### 3.384 $$\int \frac{1}{(d+e x)^{3/2} (b x+c x^2)^3} \, dx$$

Optimal. Leaf size=370 $\frac{c x (2 c d-b e) \left (-5 b^2 e^2-12 b c d e+12 c^2 d^2\right )+b \left (5 b^3 e^3-17 b c^2 d^2 e+12 c^3 d^3\right )}{4 b^4 d^2 \left (b x+c x^2\right ) \sqrt{d+e x} (c d-b e)^2}+\frac{3 e \left (-b^2 e^2-b c d e+c^2 d^2\right ) \left (5 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{4 b^4 d^3 \sqrt{d+e x} (c d-b e)^3}+\frac{3 c^{7/2} \left (33 b^2 e^2-44 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{7/2}}-\frac{3 \left (5 b^2 e^2+12 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5 d^{7/2}}-\frac{c x (2 c d-b e)+b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 \sqrt{d+e x} (c d-b e)}$

[Out]

(3*e*(c^2*d^2 - b*c*d*e - b^2*e^2)*(8*c^2*d^2 - 8*b*c*d*e + 5*b^2*e^2))/(4*b^4*d^3*(c*d - b*e)^3*Sqrt[d + e*x]
) - (b*(c*d - b*e) + c*(2*c*d - b*e)*x)/(2*b^2*d*(c*d - b*e)*Sqrt[d + e*x]*(b*x + c*x^2)^2) + (b*(12*c^3*d^3 -
17*b*c^2*d^2*e + 5*b^3*e^3) + c*(2*c*d - b*e)*(12*c^2*d^2 - 12*b*c*d*e - 5*b^2*e^2)*x)/(4*b^4*d^2*(c*d - b*e)
^2*Sqrt[d + e*x]*(b*x + c*x^2)) - (3*(16*c^2*d^2 + 12*b*c*d*e + 5*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*
b^5*d^(7/2)) + (3*c^(7/2)*(16*c^2*d^2 - 44*b*c*d*e + 33*b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*
e]])/(4*b^5*(c*d - b*e)^(7/2))

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Rubi [A]  time = 0.600412, antiderivative size = 370, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.286, Rules used = {740, 822, 828, 826, 1166, 208} $\frac{c x (2 c d-b e) \left (-5 b^2 e^2-12 b c d e+12 c^2 d^2\right )+b \left (5 b^3 e^3-17 b c^2 d^2 e+12 c^3 d^3\right )}{4 b^4 d^2 \left (b x+c x^2\right ) \sqrt{d+e x} (c d-b e)^2}+\frac{3 e \left (-b^2 e^2-b c d e+c^2 d^2\right ) \left (5 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{4 b^4 d^3 \sqrt{d+e x} (c d-b e)^3}+\frac{3 c^{7/2} \left (33 b^2 e^2-44 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{7/2}}-\frac{3 \left (5 b^2 e^2+12 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5 d^{7/2}}-\frac{c x (2 c d-b e)+b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 \sqrt{d+e x} (c d-b e)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(3*e*(c^2*d^2 - b*c*d*e - b^2*e^2)*(8*c^2*d^2 - 8*b*c*d*e + 5*b^2*e^2))/(4*b^4*d^3*(c*d - b*e)^3*Sqrt[d + e*x]
) - (b*(c*d - b*e) + c*(2*c*d - b*e)*x)/(2*b^2*d*(c*d - b*e)*Sqrt[d + e*x]*(b*x + c*x^2)^2) + (b*(12*c^3*d^3 -
17*b*c^2*d^2*e + 5*b^3*e^3) + c*(2*c*d - b*e)*(12*c^2*d^2 - 12*b*c*d*e - 5*b^2*e^2)*x)/(4*b^4*d^2*(c*d - b*e)
^2*Sqrt[d + e*x]*(b*x + c*x^2)) - (3*(16*c^2*d^2 + 12*b*c*d*e + 5*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*
b^5*d^(7/2)) + (3*c^(7/2)*(16*c^2*d^2 - 44*b*c*d*e + 33*b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*
e]])/(4*b^5*(c*d - b*e)^(7/2))

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 828

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[((
e*f - d*g)*(d + e*x)^(m + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((d
+ e*x)^(m + 1)*Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x])/(a + b*x + c*x^2), 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] && FractionQ[m] && LtQ[m, -1]

