3.3.70 \(\int \frac {\sqrt {b x+c x^2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=183 \[ -\frac {b^2 (2 c d-b e) \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{16 d^{5/2} (c d-b e)^{5/2}}+\frac {\sqrt {b x+c x^2} (2 c d-b e) (x (2 c d-b e)+b d)}{8 d^2 (d+e x)^2 (c d-b e)^2}-\frac {e \left (b x+c x^2\right )^{3/2}}{3 d (d+e x)^3 (c d-b e)} \]

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Rubi [A]  time = 0.14, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {730, 720, 724, 206} \begin {gather*} -\frac {b^2 (2 c d-b e) \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{16 d^{5/2} (c d-b e)^{5/2}}+\frac {\sqrt {b x+c x^2} (2 c d-b e) (x (2 c d-b e)+b d)}{8 d^2 (d+e x)^2 (c d-b e)^2}-\frac {e \left (b x+c x^2\right )^{3/2}}{3 d (d+e x)^3 (c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*x + c*x^2]/(d + e*x)^4,x]

[Out]

((2*c*d - b*e)*(b*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(8*d^2*(c*d - b*e)^2*(d + e*x)^2) - (e*(b*x + c*x^2)
^(3/2))/(3*d*(c*d - b*e)*(d + e*x)^3) - (b^2*(2*c*d - b*e)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d
- b*e]*Sqrt[b*x + c*x^2])])/(16*d^(5/2)*(c*d - b*e)^(5/2))

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 720

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*
(d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^p)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(p*(b^2 -
4*a*c))/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(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] && EqQ[m +
2*p + 2, 0] && GtQ[p, 0]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 730

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

Rubi steps

\begin {align*} \int \frac {\sqrt {b x+c x^2}}{(d+e x)^4} \, dx &=-\frac {e \left (b x+c x^2\right )^{3/2}}{3 d (c d-b e) (d+e x)^3}+\frac {(2 c d-b e) \int \frac {\sqrt {b x+c x^2}}{(d+e x)^3} \, dx}{2 d (c d-b e)}\\ &=\frac {(2 c d-b e) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{8 d^2 (c d-b e)^2 (d+e x)^2}-\frac {e \left (b x+c x^2\right )^{3/2}}{3 d (c d-b e) (d+e x)^3}-\frac {\left (b^2 (2 c d-b e)\right ) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{16 d^2 (c d-b e)^2}\\ &=\frac {(2 c d-b e) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{8 d^2 (c d-b e)^2 (d+e x)^2}-\frac {e \left (b x+c x^2\right )^{3/2}}{3 d (c d-b e) (d+e x)^3}+\frac {\left (b^2 (2 c d-b e)\right ) \operatorname {Subst}\left (\int \frac {1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac {-b d-(2 c d-b e) x}{\sqrt {b x+c x^2}}\right )}{8 d^2 (c d-b e)^2}\\ &=\frac {(2 c d-b e) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{8 d^2 (c d-b e)^2 (d+e x)^2}-\frac {e \left (b x+c x^2\right )^{3/2}}{3 d (c d-b e) (d+e x)^3}-\frac {b^2 (2 c d-b e) \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{16 d^{5/2} (c d-b e)^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.29, size = 191, normalized size = 1.04 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\frac {e x^{3/2} (b+c x)}{(d+e x)^3}-\frac {3 (2 c d-b e) \left (\sqrt {d} \sqrt {x} \sqrt {b+c x} \sqrt {c d-b e} (b (d-e x)+2 c d x)-b^2 (d+e x)^2 \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt {c d-b e}}{\sqrt {d} \sqrt {b+c x}}\right )\right )}{8 d^{3/2} \sqrt {b+c x} (d+e x)^2 (c d-b e)^{3/2}}\right )}{3 d \sqrt {x} (b e-c d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*x + c*x^2]/(d + e*x)^4,x]

[Out]

