3.4.60 \(\int \frac {\sqrt {d+e x}}{(b x+c x^2)^3} \, dx\)

Optimal. Leaf size=245 \[ -\frac {(b+2 c x) \sqrt {d+e x}}{2 b^2 \left (b x+c x^2\right )^2}-\frac {\left (-b^2 e^2-12 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{3/2}}+\frac {c^{3/2} \left (35 b^2 e^2-84 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{3/2}}+\frac {\sqrt {d+e x} \left (c x \left (b^2 e^2-24 b c d e+24 c^2 d^2\right )+b (c d-b e) (12 c d-b e)\right )}{4 b^4 d \left (b x+c x^2\right ) (c d-b e)} \]

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Rubi [A]  time = 0.38, antiderivative size = 245, 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 = {736, 822, 826, 1166, 208} \begin {gather*} \frac {\sqrt {d+e x} \left (c x \left (b^2 e^2-24 b c d e+24 c^2 d^2\right )+b (c d-b e) (12 c d-b e)\right )}{4 b^4 d \left (b x+c x^2\right ) (c d-b e)}+\frac {c^{3/2} \left (35 b^2 e^2-84 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{3/2}}-\frac {\left (-b^2 e^2-12 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{3/2}}-\frac {(b+2 c x) \sqrt {d+e x}}{2 b^2 \left (b x+c x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + e*x]/(b*x + c*x^2)^3,x]

[Out]

-((b + 2*c*x)*Sqrt[d + e*x])/(2*b^2*(b*x + c*x^2)^2) + (Sqrt[d + e*x]*(b*(c*d - b*e)*(12*c*d - b*e) + c*(24*c^
2*d^2 - 24*b*c*d*e + b^2*e^2)*x))/(4*b^4*d*(c*d - b*e)*(b*x + c*x^2)) - ((48*c^2*d^2 - 12*b*c*d*e - b^2*e^2)*A
rcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5*d^(3/2)) + (c^(3/2)*(48*c^2*d^2 - 84*b*c*d*e + 35*b^2*e^2)*ArcTanh[(Sqrt
[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*(c*d - b*e)^(3/2))

Rule 208

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

Rule 736

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^m*(b + 2*
c*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m
- 1)*(b*e*m + 2*c*d*(2*p + 3) + 2*c*e*(m + 2*p + 3)*x)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d
, e}, 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] && GtQ[m
, 0] && (LtQ[m, 1] || (ILtQ[m + 2*p + 3, 0] && NeQ[m, 2])) && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 826

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 1166

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 {\sqrt {d+e x}}{\left (b x+c x^2\right )^3} \, dx &=-\frac {(b+2 c x) \sqrt {d+e x}}{2 b^2 \left (b x+c x^2\right )^2}+\frac {\int \frac {-6 c d+\frac {b e}{2}-5 c e x}{\sqrt {d+e x} \left (b x+c x^2\right )^2} \, dx}{2 b^2}\\ &=-\frac {(b+2 c x) \sqrt {d+e x}}{2 b^2 \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b (c d-b e) (12 c d-b e)+c \left (24 c^2 d^2-24 b c d e+b^2 e^2\right ) x\right )}{4 b^4 d (c d-b e) \left (b x+c x^2\right )}-\frac {\int \frac {-\frac {1}{4} (c d-b e) \left (48 c^2 d^2-12 b c d e-b^2 e^2\right )-\frac {1}{4} c e \left (24 c^2 d^2-24 b c d e+b^2 e^2\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 d (c d-b e)}\\ &=-\frac {(b+2 c x) \sqrt {d+e x}}{2 b^2 \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b (c d-b e) (12 c d-b e)+c \left (24 c^2 d^2-24 b c d e+b^2 e^2\right ) x\right )}{4 b^4 d (c d-b e) \left (b x+c x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {-\frac {1}{4} e (c d-b e) \left (48 c^2 d^2-12 b c d e-b^2 e^2\right )+\frac {1}{4} c d e \left (24 c^2 d^2-24 b c d e+b^2 e^2\right )-\frac {1}{4} c e \left (24 c^2 d^2-24 b c d e+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^4 d (c d-b e)}\\ &=-\frac {(b+2 c x) \sqrt {d+e x}}{2 b^2 \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b (c d-b e) (12 c d-b e)+c \left (24 c^2 d^2-24 b c d e+b^2 e^2\right ) x\right )}{4 b^4 d (c d-b e) \left (b x+c x^2\right )}+\frac {\left (c \left (48 c^2 d^2-12 b c d e-b^2 e^2\right )\right ) \operatorname {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 )}{4 b^5 d}-\frac {\left (c^2 \left (48 c^2 d^2-84 b c d e+35 b^2 e^2\right )\right ) \operatorname {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 )}{4 b^5 (c d-b e)}\\ &=-\frac {(b+2 c x) \sqrt {d+e x}}{2 b^2 \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b (c d-b e) (12 c d-b e)+c \left (24 c^2 d^2-24 b c d e+b^2 e^2\right ) x\right )}{4 b^4 d (c d-b e) \left (b x+c x^2\right )}-\frac {\left (48 c^2 d^2-12 b c d e-b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{3/2}}+\frac {c^{3/2} \left (48 c^2 d^2-84 b c d e+35 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.96, size = 361, normalized size = 1.47 \begin {gather*} \frac {-\frac {2 c (d+e x)^{3/2} \left (-b^2 e^2-9 b c d e+12 c^2 d^2\right )}{b^2 d (b e-c d)}+\frac {(b+c x) \left ((b+c x) \left (2 c^{3/2} (c d-b e)^2 \left (-b^2 e^2-12 b c d e+48 c^2 d^2\right ) \left (\sqrt {d+e x}-\sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )-2 c^3 d^2 \left (35 b^2 e^2-84 b c d e+48 c^2 d^2\right ) \left (\sqrt {c} \sqrt {d+e x}-\sqrt {c d-b e} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )\right )\right )+2 b c^{5/2} (d+e x)^{3/2} \left (b^3 e^3+10 b^2 c d e^2-36 b c^2 d^2 e+24 c^3 d^3\right )\right )}{b^4 c^{3/2} d (c d-b e)^2}+\frac {2 (d+e x)^{3/2} (b e+8 c d)}{b d x}-\frac {4 (d+e x)^{3/2}}{x^2}}{8 b d (b+c x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d + e*x]/(b*x + c*x^2)^3,x]

[Out]

