} d^{5}\right )} \sqrt {-\frac {c}{c d - b e}} \arctan \left (-\frac {{\left (c d - b e\right )} \sqrt {x e + d} \sqrt {-\frac {c}{c d - b e}}}{c x e + c d}\right ) + 3 \, {\left (c^{2} d^{4} + b^{2} x^{2} e^{4} - 2 \, {\left (b c d x^{2} - b^{2} d x\right )} e^{3} + {\left (c^{2} d^{2} x^{2} - 4 \, b c d^{2} x + b^{2} d^{2}\right )} e^{2} + 2 \, {\left (c^{2} d^{3} x - b c d^{3}\right )} e\right )} \sqrt {d} \log \left (\frac {x e - 2 \, \sqrt {x e + d} \sqrt {d} + 2 \, d}{x}\right ) - 2 \, {\left (7 \, b c d^{3} e - 3 \, b^{2} d x e^{3} + 2 \, {\left (3 \, b c d^{2} x - 2 \, b^{2} d^{2}\right )} e^{2}\right )} \sqrt {x e + d}}{3 \, {\left (b c^{2} d^{7} + b^{3} d^{3} x^{2} e^{4} - 2 \, {\left (b^{2} c d^{4} x^{2} - b^{3} d^{4} x\right )} e^{3} + {\left (b c^{2} d^{5} x^{2} - 4 \, b^{2} c d^{5} x + b^{3} d^{5}\right )} e^{2} + 2 \, {\left (b c^{2} d^{6} x - b^{2} c d^{6}\right )} e\right )}}, \frac {6 \, {\left (c^{2} d^{4} + b^{2} x^{2} e^{4} - 2 \, {\left (b c d x^{2} - b^{2} d x\right )} e^{3} + {\left (c^{2} d^{2} x^{2} - 4 \, b c d^{2} x + b^{2} d^{2}\right )} e^{2} + 2 \, {\left (c^{2} d^{3} x - b c d^{3}\right )} e\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {x e + d} \sqrt {-d}}{d}\right ) + 3 \, {\left (c^{2} d^{3} x^{2} e^{2} + 2 \, c^{2} d^{4} x e + c^{2} d^{5}\right )} \sqrt {\frac {c}{c d - b e}} \log \left (\frac {2 \, c d + 2 \, {\left (c d - b e\right )} \sqrt {x e + d} \sqrt {\frac {c}{c d - b e}} + {\left (c x - b\right )} e}{c x + b}\right ) - 2 \, {\left (7 \, b c d^{3} e - 3 \, b^{2} d x e^{3} + 2 \, {\left (3 \, b c d^{2} x - 2 \, b^{2} d^{2}\right )} e^{2}\right )} \sqrt {x e + d}}{3 \, {\left (b c^{2} d^{7} + b^{3} d^{3} x^{2} e^{4} - 2 \, {\left (b^{2} c d^{4} x^{2} - b^{3} d^{4} x\right )} e^{3} + {\left (b c^{2} d^{5} x^{2} - 4 \, b^{2} c d^{5} x + b^{3} d^{5}\right )} e^{2} + 2 \, {\left (b c^{2} d^{6} x - b^{2} c d^{6}\right )} e\right )}}, \frac {2 \, {\left (3 \, {\left (c^{2} d^{3} x^{2} e^{2} + 2 \, c^{2} d^{4} x e + c^{2} d^{5}\right )} \sqrt {-\frac {c}{c d - b e}} \arctan \left (-\frac {{\left (c d - b e\right )} \sqrt {x e + d} \sqrt {-\frac {c}{c d - b e}}}{c x e + c d}\right ) + 3 \, {\left (c^{2} d^{4} + b^{2} x^{2} e^{4} - 2 \, {\left (b c d x^{2} - b^{2} d x\right )} e^{3} + {\left (c^{2} d^{2} x^{2} - 4 \, b c d^{2} x + b^{2} d^{2}\right )} e^{2} + 2 \, {\left (c^{2} d^{3} x - b c d^{3}\right )} e\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {x e + d} \sqrt {-d}}{d}\right ) - {\left (7 \, b c d^{3} e - 3 \, b^{2} d x e^{3} + 2 \, {\left (3 \, b c d^{2} x - 2 \, b^{2} d^{2}\right )} e^{2}\right )} \sqrt {x e + d}\right )}}{3 \, {\left (b c^{2} d^{7} + b^{3} d^{3} x^{2} e^{4} - 2 \, {\left (b^{2} c d^{4} x^{2} - b^{3} d^{4} x\right )} e^{3} + {\left (b c^{2} d^{5} x^{2} - 4 \, b^{2} c d^{5} x + b^{3} d^{5}\right )} e^{2} + 2 \, {\left (b c^{2} d^{6} x - b^{2} c d^{6}\right )} e\right )}}\right ] \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),x, algorithm="fricas")
[Out]
[1/3*(3*(c^2*d^3*x^2*e^2 + 2*c^2*d^4*x*e + c^2*d^5)*sqrt(c/(c*d - b*e))*log((2*c*d + 2*(c*d - b*e)*sqrt(x*e +
d)*sqrt(c/(c*d - b*e)) + (c*x - b)*e)/(c*x + b)) + 3*(c^2*d^4 + b^2*x^2*e^4 - 2*(b*c*d*x^2 - b^2*d*x)*e^3 + (c
^2*d^2*x^2 - 4*b*c*d^2*x + b^2*d^2)*e^2 + 2*(c^2*d^3*x - b*c*d^3)*e)*sqrt(d)*log((x*e - 2*sqrt(x*e + d)*sqrt(d
) + 2*d)/x) - 2*(7*b*c*d^3*e - 3*b^2*d*x*e^3 + 2*(3*b*c*d^2*x - 2*b^2*d^2)*e^2)*sqrt(x*e + d))/(b*c^2*d^7 + b^
3*d^3*x^2*e^4 - 2*(b^2*c*d^4*x^2 - b^3*d^4*x)*e^3 + (b*c^2*d^5*x^2 - 4*b^2*c*d^5*x + b^3*d^5)*e^2 + 2*(b*c^2*d
^6*x - b^2*c*d^6)*e), 1/3*(6*(c^2*d^3*x^2*e^2 + 2*c^2*d^4*x*e + c^2*d^5)*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*(c^2*d^4 + b^2*x^2*e^4 - 2*(b*c*d*x^2 - b^2*d*x)*e^3
+ (c^2*d^2*x^2 - 4*b*c*d^2*x + b^2*d^2)*e^2 + 2*(c^2*d^3*x - b*c*d^3)*e)*sqrt(d)*log((x*e - 2*sqrt(x*e + d)*s
qrt(d) + 2*d)/x) - 2*(7*b*c*d^3*e - 3*b^2*d*x*e^3 + 2*(3*b*c*d^2*x - 2*b^2*d^2)*e^2)*sqrt(x*e + d))/(b*c^2*d^7
+ b^3*d^3*x^2*e^4 - 2*(b^2*c*d^4*x^2 - b^3*d^4*x)*e^3 + (b*c^2*d^5*x^2 - 4*b^2*c*d^5*x + b^3*d^5)*e^2 + 2*(b*
c^2*d^6*x - b^2*c*d^6)*e), 1/3*(6*(c^2*d^4 + b^2*x^2*e^4 - 2*(b*c*d*x^2 - b^2*d*x)*e^3 + (c^2*d^2*x^2 - 4*b*c*
d^2*x + b^2*d^2)*e^2 + 2*(c^2*d^3*x - b*c*d^3)*e)*sqrt(-d)*arctan(sqrt(x*e + d)*sqrt(-d)/d) + 3*(c^2*d^3*x^2*e
^2 + 2*c^2*d^4*x*e + c^2*d^5)*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*(7*b*c*d^3*e - 3*b^2*d*x*e^3 + 2*(3*b*c*d^2*x - 2*b^2*d^2)*e^2)*sqrt(x*e + d))/
(b*c^2*d^7 + b^3*d^3*x^2*e^4 - 2*(b^2*c*d^4*x^2 - b^3*d^4*x)*e^3 + (b*c^2*d^5*x^2 - 4*b^2*c*d^5*x + b^3*d^5)*e
^2 + 2*(b*c^2*d^6*x - b^2*c*d^6)*e), 2/3*(3*(c^2*d^3*x^2*e^2 + 2*c^2*d^4*x*e + c^2*d^5)*sqrt(-c/(c*d - b*e))*a
rctan(-(c*d - b*e)*sqrt(x*e + d)*sqrt(-c/(c*d - b*e))/(c*x*e + c*d)) + 3*(c^2*d^4 + b^2*x^2*e^4 - 2*(b*c*d*x^2
- b^2*d*x)*e^3 + (c^2*d^2*x^2 - 4*b*c*d^2*x + b^2*d^2)*e^2 + 2*(c^2*d^3*x - b*c*d^3)*e)*sqrt(-d)*arctan(sqrt(
x*e + d)*sqrt(-d)/d) - (7*b*c*d^3*e - 3*b^2*d*x*e^3 + 2*(3*b*c*d^2*x - 2*b^2*d^2)*e^2)*sqrt(x*e + d))/(b*c^2*d
