3.17.17 \(\int \frac {b+2 c x}{(d+e x)^{5/2} (a+b x+c x^2)} \, dx\) [1617]

Optimal. Leaf size=518 \[ \frac {2 (2 c d-b e)}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}+\frac {2 \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )}{\left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}-\frac {\sqrt {2} \sqrt {c} \left (b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^2+2 c^2 d \left (\sqrt {b^2-4 a c} d+4 a e\right )-2 c e \left (b^2 d+b \sqrt {b^2-4 a c} d+2 a b e+a \sqrt {b^2-4 a c} e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )^2}+\frac {\sqrt {2} \sqrt {c} \left (b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c^2 d \left (\sqrt {b^2-4 a c} d-4 a e\right )-2 c e \left (b^2 d-b \sqrt {b^2-4 a c} d+2 a b e-a \sqrt {b^2-4 a c} e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )^2} \]

[Out]

2/3*(-b*e+2*c*d)/(a*e^2-b*d*e+c*d^2)/(e*x+d)^(3/2)+2*(2*c^2*d^2+b^2*e^2-2*c*e*(a*e+b*d))/(a*e^2-b*d*e+c*d^2)^2
/(e*x+d)^(1/2)-arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2))*2^(1/2)*c^(1/2)*(
b^2*e^2*(b+(-4*a*c+b^2)^(1/2))+2*c^2*d*(4*a*e+d*(-4*a*c+b^2)^(1/2))-2*c*e*(b^2*d+2*a*b*e+b*d*(-4*a*c+b^2)^(1/2
)+a*e*(-4*a*c+b^2)^(1/2)))/(a*e^2-b*d*e+c*d^2)^2/(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)+arc
tanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2))*2^(1/2)*c^(1/2)*(b^2*e^2*(b-(-4*a*c
+b^2)^(1/2))-2*c^2*d*(-4*a*e+d*(-4*a*c+b^2)^(1/2))-2*c*e*(b^2*d+2*a*b*e-b*d*(-4*a*c+b^2)^(1/2)-a*e*(-4*a*c+b^2
)^(1/2)))/(a*e^2-b*d*e+c*d^2)^2/(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)

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Rubi [A]
time = 1.43, antiderivative size = 518, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {842, 840, 1180, 214} \begin {gather*} \frac {2 \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )^2}-\frac {\sqrt {2} \sqrt {c} \left (2 c^2 d \left (d \sqrt {b^2-4 a c}+4 a e\right )-2 c e \left (b d \sqrt {b^2-4 a c}+a e \sqrt {b^2-4 a c}+2 a b e+b^2 d\right )+b^2 e^2 \left (\sqrt {b^2-4 a c}+b\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )^2}+\frac {\sqrt {2} \sqrt {c} \left (-2 c^2 d \left (d \sqrt {b^2-4 a c}-4 a e\right )-2 c e \left (-b d \sqrt {b^2-4 a c}-a e \sqrt {b^2-4 a c}+2 a b e+b^2 d\right )+b^2 e^2 \left (b-\sqrt {b^2-4 a c}\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )^2}+\frac {2 (2 c d-b e)}{3 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(2*c*d - b*e))/(3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(3/2)) + (2*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e)))/
((c*d^2 - b*d*e + a*e^2)^2*Sqrt[d + e*x]) - (Sqrt[2]*Sqrt[c]*(b^2*(b + Sqrt[b^2 - 4*a*c])*e^2 + 2*c^2*d*(Sqrt[
b^2 - 4*a*c]*d + 4*a*e) - 2*c*e*(b^2*d + b*Sqrt[b^2 - 4*a*c]*d + 2*a*b*e + a*Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sq
rt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b - Sq
rt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e + a*e^2)^2) + (Sqrt[2]*Sqrt[c]*(b^2*(b - Sqrt[b^2 - 4*a*c])*e^2 - 2*c^2*d*(
Sqrt[b^2 - 4*a*c]*d - 4*a*e) - 2*c*e*(b^2*d - b*Sqrt[b^2 - 4*a*c]*d + 2*a*b*e - a*Sqrt[b^2 - 4*a*c]*e))*ArcTan
h[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b
+ Sqrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e + a*e^2)^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 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 {b+2 c x}{(d+e x)^{5/2} \left (a+b x+c x^2\right )} \, dx &=\frac {2 (2 c d-b e)}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}+\frac {\int \frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )} \, dx}{c d^2-b d e+a e^2}\\ &=\frac {2 (2 c d-b e)}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}+\frac {2 \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )}{\left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}+\frac {\int \frac {-2 b^2 c d e+4 a c^2 d e+b^3 e^2+b c \left (c d^2-3 a e^2\right )+c \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx}{\left (c d^2-b d e+a e^2\right )^2}\\ &=\frac {2 (2 c d-b e)}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}+\frac {2 \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )}{\left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}+\frac {2 \text {Subst}\left (\int \frac {-c d \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )+e \left (-2 b^2 c d e+4 a c^2 d e+b^3 e^2+b c \left (c d^2-3 a e^2\right )\right )+c \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) x^2}{c d^2-b d e+a e^2+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{\left (c d^2-b d e+a e^2\right )^2}\\ &=\frac {2 (2 c d-b e)}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}+\frac {2 \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )}{\left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}-\frac {\left (c \left (b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c^2 d \left (\sqrt {b^2-4 a c} d-4 a e\right )-2 c e \left (b^2 d-b \sqrt {b^2-4 a c} d+2 a b e-a \sqrt {b^2-4 a c} e\right )\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2}+\frac {\left (c \left (b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^2+2 c^2 d \left (\sqrt {b^2-4 a c} d+4 a e\right )-2 c e \left (b^2 d+b \sqrt {b^2-4 a c} d+2 a b e+a \sqrt {b^2-4 a c} e\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac {2 (2 c d-b e)}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}+\frac {2 \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )}{\left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}-\frac {\sqrt {2} \sqrt {c} \left (b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^2+2 c^2 d \left (\sqrt {b^2-4 a c} d+4 a e\right )-2 c e \left (b^2 d+b \sqrt {b^2-4 a c} d+2 a b e+a \sqrt {b^2-4 a c} e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )^2}+\frac {\sqrt {2} \sqrt {c} \left (b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c^2 d \left (\sqrt {b^2-4 a c} d-4 a e\right )-2 c e \left (b^2 d-b \sqrt {b^2-4 a c} d+2 a b e-a \sqrt {b^2-4 a c} e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )^2}\\ \end {align*}

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Mathematica [A]
time = 2.20, size = 478, normalized size = 0.92 \begin {gather*} \frac {-\frac {2 \left (-2 c^2 d^2 (4 d+3 e x)+b e^2 (-4 b d+a e-3 b e x)+c e (3 b d (3 d+2 e x)+2 a e (2 d+3 e x))\right )}{(d+e x)^{3/2}}+\frac {3 \sqrt {2} \sqrt {c} \left (b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^2+2 c^2 d \left (\sqrt {b^2-4 a c} d+4 a e\right )-2 c e \left (b^2 d+b \sqrt {b^2-4 a c} d+2 a b e+a \sqrt {b^2-4 a c} e\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {3 \sqrt {2} \sqrt {c} \left (b^2 \left (-b+\sqrt {b^2-4 a c}\right ) e^2+2 c^2 d \left (\sqrt {b^2-4 a c} d-4 a e\right )-2 c e \left (-b^2 d+b \sqrt {b^2-4 a c} d-2 a b e+a \sqrt {b^2-4 a c} e\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}}{3 \left (c d^2+e (-b d+a e)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*(-2*c^2*d^2*(4*d + 3*e*x) + b*e^2*(-4*b*d + a*e - 3*b*e*x) + c*e*(3*b*d*(3*d + 2*e*x) + 2*a*e*(2*d + 3*e*
x))))/(d + e*x)^(3/2) + (3*Sqrt[2]*Sqrt[c]*(b^2*(b + Sqrt[b^2 - 4*a*c])*e^2 + 2*c^2*d*(Sqrt[b^2 - 4*a*c]*d + 4
*a*e) - 2*c*e*(b^2*d + b*Sqrt[b^2 - 4*a*c]*d + 2*a*b*e + a*Sqrt[b^2 - 4*a*c]*e))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[
d + e*x])/Sqrt[-2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[-2*c*d + (b - Sqrt[b^2 - 4*a*c])*
e]) + (3*Sqrt[2]*Sqrt[c]*(b^2*(-b + Sqrt[b^2 - 4*a*c])*e^2 + 2*c^2*d*(Sqrt[b^2 - 4*a*c]*d - 4*a*e) - 2*c*e*(-(
b^2*d) + b*Sqrt[b^2 - 4*a*c]*d - 2*a*b*e + a*Sqrt[b^2 - 4*a*c]*e))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt
[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e]))/(3*(c*d^2
+ e*(-(b*d) + a*e))^2)

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Maple [A]
time = 0.98, size = 569, normalized size = 1.10

method	result	size
derivativedivides	\(\frac {8 c \left (\frac {\left (4 c \,e^{3} a b -8 d \,e^{2} c^{2} a -b^{3} e^{3}+2 b^{2} d \,e^{2} c -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (-4 c \,e^{3} a b +8 d \,e^{2} c^{2} a +b^{3} e^{3}-2 b^{2} d \,e^{2} c -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{\left (e^{2} a -b d e +c \,d^{2}\right )^{2}}-\frac {2 \left (b e -2 c d \right )}{3 \left (e^{2} a -b d e +c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (2 a c \,e^{2}-b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}\right )}{\left (e^{2} a -b d e +c \,d^{2}\right )^{2} \sqrt {e x +d}}\)	\(569\)
default	\(\frac {8 c \left (\frac {\left (4 c \,e^{3} a b -8 d \,e^{2} c^{2} a -b^{3} e^{3}+2 b^{2} d \,e^{2} c -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (-4 c \,e^{3} a b +8 d \,e^{2} c^{2} a +b^{3} e^{3}-2 b^{2} d \,e^{2} c -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c \,e^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e^{2}-2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{2} d^{2}\right ) \sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{\left (e^{2} a -b d e +c \,d^{2}\right )^{2}}-\frac {2 \left (b e -2 c d \right )}{3 \left (e^{2} a -b d e +c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (2 a c \,e^{2}-b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}\right )}{\left (e^{2} a -b d e +c \,d^{2}\right )^{2} \sqrt {e x +d}}\)	\(569\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(e*x+d)^(5/2)/(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

integrate((2*c*x + b)/((c*x^2 + b*x + a)*(x*e + d)^(5/2)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 20771 vs. \(2 (473) = 946\).