(Sqrt[x*(b + c*x)]*((e*x^(3/2)*(b + c*x))/(d + e*x)^3 - (3*(2*c*d - b*e)*(Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[x]*Sqrt
[b + c*x]*(2*c*d*x + b*(d - e*x)) - b^2*(d + e*x)^2*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])]
))/(8*d^(3/2)*(c*d - b*e)^(3/2)*Sqrt[b + c*x]*(d + e*x)^2)))/(3*d*(-(c*d) + b*e)*Sqrt[x])

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IntegrateAlgebraic [A]  time = 1.96, size = 202, normalized size = 1.10 \begin {gather*} \frac {\sqrt {b x+c x^2} \left (-3 b^2 d^2 e+8 b^2 d e^2 x+3 b^2 e^3 x^2+6 b c d^3-14 b c d^2 e x-4 b c d e^2 x^2+12 c^2 d^3 x+4 c^2 d^2 e x^2\right )}{24 d^2 (d+e x)^3 (c d-b e)^2}+\frac {\left (b^3 e-2 b^2 c d\right ) \tanh ^{-1}\left (\frac {-e \sqrt {b x+c x^2}+\sqrt {c} d+\sqrt {c} e x}{\sqrt {d} \sqrt {c d-b e}}\right )}{8 d^{5/2} (c d-b e)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[b*x + c*x^2]/(d + e*x)^4,x]

[Out]

(Sqrt[b*x + c*x^2]*(6*b*c*d^3 - 3*b^2*d^2*e + 12*c^2*d^3*x - 14*b*c*d^2*e*x + 8*b^2*d*e^2*x + 4*c^2*d^2*e*x^2
- 4*b*c*d*e^2*x^2 + 3*b^2*e^3*x^2))/(24*d^2*(c*d - b*e)^2*(d + e*x)^3) + ((-2*b^2*c*d + b^3*e)*ArcTanh[(Sqrt[c
]*d + Sqrt[c]*e*x - e*Sqrt[b*x + c*x^2])/(Sqrt[d]*Sqrt[c*d - b*e])])/(8*d^(5/2)*(c*d - b*e)^(5/2))

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fricas [B]  time = 0.43, size = 969, normalized size = 5.30 \begin {gather*} \left [-\frac {3 \, {\left (2 \, b^{2} c d^{4} - b^{3} d^{3} e + {\left (2 \, b^{2} c d e^{3} - b^{3} e^{4}\right )} x^{3} + 3 \, {\left (2 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )} x^{2} + 3 \, {\left (2 \, b^{2} c d^{3} e - b^{3} d^{2} e^{2}\right )} x\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {b d + {\left (2 \, c d - b e\right )} x + 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{e x + d}\right ) - 2 \, {\left (6 \, b c^{2} d^{5} - 9 \, b^{2} c d^{4} e + 3 \, b^{3} d^{3} e^{2} + {\left (4 \, c^{3} d^{4} e - 8 \, b c^{2} d^{3} e^{2} + 7 \, b^{2} c d^{2} e^{3} - 3 \, b^{3} d e^{4}\right )} x^{2} + 2 \, {\left (6 \, c^{3} d^{5} - 13 \, b c^{2} d^{4} e + 11 \, b^{2} c d^{3} e^{2} - 4 \, b^{3} d^{2} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{48 \, {\left (c^{3} d^{9} - 3 \, b c^{2} d^{8} e + 3 \, b^{2} c d^{7} e^{2} - b^{3} d^{6} e^{3} + {\left (c^{3} d^{6} e^{3} - 3 \, b c^{2} d^{5} e^{4} + 3 \, b^{2} c d^{4} e^{5} - b^{3} d^{3} e^{6}\right )} x^{3} + 3 \, {\left (c^{3} d^{7} e^{2} - 3 \, b c^{2} d^{6} e^{3} + 3 \, b^{2} c d^{5} e^{4} - b^{3} d^{4} e^{5}\right )} x^{2} + 3 \, {\left (c^{3} d^{8} e - 3 \, b c^{2} d^{7} e^{2} + 3 \, b^{2} c d^{6} e^{3} - b^{3} d^{5} e^{4}\right )} x\right )}}, -\frac {3 \, {\left (2 \, b^{2} c d^{4} - b^{3} d^{3} e + {\left (2 \, b^{2} c d e^{3} - b^{3} e^{4}\right )} x^{3} + 3 \, {\left (2 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )} x^{2} + 3 \, {\left (2 \, b^{2} c d^{3} e - b^{3} d^{2} e^{2}\right )} x\right )} \sqrt {-c d^{2} + b d e} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) - {\left (6 \, b c^{2} d^{5} - 9 \, b^{2} c d^{4} e + 3 \, b^{3} d^{3} e^{2} + {\left (4 \, c^{3} d^{4} e - 8 \, b c^{2} d^{3} e^{2} + 7 \, b^{2} c d^{2} e^{3} - 3 \, b^{3} d e^{4}\right )} x^{2} + 2 \, {\left (6 \, c^{3} d^{5} - 13 \, b c^{2} d^{4} e + 11 \, b^{2} c d^{3} e^{2} - 4 \, b^{3} d^{2} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{24 \, {\left (c^{3} d^{9} - 3 \, b c^{2} d^{8} e + 3 \, b^{2} c d^{7} e^{2} - b^{3} d^{6} e^{3} + {\left (c^{3} d^{6} e^{3} - 3 \, b c^{2} d^{5} e^{4} + 3 \, b^{2} c d^{4} e^{5} - b^{3} d^{3} e^{6}\right )} x^{3} + 3 \, {\left (c^{3} d^{7} e^{2} - 3 \, b c^{2} d^{6} e^{3} + 3 \, b^{2} c d^{5} e^{4} - b^{3} d^{4} e^{5}\right )} x^{2} + 3 \, {\left (c^{3} d^{8} e - 3 \, b c^{2} d^{7} e^{2} + 3 \, b^{2} c d^{6} e^{3} - b^{3} d^{5} e^{4}\right )} x\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(1/2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