((-2*c*(12*c^2*d^2 - 9*b*c*d*e - b^2*e^2)*(d + e*x)^(3/2))/(b^2*d*(-(c*d) + b*e)) - (4*(d + e*x)^(3/2))/x^2 +
(2*(8*c*d + b*e)*(d + e*x)^(3/2))/(b*d*x) + ((b + c*x)*(2*b*c^(5/2)*(24*c^3*d^3 - 36*b*c^2*d^2*e + 10*b^2*c*d*
e^2 + b^3*e^3)*(d + e*x)^(3/2) + (b + c*x)*(2*c^(3/2)*(c*d - b*e)^2*(48*c^2*d^2 - 12*b*c*d*e - b^2*e^2)*(Sqrt[
d + e*x] - Sqrt[d]*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]) - 2*c^3*d^2*(48*c^2*d^2 - 84*b*c*d*e + 35*b^2*e^2)*(Sqrt[c]
*Sqrt[d + e*x] - Sqrt[c*d - b*e]*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]]))))/(b^4*c^(3/2)*d*(c*d - b*
e)^2))/(8*b*d*(b + c*x)^2)

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IntegrateAlgebraic [A]  time = 1.79, size = 435, normalized size = 1.78 \begin {gather*} \frac {\left (-35 b^2 c^{3/2} e^2+84 b c^{5/2} d e-48 c^{7/2} d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x} \sqrt {b e-c d}}{c d-b e}\right )}{4 b^5 (b e-c d)^{3/2}}+\frac {\left (b^2 e^2+12 b c d e-48 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{3/2}}-\frac {\sqrt {d+e x} \left (b^4 e^4 (d+e x)+b^4 d e^4+9 b^3 c d^2 e^3-13 b^3 c d e^3 (d+e x)+2 b^3 c e^3 (d+e x)^2-46 b^2 c^2 d^3 e^2+85 b^2 c^2 d^2 e^2 (d+e x)-40 b^2 c^2 d e^2 (d+e x)^2+b^2 c^2 e^2 (d+e x)^3+60 b c^3 d^4 e-144 b c^3 d^3 e (d+e x)+108 b c^3 d^2 e (d+e x)^2-24 b c^3 d e (d+e x)^3-24 c^4 d^5+72 c^4 d^4 (d+e x)-72 c^4 d^3 (d+e x)^2+24 c^4 d^2 (d+e x)^3\right )}{4 b^4 d e x^2 (b e-c d) (b e+c (d+e x)-c d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[d + e*x]/(b*x + c*x^2)^3,x]

[Out]

-1/4*(Sqrt[d + e*x]*(-24*c^4*d^5 + 60*b*c^3*d^4*e - 46*b^2*c^2*d^3*e^2 + 9*b^3*c*d^2*e^3 + b^4*d*e^4 + 72*c^4*
d^4*(d + e*x) - 144*b*c^3*d^3*e*(d + e*x) + 85*b^2*c^2*d^2*e^2*(d + e*x) - 13*b^3*c*d*e^3*(d + e*x) + b^4*e^4*
(d + e*x) - 72*c^4*d^3*(d + e*x)^2 + 108*b*c^3*d^2*e*(d + e*x)^2 - 40*b^2*c^2*d*e^2*(d + e*x)^2 + 2*b^3*c*e^3*
(d + e*x)^2 + 24*c^4*d^2*(d + e*x)^3 - 24*b*c^3*d*e*(d + e*x)^3 + b^2*c^2*e^2*(d + e*x)^3))/(b^4*d*e*(-(c*d) +
b*e)*x^2*(-(c*d) + b*e + c*(d + e*x))^2) + ((-48*c^(7/2)*d^2 + 84*b*c^(5/2)*d*e - 35*b^2*c^(3/2)*e^2)*ArcTan[
(Sqrt[c]*Sqrt[-(c*d) + b*e]*Sqrt[d + e*x])/(c*d - b*e)])/(4*b^5*(-(c*d) + b*e)^(3/2)) + ((-48*c^2*d^2 + 12*b*c
*d*e + b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5*d^(3/2))

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fricas [B]  time = 0.75, size = 2265, normalized size = 9.24