^7 + b^3*d^3*x^2*e^4 - 2*(b^2*c*d^4*x^2 - b^3*d^4*x)*e^3 + (b*c^2*d^5*x^2 - 4*b^2*c*d^5*x + b^3*d^5)*e^2 + 2*(
b*c^2*d^6*x - b^2*c*d^6)*e)]
________________________________________________________________________________________
Sympy [A]
time = 7.54, size = 133, normalized size = 0.96 \begin {gather*} \frac {2 e}{3 d \left (d + e x\right )^{\frac {3}{2}} \left (b e - c d\right )} + \frac {2 e \left (b e - 2 c d\right )}{d^{2} \sqrt {d + e x} \left (b e - c d\right )^{2}} - \frac {2 c^{2} \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {b e - c d}{c}}} \right )}}{b \sqrt {\frac {b e - c d}{c}} \left (b e - c d\right )^{2}} + \frac {2 \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d}} \right )}}{b d^{2} \sqrt {- d}} \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),x)
[Out]
2*e/(3*d*(d + e*x)**(3/2)*(b*e - c*d)) + 2*e*(b*e - 2*c*d)/(d**2*sqrt(d + e*x)*(b*e - c*d)**2) - 2*c**2*atan(s
qrt(d + e*x)/sqrt((b*e - c*d)/c))/(b*sqrt((b*e - c*d)/c)*(b*e - c*d)**2) + 2*atan(sqrt(d + e*x)/sqrt(-d))/(b*d
**2*sqrt(-d))
________________________________________________________________________________________
Giac [A]
time = 1.70, size = 174, normalized size = 1.26 \begin {gather*} -\frac {2 \, c^{3} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{{\left (b c^{2} d^{2} - 2 \, b^{2} c d e + b^{3} e^{2}\right )} \sqrt {-c^{2} d + b c e}} - \frac {2 \, {\left (6 \, {\left (x e + d\right )} c d e + c d^{2} e - 3 \, {\left (x e + d\right )} b e^{2} - b d e^{2}\right )}}{3 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} {\left (x e + d\right )}^{\frac {3}{2}}} + \frac {2 \, \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{b \sqrt {-d} d^{2}} \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),x, algorithm="giac")
[Out]
-2*c^3*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b*c^2*d^2 - 2*b^2*c*d*e + b^3*e^2)*sqrt(-c^2*d + b*c*e))
- 2/3*(6*(x*e + d)*c*d*e + c*d^2*e - 3*(x*e + d)*b*e^2 - b*d*e^2)/((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2)*(x*e
+ d)^(3/2)) + 2*arctan(sqrt(x*e + d)/sqrt(-d))/(b*sqrt(-d)*d^2)
________________________________________________________________________________________
Mupad [B]
time = 1.17, size = 2500, normalized size = 18.12 \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)*(d + e*x)^(5/2)),x)
[Out]
(atan(((((d + e*x)^(1/2)*(16*c^13*d^16*e^2 - 128*b*c^12*d^15*e^3 + 480*b^2*c^11*d^14*e^4 - 1120*b^3*c^10*d^13*
