3.321 \(\int \frac{1}{(d+e x)^3 \sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=180 \[ \frac{\left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 d^{5/2} (c d-b e)^{5/2}}-\frac{3 e \sqrt{b x+c x^2} (2 c d-b e)}{4 d^2 (d+e x) (c d-b e)^2}-\frac{e \sqrt{b x+c x^2}}{2 d (d+e x)^2 (c d-b e)} \]

[Out]

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

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Rubi [A]  time = 0.172979, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {744, 806, 724, 206} \[ \frac{\left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 d^{5/2} (c d-b e)^{5/2}}-\frac{3 e \sqrt{b x+c x^2} (2 c d-b e)}{4 d^2 (d+e x) (c d-b e)^2}-\frac{e \sqrt{b x+c x^2}}{2 d (d+e x)^2 (c d-b e)} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^3*Sqrt[b*x + c*x^2]),x]

[Out]

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

Rule 744

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[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)),
Int[(d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, x]*(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] && NeQ[m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ
[p]) || ILtQ[Simplify[m + 2*p + 3], 0])

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(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, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 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 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])

Rubi steps

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

Mathematica [A]  time = 0.257207, size = 183, normalized size = 1.02 \[ \frac{\sqrt{x} \left (\frac{\sqrt{b+c x} \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{2 d^{3/2} (b e-c d)^{3/2}}+\frac{3 e \sqrt{x} (b+c x) (2 c d-b e)}{2 d (d+e x) (c d-b e)}+\frac{e \sqrt{x} (b+c x)}{(d+e x)^2}\right )}{2 d \sqrt{x (b+c x)} (b e-c d)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^3*Sqrt[b*x + c*x^2]),x]

[Out]