time = 6.66, size = 20771, normalized size = 40.10 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(e*x+d)^(5/2)/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

-1/6*(3*sqrt(2)*(c^2*d^6 + a^2*x^2*e^6 - 2*(a*b*d*x^2 - a^2*d*x)*e^5 - (4*a*b*d^2*x - (b^2 + 2*a*c)*d^2*x^2 -
a^2*d^2)*e^4 - 2*(b*c*d^3*x^2 + a*b*d^3 - (b^2 + 2*a*c)*d^3*x)*e^3 + (c^2*d^4*x^2 - 4*b*c*d^4*x + (b^2 + 2*a*c
)*d^4)*e^2 + 2*(c^2*d^5*x - b*c*d^5)*e)*sqrt((2*c^5*d^5 - 5*b*c^4*d^4*e + 10*(b^2*c^3 - 2*a*c^4)*d^3*e^2 - 10*
(b^3*c^2 - 3*a*b*c^3)*d^2*e^3 + 5*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*e^4 - (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e^
5 + (c^5*d^10 - 5*b*c^4*d^9*e + 5*(2*b^2*c^3 + a*c^4)*d^8*e^2 - 10*(b^3*c^2 + 2*a*b*c^3)*d^7*e^3 + 5*(b^4*c +
6*a*b^2*c^2 + 2*a^2*c^3)*d^6*e^4 - 5*a^4*b*d*e^9 - (b^5 + 20*a*b^3*c + 30*a^2*b*c^2)*d^5*e^5 + a^5*e^10 + 5*(a
*b^4 + 6*a^2*b^2*c + 2*a^3*c^2)*d^4*e^6 - 10*(a^2*b^3 + 2*a^3*b*c)*d^3*e^7 + 5*(2*a^3*b^2 + a^4*c)*d^2*e^8)*sq
rt((25*(b^2*c^8 - 4*a*c^9)*d^8*e^2 - 100*(b^3*c^7 - 4*a*b*c^8)*d^7*e^3 + 100*(2*b^4*c^6 - 9*a*b^2*c^7 + 4*a^2*
c^8)*d^6*e^4 - 50*(5*b^5*c^5 - 26*a*b^3*c^6 + 24*a^2*b*c^7)*d^5*e^5 + 10*(21*b^6*c^4 - 127*a*b^4*c^5 + 183*a^2
*b^2*c^6 - 44*a^3*c^7)*d^4*e^6 - 20*(6*b^7*c^3 - 42*a*b^5*c^4 + 83*a^2*b^3*c^5 - 44*a^3*b*c^6)*d^3*e^7 + 5*(9*
b^8*c^2 - 72*a*b^6*c^3 + 180*a^2*b^4*c^4 - 148*a^3*b^2*c^5 + 16*a^4*c^6)*d^2*e^8 - 10*(b^9*c - 9*a*b^7*c^2 + 2
7*a^2*b^5*c^3 - 30*a^3*b^3*c^4 + 8*a^4*b*c^5)*d*e^9 + (b^10 - 10*a*b^8*c + 35*a^2*b^6*c^2 - 50*a^3*b^4*c^3 + 2
5*a^4*b^2*c^4 - 4*a^5*c^5)*e^10)/(c^10*d^20 - 10*b*c^9*d^19*e + 5*(9*b^2*c^8 + 2*a*c^9)*d^18*e^2 - 30*(4*b^3*c
^7 + 3*a*b*c^8)*d^17*e^3 + 15*(14*b^4*c^6 + 24*a*b^2*c^7 + 3*a^2*c^8)*d^16*e^4 - 12*(21*b^5*c^5 + 70*a*b^3*c^6
+ 30*a^2*b*c^7)*d^15*e^5 + 30*(7*b^6*c^4 + 42*a*b^4*c^5 + 42*a^2*b^2*c^6 + 4*a^3*c^7)*d^14*e^6 - 60*(2*b^7*c^
3 + 21*a*b^5*c^4 + 42*a^2*b^3*c^5 + 14*a^3*b*c^6)*d^13*e^7 + 15*(3*b^8*c^2 + 56*a*b^6*c^3 + 210*a^2*b^4*c^4 +
168*a^3*b^2*c^5 + 14*a^4*c^6)*d^12*e^8 - 10*(b^9*c + 36*a*b^7*c^2 + 252*a^2*b^5*c^3 + 420*a^3*b^3*c^4 + 126*a^
4*b*c^5)*d^11*e^9 - 10*a^9*b*d*e^19 + (b^10 + 90*a*b^8*c + 1260*a^2*b^6*c^2 + 4200*a^3*b^4*c^3 + 3150*a^4*b^2*
c^4 + 252*a^5*c^5)*d^10*e^10 + a^10*e^20 - 10*(a*b^9 + 36*a^2*b^7*c + 252*a^3*b^5*c^2 + 420*a^4*b^3*c^3 + 126*
a^5*b*c^4)*d^9*e^11 + 15*(3*a^2*b^8 + 56*a^3*b^6*c + 210*a^4*b^4*c^2 + 168*a^5*b^2*c^3 + 14*a^6*c^4)*d^8*e^12
- 60*(2*a^3*b^7 + 21*a^4*b^5*c + 42*a^5*b^3*c^2 + 14*a^6*b*c^3)*d^7*e^13 + 30*(7*a^4*b^6 + 42*a^5*b^4*c + 42*a
^6*b^2*c^2 + 4*a^7*c^3)*d^6*e^14 - 12*(21*a^5*b^5 + 70*a^6*b^3*c + 30*a^7*b*c^2)*d^5*e^15 + 15*(14*a^6*b^4 + 2
