### 3.383 $$\int \frac{1}{\sqrt{d+e x} (b x+c x^2)^3} \, dx$$

Optimal. Leaf size=299 $\frac{\sqrt{d+e x} \left (3 c x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b (c d-b e) \left (-3 b^2 e^2-7 b c d e+12 c^2 d^2\right )\right )}{4 b^4 d^2 \left (b x+c x^2\right ) (c d-b e)^2}+\frac{3 c^{5/2} \left (21 b^2 e^2-36 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{5/2}}-\frac{3 \left (b^2 e^2+4 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5 d^{5/2}}-\frac{\sqrt{d+e x} (c x (2 c d-b e)+b (c d-b e))}{2 b^2 d \left (b x+c x^2\right )^2 (c d-b e)}$

[Out]

-(Sqrt[d + e*x]*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(2*b^2*d*(c*d - b*e)*(b*x + c*x^2)^2) + (Sqrt[d + e*x]*(b
*(c*d - b*e)*(12*c^2*d^2 - 7*b*c*d*e - 3*b^2*e^2) + 3*c*(2*c*d - b*e)*(4*c^2*d^2 - 4*b*c*d*e - b^2*e^2)*x))/(4
*b^4*d^2*(c*d - b*e)^2*(b*x + c*x^2)) - (3*(16*c^2*d^2 + 4*b*c*d*e + b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/
(4*b^5*d^(5/2)) + (3*c^(5/2)*(16*c^2*d^2 - 36*b*c*d*e + 21*b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d -
b*e]])/(4*b^5*(c*d - b*e)^(5/2))

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Rubi [A]  time = 0.429534, antiderivative size = 299, normalized size of antiderivative = 1., 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 = {740, 822, 826, 1166, 208} $\frac{\sqrt{d+e x} \left (3 c x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b (c d-b e) \left (-3 b^2 e^2-7 b c d e+12 c^2 d^2\right )\right )}{4 b^4 d^2 \left (b x+c x^2\right ) (c d-b e)^2}+\frac{3 c^{5/2} \left (21 b^2 e^2-36 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{5/2}}-\frac{3 \left (b^2 e^2+4 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5 d^{5/2}}-\frac{\sqrt{d+e x} (c x (2 c d-b e)+b (c d-b e))}{2 b^2 d \left (b x+c x^2\right )^2 (c d-b e)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(Sqrt[d + e*x]*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(2*b^2*d*(c*d - b*e)*(b*x + c*x^2)^2) + (Sqrt[d + e*x]*(b
*(c*d - b*e)*(12*c^2*d^2 - 7*b*c*d*e - 3*b^2*e^2) + 3*c*(2*c*d - b*e)*(4*c^2*d^2 - 4*b*c*d*e - b^2*e^2)*x))/(4
*b^4*d^2*(c*d - b*e)^2*(b*x + c*x^2)) - (3*(16*c^2*d^2 + 4*b*c*d*e + b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/
(4*b^5*d^(5/2)) + (3*c^(5/2)*(16*c^2*d^2 - 36*b*c*d*e + 21*b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d -
b*e]])/(4*b^5*(c*d - b*e)^(5/2))

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 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]