[-1/48*(3*(2*b^2*c*d^4 - b^3*d^3*e + (2*b^2*c*d*e^3 - b^3*e^4)*x^3 + 3*(2*b^2*c*d^2*e^2 - b^3*d*e^3)*x^2 + 3*(
2*b^2*c*d^3*e - b^3*d^2*e^2)*x)*sqrt(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*
x^2 + b*x))/(e*x + d)) - 2*(6*b*c^2*d^5 - 9*b^2*c*d^4*e + 3*b^3*d^3*e^2 + (4*c^3*d^4*e - 8*b*c^2*d^3*e^2 + 7*b
^2*c*d^2*e^3 - 3*b^3*d*e^4)*x^2 + 2*(6*c^3*d^5 - 13*b*c^2*d^4*e + 11*b^2*c*d^3*e^2 - 4*b^3*d^2*e^3)*x)*sqrt(c*
x^2 + b*x))/(c^3*d^9 - 3*b*c^2*d^8*e + 3*b^2*c*d^7*e^2 - b^3*d^6*e^3 + (c^3*d^6*e^3 - 3*b*c^2*d^5*e^4 + 3*b^2*
c*d^4*e^5 - b^3*d^3*e^6)*x^3 + 3*(c^3*d^7*e^2 - 3*b*c^2*d^6*e^3 + 3*b^2*c*d^5*e^4 - b^3*d^4*e^5)*x^2 + 3*(c^3*
d^8*e - 3*b*c^2*d^7*e^2 + 3*b^2*c*d^6*e^3 - b^3*d^5*e^4)*x), -1/24*(3*(2*b^2*c*d^4 - b^3*d^3*e + (2*b^2*c*d*e^
3 - b^3*e^4)*x^3 + 3*(2*b^2*c*d^2*e^2 - b^3*d*e^3)*x^2 + 3*(2*b^2*c*d^3*e - b^3*d^2*e^2)*x)*sqrt(-c*d^2 + b*d*
e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) - (6*b*c^2*d^5 - 9*b^2*c*d^4*e + 3*b^3*d^3*
e^2 + (4*c^3*d^4*e - 8*b*c^2*d^3*e^2 + 7*b^2*c*d^2*e^3 - 3*b^3*d*e^4)*x^2 + 2*(6*c^3*d^5 - 13*b*c^2*d^4*e + 11
*b^2*c*d^3*e^2 - 4*b^3*d^2*e^3)*x)*sqrt(c*x^2 + b*x))/(c^3*d^9 - 3*b*c^2*d^8*e + 3*b^2*c*d^7*e^2 - b^3*d^6*e^3
+ (c^3*d^6*e^3 - 3*b*c^2*d^5*e^4 + 3*b^2*c*d^4*e^5 - b^3*d^3*e^6)*x^3 + 3*(c^3*d^7*e^2 - 3*b*c^2*d^6*e^3 + 3*
b^2*c*d^5*e^4 - b^3*d^4*e^5)*x^2 + 3*(c^3*d^8*e - 3*b*c^2*d^7*e^2 + 3*b^2*c*d^6*e^3 - b^3*d^5*e^4)*x)]