4*a^7*b^2*c + 3*a^8*c^2)*d^4*e^16 - 30*(4*a^7*b^3 + 3*a^8*b*c)*d^3*e^17 + 5*(9*a^8*b^2 + 2*a^9*c)*d^2*e^18)))/
(c^5*d^10 - 5*b*c^4*d^9*e + 5*(2*b^2*c^3 + a*c^4)*d^8*e^2 - 10*(b^3*c^2 + 2*a*b*c^3)*d^7*e^3 + 5*(b^4*c + 6*a*
b^2*c^2 + 2*a^2*c^3)*d^6*e^4 - 5*a^4*b*d*e^9 - (b^5 + 20*a*b^3*c + 30*a^2*b*c^2)*d^5*e^5 + a^5*e^10 + 5*(a*b^4
+ 6*a^2*b^2*c + 2*a^3*c^2)*d^4*e^6 - 10*(a^2*b^3 + 2*a^3*b*c)*d^3*e^7 + 5*(2*a^3*b^2 + a^4*c)*d^2*e^8))*log(s
qrt(2)*(10*c^7*d^7 - 35*b*c^6*d^6*e + 5*(13*b^2*c^5 - 10*a*c^6)*d^5*e^2 - 25*(3*b^3*c^4 - 5*a*b*c^5)*d^4*e^3 +
(57*b^4*c^3 - 156*a*b^2*c^4 + 62*a^2*c^5)*d^3*e^4 - (28*b^5*c^2 - 109*a*b^3*c^3 + 93*a^2*b*c^4)*d^2*e^5 + (8*
b^6*c - 40*a*b^4*c^2 + 51*a^2*b^2*c^3 - 6*a^3*c^4)*d*e^6 - (b^7 - 6*a*b^5*c + 10*a^2*b^3*c^2 - 3*a^3*b*c^3)*e^
7 - (3*c^7*d^12 - 18*b*c^6*d^11*e + 2*(23*b^2*c^5 + 7*a*c^6)*d^10*e^2 - 5*(13*b^3*c^4 + 14*a*b*c^5)*d^9*e^3 +
5*(11*b^4*c^3 + 29*a*b^2*c^4 + 5*a^2*c^5)*d^8*e^4 - 4*(7*b^5*c^2 + 40*a*b^3*c^3 + 25*a^2*b*c^4)*d^7*e^5 + 4*(2
*b^6*c + 25*a*b^4*c^2 + 40*a^2*b^2*c^3 + 5*a^3*c^4)*d^6*e^6 - (b^7 + 34*a*b^5*c + 130*a^2*b^3*c^2 + 60*a^3*b*c
^3)*d^5*e^7 + 5*(a*b^6 + 11*a^2*b^4*c + 14*a^3*b^2*c^2 + a^4*c^3)*d^4*e^8 - 10*(a^2*b^5 + 4*a^3*b^3*c + a^4*b*
c^2)*d^3*e^9 + 2*(5*a^3*b^4 + 5*a^4*b^2*c - a^5*c^2)*d^2*e^10 - (5*a^4*b^3 - 2*a^5*b*c)*d*e^11 + (a^5*b^2 - a^
6*c)*e^12)*sqrt((25*(b^2*c^8 - 4*a*c^9)*d^8*e^2 - 100*(b^3*c^7 - 4*a*b*c^8)*d^7*e^3 + 100*(2*b^4*c^6 - 9*a*b^2
*c^7 + 4*a^2*c^8)*d^6*e^4 - 50*(5*b^5*c^5 - 26*a*b^3*c^6 + 24*a^2*b*c^7)*d^5*e^5 + 10*(21*b^6*c^4 - 127*a*b^4*
c^5 + 183*a^2*b^2*c^6 - 44*a^3*c^7)*d^4*e^6 - 20*(6*b^7*c^3 - 42*a*b^5*c^4 + 83*a^2*b^3*c^5 - 44*a^3*b*c^6)*d^
3*e^7 + 5*(9*b^8*c^2 - 72*a*b^6*c^3 + 180*a^2*b^4*c^4 - 148*a^3*b^2*c^5 + 16*a^4*c^6)*d^2*e^8 - 10*(b^9*c - 9*
a*b^7*c^2 + 27*a^2*b^5*c^3 - 30*a^3*b^3*c^4 + 8*a^4*b*c^5)*d*e^9 + (b^10 - 10*a*b^8*c + 35*a^2*b^6*c^2 - 50*a^
3*b^4*c^3 + 25*a^4*b^2*c^4 - 4*a^5*c^5)*e^10)/(c^10*d^20 - 10*b*c^9*d^19*e + 5*(9*b^2*c^8 + 2*a*c^9)*d^18*e^2
- 30*(4*b^3*c^7 + 3*a*b*c^8)*d^17*e^3 + 15*(14*b^4*c^6 + 24*a*b^2*c^7 + 3*a^2*c^8)*d^16*e^4 - 12*(21*b^5*c^5 +
70*a*b^3*c^6 + 30*a^2*b*c^7)*d^15*e^5 + 30*(7*b^6*c^4 + 42*a*b^4*c^5 + 42*a^2*b^2*c^6 + 4*a^3*c^7)*d^14*e^6 -
60*(2*b^7*c^3 + 21*a*b^5*c^4 + 42*a^2*b^3*c^5 + 14*a^3*b*c^6)*d^13*e^7 + 15*(3*b^8*c^2 + 56*a*b^6*c^3 + 210*a
^2*b^4*c^4 + 168*a^3*b^2*c^5 + 14*a^4*c^6)*d^12*e^8 - 10*(b^9*c + 36*a*b^7*c^2 + 252*a^2*b^5*c^3 + 420*a^3*b^3
*c^4 + 126*a^4*b*c^5)*d^11*e^9 - 10*a^9*b*d*e^19 + (b^10 + 90*a*b^8*c + 1260*a^2*b^6*c^2 + 4200*a^3*b^4*c^3 +