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}{(d+e x)^{3/2} \left (b x+c x^2\right )^3} \, dx &=-\frac{b (c d-b e)+c (2 c d-b e) x}{2 b^2 d (c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^2}-\frac{\int \frac{\frac{1}{2} \left (12 c^2 d^2-5 b c d e-5 b^2 e^2\right )+\frac{7}{2} c e (2 c d-b e) x}{(d+e x)^{3/2} \left (b x+c x^2\right )^2} \, dx}{2 b^2 d (c d-b e)}\\ &=-\frac{b (c d-b e)+c (2 c d-b e) x}{2 b^2 d (c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^2}+\frac{b \left (12 c^3 d^3-17 b c^2 d^2 e+5 b^3 e^3\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-5 b^2 e^2\right ) x}{4 b^4 d^2 (c d-b e)^2 \sqrt{d+e x} \left (b x+c x^2\right )}+\frac{\int \frac{\frac{3}{4} (c d-b e)^2 \left (16 c^2 d^2+12 b c d e+5 b^2 e^2\right )+\frac{3}{4} c e (2 c d-b e) \left (12 c^2 d^2-12 b c d e-5 b^2 e^2\right ) x}{(d+e x)^{3/2} \left (b x+c x^2\right )} \, dx}{2 b^4 d^2 (c d-b e)^2}\\ &=\frac{3 e \left (c^2 d^2-b c d e-b^2 e^2\right ) \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right )}{4 b^4 d^3 (c d-b e)^3 \sqrt{d+e x}}-\frac{b (c d-b e)+c (2 c d-b e) x}{2 b^2 d (c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^2}+\frac{b \left (12 c^3 d^3-17 b c^2 d^2 e+5 b^3 e^3\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-5 b^2 e^2\right ) x}{4 b^4 d^2 (c d-b e)^2 \sqrt{d+e x} \left (b x+c x^2\right )}+\frac{\int \frac{\frac{3}{4} (c d-b e)^3 \left (16 c^2 d^2+12 b c d e+5 b^2 e^2\right )+\frac{3}{4} c e \left (c^2 d^2-b c d e-b^2 e^2\right ) \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right ) x}{\sqrt{d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 d^3 (c d-b e)^3}\\ &=\frac{3 e \left (c^2 d^2-b c d e-b^2 e^2\right ) \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right )}{4 b^4 d^3 (c d-b e)^3 \sqrt{d+e x}}-\frac{b (c d-b e)+c (2 c d-b e) x}{2 b^2 d (c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^2}+\frac{b \left (12 c^3 d^3-17 b c^2 d^2 e+5 b^3 e^3\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-5 b^2 e^2\right ) x}{4 b^4 d^2 (c d-b e)^2 \sqrt{d+e x} \left (b x+c x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{3}{4} c d e \left (c^2 d^2-b c d e-b^2 e^2\right ) \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right )+\frac{3}{4} e (c d-b e)^3 \left (16 c^2 d^2+12 b c d e+5 b^2 e^2\right )+\frac{3}{4} c e \left (c^2 d^2-b c d e-b^2 e^2\right ) \left (8 c^2 d^2-8 b c d e+5 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^3 (c d-b e)^3}\\ &=\frac{3 e \left (c^2 d^2-b c d e-b^2 e^2\right ) \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right )}{4 b^4 d^3 (c d-b e)^3 \sqrt{d+e x}}-\frac{b (c d-b e)+c (2 c d-b e) x}{2 b^2 d (c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^2}+\frac{b \left (12 c^3 d^3-17 b c^2 d^2 e+5 b^3 e^3\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-5 b^2 e^2\right ) x}{4 b^4 d^2 (c d-b e)^2 \sqrt{d+e x} \left (b x+c x^2\right )}+\frac{\left (3 c \left (16 c^2 d^2+12 b c d e+5 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^3}-\frac{\left (3 c^4 \left (16 c^2 d^2-44 b c d e+33 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)^3}\\ &=\frac{3 e \left (c^2 d^2-b c d e-b^2 e^2\right ) \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right )}{4 b^4 d^3 (c d-b e)^3 \sqrt{d+e x}}-\frac{b (c d-b e)+c (2 c d-b e) x}{2 b^2 d (c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^2}+\frac{b \left (12 c^3 d^3-17 b c^2 d^2 e+5 b^3 e^3\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-5 b^2 e^2\right ) x}{4 b^4 d^2 (c d-b e)^2 \sqrt{d+e x} \left (b x+c x^2\right )}-\frac{3 \left (16 c^2 d^2+12 b c d e+5 b^2 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5 d^{7/2}}+\frac{3 c^{7/2} \left (16 c^2 d^2-44 b c d e+33 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)^{7/2}}\\ \end{align*}