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

[-1/8*(((48*c^5*d^4 - 84*b*c^4*d^3*e + 35*b^2*c^3*d^2*e^2)*x^4 + 2*(48*b*c^4*d^4 - 84*b^2*c^3*d^3*e + 35*b^3*c
^2*d^2*e^2)*x^3 + (48*b^2*c^3*d^4 - 84*b^3*c^2*d^3*e + 35*b^4*c*d^2*e^2)*x^2)*sqrt(c/(c*d - b*e))*log((c*e*x +
2*c*d - b*e - 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x + b)) + ((48*c^5*d^3 - 60*b*c^4*d^2*e + 1
1*b^2*c^3*d*e^2 + b^3*c^2*e^3)*x^4 + 2*(48*b*c^4*d^3 - 60*b^2*c^3*d^2*e + 11*b^3*c^2*d*e^2 + b^4*c*e^3)*x^3 +
(48*b^2*c^3*d^3 - 60*b^3*c^2*d^2*e + 11*b^4*c*d*e^2 + b^5*e^3)*x^2)*sqrt(d)*log((e*x + 2*sqrt(e*x + d)*sqrt(d)
+ 2*d)/x) + 2*(2*b^4*c*d^3 - 2*b^5*d^2*e - (24*b*c^4*d^3 - 24*b^2*c^3*d^2*e + b^3*c^2*d*e^2)*x^3 - (36*b^2*c^
3*d^3 - 37*b^3*c^2*d^2*e + 2*b^4*c*d*e^2)*x^2 - (8*b^3*c^2*d^3 - 9*b^4*c*d^2*e + b^5*d*e^2)*x)*sqrt(e*x + d))/
((b^5*c^3*d^3 - b^6*c^2*d^2*e)*x^4 + 2*(b^6*c^2*d^3 - b^7*c*d^2*e)*x^3 + (b^7*c*d^3 - b^8*d^2*e)*x^2), 1/8*(2*
((48*c^5*d^4 - 84*b*c^4*d^3*e + 35*b^2*c^3*d^2*e^2)*x^4 + 2*(48*b*c^4*d^4 - 84*b^2*c^3*d^3*e + 35*b^3*c^2*d^2*
e^2)*x^3 + (48*b^2*c^3*d^4 - 84*b^3*c^2*d^3*e + 35*b^4*c*d^2*e^2)*x^2)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e
)*sqrt(e*x + d)*sqrt(-c/(c*d - b*e))/(c*e*x + c*d)) - ((48*c^5*d^3 - 60*b*c^4*d^2*e + 11*b^2*c^3*d*e^2 + b^3*c
^2*e^3)*x^4 + 2*(48*b*c^4*d^3 - 60*b^2*c^3*d^2*e + 11*b^3*c^2*d*e^2 + b^4*c*e^3)*x^3 + (48*b^2*c^3*d^3 - 60*b^
3*c^2*d^2*e + 11*b^4*c*d*e^2 + b^5*e^3)*x^2)*sqrt(d)*log((e*x + 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*b^4*c
*d^3 - 2*b^5*d^2*e - (24*b*c^4*d^3 - 24*b^2*c^3*d^2*e + b^3*c^2*d*e^2)*x^3 - (36*b^2*c^3*d^3 - 37*b^3*c^2*d^2*
e + 2*b^4*c*d*e^2)*x^2 - (8*b^3*c^2*d^3 - 9*b^4*c*d^2*e + b^5*d*e^2)*x)*sqrt(e*x + d))/((b^5*c^3*d^3 - b^6*c^2
*d^2*e)*x^4 + 2*(b^6*c^2*d^3 - b^7*c*d^2*e)*x^3 + (b^7*c*d^3 - b^8*d^2*e)*x^2), 1/8*(2*((48*c^5*d^3 - 60*b*c^4
*d^2*e + 11*b^2*c^3*d*e^2 + b^3*c^2*e^3)*x^4 + 2*(48*b*c^4*d^3 - 60*b^2*c^3*d^2*e + 11*b^3*c^2*d*e^2 + b^4*c*e
^3)*x^3 + (48*b^2*c^3*d^3 - 60*b^3*c^2*d^2*e + 11*b^4*c*d*e^2 + b^5*e^3)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)*sq
rt(-d)/d) - ((48*c^5*d^4 - 84*b*c^4*d^3*e + 35*b^2*c^3*d^2*e^2)*x^4 + 2*(48*b*c^4*d^4 - 84*b^2*c^3*d^3*e + 35*
b^3*c^2*d^2*e^2)*x^3 + (48*b^2*c^3*d^4 - 84*b^3*c^2*d^3*e + 35*b^4*c*d^2*e^2)*x^2)*sqrt(c/(c*d - b*e))*log((c*
e*x + 2*c*d - b*e - 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x + b)) - 2*(2*b^4*c*d^3 - 2*b^5*d^2*e
- (24*b*c^4*d^3 - 24*b^2*c^3*d^2*e + b^3*c^2*d*e^2)*x^3 - (36*b^2*c^3*d^3 - 37*b^3*c^2*d^2*e + 2*b^4*c*d*e^2)
*x^2 - (8*b^3*c^2*d^3 - 9*b^4*c*d^2*e + b^5*d*e^2)*x)*sqrt(e*x + d))/((b^5*c^3*d^3 - b^6*c^2*d^2*e)*x^4 + 2*(b
^6*c^2*d^3 - b^7*c*d^2*e)*x^3 + (b^7*c*d^3 - b^8*d^2*e)*x^2), 1/4*(((48*c^5*d^4 - 84*b*c^4*d^3*e + 35*b^2*c^3*
d^2*e^2)*x^4 + 2*(48*b*c^4*d^4 - 84*b^2*c^3*d^3*e + 35*b^3*c^2*d^2*e^2)*x^3 + (48*b^2*c^3*d^4 - 84*b^3*c^2*d^3
*e + 35*b^4*c*d^2*e^2)*x^2)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(e*x + d)*sqrt(-c/(c*d - b*e))/(c*e*x
+ c*d)) + ((48*c^5*d^3 - 60*b*c^4*d^2*e + 11*b^2*c^3*d*e^2 + b^3*c^2*e^3)*x^4 + 2*(48*b*c^4*d^3 - 60*b^2*c^3*
d^2*e + 11*b^3*c^2*d*e^2 + b^4*c*e^3)*x^3 + (48*b^2*c^3*d^3 - 60*b^3*c^2*d^2*e + 11*b^4*c*d*e^2 + b^5*e^3)*x^2
)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) - (2*b^4*c*d^3 - 2*b^5*d^2*e - (24*b*c^4*d^3 - 24*b^2*c^3*d^2*e +
b^3*c^2*d*e^2)*x^3 - (36*b^2*c^3*d^3 - 37*b^3*c^2*d^2*e + 2*b^4*c*d*e^2)*x^2 - (8*b^3*c^2*d^3 - 9*b^4*c*d^2*e
+ b^5*d*e^2)*x)*sqrt(e*x + d))/((b^5*c^3*d^3 - b^6*c^2*d^2*e)*x^4 + 2*(b^6*c^2*d^3 - b^7*c*d^2*e)*x^3 + (b^7*c
*d^3 - b^8*d^2*e)*x^2)]

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giac [B]  time = 0.25, size = 508, normalized size = 2.07 \begin {gather*} -\frac {{\left (48 \, c^{4} d^{2} - 84 \, b c^{3} d e + 35 \, b^{2} c^{2} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{4 \, {\left (b^{5} c d - b^{6} e\right )} \sqrt {-c^{2} d + b c e}} + \frac {24 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{4} d^{2} e - 72 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{4} d^{3} e + 72 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{4} d^{4} e - 24 \, \sqrt {x e + d} c^{4} d^{5} e - 24 \, {\left (x e + d\right )}^{\frac {7}{2}} b c^{3} d e^{2} + 108 \, {\left (x e + d\right )}^{\frac {5}{2}} b c^{3} d^{2} e^{2} - 144 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{3} d^{3} e^{2} + 60 \, \sqrt {x e + d} b c^{3} d^{4} e^{2} + {\left (x e + d\right )}^{\frac {7}{2}} b^{2} c^{2} e^{3} - 40 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{2} c^{2} d e^{3} + 85 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c^{2} d^{2} e^{3} - 46 \, \sqrt {x e + d} b^{2} c^{2} d^{3} e^{3} + 2 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{3} c e^{4} - 13 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} c d e^{4} + 9 \, \sqrt {x e + d} b^{3} c d^{2} e^{4} + {\left (x e + d\right )}^{\frac {3}{2}} b^{4} e^{5} + \sqrt {x e + d} b^{4} d e^{5}}{4 \, {\left (b^{4} c d^{2} - b^{5} d e\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 )}^{2}} + \frac {{\left (48 \, c^{2} d^{2} - 12 \, b c d e - b^{2} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{4 \, b^{5} \sqrt {-d} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