3150*a^4*b^2*c^4 + 252*a^5*c^5)*d^10*e^10 + a^1...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(e*x+d)**(5/2)/(c*x**2+b*x+a),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1255 vs. \(2 (473) = 946\).
time = 3.93, size = 1255, normalized size = 2.42 \begin {gather*} -\frac {2 \, \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} c^{3} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} + 4 \, a c^{2} d^{3} e^{2} - b^{3} d^{2} e^{3} - 6 \, a b c d^{2} e^{3} + 2 \, a b^{2} d e^{4} + 2 \, a^{2} c d e^{4} - a^{2} b e^{5} + \sqrt {{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} + 4 \, a c^{2} d^{3} e^{2} - b^{3} d^{2} e^{3} - 6 \, a b c d^{2} e^{3} + 2 \, a b^{2} d e^{4} + 2 \, a^{2} c d e^{4} - a^{2} b e^{5}\right )}^{2} - 4 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} + 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} c d^{2} e^{4} - 3 \, a^{2} b d e^{5} + a^{3} e^{6}\right )} {\left (c^{3} d^{4} - 2 \, b c^{2} d^{3} e + b^{2} c d^{2} e^{2} + 2 \, a c^{2} d^{2} e^{2} - 2 \, a b c d e^{3} + a^{2} c e^{4}\right )}}}{c^{3} d^{4} - 2 \, b c^{2} d^{3} e + b^{2} c d^{2} e^{2} + 2 \, a c^{2} d^{2} e^{2} - 2 \, a b c d e^{3} + a^{2} c e^{4}}}}\right )}{{\left (2 \, c^{3} d^{3} - 3 \, {\left (b c^{2} - \sqrt {b^{2} - 4 \, a c} c^{2}\right )} d^{2} e + 3 \, {\left (b^{2} c - 2 \, a c^{2} - \sqrt {b^{2} - 4 \, a c} b c\right )} d e^{2} - {\left (b^{3} - 3 \, a b c - {\left (b^{2} - a c\right )} \sqrt {b^{2} - 4 \, a c}\right )} e^{3}\right )} {\left | c \right |}} - \frac {2 \, \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} c^{3} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} + 4 \, a c^{2} d^{3} e^{2} - b^{3} d^{2} e^{3} - 6 \, a b c d^{2} e^{3} + 2 \, a b^{2} d e^{4} + 2 \, a^{2} c d e^{4} - a^{2} b e^{5} - \sqrt {{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} + 4 \, a c^{2} d^{3} e^{2} - b^{3} d^{2} e^{3} - 6 \, a b c d^{2} e^{3} + 2 \, a b^{2} d e^{4} + 2 \, a^{2} c d e^{4} - a^{2} b e^{5}\right )}^{2} - 4 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} + 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} c d^{2} e^{4} - 3 \, a^{2} b d e^{5} + a^{3} e^{6}\right )} {\left (c^{3} d^{4} - 2 \, b c^{2} d^{3} e + b^{2} c d^{2} e^{2} + 2 \, a c^{2} d^{2} e^{2} - 2 \, a b c d e^{3} + a^{2} c e^{4}\right )}}}{c^{3} d^{4} - 2 \, b c^{2} d^{3} e + b^{2} c d^{2} e^{2} + 2 \, a c^{2} d^{2} e^{2} - 2 \, a b c d e^{3} + a^{2} c e^{4}}}}\right )}{{\left (2 \, c^{3} d^{3} - 3 \, {\left (b c^{2} + \sqrt {b^{2} - 4 \, a c} c^{2}\right )} d^{2} e + 3 \, {\left (b^{2} c - 2 \, a c^{2} + \sqrt {b^{2} - 4 \, a c} b c\right )} d e^{2} - {\left (b^{3} - 3 \, a