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 )^3} \, dx &=-\frac{\sqrt{d+e x} (b (c d-b e)+c (2 c d-b e) x)}{2 b^2 d (c d-b e) \left (b x+c x^2\right )^2}-\frac{\int \frac{\frac{1}{2} \left (12 c^2 d^2-7 b c d e-3 b^2 e^2\right )+\frac{5}{2} c e (2 c d-b e) x}{\sqrt{d+e x} \left (b x+c x^2\right )^2} \, dx}{2 b^2 d (c d-b e)}\\ &=-\frac{\sqrt{d+e x} (b (c d-b e)+c (2 c d-b e) x)}{2 b^2 d (c d-b e) \left (b x+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (b (c d-b e) \left (12 c^2 d^2-7 b c d e-3 b^2 e^2\right )+3 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{4 b^4 d^2 (c d-b e)^2 \left (b x+c x^2\right )}+\frac{\int \frac{\frac{3}{4} (c d-b e)^2 \left (16 c^2 d^2+4 b c d e+b^2 e^2\right )+\frac{3}{4} c e (2 c d-b e) \left (4 c^2 d^2-4 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^2 (c d-b e)^2}\\ &=-\frac{\sqrt{d+e x} (b (c d-b e)+c (2 c d-b e) x)}{2 b^2 d (c d-b e) \left (b x+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (b (c d-b e) \left (12 c^2 d^2-7 b c d e-3 b^2 e^2\right )+3 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{4 b^4 d^2 (c d-b e)^2 \left (b x+c x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{3}{4} c d e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right )+\frac{3}{4} e (c d-b e)^2 \left (16 c^2 d^2+4 b c d e+b^2 e^2\right )+\frac{3}{4} c e (2 c d-b e) \left (4 c^2 d^2-4 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^2 (c d-b e)^2}\\ &=-\frac{\sqrt{d+e x} (b (c d-b e)+c (2 c d-b e) x)}{2 b^2 d (c d-b e) \left (b x+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (b (c d-b e) \left (12 c^2 d^2-7 b c d e-3 b^2 e^2\right )+3 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{4 b^4 d^2 (c d-b e)^2 \left (b x+c x^2\right )}+\frac{\left (3 c \left (16 c^2 d^2+4 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^2}-\frac{\left (3 c^3 \left (16 c^2 d^2-36 b c d e+21 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)^2}\\ &=-\frac{\sqrt{d+e x} (b (c d-b e)+c (2 c d-b e) x)}{2 b^2 d (c d-b e) \left (b x+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (b (c d-b e) \left (12 c^2 d^2-7 b c d e-3 b^2 e^2\right )+3 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{4 b^4 d^2 (c d-b e)^2 \left (b x+c x^2\right )}-\frac{3 \left (16 c^2 d^2+4 b c d e+b^2 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5 d^{5/2}}+\frac{3 c^{5/2} \left (16 c^2 d^2-36 b c d e+21 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)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 1.22247, size = 299, normalized size = 1. $\frac{\frac{b \sqrt{d} \sqrt{d+e x} \left (b^3 c^2 \left (-13 d^2 e x-2 d^3+10 d e^2 x^2+3 e^3 x^3\right )+b^2 c^3 d x \left (8 d^2-55 d e x+6 e^2 x^2\right )+2 b^4 c e \left (2 d^2+d e x+3 e^2 x^2\right )+b^5 e^2 (3 e x-2 d)+36 b c^4 d^2 x^2 (d-e x)+24 c^5 d^3 x^3\right )}{x^2 (b+c x)^2 (c d-b e)^2}+\frac{3 c^{5/2} d^{5/2} \left (21 b^2 e^2-36 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{(c d-b e)^{5/2}}-3 \left (b^2 e^2+4 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5 d^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((b*Sqrt[d]*Sqrt[d + e*x]*(24*c^5*d^3*x^3 + 36*b*c^4*d^2*x^2*(d - e*x) + b^5*e^2*(-2*d + 3*e*x) + 2*b^4*c*e*(2
*d^2 + d*e*x + 3*e^2*x^2) + b^2*c^3*d*x*(8*d^2 - 55*d*e*x + 6*e^2*x^2) + b^3*c^2*(-2*d^3 - 13*d^2*e*x + 10*d*e
^2*x^2 + 3*e^3*x^3)))/((c*d - b*e)^2*x^2*(b + c*x)^2) - 3*(16*c^2*d^2 + 4*b*c*d*e + b^2*e^2)*ArcTanh[Sqrt[d +
e*x]/Sqrt[d]] + (3*c^(5/2)*d^(5/2)*(16*c^2*d^2 - 36*b*c*d*e + 21*b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt
[c*d - b*e]])/(c*d - b*e)^(5/2))/(4*b^5*d^(5/2))