Mathematica [C]  time = 0.2893, size = 299, normalized size = 0.81 $\frac{-x^2 \left ((b+c x) \left (b c d (b e-c d) \left (2 b^2 c d e^2+5 b^3 e^3-36 b c^2 d^2 e+24 c^3 d^3\right )-(b+c x) \left (3 (c d-b e)^3 \left (5 b^2 e^2+12 b c d e+16 c^2 d^2\right ) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{e x}{d}+1\right )-3 c^3 d^3 \left (33 b^2 e^2-44 b c d e+16 c^2 d^2\right ) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{c (d+e x)}{c d-b e}\right )\right )\right )+b^2 c d \left (5 b^2 e^2+5 b c d e-12 c^2 d^2\right ) (c d-b e)^2\right )+2 b^4 d^2 (b e-c d)^3+b^3 d x (c d-b e)^3 (5 b e+8 c d)}{4 b^5 d^3 x^2 (b+c x)^2 \sqrt{d+e x} (c d-b e)^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*b^4*d^2*(-(c*d) + b*e)^3 + b^3*d*(c*d - b*e)^3*(8*c*d + 5*b*e)*x - x^2*(b^2*c*d*(c*d - b*e)^2*(-12*c^2*d^2
+ 5*b*c*d*e + 5*b^2*e^2) + (b + c*x)*(b*c*d*(-(c*d) + b*e)*(24*c^3*d^3 - 36*b*c^2*d^2*e + 2*b^2*c*d*e^2 + 5*b^
3*e^3) - (b + c*x)*(-3*c^3*d^3*(16*c^2*d^2 - 44*b*c*d*e + 33*b^2*e^2)*Hypergeometric2F1[-1/2, 1, 1/2, (c*(d +
e*x))/(c*d - b*e)] + 3*(c*d - b*e)^3*(16*c^2*d^2 + 12*b*c*d*e + 5*b^2*e^2)*Hypergeometric2F1[-1/2, 1, 1/2, 1 +
(e*x)/d]))))/(4*b^5*d^3*(c*d - b*e)^3*x^2*(b + c*x)^2*Sqrt[d + e*x])

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Maple [A]  time = 0.399, size = 530, normalized size = 1.4 \begin{align*} 2\,{\frac{{e}^{5}}{{d}^{3} \left ( be-cd \right ) ^{3}\sqrt{ex+d}}}+{\frac{19\,{e}^{2}{c}^{5}}{4\,{b}^{3} \left ( be-cd \right ) ^{3} \left ( cex+be \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-3\,{\frac{e{c}^{6} \left ( ex+d \right ) ^{3/2}d}{{b}^{4} \left ( be-cd \right ) ^{3} \left ( cex+be \right ) ^{2}}}+{\frac{21\,{e}^{3}{c}^{4}}{4\,{b}^{2} \left ( be-cd \right ) ^{3} \left ( cex+be \right ) ^{2}}\sqrt{ex+d}}-{\frac{33\,{e}^{2}{c}^{5}d}{4\,{b}^{3} \left ( be-cd \right ) ^{3} \left ( cex+be \right ) ^{2}}\sqrt{ex+d}}+3\,{\frac{e{c}^{6}\sqrt{ex+d}{d}^{2}}{{b}^{4} \left ( be-cd \right ) ^{3} \left ( cex+be \right ) ^{2}}}+{\frac{99\,{e}^{2}{c}^{4}}{4\,{b}^{3} \left ( be-cd \right ) ^{3}}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}}-33\,{\frac{e{c}^{5}d}{{b}^{4} \left ( be-cd \right ) ^{3}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+12\,{\frac{{c}^{6}{d}^{2}}{{b}^{5} \left ( be-cd \right ) ^{3}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+{\frac{7}{4\,{d}^{3}{b}^{3}{x}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+3\,{\frac{ \left ( ex+d \right ) ^{3/2}c}{e{d}^{2}{b}^{4}{x}^{2}}}-{\frac{9}{4\,{d}^{2}{b}^{3}{x}^{2}}\sqrt{ex+d}}-3\,{\frac{\sqrt{ex+d}c}{de{b}^{4}{x}^{2}}}-{\frac{15\,{e}^{2}}{4\,{b}^{3}}{\it Artanh} \left ({\sqrt{ex+d}{\frac{1}{\sqrt{d}}}} \right ){d}^{-{\frac{7}{2}}}}-9\,{\frac{ce}{{d}^{5/2}{b}^{4}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }-12\,{\frac{{c}^{2}}{{d}^{3/2}{b}^{5}}{\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)^(3/2)/(c*x^2+b*x)^3,x)

[Out]