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giac [B]  time = 0.27, size = 825, normalized size = 4.51 \begin {gather*} \frac {{\left (2 \, b^{2} c d - b^{3} e\right )} \arctan \left (\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e}}\right )}{8 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \sqrt {-c d^{2} + b d e}} + \frac {48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} c^{\frac {7}{2}} d^{4} e + 32 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} c^{4} d^{5} + 16 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b c^{3} d^{4} e + 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b c^{\frac {7}{2}} d^{5} - 96 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} b c^{\frac {5}{2}} d^{3} e^{2} - 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b^{2} c^{\frac {5}{2}} d^{4} e + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{2} c^{3} d^{5} - 84 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b^{2} c^{2} d^{3} e^{2} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{3} c^{2} d^{4} e + 4 \, b^{3} c^{\frac {5}{2}} d^{5} + 78 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} b^{2} c^{\frac {3}{2}} d^{2} e^{3} - 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b^{3} c^{\frac {3}{2}} d^{3} e^{2} - 4 \, b^{4} c^{\frac {3}{2}} d^{4} e + 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} b^{2} c d e^{4} + 74 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b^{3} c d^{2} e^{3} + 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{4} c d^{3} e^{2} - 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} b^{3} \sqrt {c} d e^{4} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b^{4} \sqrt {c} d^{2} e^{3} + 3 \, b^{5} \sqrt {c} d^{3} e^{2} - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} b^{3} e^{5} - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b^{4} d e^{4} + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{5} d^{2} e^{3}}{24 \, {\left (c^{2} d^{4} e^{2} - 2 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4}\right )} {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} d + b d\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(1/2)/(e*x+d)^4,x, algorithm="giac")

[Out]

1/8*(2*b^2*c*d - b^3*e)*arctan(((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))/((c^2*d^4
- 2*b*c*d^3*e + b^2*d^2*e^2)*sqrt(-c*d^2 + b*d*e)) + 1/24*(48*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*c^(7/2)*d^4*e
+ 32*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*c^4*d^5 + 16*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b*c^3*d^4*e + 48*(sqrt(
c)*x - sqrt(c*x^2 + b*x))^2*b*c^(7/2)*d^5 - 96*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*b*c^(5/2)*d^3*e^2 - 36*(sqrt(
c)*x - sqrt(c*x^2 + b*x))^2*b^2*c^(5/2)*d^4*e + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^2*c^3*d^5 - 84*(sqrt(c)*x
- sqrt(c*x^2 + b*x))^3*b^2*c^2*d^3*e^2 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^3*c^2*d^4*e + 4*b^3*c^(5/2)*d^5
+ 78*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*b^2*c^(3/2)*d^2*e^3 - 6*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^3*c^(3/2)*
d^3*e^2 - 4*b^4*c^(3/2)*d^4*e + 6*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*b^2*c*d*e^4 + 74*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^3*b^3*c*d^2*e^3 + 12*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^4*c*d^3*e^2 - 15*(sqrt(c)*x - sqrt(c*x^2 + b*x))
^4*b^3*sqrt(c)*d*e^4 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^4*sqrt(c)*d^2*e^3 + 3*b^5*sqrt(c)*d^3*e^2 - 3*(s
qrt(c)*x - sqrt(c*x^2 + b*x))^5*b^3*e^5 - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b^4*d*e^4 + 3*(sqrt(c)*x - sqrt(
c*x^2 + b*x))*b^5*d^2*e^3)/((c^2*d^4*e^2 - 2*b*c*d^3*e^3 + b^2*d^2*e^4)*((sqrt(c)*x - sqrt(c*x^2 + b*x))^2*e +
2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c)*d + b*d)^3)