e^5 + 1800*b^4*c^9*d^12*e^6 - 2064*b^5*c^8*d^11*e^7 + 1688*b^6*c^7*d^10*e^8 - 960*b^7*c^6*d^9*e^9 + 360*b^8*c^
5*d^8*e^10 - 80*b^9*c^4*d^7*e^11 + 8*b^10*c^3*d^6*e^12) + ((-c^5*(b*e - c*d)^5)^(1/2)*(24*b^2*c^12*d^18*e^3 -
216*b^3*c^11*d^17*e^4 + 872*b^4*c^10*d^16*e^5 - 2080*b^5*c^9*d^15*e^6 + 3248*b^6*c^8*d^14*e^7 - 3472*b^7*c^7*d
^13*e^8 + 2576*b^8*c^6*d^12*e^9 - 1312*b^9*c^5*d^11*e^10 + 440*b^10*c^4*d^10*e^11 - 88*b^11*c^3*d^9*e^12 + 8*b
^12*c^2*d^8*e^13 - ((-c^5*(b*e - c*d)^5)^(1/2)*(d + e*x)^(1/2)*(16*b^2*c^13*d^21*e^2 - 168*b^3*c^12*d^20*e^3 +
800*b^4*c^11*d^19*e^4 - 2280*b^5*c^10*d^18*e^5 + 4320*b^6*c^9*d^17*e^6 - 5712*b^7*c^8*d^16*e^7 + 5376*b^8*c^7
*d^15*e^8 - 3600*b^9*c^6*d^14*e^9 + 1680*b^10*c^5*d^13*e^10 - 520*b^11*c^4*d^12*e^11 + 96*b^12*c^3*d^11*e^12 -
8*b^13*c^2*d^10*e^13))/(b*(b*e - c*d)^5)))/(b*(b*e - c*d)^5))*(-c^5*(b*e - c*d)^5)^(1/2)*1i)/(b*(b*e - c*d)^5
) + (((d + e*x)^(1/2)*(16*c^13*d^16*e^2 - 128*b*c^12*d^15*e^3 + 480*b^2*c^11*d^14*e^4 - 1120*b^3*c^10*d^13*e^5
+ 1800*b^4*c^9*d^12*e^6 - 2064*b^5*c^8*d^11*e^7 + 1688*b^6*c^7*d^10*e^8 - 960*b^7*c^6*d^9*e^9 + 360*b^8*c^5*d
^8*e^10 - 80*b^9*c^4*d^7*e^11 + 8*b^10*c^3*d^6*e^12) - ((-c^5*(b*e - c*d)^5)^(1/2)*(24*b^2*c^12*d^18*e^3 - 216
*b^3*c^11*d^17*e^4 + 872*b^4*c^10*d^16*e^5 - 2080*b^5*c^9*d^15*e^6 + 3248*b^6*c^8*d^14*e^7 - 3472*b^7*c^7*d^13
*e^8 + 2576*b^8*c^6*d^12*e^9 - 1312*b^9*c^5*d^11*e^10 + 440*b^10*c^4*d^10*e^11 - 88*b^11*c^3*d^9*e^12 + 8*b^12
*c^2*d^8*e^13 + ((-c^5*(b*e - c*d)^5)^(1/2)*(d + e*x)^(1/2)*(16*b^2*c^13*d^21*e^2 - 168*b^3*c^12*d^20*e^3 + 80
0*b^4*c^11*d^19*e^4 - 2280*b^5*c^10*d^18*e^5 + 4320*b^6*c^9*d^17*e^6 - 5712*b^7*c^8*d^16*e^7 + 5376*b^8*c^7*d^
15*e^8 - 3600*b^9*c^6*d^14*e^9 + 1680*b^10*c^5*d^13*e^10 - 520*b^11*c^4*d^12*e^11 + 96*b^12*c^3*d^11*e^12 - 8*
b^13*c^2*d^10*e^13))/(b*(b*e - c*d)^5)))/(b*(b*e - c*d)^5))*(-c^5*(b*e - c*d)^5)^(1/2)*1i)/(b*(b*e - c*d)^5))/
(32*c^12*d^13*e^3 - 208*b*c^11*d^12*e^4 + 576*b^2*c^10*d^11*e^5 - 880*b^3*c^9*d^10*e^6 + 800*b^4*c^8*d^9*e^7 -
432*b^5*c^7*d^8*e^8 + 128*b^6*c^6*d^7*e^9 - 16*b^7*c^5*d^6*e^10 + (((d + e*x)^(1/2)*(16*c^13*d^16*e^2 - 128*b
*c^12*d^15*e^3 + 480*b^2*c^11*d^14*e^4 - 1120*b^3*c^10*d^13*e^5 + 1800*b^4*c^9*d^12*e^6 - 2064*b^5*c^8*d^11*e^
7 + 1688*b^6*c^7*d^10*e^8 - 960*b^7*c^6*d^9*e^9 + 360*b^8*c^5*d^8*e^10 - 80*b^9*c^4*d^7*e^11 + 8*b^10*c^3*d^6*
e^12) + ((-c^5*(b*e - c*d)^5)^(1/2)*(24*b^2*c^12*d^18*e^3 - 216*b^3*c^11*d^17*e^4 + 872*b^4*c^10*d^16*e^5 - 20
80*b^5*c^9*d^15*e^6 + 3248*b^6*c^8*d^14*e^7 - 3472*b^7*c^7*d^13*e^8 + 2576*b^8*c^6*d^12*e^9 - 1312*b^9*c^5*d^1