-1/4*(48*c^4*d^2 - 84*b*c^3*d*e + 35*b^2*c^2*e^2)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b^5*c*d - b^6
*e)*sqrt(-c^2*d + b*c*e)) + 1/4*(24*(x*e + d)^(7/2)*c^4*d^2*e - 72*(x*e + d)^(5/2)*c^4*d^3*e + 72*(x*e + d)^(3
/2)*c^4*d^4*e - 24*sqrt(x*e + d)*c^4*d^5*e - 24*(x*e + d)^(7/2)*b*c^3*d*e^2 + 108*(x*e + d)^(5/2)*b*c^3*d^2*e^
2 - 144*(x*e + d)^(3/2)*b*c^3*d^3*e^2 + 60*sqrt(x*e + d)*b*c^3*d^4*e^2 + (x*e + d)^(7/2)*b^2*c^2*e^3 - 40*(x*e
+ d)^(5/2)*b^2*c^2*d*e^3 + 85*(x*e + d)^(3/2)*b^2*c^2*d^2*e^3 - 46*sqrt(x*e + d)*b^2*c^2*d^3*e^3 + 2*(x*e + d
)^(5/2)*b^3*c*e^4 - 13*(x*e + d)^(3/2)*b^3*c*d*e^4 + 9*sqrt(x*e + d)*b^3*c*d^2*e^4 + (x*e + d)^(3/2)*b^4*e^5 +
sqrt(x*e + d)*b^4*d*e^5)/((b^4*c*d^2 - b^5*d*e)*((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*
d*e)^2) + 1/4*(48*c^2*d^2 - 12*b*c*d*e - b^2*e^2)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^5*sqrt(-d)*d)

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maple [B]  time = 0.21, size = 436, normalized size = 1.78 \begin {gather*} \frac {11 \left (e x +d \right )^{\frac {3}{2}} c^{3} e^{2}}{4 \left (c e x +b e \right )^{2} \left (b e -c d \right ) b^{3}}+\frac {35 c^{2} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{4 \left (b e -c d \right ) \sqrt {\left (b e -c d \right ) c}\, b^{3}}-\frac {3 \left (e x +d \right )^{\frac {3}{2}} c^{4} d e}{\left (c e x +b e \right )^{2} \left (b e -c d \right ) b^{4}}-\frac {21 c^{3} d e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right ) \sqrt {\left (b e -c d \right ) c}\, b^{4}}+\frac {12 c^{4} d^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right ) \sqrt {\left (b e -c d \right ) c}\, b^{5}}+\frac {13 \sqrt {e x +d}\, c^{2} e^{2}}{4 \left (c e x +b e \right )^{2} b^{3}}-\frac {3 \sqrt {e x +d}\, c^{3} d e}{\left (c e x +b e \right )^{2} b^{4}}+\frac {e^{2} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{4 b^{3} d^{\frac {3}{2}}}+\frac {3 c e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{4} \sqrt {d}}-\frac {12 c^{2} \sqrt {d}\, \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{5}}-\frac {\sqrt {e x +d}}{4 b^{3} x^{2}}-\frac {3 \sqrt {e x +d}\, c d}{b^{4} e \,x^{2}}-\frac {\left (e x +d \right )^{\frac {3}{2}}}{4 b^{3} d \,x^{2}}+\frac {3 \left (e x +d \right )^{\frac {3}{2}} c}{b^{4} e \,x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(1/2)/(c*x^2+b*x)^3,x)

[Out]

11/4*e^2/b^3*c^3/(c*e*x+b*e)^2/(b*e-c*d)*(e*x+d)^(3/2)-3*e/b^4*c^4/(c*e*x+b*e)^2/(b*e-c*d)*(e*x+d)^(3/2)*d+13/
4*e^2/b^3*c^2/(c*e*x+b*e)^2*(e*x+d)^(1/2)-3*e/b^4*c^3/(c*e*x+b*e)^2*(e*x+d)^(1/2)*d+35/4*e^2/b^3*c^2/(b*e-c*d)
/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)-21*e/b^4*c^3/(b*e-c*d)/((b*e-c*d)*c)^(1/2)*ar
ctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*d+12/b^5*c^4/(b*e-c*d)/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*
e-c*d)*c)^(1/2)*c)*d^2-1/4/b^3/x^2/d*(e*x+d)^(3/2)+3/e/b^4/x^2*(e*x+d)^(3/2)*c-3/e/b^4/x^2*(e*x+d)^(1/2)*c*d-1
/4/b^3/x^2*(e*x+d)^(1/2)+1/4*e^2/b^3/d^(3/2)*arctanh((e*x+d)^(1/2)/d^(1/2))+3*e/b^4/d^(1/2)*arctanh((e*x+d)^(1
/2)/d^(1/2))*c-12/b^5*d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*c^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((e*x+d)^(1/2)/(c*x^2+b*x)^3,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(b*e-c*d>0)', see `assume?` for
more details)Is b*e-c*d positive or negative?

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mupad [B]  time = 2.01, size = 4815, normalized size = 19.65

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(1/2)/(b*x + c*x^2)^3,x)

[Out]

