### 3.374 $$\int \frac{1}{\sqrt{d+e x} (b x+c x^2)^2} \, dx$$

Optimal. Leaf size=154 $-\frac{c^{3/2} (4 c d-5 b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 (c d-b e)^{3/2}}+\frac{(b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3 d^{3/2}}-\frac{\sqrt{d+e x} (c x (2 c d-b e)+b (c d-b e))}{b^2 d \left (b x+c x^2\right ) (c d-b e)}$

[Out]

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

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Rubi [A]  time = 0.260734, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.19, Rules used = {740, 826, 1166, 208} $-\frac{c^{3/2} (4 c d-5 b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 (c d-b e)^{3/2}}+\frac{(b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3 d^{3/2}}-\frac{\sqrt{d+e x} (c x (2 c d-b e)+b (c d-b e))}{b^2 d \left (b x+c x^2\right ) (c d-b e)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 740

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

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]

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]

Rubi steps

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

Mathematica [A]  time = 0.499211, size = 148, normalized size = 0.96 $\frac{\frac{b \sqrt{d+e x} \left (b^2 e+b c (e x-d)-2 c^2 d x\right )}{x (b+c x) (c d-b e)}+\frac{c^{3/2} d (5 b e-4 c d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{(c d-b e)^{3/2}}+\frac{(b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{\sqrt{d}}}{b^3 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.227, size = 202, normalized size = 1.3 \begin{align*}{\frac{{c}^{2}e}{{b}^{2} \left ( be-cd \right ) \left ( cex+be \right ) }\sqrt{ex+d}}+5\,{\frac{{c}^{2}e}{{b}^{2} \left ( be-cd \right ) \sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-4\,{\frac{{c}^{3}d}{{b}^{3} \left ( be-cd \right ) \sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-{\frac{1}{{b}^{2}dx}\sqrt{ex+d}}+{\frac{e}{{b}^{2}}{\it Artanh} \left ({\sqrt{ex+d}{\frac{1}{\sqrt{d}}}} \right ){d}^{-{\frac{3}{2}}}}+4\,{\frac{c}{{b}^{3}\sqrt{d}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e*c^2/b^2/(b*e-c*d)*(e*x+d)^(1/2)/(c*e*x+b*e)+5*e*c^2/b^2/(b*e-c*d)/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c
/((b*e-c*d)*c)^(1/2))-4*c^3/b^3/(b*e-c*d)/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*d-1/
b^2/d*(e*x+d)^(1/2)/x+e/b^2/d^(3/2)*arctanh((e*x+d)^(1/2)/d^(1/2))+4/b^3/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2)
)*c

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 7.38875, size = 2396, normalized size = 15.56 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(((4*c^3*d^3 - 5*b*c^2*d^2*e)*x^2 + (4*b*c^2*d^3 - 5*b^2*c*d^2*e)*x)*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)) + ((4*c^3*d^2 - 3*b*c^2*d*e - b^2*c*e^2
)*x^2 + (4*b*c^2*d^2 - 3*b^2*c*d*e - b^3*e^2)*x)*sqrt(d)*log((e*x + 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(b^2
*c*d^2 - b^3*d*e + (2*b*c^2*d^2 - b^2*c*d*e)*x)*sqrt(e*x + d))/((b^3*c^2*d^3 - b^4*c*d^2*e)*x^2 + (b^4*c*d^3 -
b^5*d^2*e)*x), -1/2*(2*((4*c^3*d^3 - 5*b*c^2*d^2*e)*x^2 + (4*b*c^2*d^3 - 5*b^2*c*d^2*e)*x)*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)) - ((4*c^3*d^2 - 3*b*c^2*d*e - b^2*c*e
^2)*x^2 + (4*b*c^2*d^2 - 3*b^2*c*d*e - b^3*e^2)*x)*sqrt(d)*log((e*x + 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(b
^2*c*d^2 - b^3*d*e + (2*b*c^2*d^2 - b^2*c*d*e)*x)*sqrt(e*x + d))/((b^3*c^2*d^3 - b^4*c*d^2*e)*x^2 + (b^4*c*d^3
- b^5*d^2*e)*x), -1/2*(2*((4*c^3*d^2 - 3*b*c^2*d*e - b^2*c*e^2)*x^2 + (4*b*c^2*d^2 - 3*b^2*c*d*e - b^3*e^2)*x
)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) - ((4*c^3*d^3 - 5*b*c^2*d^2*e)*x^2 + (4*b*c^2*d^3 - 5*b^2*c*d^2*e)
*x)*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*(b^2*c*d^2 - b^3*d*e + (2*b*c^2*d^2 - b^2*c*d*e)*x)*sqrt(e*x + d))/((b^3*c^2*d^3 - b^4*c*d^2*e)*x^2 + (b^
4*c*d^3 - b^5*d^2*e)*x), -(((4*c^3*d^3 - 5*b*c^2*d^2*e)*x^2 + (4*b*c^2*d^3 - 5*b^2*c*d^2*e)*x)*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)) + ((4*c^3*d^2 - 3*b*c^2*d*e - b^2*
c*e^2)*x^2 + (4*b*c^2*d^2 - 3*b^2*c*d*e - b^3*e^2)*x)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + (b^2*c*d^2 -
b^3*d*e + (2*b*c^2*d^2 - b^2*c*d*e)*x)*sqrt(e*x + d))/((b^3*c^2*d^3 - b^4*c*d^2*e)*x^2 + (b^4*c*d^3 - b^5*d^2
*e)*x)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \left (b + c x\right )^{2} \sqrt{d + e x}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x**2*(b + c*x)**2*sqrt(d + e*x)), x)

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Giac [A]  time = 1.32119, size = 342, normalized size = 2.22 \begin{align*} \frac{{\left (4 \, c^{3} d - 5 \, b c^{2} e\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{{\left (b^{3} c d - b^{4} e\right )} \sqrt{-c^{2} d + b c e}} - \frac{2 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{2} d e - 2 \, \sqrt{x e + d} c^{2} d^{2} e -{\left (x e + d\right )}^{\frac{3}{2}} b c e^{2} + 2 \, \sqrt{x e + d} b c d e^{2} - \sqrt{x e + d} b^{2} e^{3}}{{\left (b^{2} c d^{2} - b^{3} 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 )}} - \frac{{\left (4 \, c d + b e\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b^{3} \sqrt{-d} d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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