1*e^10 + 440*b^10*c^4*d^10*e^11 - 88*b^11*c^3*d^9*e^12 + 8*b^12*c^2*d^8*e^13 - ((-c^5*(b*e - c*d)^5)^(1/2)*(d
+ e*x)^(1/2)*(16*b^2*c^13*d^21*e^2 - 168*b^3*c^12*d^20*e^3 + 800*b^4*c^11*d^19*e^4 - 2280*b^5*c^10*d^18*e^5 +
4320*b^6*c^9*d^17*e^6 - 5712*b^7*c^8*d^16*e^7 + 5376*b^8*c^7*d^15*e^8 - 3600*b^9*c^6*d^14*e^9 + 1680*b^10*c^5*
d^13*e^10 - 520*b^11*c^4*d^12*e^11 + 96*b^12*c^3*d^11*e^12 - 8*b^13*c^2*d^10*e^13))/(b*(b*e - c*d)^5)))/(b*(b*
e - c*d)^5))*(-c^5*(b*e - c*d)^5)^(1/2))/(b*(b*e - c*d)^5) - (((d + e*x)^(1/2)*(16*c^13*d^16*e^2 - 128*b*c^12*
d^15*e^3 + 480*b^2*c^11*d^14*e^4 - 1120*b^3*c^10*d^13*e^5 + 1800*b^4*c^9*d^12*e^6 - 2064*b^5*c^8*d^11*e^7 + 16
88*b^6*c^7*d^10*e^8 - 960*b^7*c^6*d^9*e^9 + 360*b^8*c^5*d^8*e^10 - 80*b^9*c^4*d^7*e^11 + 8*b^10*c^3*d^6*e^12)
- ((-c^5*(b*e - c*d)^5)^(1/2)*(24*b^2*c^12*d^18*e^3 - 216*b^3*c^11*d^17*e^4 + 872*b^4*c^10*d^16*e^5 - 2080*b^5
*c^9*d^15*e^6 + 3248*b^6*c^8*d^14*e^7 - 3472*b^7*c^7*d^13*e^8 + 2576*b^8*c^6*d^12*e^9 - 1312*b^9*c^5*d^11*e^10
+ 440*b^10*c^4*d^10*e^11 - 88*b^11*c^3*d^9*e^12 + 8*b^12*c^2*d^8*e^13 + ((-c^5*(b*e - c*d)^5)^(1/2)*(d + e*x)
^(1/2)*(16*b^2*c^13*d^21*e^2 - 168*b^3*c^12*d^20*e^3 + 800*b^4*c^11*d^19*e^4 - 2280*b^5*c^10*d^18*e^5 + 4320*b
^6*c^9*d^17*e^6 - 5712*b^7*c^8*d^16*e^7 + 5376*b^8*c^7*d^15*e^8 - 3600*b^9*c^6*d^14*e^9 + 1680*b^10*c^5*d^13*e
^10 - 520*b^11*c^4*d^12*e^11 + 96*b^12*c^3*d^11*e^12 - 8*b^13*c^2*d^10*e^13))/(b*(b*e - c*d)^5)))/(b*(b*e - c*
d)^5))*(-c^5*(b*e - c*d)^5)^(1/2))/(b*(b*e - c*d)^5)))*(-c^5*(b*e - c*d)^5)^(1/2)*2i)/(b*(b*e - c*d)^5) - (2*a
tanh((80*c^12*d^15*e^3*(d + e*x)^(1/2))/((d^5)^(1/2)*(80*c^12*d^13*e^3 - 640*b*c^11*d^12*e^4 + 2320*b^2*c^10*d
^11*e^5 - 5040*b^3*c^9*d^10*e^6 + 7296*b^4*c^8*d^9*e^7 - 7376*b^5*c^7*d^8*e^8 + 5280*b^6*c^6*d^7*e^9 - 2640*b^
7*c^5*d^6*e^10 + 880*b^8*c^4*d^5*e^11 - 176*b^9*c^3*d^4*e^12 + 16*b^10*c^2*d^3*e^13)) + (2320*b^2*c^10*d^13*e^
5*(d + e*x)^(1/2))/((d^5)^(1/2)*(80*c^12*d^13*e^3 - 640*b*c^11*d^12*e^4 + 2320*b^2*c^10*d^11*e^5 - 5040*b^3*c^
9*d^10*e^6 + 7296*b^4*c^8*d^9*e^7 - 7376*b^5*c^7*d^8*e^8 + 5280*b^6*c^6*d^7*e^9 - 2640*b^7*c^5*d^6*e^10 + 880*
b^8*c^4*d^5*e^11 - 176*b^9*c^3*d^4*e^12 + 16*b^10*c^2*d^3*e^13)) - (5040*b^3*c^9*d^12*e^6*(d + e*x)^(1/2))/((d
^5)^(1/2)*(80*c^12*d^13*e^3 - 640*b*c^11*d^12*e^4 + 2320*b^2*c^10*d^11*e^5 - 5040*b^3*c^9*d^10*e^6 + 7296*b^4*
c^8*d^9*e^7 - 7376*b^5*c^7*d^8*e^8 + 5280*b^6*c...
________________________________________________________________________________________