b c + {\left (b^{2} - a c\right )} \sqrt {b^{2} - 4 \, a c}\right )} e^{3}\right )} {\left | c \right |}} + \frac {2 \, {\left (6 \, {\left (x e + d\right )} c^{2} d^{2} + 2 \, c^{2} d^{3} - 6 \, {\left (x e + d\right )} b c d e - 3 \, b c d^{2} e + 3 \, {\left (x e + d\right )} b^{2} e^{2} - 6 \, {\left (x e + d\right )} a c e^{2} + b^{2} d e^{2} + 2 \, a c d e^{2} - a b e^{3}\right )}}{3 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} {\left (x e + d\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(e*x+d)^(5/2)/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

-2*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*c^3*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*c^3*d^5 - 5*
b*c^2*d^4*e + 4*b^2*c*d^3*e^2 + 4*a*c^2*d^3*e^2 - b^3*d^2*e^3 - 6*a*b*c*d^2*e^3 + 2*a*b^2*d*e^4 + 2*a^2*c*d*e^
4 - a^2*b*e^5 + sqrt((2*c^3*d^5 - 5*b*c^2*d^4*e + 4*b^2*c*d^3*e^2 + 4*a*c^2*d^3*e^2 - b^3*d^2*e^3 - 6*a*b*c*d^
2*e^3 + 2*a*b^2*d*e^4 + 2*a^2*c*d*e^4 - a^2*b*e^5)^2 - 4*(c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 + 3*a*c^2*
d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e^4 - 3*a^2*b*d*e^5 + a^3*e^6)*(c^3*d^
4 - 2*b*c^2*d^3*e + b^2*c*d^2*e^2 + 2*a*c^2*d^2*e^2 - 2*a*b*c*d*e^3 + a^2*c*e^4)))/(c^3*d^4 - 2*b*c^2*d^3*e +
b^2*c*d^2*e^2 + 2*a*c^2*d^2*e^2 - 2*a*b*c*d*e^3 + a^2*c*e^4)))/((2*c^3*d^3 - 3*(b*c^2 - sqrt(b^2 - 4*a*c)*c^2)
*d^2*e + 3*(b^2*c - 2*a*c^2 - sqrt(b^2 - 4*a*c)*b*c)*d*e^2 - (b^3 - 3*a*b*c - (b^2 - a*c)*sqrt(b^2 - 4*a*c))*e
^3)*abs(c)) - 2*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*c^3*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2
*c^3*d^5 - 5*b*c^2*d^4*e + 4*b^2*c*d^3*e^2 + 4*a*c^2*d^3*e^2 - b^3*d^2*e^3 - 6*a*b*c*d^2*e^3 + 2*a*b^2*d*e^4 +
2*a^2*c*d*e^4 - a^2*b*e^5 - sqrt((2*c^3*d^5 - 5*b*c^2*d^4*e + 4*b^2*c*d^3*e^2 + 4*a*c^2*d^3*e^2 - b^3*d^2*e^3
- 6*a*b*c*d^2*e^3 + 2*a*b^2*d*e^4 + 2*a^2*c*d*e^4 - a^2*b*e^5)^2 - 4*(c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e
^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e^4 - 3*a^2*b*d*e^5 + a^3
*e^6)*(c^3*d^4 - 2*b*c^2*d^3*e + b^2*c*d^2*e^2 + 2*a*c^2*d^2*e^2 - 2*a*b*c*d*e^3 + a^2*c*e^4)))/(c^3*d^4 - 2*b
*c^2*d^3*e + b^2*c*d^2*e^2 + 2*a*c^2*d^2*e^2 - 2*a*b*c*d*e^3 + a^2*c*e^4)))/((2*c^3*d^3 - 3*(b*c^2 + sqrt(b^2
- 4*a*c)*c^2)*d^2*e + 3*(b^2*c - 2*a*c^2 + sqrt(b^2 - 4*a*c)*b*c)*d*e^2 - (b^3 - 3*a*b*c + (b^2 - a*c)*sqrt(b^
2 - 4*a*c))*e^3)*abs(c)) + 2/3*(6*(x*e + d)*c^2*d^2 + 2*c^2*d^3 - 6*(x*e + d)*b*c*d*e - 3*b*c*d^2*e + 3*(x*e +
d)*b^2*e^2 - 6*(x*e + d)*a*c*e^2 + b^2*d*e^2 + 2*a*c*d*e^2 - a*b*e^3)/((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 +