(((d + e*x)^(3/2)*(b^4*e^5 + 72*c^4*d^4*e - 144*b*c^3*d^3*e^2 + 85*b^2*c^2*d^2*e^3 - 13*b^3*c*d*e^4))/(4*b^4*(
c*d^2 - b*d*e)) - ((d + e*x)^(1/2)*(b^3*e^4 + 24*c^3*d^3*e - 36*b*c^2*d^2*e^2 + 10*b^2*c*d*e^3))/(4*b^4) + ((b
*e - 2*c*d)*(d + e*x)^(5/2)*(b^2*c*e^3 + 18*c^3*d^2*e - 18*b*c^2*d*e^2))/(2*b^4*(c*d^2 - b*d*e)) + (c*e*(d + e
*x)^(7/2)*(24*c^3*d^2 + b^2*c*e^2 - 24*b*c^2*d*e))/(4*b^4*(c*d^2 - b*d*e)))/(c^2*(d + e*x)^4 - (d + e*x)*(4*c^
2*d^3 + 2*b^2*d*e^2 - 6*b*c*d^2*e) - (4*c^2*d - 2*b*c*e)*(d + e*x)^3 + (d + e*x)^2*(b^2*e^2 + 6*c^2*d^2 - 6*b*
c*d*e) + c^2*d^4 + b^2*d^2*e^2 - 2*b*c*d^3*e) - (atan((((((d + e*x)^(1/2)*(b^6*c^3*e^8 + 4608*c^9*d^6*e^2 - 13
824*b*c^8*d^5*e^3 + 22*b^5*c^4*d*e^7 + 15072*b^2*c^7*d^4*e^4 - 7104*b^3*c^6*d^3*e^5 + 1226*b^4*c^5*d^2*e^6))/(
8*(b^8*c^2*d^4 + b^10*d^2*e^2 - 2*b^9*c*d^3*e)) + (((b^14*c^2*d*e^7 - 24*b^10*c^6*d^5*e^3 + 60*b^11*c^5*d^4*e^
4 - 46*b^12*c^4*d^3*e^5 + 9*b^13*c^3*d^2*e^6)/(b^12*c^2*d^4 + b^14*d^2*e^2 - 2*b^13*c*d^3*e) - ((d + e*x)^(1/2
)*(b^2*e^2 - 48*c^2*d^2 + 12*b*c*d*e)*(128*b^10*c^5*d^5*e^2 - 320*b^11*c^4*d^4*e^3 + 256*b^12*c^3*d^3*e^4 - 64
*b^13*c^2*d^2*e^5))/(64*b^5*(d^3)^(1/2)*(b^8*c^2*d^4 + b^10*d^2*e^2 - 2*b^9*c*d^3*e)))*(b^2*e^2 - 48*c^2*d^2 +
12*b*c*d*e))/(8*b^5*(d^3)^(1/2)))*(b^2*e^2 - 48*c^2*d^2 + 12*b*c*d*e)*1i)/(8*b^5*(d^3)^(1/2)) + ((((d + e*x)^
(1/2)*(b^6*c^3*e^8 + 4608*c^9*d^6*e^2 - 13824*b*c^8*d^5*e^3 + 22*b^5*c^4*d*e^7 + 15072*b^2*c^7*d^4*e^4 - 7104*
b^3*c^6*d^3*e^5 + 1226*b^4*c^5*d^2*e^6))/(8*(b^8*c^2*d^4 + b^10*d^2*e^2 - 2*b^9*c*d^3*e)) - (((b^14*c^2*d*e^7
- 24*b^10*c^6*d^5*e^3 + 60*b^11*c^5*d^4*e^4 - 46*b^12*c^4*d^3*e^5 + 9*b^13*c^3*d^2*e^6)/(b^12*c^2*d^4 + b^14*d
^2*e^2 - 2*b^13*c*d^3*e) + ((d + e*x)^(1/2)*(b^2*e^2 - 48*c^2*d^2 + 12*b*c*d*e)*(128*b^10*c^5*d^5*e^2 - 320*b^
11*c^4*d^4*e^3 + 256*b^12*c^3*d^3*e^4 - 64*b^13*c^2*d^2*e^5))/(64*b^5*(d^3)^(1/2)*(b^8*c^2*d^4 + b^10*d^2*e^2
- 2*b^9*c*d^3*e)))*(b^2*e^2 - 48*c^2*d^2 + 12*b*c*d*e))/(8*b^5*(d^3)^(1/2)))*(b^2*e^2 - 48*c^2*d^2 + 12*b*c*d*
e)*1i)/(8*b^5*(d^3)^(1/2)))/((1728*c^10*d^6*e^3 - (35*b^6*c^4*e^9)/32 - 5184*b*c^9*d^5*e^4 + (63*b^5*c^5*d*e^8
)/4 + 5508*b^2*c^8*d^4*e^5 - 2376*b^3*c^7*d^3*e^6 + (1233*b^4*c^6*d^2*e^7)/4)/(b^12*c^2*d^4 + b^14*d^2*e^2 - 2
*b^13*c*d^3*e) + ((((d + e*x)^(1/2)*(b^6*c^3*e^8 + 4608*c^9*d^6*e^2 - 13824*b*c^8*d^5*e^3 + 22*b^5*c^4*d*e^7 +
15072*b^2*c^7*d^4*e^4 - 7104*b^3*c^6*d^3*e^5 + 1226*b^4*c^5*d^2*e^6))/(8*(b^8*c^2*d^4 + b^10*d^2*e^2 - 2*b^9*
c*d^3*e)) + (((b^14*c^2*d*e^7 - 24*b^10*c^6*d^5*e^3 + 60*b^11*c^5*d^4*e^4 - 46*b^12*c^4*d^3*e^5 + 9*b^13*c^3*d