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maple [B]  time = 0.06, size = 2891, normalized size = 15.80 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^(1/2)/(e*x+d)^4,x)

[Out]

1/2/e/(b*e-c*d)^3*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*c^3-1/2/e^2/(b*e-c*d)^2*c^(5/2)*ln
(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))+1/8*e^2/(b*e
-c*d)^3/d^3/(x+d/e)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)*b^2+5/8*e/(b*e-c*d)^3/d^2*((x+d/
e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*b^2*c+1/8*e/(b*e-c*d)^3/d^2*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/
e)/c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*c^(1/2)*b^3-1/16*e/(b*e-c*d)^3/d^2/(-(b*
e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-
c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b^4+3/4/e^2/(b*e-c*d)^2*c^2/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*
(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d
/e)/e)^(1/2))/(x+d/e))*b-9/8/e/(b*e-c*d)^3/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)
/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b^2*c^2+7/16
/(b*e-c*d)^3/d/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2
)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b^3*c-1/2/e/(b*e-c*d)^2/d/(x+d/e)^2*((x+
d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)*c-1/2/e^3/(b*e-c*d)^3*d^2/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-
2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x
+d/e)/e)^(1/2))/(x+d/e))*c^4+1/2/e/(b*e-c*d)^2/d*c^(3/2)*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c
-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*b-1/2/e^3/(b*e-c*d)^2*d*c^3/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*
e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)
/e)^(1/2))/(x+d/e))+5/4/e^2/(b*e-c*d)^3*d/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/
e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b*c^3-1/8*e^2
/(b*e-c*d)^3/d^3*c*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*b^2+1/2*e/(b*e-c*d)^3/d^2*c^2*(
(x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*b-1/4/e/(b*e-c*d)^2/d*c/(-(b*e-c*d)*d/e^2)^(1/2)*ln
((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)
*(x+d/e)/e)^(1/2))/(x+d/e))*b^2-1/2*e/(b*e-c*d)^3/d^2/(x+d/e)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)
/e)^(3/2)*b*c-1/8/(b*e-c*d)^2/d^2*c^(1/2)*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^
2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*b^2+1/3/e^2/(b*e-c*d)/d/(x+d/e)^3*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+
d/e)/e)^(3/2)-1/8*e^2/(b*e-c*d)^3/d^3*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*b^3+1/2/e/(b*e
-c*d)^2/d*c^2*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)+1/4/(b*e-c*d)^2/d^2/(x+d/e)^2*((x+d/e)
^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)*b-5/8/(b*e-c*d)^3/d*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)
+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*c^(3/2)*b^2-1/4/(b*e-c*d)^2/d^2*c*((x+d/e)^2*c-(b*
e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*b+1/2/(b*e-c*d)^3/d/(x+d/e)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)
*(x+d/e)/e)^(3/2)*c^2-1/(b*e-c*d)^3/d*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*b*c^2-1/2/(b*e
-c*d)^3/d*c^3*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x+1/e/(b*e-c*d)^3*ln(((x+d/e)*c+1/2*(b
*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*c^(5/2)*b-1/2/e^2/(b*e-c*d)^3*
d*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*c^(7/2)

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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((c*x^2+b*x)^(1/2)/(e*x+d)^4,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 zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {c\,x^2+b\,x}}{{\left (d+e\,x\right )}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^(1/2)/(d + e*x)^4,x)

[Out]

int((b*x + c*x^2)^(1/2)/(d + e*x)^4, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x \left (b + c x\right )}}{\left (d + e x\right )^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x)**(1/2)/(e*x+d)**4,x)

[Out]

Integral(sqrt(x*(b + c*x))/(d + e*x)**4, x)

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