2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*(x*e + d)^(3/2))

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Mupad [B]
time = 11.38, size = 2500, normalized size = 4.83 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b + 2*c*x)/((d + e*x)^(5/2)*(a + b*x + c*x^2)),x)

[Out]

atan((((-(b^5*e^5 - 2*c^5*d^5 + b^4*e^5*(b^2 - 4*a*c)^(1/2) + 5*a^2*b*c^2*e^5 + 20*a*c^4*d^3*e^2 - 10*a^2*c^3*
d*e^4 + a^2*c^2*e^5*(b^2 - 4*a*c)^(1/2) - 10*b^2*c^3*d^3*e^2 + 10*b^3*c^2*d^2*e^3 - 5*a*b^3*c*e^5 + 5*b*c^4*d^
4*e - 5*b^4*c*d*e^4 + 5*c^4*d^4*e*(b^2 - 4*a*c)^(1/2) + 10*b^2*c^2*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b^2*c*e^5
*(b^2 - 4*a*c)^(1/2) - 5*b^3*c*d*e^4*(b^2 - 4*a*c)^(1/2) - 30*a*b*c^3*d^2*e^3 + 20*a*b^2*c^2*d*e^4 - 10*a*c^3*
d^2*e^3*(b^2 - 4*a*c)^(1/2) - 10*b*c^3*d^3*e^2*(b^2 - 4*a*c)^(1/2) + 10*a*b*c^2*d*e^4*(b^2 - 4*a*c)^(1/2))/(2*
(a^5*e^10 + c^5*d^10 - b^5*d^5*e^5 + 5*a*b^4*d^4*e^6 + 5*a*c^4*d^8*e^2 + 5*a^4*c*d^2*e^8 + 5*b^4*c*d^6*e^4 - 1
0*a^2*b^3*d^3*e^7 + 10*a^3*b^2*d^2*e^8 + 10*a^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6 + 10*b^2*c^3*d^8*e^2 - 10*b^3
*c^2*d^7*e^3 - 5*a^4*b*d*e^9 - 5*b*c^4*d^9*e - 20*a*b*c^3*d^7*e^3 - 20*a*b^3*c*d^5*e^5 - 20*a^3*b*c*d^3*e^7 +
30*a*b^2*c^2*d^6*e^4 - 30*a^2*b*c^2*d^5*e^5 + 30*a^2*b^2*c*d^4*e^6)))^(1/2)*((d + e*x)^(1/2)*(-(b^5*e^5 - 2*c^
5*d^5 + b^4*e^5*(b^2 - 4*a*c)^(1/2) + 5*a^2*b*c^2*e^5 + 20*a*c^4*d^3*e^2 - 10*a^2*c^3*d*e^4 + a^2*c^2*e^5*(b^2
- 4*a*c)^(1/2) - 10*b^2*c^3*d^3*e^2 + 10*b^3*c^2*d^2*e^3 - 5*a*b^3*c*e^5 + 5*b*c^4*d^4*e - 5*b^4*c*d*e^4 + 5*
c^4*d^4*e*(b^2 - 4*a*c)^(1/2) + 10*b^2*c^2*d^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b^2*c*e^5*(b^2 - 4*a*c)^(1/2) - 5
*b^3*c*d*e^4*(b^2 - 4*a*c)^(1/2) - 30*a*b*c^3*d^2*e^3 + 20*a*b^2*c^2*d*e^4 - 10*a*c^3*d^2*e^3*(b^2 - 4*a*c)^(1
/2) - 10*b*c^3*d^3*e^2*(b^2 - 4*a*c)^(1/2) + 10*a*b*c^2*d*e^4*(b^2 - 4*a*c)^(1/2))/(2*(a^5*e^10 + c^5*d^10 - b
^5*d^5*e^5 + 5*a*b^4*d^4*e^6 + 5*a*c^4*d^8*e^2 + 5*a^4*c*d^2*e^8 + 5*b^4*c*d^6*e^4 - 10*a^2*b^3*d^3*e^7 + 10*a
^3*b^2*d^2*e^8 + 10*a^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6 + 10*b^2*c^3*d^8*e^2 - 10*b^3*c^2*d^7*e^3 - 5*a^4*b*d
*e^9 - 5*b*c^4*d^9*e - 20*a*b*c^3*d^7*e^3 - 20*a*b^3*c*d^5*e^5 - 20*a^3*b*c*d^3*e^7 + 30*a*b^2*c^2*d^6*e^4 - 3
0*a^2*b*c^2*d^5*e^5 + 30*a^2*b^2*c*d^4*e^6)))^(1/2)*(64*a*c^14*d^21*e^2 - 32*a^11*b*c^3*e^23 + 64*a^11*c^4*d*e
^22 + 8*a^10*b^3*c^2*e^23 + 640*a^2*c^13*d^19*e^4 + 2880*a^3*c^12*d^17*e^6 + 7680*a^4*c^11*d^15*e^8 + 13440*a^
5*c^10*d^13*e^10 + 16128*a^6*c^9*d^11*e^12 + 13440*a^7*c^8*d^9*e^14 + 7680*a^8*c^7*d^7*e^16 + 2880*a^9*c^6*d^5