^2*e^6)/(b^12*c^2*d^4 + b^14*d^2*e^2 - 2*b^13*c*d^3*e) - ((d + e*x)^(1/2)*(b^2*e^2 - 48*c^2*d^2 + 12*b*c*d*e)*
(128*b^10*c^5*d^5*e^2 - 320*b^11*c^4*d^4*e^3 + 256*b^12*c^3*d^3*e^4 - 64*b^13*c^2*d^2*e^5))/(64*b^5*(d^3)^(1/2
)*(b^8*c^2*d^4 + b^10*d^2*e^2 - 2*b^9*c*d^3*e)))*(b^2*e^2 - 48*c^2*d^2 + 12*b*c*d*e))/(8*b^5*(d^3)^(1/2)))*(b^
2*e^2 - 48*c^2*d^2 + 12*b*c*d*e))/(8*b^5*(d^3)^(1/2)) - ((((d + e*x)^(1/2)*(b^6*c^3*e^8 + 4608*c^9*d^6*e^2 - 1
3824*b*c^8*d^5*e^3 + 22*b^5*c^4*d*e^7 + 15072*b^2*c^7*d^4*e^4 - 7104*b^3*c^6*d^3*e^5 + 1226*b^4*c^5*d^2*e^6))/
(8*(b^8*c^2*d^4 + b^10*d^2*e^2 - 2*b^9*c*d^3*e)) - (((b^14*c^2*d*e^7 - 24*b^10*c^6*d^5*e^3 + 60*b^11*c^5*d^4*e
^4 - 46*b^12*c^4*d^3*e^5 + 9*b^13*c^3*d^2*e^6)/(b^12*c^2*d^4 + b^14*d^2*e^2 - 2*b^13*c*d^3*e) + ((d + e*x)^(1/
2)*(b^2*e^2 - 48*c^2*d^2 + 12*b*c*d*e)*(128*b^10*c^5*d^5*e^2 - 320*b^11*c^4*d^4*e^3 + 256*b^12*c^3*d^3*e^4 - 6
4*b^13*c^2*d^2*e^5))/(64*b^5*(d^3)^(1/2)*(b^8*c^2*d^4 + b^10*d^2*e^2 - 2*b^9*c*d^3*e)))*(b^2*e^2 - 48*c^2*d^2
+ 12*b*c*d*e))/(8*b^5*(d^3)^(1/2)))*(b^2*e^2 - 48*c^2*d^2 + 12*b*c*d*e))/(8*b^5*(d^3)^(1/2))))*(b^2*e^2 - 48*c
^2*d^2 + 12*b*c*d*e)*1i)/(4*b^5*(d^3)^(1/2)) - (atan((((-c^3*(b*e - c*d)^3)^(1/2)*(((d + e*x)^(1/2)*(b^6*c^3*e
^8 + 4608*c^9*d^6*e^2 - 13824*b*c^8*d^5*e^3 + 22*b^5*c^4*d*e^7 + 15072*b^2*c^7*d^4*e^4 - 7104*b^3*c^6*d^3*e^5
+ 1226*b^4*c^5*d^2*e^6))/(8*(b^8*c^2*d^4 + b^10*d^2*e^2 - 2*b^9*c*d^3*e)) + ((-c^3*(b*e - c*d)^3)^(1/2)*((b^14
*c^2*d*e^7 - 24*b^10*c^6*d^5*e^3 + 60*b^11*c^5*d^4*e^4 - 46*b^12*c^4*d^3*e^5 + 9*b^13*c^3*d^2*e^6)/(b^12*c^2*d
^4 + b^14*d^2*e^2 - 2*b^13*c*d^3*e) - ((-c^3*(b*e - c*d)^3)^(1/2)*(d + e*x)^(1/2)*(35*b^2*e^2 + 48*c^2*d^2 - 8
4*b*c*d*e)*(128*b^10*c^5*d^5*e^2 - 320*b^11*c^4*d^4*e^3 + 256*b^12*c^3*d^3*e^4 - 64*b^13*c^2*d^2*e^5))/(64*(b^
8*c^2*d^4 + b^10*d^2*e^2 - 2*b^9*c*d^3*e)*(b^8*e^3 - b^5*c^3*d^3 + 3*b^6*c^2*d^2*e - 3*b^7*c*d*e^2)))*(35*b^2*
e^2 + 48*c^2*d^2 - 84*b*c*d*e))/(8*(b^8*e^3 - b^5*c^3*d^3 + 3*b^6*c^2*d^2*e - 3*b^7*c*d*e^2)))*(35*b^2*e^2 + 4
8*c^2*d^2 - 84*b*c*d*e)*1i)/(8*(b^8*e^3 - b^5*c^3*d^3 + 3*b^6*c^2*d^2*e - 3*b^7*c*d*e^2)) + ((-c^3*(b*e - c*d)
^3)^(1/2)*(((d + e*x)^(1/2)*(b^6*c^3*e^8 + 4608*c^9*d^6*e^2 - 13824*b*c^8*d^5*e^3 + 22*b^5*c^4*d*e^7 + 15072*b
^2*c^7*d^4*e^4 - 7104*b^3*c^6*d^3*e^5 + 1226*b^4*c^5*d^2*e^6))/(8*(b^8*c^2*d^4 + b^10*d^2*e^2 - 2*b^9*c*d^3*e)
) - ((-c^3*(b*e - c*d)^3)^(1/2)*((b^14*c^2*d*e^7 - 24*b^10*c^6*d^5*e^3 + 60*b^11*c^5*d^4*e^4 - 46*b^12*c^4*d^3
*e^5 + 9*b^13*c^3*d^2*e^6)/(b^12*c^2*d^4 + b^14*d^2*e^2 - 2*b^13*c*d^3*e) + ((-c^3*(b*e - c*d)^3)^(1/2)*(d + e
*x)^(1/2)*(35*b^2*e^2 + 48*c^2*d^2 - 84*b*c*d*e)*(128*b^10*c^5*d^5*e^2 - 320*b^11*c^4*d^4*e^3 + 256*b^12*c^3*d