2*e^5/d^3/(b*e-c*d)^3/(e*x+d)^(1/2)+19/4*e^2*c^5/b^3/(b*e-c*d)^3/(c*e*x+b*e)^2*(e*x+d)^(3/2)-3*e*c^6/b^4/(b*e-
c*d)^3/(c*e*x+b*e)^2*(e*x+d)^(3/2)*d+21/4*e^3*c^4/b^2/(b*e-c*d)^3/(c*e*x+b*e)^2*(e*x+d)^(1/2)-33/4*e^2*c^5/b^3
/(b*e-c*d)^3/(c*e*x+b*e)^2*(e*x+d)^(1/2)*d+3*e*c^6/b^4/(b*e-c*d)^3/(c*e*x+b*e)^2*(e*x+d)^(1/2)*d^2+99/4*e^2*c^
4/b^3/(b*e-c*d)^3/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))-33*e*c^5/b^4/(b*e-c*d)^3/((b
*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*d+12*c^6/b^5/(b*e-c*d)^3/((b*e-c*d)*c)^(1/2)*arct
an((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*d^2+7/4/d^3/b^3/x^2*(e*x+d)^(3/2)+3/e/d^2/b^4/x^2*(e*x+d)^(3/2)*c-9/4/
d^2/b^3/x^2*(e*x+d)^(1/2)-3/e/d/b^4/x^2*(e*x+d)^(1/2)*c-15/4*e^2/d^(7/2)/b^3*arctanh((e*x+d)^(1/2)/d^(1/2))-9*
e/d^(5/2)/b^4*arctanh((e*x+d)^(1/2)/d^(1/2))*c-12/d^(3/2)/b^5*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)^(3/2)/(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 39.5685, size = 9646, normalized size = 26.07 \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)^(3/2)/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