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Maple [A]  time = 0.358, size = 526, normalized size = 1.8 \begin{align*} -{\frac{15\,{e}^{2}{c}^{4}}{4\,{b}^{3} \left ( cex+be \right ) ^{2} \left ({b}^{2}{e}^{2}-2\,bcde+{c}^{2}{d}^{2} \right ) } \left ( ex+d \right ) ^{{\frac{3}{2}}}}+3\,{\frac{e{c}^{5} \left ( ex+d \right ) ^{3/2}d}{{b}^{4} \left ( cex+be \right ) ^{2} \left ({b}^{2}{e}^{2}-2\,bcde+{c}^{2}{d}^{2} \right ) }}-{\frac{17\,{e}^{2}{c}^{3}}{4\,{b}^{3} \left ( cex+be \right ) ^{2} \left ( be-cd \right ) }\sqrt{ex+d}}+3\,{\frac{e{c}^{4}\sqrt{ex+d}d}{{b}^{4} \left ( cex+be \right ) ^{2} \left ( be-cd \right ) }}-{\frac{63\,{e}^{2}{c}^{3}}{4\,{b}^{3} \left ({b}^{2}{e}^{2}-2\,bcde+{c}^{2}{d}^{2} \right ) }\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}}+27\,{\frac{e{c}^{4}d}{{b}^{4} \left ({b}^{2}{e}^{2}-2\,bcde+{c}^{2}{d}^{2} \right ) \sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-12\,{\frac{{c}^{5}{d}^{2}}{{b}^{5} \left ({b}^{2}{e}^{2}-2\,bcde+{c}^{2}{d}^{2} \right ) \sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+{\frac{3}{4\,{b}^{3}{x}^{2}{d}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+3\,{\frac{ \left ( ex+d \right ) ^{3/2}c}{e{b}^{4}{x}^{2}d}}-{\frac{5}{4\,{b}^{3}{x}^{2}d}\sqrt{ex+d}}-3\,{\frac{\sqrt{ex+d}c}{e{b}^{4}{x}^{2}}}-{\frac{3\,{e}^{2}}{4\,{b}^{3}}{\it Artanh} \left ({\sqrt{ex+d}{\frac{1}{\sqrt{d}}}} \right ){d}^{-{\frac{5}{2}}}}-3\,{\frac{ce}{{b}^{4}{d}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }-12\,{\frac{{c}^{2}}{{b}^{5}\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)^3,x)

[Out]

-15/4*e^2*c^4/b^3/(c*e*x+b*e)^2/(b^2*e^2-2*b*c*d*e+c^2*d^2)*(e*x+d)^(3/2)+3*e*c^5/b^4/(c*e*x+b*e)^2/(b^2*e^2-2
*b*c*d*e+c^2*d^2)*(e*x+d)^(3/2)*d-17/4*e^2*c^3/b^3/(c*e*x+b*e)^2/(b*e-c*d)*(e*x+d)^(1/2)+3*e*c^4/b^4/(c*e*x+b*
e)^2/(b*e-c*d)*(e*x+d)^(1/2)*d-63/4*e^2*c^3/b^3/(b^2*e^2-2*b*c*d*e+c^2*d^2)/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)
^(1/2)*c/((b*e-c*d)*c)^(1/2))+27*e*c^4/b^4/(b^2*e^2-2*b*c*d*e+c^2*d^2)/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2
)*c/((b*e-c*d)*c)^(1/2))*d-12*c^5/b^5/(b^2*e^2-2*b*c*d*e+c^2*d^2)/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/(
(b*e-c*d)*c)^(1/2))*d^2+3/4/b^3/x^2/d^2*(e*x+d)^(3/2)+3/e/b^4/x^2/d*(e*x+d)^(3/2)*c-5/4/b^3/x^2/d*(e*x+d)^(1/2
)-3/e/b^4/x^2*(e*x+d)^(1/2)*c-3/4*e^2/b^3/d^(5/2)*arctanh((e*x+d)^(1/2)/d^(1/2))-3*e/b^4/d^(3/2)*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., 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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 11.389, size = 5762, normalized size = 19.27 \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)^3,x, algorithm="fricas")

[Out]