^3*e^4 - 64*b^13*c^2*d^2*e^5))/(64*(b^8*c^2*d^4 + b^10*d^2*e^2 - 2*b^9*c*d^3*e)*(b^8*e^3 - b^5*c^3*d^3 + 3*b^6
*c^2*d^2*e - 3*b^7*c*d*e^2)))*(35*b^2*e^2 + 48*c^2*d^2 - 84*b*c*d*e))/(8*(b^8*e^3 - b^5*c^3*d^3 + 3*b^6*c^2*d^
2*e - 3*b^7*c*d*e^2)))*(35*b^2*e^2 + 48*c^2*d^2 - 84*b*c*d*e)*1i)/(8*(b^8*e^3 - b^5*c^3*d^3 + 3*b^6*c^2*d^2*e
- 3*b^7*c*d*e^2)))/((1728*c^10*d^6*e^3 - (35*b^6*c^4*e^9)/32 - 5184*b*c^9*d^5*e^4 + (63*b^5*c^5*d*e^8)/4 + 550
8*b^2*c^8*d^4*e^5 - 2376*b^3*c^7*d^3*e^6 + (1233*b^4*c^6*d^2*e^7)/4)/(b^12*c^2*d^4 + b^14*d^2*e^2 - 2*b^13*c*d
^3*e) + ((-c^3*(b*e - c*d)^3)^(1/2)*(((d + e*x)^(1/2)*(b^6*c^3*e^8 + 4608*c^9*d^6*e^2 - 13824*b*c^8*d^5*e^3 +
22*b^5*c^4*d*e^7 + 15072*b^2*c^7*d^4*e^4 - 7104*b^3*c^6*d^3*e^5 + 1226*b^4*c^5*d^2*e^6))/(8*(b^8*c^2*d^4 + b^1
0*d^2*e^2 - 2*b^9*c*d^3*e)) + ((-c^3*(b*e - c*d)^3)^(1/2)*((b^14*c^2*d*e^7 - 24*b^10*c^6*d^5*e^3 + 60*b^11*c^5
*d^4*e^4 - 46*b^12*c^4*d^3*e^5 + 9*b^13*c^3*d^2*e^6)/(b^12*c^2*d^4 + b^14*d^2*e^2 - 2*b^13*c*d^3*e) - ((-c^3*(
b*e - c*d)^3)^(1/2)*(d + e*x)^(1/2)*(35*b^2*e^2 + 48*c^2*d^2 - 84*b*c*d*e)*(128*b^10*c^5*d^5*e^2 - 320*b^11*c^
4*d^4*e^3 + 256*b^12*c^3*d^3*e^4 - 64*b^13*c^2*d^2*e^5))/(64*(b^8*c^2*d^4 + b^10*d^2*e^2 - 2*b^9*c*d^3*e)*(b^8
*e^3 - b^5*c^3*d^3 + 3*b^6*c^2*d^2*e - 3*b^7*c*d*e^2)))*(35*b^2*e^2 + 48*c^2*d^2 - 84*b*c*d*e))/(8*(b^8*e^3 -
b^5*c^3*d^3 + 3*b^6*c^2*d^2*e - 3*b^7*c*d*e^2)))*(35*b^2*e^2 + 48*c^2*d^2 - 84*b*c*d*e))/(8*(b^8*e^3 - b^5*c^3
*d^3 + 3*b^6*c^2*d^2*e - 3*b^7*c*d*e^2)) - ((-c^3*(b*e - c*d)^3)^(1/2)*(((d + e*x)^(1/2)*(b^6*c^3*e^8 + 4608*c
^9*d^6*e^2 - 13824*b*c^8*d^5*e^3 + 22*b^5*c^4*d*e^7 + 15072*b^2*c^7*d^4*e^4 - 7104*b^3*c^6*d^3*e^5 + 1226*b^4*
c^5*d^2*e^6))/(8*(b^8*c^2*d^4 + b^10*d^2*e^2 - 2*b^9*c*d^3*e)) - ((-c^3*(b*e - c*d)^3)^(1/2)*((b^14*c^2*d*e^7
- 24*b^10*c^6*d^5*e^3 + 60*b^11*c^5*d^4*e^4 - 46*b^12*c^4*d^3*e^5 + 9*b^13*c^3*d^2*e^6)/(b^12*c^2*d^4 + b^14*d
^2*e^2 - 2*b^13*c*d^3*e) + ((-c^3*(b*e - c*d)^3)^(1/2)*(d + e*x)^(1/2)*(35*b^2*e^2 + 48*c^2*d^2 - 84*b*c*d*e)*
(128*b^10*c^5*d^5*e^2 - 320*b^11*c^4*d^4*e^3 + 256*b^12*c^3*d^3*e^4 - 64*b^13*c^2*d^2*e^5))/(64*(b^8*c^2*d^4 +
b^10*d^2*e^2 - 2*b^9*c*d^3*e)*(b^8*e^3 - b^5*c^3*d^3 + 3*b^6*c^2*d^2*e - 3*b^7*c*d*e^2)))*(35*b^2*e^2 + 48*c^
2*d^2 - 84*b*c*d*e))/(8*(b^8*e^3 - b^5*c^3*d^3 + 3*b^6*c^2*d^2*e - 3*b^7*c*d*e^2)))*(35*b^2*e^2 + 48*c^2*d^2 -
84*b*c*d*e))/(8*(b^8*e^3 - b^5*c^3*d^3 + 3*b^6*c^2*d^2*e - 3*b^7*c*d*e^2))))*(-c^3*(b*e - c*d)^3)^(1/2)*(35*b
^2*e^2 + 48*c^2*d^2 - 84*b*c*d*e)*1i)/(4*(b^8*e^3 - b^5*c^3*d^3 + 3*b^6*c^2*d^2*e - 3*b^7*c*d*e^2))