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

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Maple [B]  time = 0.265, size = 798, normalized size = 4.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.03269, size = 1501, normalized size = 8.34 \begin{align*} \left [\frac{{\left (8 \, c^{2} d^{4} - 8 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2} +{\left (8 \, c^{2} d^{2} e^{2} - 8 \, b c d e^{3} + 3 \, b^{2} e^{4}\right )} x^{2} + 2 \,{\left (8 \, c^{2} d^{3} e - 8 \, b c d^{2} e^{2} + 3 \, b^{2} d e^{3}\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 (8 \, c^{2} d^{4} e - 13 \, b c d^{3} e^{2} + 5 \, b^{2} d^{2} e^{3} + 3 \,{\left (2 \, c^{2} d^{3} e^{2} - 3 \, b c d^{2} e^{3} + b^{2} d e^{4}\right )} x\right )} \sqrt{c x^{2} + b x}}{8 \,{\left (c^{3} d^{8} - 3 \, b c^{2} d^{7} e + 3 \, b^{2} c d^{6} e^{2} - b^{3} d^{5} e^{3} +{\left (c^{3} d^{6} e^{2} - 3 \, b c^{2} d^{5} e^{3} + 3 \, b^{2} c d^{4} e^{4} - b^{3} d^{3} e^{5}\right )} x^{2} + 2 \,{\left (c^{3} d^{7} e - 3 \, b c^{2} d^{6} e^{2} + 3 \, b^{2} c d^{5} e^{3} - b^{3} d^{4} e^{4}\right )} x\right )}}, \frac{{\left (8 \, c^{2} d^{4} - 8 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2} +{\left (8 \, c^{2} d^{2} e^{2} - 8 \, b c d e^{3} + 3 \, b^{2} e^{4}\right )} x^{2} + 2 \,{\left (8 \, c^{2} d^{3} e - 8 \, b c d^{2} e^{2} + 3 \, b^{2} d e^{3}\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 (8 \, c^{2} d^{4} e - 13 \, b c d^{3} e^{2} + 5 \, b^{2} d^{2} e^{3} + 3 \,{\left (2 \, c^{2} d^{3} e^{2} - 3 \, b c d^{2} e^{3} + b^{2} d e^{4}\right )} x\right )} \sqrt{c x^{2} + b x}}{4 \,{\left (c^{3} d^{8} - 3 \, b c^{2} d^{7} e + 3 \, b^{2} c d^{6} e^{2} - b^{3} d^{5} e^{3} +{\left (c^{3} d^{6} e^{2} - 3 \, b c^{2} d^{5} e^{3} + 3 \, b^{2} c d^{4} e^{4} - b^{3} d^{3} e^{5}\right )} x^{2} + 2 \,{\left (c^{3} d^{7} e - 3 \, b c^{2} d^{6} e^{2} + 3 \, b^{2} c d^{5} e^{3} - b^{3} d^{4} e^{4}\right )} x\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*((8*c^2*d^4 - 8*b*c*d^3*e + 3*b^2*d^2*e^2 + (8*c^2*d^2*e^2 - 8*b*c*d*e^3 + 3*b^2*e^4)*x^2 + 2*(8*c^2*d^3*
e - 8*b*c*d^2*e^2 + 3*b^2*d*e^3)*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)*sqr
t(c*x^2 + b*x))/(e*x + d)) - 2*(8*c^2*d^4*e - 13*b*c*d^3*e^2 + 5*b^2*d^2*e^3 + 3*(2*c^2*d^3*e^2 - 3*b*c*d^2*e^
3 + b^2*d*e^4)*x)*sqrt(c*x^2 + b*x))/(c^3*d^8 - 3*b*c^2*d^7*e + 3*b^2*c*d^6*e^2 - b^3*d^5*e^3 + (c^3*d^6*e^2 -
3*b*c^2*d^5*e^3 + 3*b^2*c*d^4*e^4 - b^3*d^3*e^5)*x^2 + 2*(c^3*d^7*e - 3*b*c^2*d^6*e^2 + 3*b^2*c*d^5*e^3 - b^3
*d^4*e^4)*x), 1/4*((8*c^2*d^4 - 8*b*c*d^3*e + 3*b^2*d^2*e^2 + (8*c^2*d^2*e^2 - 8*b*c*d*e^3 + 3*b^2*e^4)*x^2 +
2*(8*c^2*d^3*e - 8*b*c*d^2*e^2 + 3*b^2*d*e^3)*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)) - (8*c^2*d^4*e - 13*b*c*d^3*e^2 + 5*b^2*d^2*e^3 + 3*(2*c^2*d^3*e^2 - 3*b*c*d^2*e^3 + b
^2*d*e^4)*x)*sqrt(c*x^2 + b*x))/(c^3*d^8 - 3*b*c^2*d^7*e + 3*b^2*c*d^6*e^2 - b^3*d^5*e^3 + (c^3*d^6*e^2 - 3*b*
c^2*d^5*e^3 + 3*b^2*c*d^4*e^4 - b^3*d^3*e^5)*x^2 + 2*(c^3*d^7*e - 3*b*c^2*d^6*e^2 + 3*b^2*c*d^5*e^3 - b^3*d^4*
e^4)*x)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x \left (b + c x\right )} \left (d + e x\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(x*(b + c*x))*(d + e*x)**3), x)

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Giac [B]  time = 1.3162, size = 657, normalized size = 3.65 \begin{align*} -\frac{{\left (8 \, c^{2} d^{2} - 8 \, b c d e + 3 \, b^{2} e^{2}\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 )}{4 \,{\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{8 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} c^{2} d^{2} e + 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} c^{\frac{5}{2}} d^{3} - 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} b c^{\frac{3}{2}} d^{2} e + 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b c^{2} d^{3} - 8 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} b c d e^{2} - 20 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b^{2} c d^{2} e + 6 \, b^{2} c^{\frac{3}{2}} d^{3} + 9 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} b^{2} \sqrt{c} d e^{2} - 3 \, b^{3} \sqrt{c} d^{2} e + 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} b^{2} e^{3} + 5 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b^{3} d e^{2}}{4 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\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 )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(8*c^2*d^2 - 8*b*c*d*e + 3*b^2*e^2)*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)) - 1/4*(8*(sqrt(c)*x - sqrt(c*x^2 + b*x))^
3*c^2*d^2*e + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*c^(5/2)*d^3 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b*c^(3/2
)*d^2*e + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b*c^2*d^3 - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b*c*d*e^2 - 20*(s
qrt(c)*x - sqrt(c*x^2 + b*x))*b^2*c*d^2*e + 6*b^2*c^(3/2)*d^3 + 9*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^2*sqrt(c
)*d*e^2 - 3*b^3*sqrt(c)*d^2*e + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b^2*e^3 + 5*(sqrt(c)*x - sqrt(c*x^2 + b*x)
)*b^3*d*e^2)/((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2)*((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)^2)