[1/8*(3*((16*c^6*d^5 - 36*b*c^5*d^4*e + 21*b^2*c^4*d^3*e^2)*x^4 + 2*(16*b*c^5*d^5 - 36*b^2*c^4*d^4*e + 21*b^3*
c^3*d^3*e^2)*x^3 + (16*b^2*c^4*d^5 - 36*b^3*c^3*d^4*e + 21*b^4*c^2*d^3*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)) + 3*((16*c^6*d^4 - 28*b*c^5*d^3*
e + 9*b^2*c^4*d^2*e^2 + 2*b^3*c^3*d*e^3 + b^4*c^2*e^4)*x^4 + 2*(16*b*c^5*d^4 - 28*b^2*c^4*d^3*e + 9*b^3*c^3*d^
2*e^2 + 2*b^4*c^2*d*e^3 + b^5*c*e^4)*x^3 + (16*b^2*c^4*d^4 - 28*b^3*c^3*d^3*e + 9*b^4*c^2*d^2*e^2 + 2*b^5*c*d*
e^3 + b^6*e^4)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*b^4*c^2*d^4 - 4*b^5*c*d^3*e +
2*b^6*d^2*e^2 - 3*(8*b*c^5*d^4 - 12*b^2*c^4*d^3*e + 2*b^3*c^3*d^2*e^2 + b^4*c^2*d*e^3)*x^3 - (36*b^2*c^4*d^4 -
55*b^3*c^3*d^3*e + 10*b^4*c^2*d^2*e^2 + 6*b^5*c*d*e^3)*x^2 - (8*b^3*c^3*d^4 - 13*b^4*c^2*d^3*e + 2*b^5*c*d^2*
e^2 + 3*b^6*d*e^3)*x)*sqrt(e*x + d))/((b^5*c^4*d^5 - 2*b^6*c^3*d^4*e + b^7*c^2*d^3*e^2)*x^4 + 2*(b^6*c^3*d^5 -
2*b^7*c^2*d^4*e + b^8*c*d^3*e^2)*x^3 + (b^7*c^2*d^5 - 2*b^8*c*d^4*e + b^9*d^3*e^2)*x^2), 1/8*(6*((16*c^6*d^5
- 36*b*c^5*d^4*e + 21*b^2*c^4*d^3*e^2)*x^4 + 2*(16*b*c^5*d^5 - 36*b^2*c^4*d^4*e + 21*b^3*c^3*d^3*e^2)*x^3 + (1
6*b^2*c^4*d^5 - 36*b^3*c^3*d^4*e + 21*b^4*c^2*d^3*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)) + 3*((16*c^6*d^4 - 28*b*c^5*d^3*e + 9*b^2*c^4*d^2*e^2 + 2*b^3*c^3*d*e
^3 + b^4*c^2*e^4)*x^4 + 2*(16*b*c^5*d^4 - 28*b^2*c^4*d^3*e + 9*b^3*c^3*d^2*e^2 + 2*b^4*c^2*d*e^3 + b^5*c*e^4)*
x^3 + (16*b^2*c^4*d^4 - 28*b^3*c^3*d^3*e + 9*b^4*c^2*d^2*e^2 + 2*b^5*c*d*e^3 + b^6*e^4)*x^2)*sqrt(d)*log((e*x
- 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*b^4*c^2*d^4 - 4*b^5*c*d^3*e + 2*b^6*d^2*e^2 - 3*(8*b*c^5*d^4 - 12*b
^2*c^4*d^3*e + 2*b^3*c^3*d^2*e^2 + b^4*c^2*d*e^3)*x^3 - (36*b^2*c^4*d^4 - 55*b^3*c^3*d^3*e + 10*b^4*c^2*d^2*e^
2 + 6*b^5*c*d*e^3)*x^2 - (8*b^3*c^3*d^4 - 13*b^4*c^2*d^3*e + 2*b^5*c*d^2*e^2 + 3*b^6*d*e^3)*x)*sqrt(e*x + d))/
((b^5*c^4*d^5 - 2*b^6*c^3*d^4*e + b^7*c^2*d^3*e^2)*x^4 + 2*(b^6*c^3*d^5 - 2*b^7*c^2*d^4*e + b^8*c*d^3*e^2)*x^3
+ (b^7*c^2*d^5 - 2*b^8*c*d^4*e + b^9*d^3*e^2)*x^2), 1/8*(6*((16*c^6*d^4 - 28*b*c^5*d^3*e + 9*b^2*c^4*d^2*e^2
+ 2*b^3*c^3*d*e^3 + b^4*c^2*e^4)*x^4 + 2*(16*b*c^5*d^4 - 28*b^2*c^4*d^3*e + 9*b^3*c^3*d^2*e^2 + 2*b^4*c^2*d*e^
3 + b^5*c*e^4)*x^3 + (16*b^2*c^4*d^4 - 28*b^3*c^3*d^3*e + 9*b^4*c^2*d^2*e^2 + 2*b^5*c*d*e^3 + b^6*e^4)*x^2)*sq
rt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + 3*((16*c^6*d^5 - 36*b*c^5*d^4*e + 21*b^2*c^4*d^3*e^2)*x^4 + 2*(16*b*
c^5*d^5 - 36*b^2*c^4*d^4*e + 21*b^3*c^3*d^3*e^2)*x^3 + (16*b^2*c^4*d^5 - 36*b^3*c^3*d^4*e + 21*b^4*c^2*d^3*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^2*d^4 - 4*b^5*c*d^3*e + 2*b^6*d^2*e^2 - 3*(8*b*c^5*d^4 - 12*b^2*c^4*d^3*e + 2*b^3*c^3*d^2*e^2
+ b^4*c^2*d*e^3)*x^3 - (36*b^2*c^4*d^4 - 55*b^3*c^3*d^3*e + 10*b^4*c^2*d^2*e^2 + 6*b^5*c*d*e^3)*x^2 - (8*b^3*
c^3*d^4 - 13*b^4*c^2*d^3*e + 2*b^5*c*d^2*e^2 + 3*b^6*d*e^3)*x)*sqrt(e*x + d))/((b^5*c^4*d^5 - 2*b^6*c^3*d^4*e
+ b^7*c^2*d^3*e^2)*x^4 + 2*(b^6*c^3*d^5 - 2*b^7*c^2*d^4*e + b^8*c*d^3*e^2)*x^3 + (b^7*c^2*d^5 - 2*b^8*c*d^4*e