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sympy [B]  time = 177.27, size = 2584, normalized size = 10.55

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(1/2)/(c*x**2+b*x)**3,x)

[Out]

10*c**4*d**2*e**2*sqrt(d + e*x)/(8*b**7*e**4 - 16*b**6*c*d*e**3 + 16*b**6*c*e**4*x - 48*b**5*c**2*d*e**3*x + 8
*b**5*c**2*e**2*(d + e*x)**2 + 16*b**4*c**3*d**3*e + 48*b**4*c**3*d**2*e**2*x - 16*b**4*c**3*d*e*(d + e*x)**2
- 8*b**3*c**4*d**4 - 16*b**3*c**4*d**3*e*x + 8*b**3*c**4*d**2*(d + e*x)**2) - 6*c**4*d*e**2*(d + e*x)**(3/2)/(
8*b**7*e**4 - 16*b**6*c*d*e**3 + 16*b**6*c*e**4*x - 48*b**5*c**2*d*e**3*x + 8*b**5*c**2*e**2*(d + e*x)**2 + 16
*b**4*c**3*d**3*e + 48*b**4*c**3*d**2*e**2*x - 16*b**4*c**3*d*e*(d + e*x)**2 - 8*b**3*c**4*d**4 - 16*b**3*c**4
*d**3*e*x + 8*b**3*c**4*d**2*(d + e*x)**2) - 20*c**3*d*e**3*sqrt(d + e*x)/(8*b**6*e**4 - 16*b**5*c*d*e**3 + 16
*b**5*c*e**4*x - 48*b**4*c**2*d*e**3*x + 8*b**4*c**2*e**2*(d + e*x)**2 + 16*b**3*c**3*d**3*e + 48*b**3*c**3*d*
*2*e**2*x - 16*b**3*c**3*d*e*(d + e*x)**2 - 8*b**2*c**4*d**4 - 16*b**2*c**4*d**3*e*x + 8*b**2*c**4*d**2*(d + e
*x)**2) - 6*c**3*d*e*sqrt(d + e*x)/(2*b**6*e**2 - 2*b**5*c*d*e + 2*b**5*c*e**2*x - 2*b**4*c**2*d*e*x) + 6*c**3
*e**3*(d + e*x)**(3/2)/(8*b**6*e**4 - 16*b**5*c*d*e**3 + 16*b**5*c*e**4*x - 48*b**4*c**2*d*e**3*x + 8*b**4*c**
2*e**2*(d + e*x)**2 + 16*b**3*c**3*d**3*e + 48*b**3*c**3*d**2*e**2*x - 16*b**3*c**3*d*e*(d + e*x)**2 - 8*b**2*
c**4*d**4 - 16*b**2*c**4*d**3*e*x + 8*b**2*c**4*d**2*(d + e*x)**2) + 10*c**2*e**4*sqrt(d + e*x)/(8*b**5*e**4 -
16*b**4*c*d*e**3 + 16*b**4*c*e**4*x - 48*b**3*c**2*d*e**3*x + 8*b**3*c**2*e**2*(d + e*x)**2 + 16*b**2*c**3*d*
*3*e + 48*b**2*c**3*d**2*e**2*x - 16*b**2*c**3*d*e*(d + e*x)**2 - 8*b*c**4*d**4 - 16*b*c**4*d**3*e*x + 8*b*c**
4*d**2*(d + e*x)**2) + 4*c**2*e**2*sqrt(d + e*x)/(2*b**5*e**2 - 2*b**4*c*d*e + 2*b**4*c*e**2*x - 2*b**3*c**2*d
*e*x) - 10*d**2*e**2*sqrt(d + e*x)/(-8*b**3*d**4 - 16*b**3*d**3*e*x + 8*b**3*d**2*(d + e*x)**2) + 6*d*e**2*(d
+ e*x)**(3/2)/(-8*b**3*d**4 - 16*b**3*d**3*e*x + 8*b**3*d**2*(d + e*x)**2) - 3*c**2*e**3*sqrt(-1/(c*(b*e - c*d
)**5))*log(-b**3*e**3*sqrt(-1/(c*(b*e - c*d)**5)) + 3*b**2*c*d*e**2*sqrt(-1/(c*(b*e - c*d)**5)) - 3*b*c**2*d**
2*e*sqrt(-1/(c*(b*e - c*d)**5)) + c**3*d**3*sqrt(-1/(c*(b*e - c*d)**5)) + sqrt(d + e*x))/(8*b**2) + 3*c**2*e**
3*sqrt(-1/(c*(b*e - c*d)**5))*log(b**3*e**3*sqrt(-1/(c*(b*e - c*d)**5)) - 3*b**2*c*d*e**2*sqrt(-1/(c*(b*e - c*
d)**5)) + 3*b*c**2*d**2*e*sqrt(-1/(c*(b*e - c*d)**5)) - c**3*d**3*sqrt(-1/(c*(b*e - c*d)**5)) + sqrt(d + e*x))
/(8*b**2) + 3*c**3*d*e**2*sqrt(-1/(c*(b*e - c*d)**5))*log(-b**3*e**3*sqrt(-1/(c*(b*e - c*d)**5)) + 3*b**2*c*d*
e**2*sqrt(-1/(c*(b*e - c*d)**5)) - 3*b*c**2*d**2*e*sqrt(-1/(c*(b*e - c*d)**5)) + c**3*d**3*sqrt(-1/(c*(b*e - c
*d)**5)) + sqrt(d + e*x))/(8*b**3) - 3*c**3*d*e**2*sqrt(-1/(c*(b*e - c*d)**5))*log(b**3*e**3*sqrt(-1/(c*(b*e -
c*d)**5)) - 3*b**2*c*d*e**2*sqrt(-1/(c*(b*e - c*d)**5)) + 3*b*c**2*d**2*e*sqrt(-1/(c*(b*e - c*d)**5)) - c**3*
d**3*sqrt(-1/(c*(b*e - c*d)**5)) + sqrt(d + e*x))/(8*b**3) - c**2*e**2*sqrt(-1/(c*(b*e - c*d)**3))*log(-b**2*e
**2*sqrt(-1/(c*(b*e - c*d)**3)) + 2*b*c*d*e*sqrt(-1/(c*(b*e - c*d)**3)) - c**2*d**2*sqrt(-1/(c*(b*e - c*d)**3)
) + sqrt(d + e*x))/b**3 + c**2*e**2*sqrt(-1/(c*(b*e - c*d)**3))*log(b**2*e**2*sqrt(-1/(c*(b*e - c*d)**3)) - 2*
b*c*d*e*sqrt(-1/(c*(b*e - c*d)**3)) + c**2*d**2*sqrt(-1/(c*(b*e - c*d)**3)) + sqrt(d + e*x))/b**3 + 3*d*e**2*s
qrt(d**(-5))*log(-d**3*sqrt(d**(-5)) + sqrt(d + e*x))/(8*b**3) - 3*d*e**2*sqrt(d**(-5))*log(d**3*sqrt(d**(-5))
+ sqrt(d + e*x))/(8*b**3) - e**2*sqrt(d**(-3))*log(-d**2*sqrt(d**(-3)) + sqrt(d + e*x))/(2*b**3) + e**2*sqrt(
d**(-3))*log(d**2*sqrt(d**(-3)) + sqrt(d + e*x))/(2*b**3) - e*sqrt(d + e*x)/(b**3*d*x) + 3*c**3*d*e*sqrt(-1/(c
*(b*e - c*d)**3))*log(-b**2*e**2*sqrt(-1/(c*(b*e - c*d)**3)) + 2*b*c*d*e*sqrt(-1/(c*(b*e - c*d)**3)) - c**2*d*
*2*sqrt(-1/(c*(b*e - c*d)**3)) + sqrt(d + e*x))/(2*b**4) - 3*c**3*d*e*sqrt(-1/(c*(b*e - c*d)**3))*log(b**2*e**
2*sqrt(-1/(c*(b*e - c*d)**3)) - 2*b*c*d*e*sqrt(-1/(c*(b*e - c*d)**3)) + c**2*d**2*sqrt(-1/(c*(b*e - c*d)**3))
+ sqrt(d + e*x))/(2*b**4) + 3*c*d*e*sqrt(d**(-3))*log(-d**2*sqrt(d**(-3)) + sqrt(d + e*x))/(2*b**4) - 3*c*d*e*
sqrt(d**(-3))*log(d**2*sqrt(d**(-3)) + sqrt(d + e*x))/(2*b**4) + 6*c*e*atan(sqrt(d + e*x)/sqrt(b*e/c - d))/(b*
*4*sqrt(b*e/c - d)) - 6*c*e*atan(sqrt(d + e*x)/sqrt(-d))/(b**4*sqrt(-d)) + 3*c*sqrt(d + e*x)/(b**4*x) - 12*c**
2*d*atan(sqrt(d + e*x)/sqrt(b*e/c - d))/(b**5*sqrt(b*e/c - d)) + 12*c**2*d*atan(sqrt(d + e*x)/sqrt(-d))/(b**5*
sqrt(-d))

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