*e^18 + 640*a^10*c^5*d^3*e^20 - 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 + 25200*a^
2*b^2*c^11*d^17*e^6 - 59160*a^2*b^3*c^10*d^16*e^7 + 84480*a^2*b^4*c^9*d^15*e^8 - 70560*a^2*b^5*c^8*d^14*e^9 +
23520*a^2*b^6*c^7*d^13*e^10 + 15600*a^2*b^7*c^6*d^12*e^11 - 23040*a^2*b^8*c^5*d^11*e^12 + 12320*a^2*b^9*c^4*d^
10*e^13 - 3280*a^2*b^10*c^3*d^9*e^14 + 360*a^2*b^11*c^2*d^8*e^15 + 90240*a^3*b^2*c^10*d^15*e^8 - 187200*a^3*b^
3*c^9*d^14*e^9 + 235200*a^3*b^4*c^8*d^13*e^10 - 174720*a^3*b^5*c^7*d^12*e^11 + 60480*a^3*b^6*c^6*d^11*e^12 + 1
0560*a^3*b^7*c^5*d^10*e^13 - 19200*a^3*b^8*c^4*d^9*e^14 + 7200*a^3*b^9*c^3*d^8*e^15 - 960*a^3*b^10*c^2*d^7*e^1
6 + 184800*a^4*b^2*c^9*d^13*e^10 - 327600*a^4*b^3*c^8*d^12*e^11 + 342720*a^4*b^4*c^7*d^11*e^12 - 203280*a^4*b^
5*c^6*d^10*e^13 + 50400*a^4*b^6*c^5*d^9*e^14 + 10800*a^4*b^7*c^4*d^8*e^15 - 9600*a^4*b^8*c^3*d^7*e^16 + 1680*a
^4*b^9*c^2*d^6*e^17 + 237888*a^5*b^2*c^8*d^11*e^12 - 347424*a^5*b^3*c^7*d^10*e^13 + 285600*a^5*b^4*c^6*d^9*e^1
4 - 120960*a^5*b^5*c^5*d^8*e^15 + 13440*a^5*b^6*c^4*d^7*e^16 + 7392*a^5*b^7*c^3*d^6*e^17 - 2016*a^5*b^8*c^2*d^
5*e^18 + 198240*a^6*b^2*c^7*d^9*e^14 - 226800*a^6*b^3*c^6*d^8*e^15 + 134400*a^6*b^4*c^5*d^7*e^16 - 32928*a^6*b
^5*c^4*d^6*e^17 - 2016*a^6*b^6*c^3*d^5*e^18 + 1680*a^6*b^7*c^2*d^4*e^19 + 105600*a^7*b^2*c^6*d^7*e^16 - 87360*
a^7*b^3*c^5*d^6*e^17 + 31680*a^7*b^4*c^4*d^5*e^18 - 1920*a^7*b^5*c^3*d^4*e^19 - 960*a^7*b^6*c^2*d^3*e^20 + 338
40*a^8*b^2*c^5*d^5*e^18 - 17400*a^8*b^3*c^4*d^4*e^19 + 2400*a^8*b^4*c^3*d^3*e^20 + 360*a^8*b^5*c^2*d^2*e^21 +
5600*a^9*b^2*c^4*d^3*e^20 - 1200*a^9*b^3*c^3*d^2*e^21 - 672*a*b*c^13*d^20*e^3 + 3040*a*b^2*c^12*d^19*e^4 - 760
0*a*b^3*c^11*d^18*e^5 + 10800*a*b^4*c^10*d^17*e^6 - 6528*a*b^5*c^9*d^16*e^7 - 5376*a*b^6*c^8*d^15*e^8 + 15840*
a*b^7*c^7*d^14*e^9 - 16800*a*b^8*c^6*d^13*e^10 + 10400*a*b^9*c^5*d^12*e^11 - 3936*a*b^10*c^4*d^11*e^12 + 848*a
*b^11*c^3*d^10*e^13 - 80*a*b^12*c^2*d^9*e^14 - 6080*a^2*b*c^12*d^18*e^5 - 24480*a^3*b*c^11*d^16*e^7 - 57600*a^
4*b*c^10*d^14*e^9 - 87360*a^5*b*c^9*d^12*e^11 - 88704*a^6*b*c^8*d^10*e^13 - 60480*a^7*b*c^7*d^8*e^15 - 26880*a
^8*b*c^6*d^6*e^17 - 7200*a^9*b*c^5*d^4*e^19 - 80*a^9*b^4*c^2*d*e^22 - 960*a^10*b*c^4*d^2*e^21 + 304*a^10*b^2*c
^3*d*e^22) - 96*a^10*b*c^4*e^21 - 64*a*c^14*d^19*e^2 + 192*a^10*c^5*d*e^20 - 8*a^8*b^5*c^2*e^21 + 56*a^9*b^3*c
^3*e^21 - 320*a^2*c^13*d^17*e^4 - 256*a^3*c^12*d^15*e^6 + 1792*a^4*c^11*d^13*e^8 + 6272*a^5*c^10*d^11*e^10 + 9
856*a^6*c^9*d^9*e^12 + 8960*a^7*c^8*d^7*e^14 + 4864*a^8*c^7*d^5*e^16 + 1472*a^9*c^6*d^3*e^18 + 16*b^2*c^13*d^1
9*e^2 - 152*b^3*c^12*d^18*e^3 + 664*b^4*c^11*d^...

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