+ b^9*d^3*e^2)*x^2), 1/4*(3*((16*c^6*d^5 - 36*b*c^5*d^4*e + 21*b^2*c^4*d^3*e^2)*x^4 + 2*(16*b*c^5*d^5 - 36*b^2
*c^4*d^4*e + 21*b^3*c^3*d^3*e^2)*x^3 + (16*b^2*c^4*d^5 - 36*b^3*c^3*d^4*e + 21*b^4*c^2*d^3*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)) + 3*((16*c^6*d^4 - 28*b*c^5*
d^3*e + 9*b^2*c^4*d^2*e^2 + 2*b^3*c^3*d*e^3 + b^4*c^2*e^4)*x^4 + 2*(16*b*c^5*d^4 - 28*b^2*c^4*d^3*e + 9*b^3*c^
3*d^2*e^2 + 2*b^4*c^2*d*e^3 + b^5*c*e^4)*x^3 + (16*b^2*c^4*d^4 - 28*b^3*c^3*d^3*e + 9*b^4*c^2*d^2*e^2 + 2*b^5*
c*d*e^3 + b^6*e^4)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) - (2*b^4*c^2*d^4 - 4*b^5*c*d^3*e + 2*b^6*d^2
*e^2 - 3*(8*b*c^5*d^4 - 12*b^2*c^4*d^3*e + 2*b^3*c^3*d^2*e^2 + b^4*c^2*d*e^3)*x^3 - (36*b^2*c^4*d^4 - 55*b^3*c
^3*d^3*e + 10*b^4*c^2*d^2*e^2 + 6*b^5*c*d*e^3)*x^2 - (8*b^3*c^3*d^4 - 13*b^4*c^2*d^3*e + 2*b^5*c*d^2*e^2 + 3*b
^6*d*e^3)*x)*sqrt(e*x + d))/((b^5*c^4*d^5 - 2*b^6*c^3*d^4*e + b^7*c^2*d^3*e^2)*x^4 + 2*(b^6*c^3*d^5 - 2*b^7*c^
2*d^4*e + b^8*c*d^3*e^2)*x^3 + (b^7*c^2*d^5 - 2*b^8*c*d^4*e + b^9*d^3*e^2)*x^2)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.28625, size = 836, normalized size = 2.8 \begin{align*} -\frac{3 \,{\left (16 \, c^{5} d^{2} - 36 \, b c^{4} d e + 21 \, b^{2} c^{3} e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{4 \,{\left (b^{5} c^{2} d^{2} - 2 \, b^{6} c d e + b^{7} e^{2}\right )} \sqrt{-c^{2} d + b c e}} + \frac{24 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{5} d^{3} e - 72 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{5} d^{4} e + 72 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{5} d^{5} e - 24 \, \sqrt{x e + d} c^{5} d^{6} e - 36 \,{\left (x e + d\right )}^{\frac{7}{2}} b c^{4} d^{2} e^{2} + 144 \,{\left (x e + d\right )}^{\frac{5}{2}} b c^{4} d^{3} e^{2} - 180 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{4} d^{4} e^{2} + 72 \, \sqrt{x e + d} b c^{4} d^{5} e^{2} + 6 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{2} c^{3} d e^{3} - 73 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{2} c^{3} d^{2} e^{3} + 136 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} c^{3} d^{3} e^{3} - 69 \, \sqrt{x e + d} b^{2} c^{3} d^{4} e^{3} + 3 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{3} c^{2} e^{4} +{\left (x e + d\right )}^{\frac{5}{2}} b^{3} c^{2} d e^{4} - 24 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} c^{2} d^{2} e^{4} + 18 \, \sqrt{x e + d} b^{3} c^{2} d^{3} e^{4} + 6 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} c e^{5} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} c d e^{5} + 8 \, \sqrt{x e + d} b^{4} c d^{2} e^{5} + 3 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{5} e^{6} - 5 \, \sqrt{x e + d} b^{5} d e^{6}}{4 \,{\left (b^{4} c^{2} d^{4} - 2 \, b^{5} c d^{3} e + b^{6} d^{2} e^{2}\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{3 \,{\left (16 \, c^{2} d^{2} + 4 \, 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^{2}} \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)^3,x, algorithm="giac")

[Out]

-3/4*(16*c^5*d^2 - 36*b*c^4*d*e + 21*b^2*c^3*e^2)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b^5*c^2*d^2 -
2*b^6*c*d*e + b^7*e^2)*sqrt(-c^2*d + b*c*e)) + 1/4*(24*(x*e + d)^(7/2)*c^5*d^3*e - 72*(x*e + d)^(5/2)*c^5*d^4
*e + 72*(x*e + d)^(3/2)*c^5*d^5*e - 24*sqrt(x*e + d)*c^5*d^6*e - 36*(x*e + d)^(7/2)*b*c^4*d^2*e^2 + 144*(x*e +
d)^(5/2)*b*c^4*d^3*e^2 - 180*(x*e + d)^(3/2)*b*c^4*d^4*e^2 + 72*sqrt(x*e + d)*b*c^4*d^5*e^2 + 6*(x*e + d)^(7/
2)*b^2*c^3*d*e^3 - 73*(x*e + d)^(5/2)*b^2*c^3*d^2*e^3 + 136*(x*e + d)^(3/2)*b^2*c^3*d^3*e^3 - 69*sqrt(x*e + d)
*b^2*c^3*d^4*e^3 + 3*(x*e + d)^(7/2)*b^3*c^2*e^4 + (x*e + d)^(5/2)*b^3*c^2*d*e^4 - 24*(x*e + d)^(3/2)*b^3*c^2*
d^2*e^4 + 18*sqrt(x*e + d)*b^3*c^2*d^3*e^4 + 6*(x*e + d)^(5/2)*b^4*c*e^5 - 10*(x*e + d)^(3/2)*b^4*c*d*e^5 + 8*
sqrt(x*e + d)*b^4*c*d^2*e^5 + 3*(x*e + d)^(3/2)*b^5*e^6 - 5*sqrt(x*e + d)*b^5*d*e^6)/((b^4*c^2*d^4 - 2*b^5*c*d
^3*e + b^6*d^2*e^2)*((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e)^2) + 3/4*(16*c^2*d^2 + 4
*b*c*d*e + b^2*e^2)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^5*sqrt(-d)*d^2)