[-1/8*(3*((16*c^7*d^6*e - 44*b*c^6*d^5*e^2 + 33*b^2*c^5*d^4*e^3)*x^5 + (16*c^7*d^7 - 12*b*c^6*d^6*e - 55*b^2*c
^5*d^5*e^2 + 66*b^3*c^4*d^4*e^3)*x^4 + (32*b*c^6*d^7 - 72*b^2*c^5*d^6*e + 22*b^3*c^4*d^5*e^2 + 33*b^4*c^3*d^4*
e^3)*x^3 + (16*b^2*c^5*d^7 - 44*b^3*c^4*d^6*e + 33*b^4*c^3*d^5*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^7*d^5*e - 36*b*c^6*d^4*e^2 +
17*b^2*c^5*d^3*e^3 + 5*b^3*c^4*d^2*e^4 + 3*b^4*c^3*d*e^5 - 5*b^5*c^2*e^6)*x^5 + (16*c^7*d^6 - 4*b*c^6*d^5*e -
55*b^2*c^5*d^4*e^2 + 39*b^3*c^4*d^3*e^3 + 13*b^4*c^3*d^2*e^4 + b^5*c^2*d*e^5 - 10*b^6*c*e^6)*x^4 + (32*b*c^6*d
^6 - 56*b^2*c^5*d^5*e - 2*b^3*c^4*d^4*e^2 + 27*b^4*c^3*d^3*e^3 + 11*b^5*c^2*d^2*e^4 - 7*b^6*c*d*e^5 - 5*b^7*e^
6)*x^3 + (16*b^2*c^5*d^6 - 36*b^3*c^4*d^5*e + 17*b^4*c^3*d^4*e^2 + 5*b^5*c^2*d^3*e^3 + 3*b^6*c*d^2*e^4 - 5*b^7
*d*e^5)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(2*b^4*c^3*d^6 - 6*b^5*c^2*d^5*e + 6*b^6
*c*d^4*e^2 - 2*b^7*d^3*e^3 - 3*(8*b*c^6*d^5*e - 16*b^2*c^5*d^4*e^2 + 5*b^3*c^4*d^3*e^3 + 3*b^4*c^3*d^2*e^4 - 5
*b^5*c^2*d*e^5)*x^4 - (24*b*c^6*d^6 - 12*b^2*c^5*d^5*e - 58*b^3*c^4*d^4*e^2 + 33*b^4*c^3*d^3*e^3 + 13*b^5*c^2*
d^2*e^4 - 30*b^6*c*d*e^5)*x^3 - (36*b^2*c^5*d^6 - 65*b^3*c^4*d^5*e + 7*b^4*c^3*d^4*e^2 + 23*b^5*c^2*d^3*e^3 -
b^6*c*d^2*e^4 - 15*b^7*d*e^5)*x^2 - (8*b^3*c^4*d^6 - 19*b^4*c^3*d^5*e + 9*b^5*c^2*d^4*e^2 + 7*b^6*c*d^3*e^3 -
5*b^7*d^2*e^4)*x)*sqrt(e*x + d))/((b^5*c^5*d^7*e - 3*b^6*c^4*d^6*e^2 + 3*b^7*c^3*d^5*e^3 - b^8*c^2*d^4*e^4)*x^
5 + (b^5*c^5*d^8 - b^6*c^4*d^7*e - 3*b^7*c^3*d^6*e^2 + 5*b^8*c^2*d^5*e^3 - 2*b^9*c*d^4*e^4)*x^4 + (2*b^6*c^4*d
^8 - 5*b^7*c^3*d^7*e + 3*b^8*c^2*d^6*e^2 + b^9*c*d^5*e^3 - b^10*d^4*e^4)*x^3 + (b^7*c^3*d^8 - 3*b^8*c^2*d^7*e
+ 3*b^9*c*d^6*e^2 - b^10*d^5*e^3)*x^2), 1/8*(6*((16*c^7*d^6*e - 44*b*c^6*d^5*e^2 + 33*b^2*c^5*d^4*e^3)*x^5 + (
16*c^7*d^7 - 12*b*c^6*d^6*e - 55*b^2*c^5*d^5*e^2 + 66*b^3*c^4*d^4*e^3)*x^4 + (32*b*c^6*d^7 - 72*b^2*c^5*d^6*e
+ 22*b^3*c^4*d^5*e^2 + 33*b^4*c^3*d^4*e^3)*x^3 + (16*b^2*c^5*d^7 - 44*b^3*c^4*d^6*e + 33*b^4*c^3*d^5*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^7*d^5*e
- 36*b*c^6*d^4*e^2 + 17*b^2*c^5*d^3*e^3 + 5*b^3*c^4*d^2*e^4 + 3*b^4*c^3*d*e^5 - 5*b^5*c^2*e^6)*x^5 + (16*c^7*
d^6 - 4*b*c^6*d^5*e - 55*b^2*c^5*d^4*e^2 + 39*b^3*c^4*d^3*e^3 + 13*b^4*c^3*d^2*e^4 + b^5*c^2*d*e^5 - 10*b^6*c*
e^6)*x^4 + (32*b*c^6*d^6 - 56*b^2*c^5*d^5*e - 2*b^3*c^4*d^4*e^2 + 27*b^4*c^3*d^3*e^3 + 11*b^5*c^2*d^2*e^4 - 7*
b^6*c*d*e^5 - 5*b^7*e^6)*x^3 + (16*b^2*c^5*d^6 - 36*b^3*c^4*d^5*e + 17*b^4*c^3*d^4*e^2 + 5*b^5*c^2*d^3*e^3 + 3
*b^6*c*d^2*e^4 - 5*b^7*d*e^5)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*b^4*c^3*d^6 - 6
*b^5*c^2*d^5*e + 6*b^6*c*d^4*e^2 - 2*b^7*d^3*e^3 - 3*(8*b*c^6*d^5*e - 16*b^2*c^5*d^4*e^2 + 5*b^3*c^4*d^3*e^3 +
3*b^4*c^3*d^2*e^4 - 5*b^5*c^2*d*e^5)*x^4 - (24*b*c^6*d^6 - 12*b^2*c^5*d^5*e - 58*b^3*c^4*d^4*e^2 + 33*b^4*c^3
*d^3*e^3 + 13*b^5*c^2*d^2*e^4 - 30*b^6*c*d*e^5)*x^3 - (36*b^2*c^5*d^6 - 65*b^3*c^4*d^5*e + 7*b^4*c^3*d^4*e^2 +
23*b^5*c^2*d^3*e^3 - b^6*c*d^2*e^4 - 15*b^7*d*e^5)*x^2 - (8*b^3*c^4*d^6 - 19*b^4*c^3*d^5*e + 9*b^5*c^2*d^4*e^
2 + 7*b^6*c*d^3*e^3 - 5*b^7*d^2*e^4)*x)*sqrt(e*x + d))/((b^5*c^5*d^7*e - 3*b^6*c^4*d^6*e^2 + 3*b^7*c^3*d^5*e^3
- b^8*c^2*d^4*e^4)*x^5 + (b^5*c^5*d^8 - b^6*c^4*d^7*e - 3*b^7*c^3*d^6*e^2 + 5*b^8*c^2*d^5*e^3 - 2*b^9*c*d^4*e
^4)*x^4 + (2*b^6*c^4*d^8 - 5*b^7*c^3*d^7*e + 3*b^8*c^2*d^6*e^2 + b^9*c*d^5*e^3 - b^10*d^4*e^4)*x^3 + (b^7*c^3*
d^8 - 3*b^8*c^2*d^7*e + 3*b^9*c*d^6*e^2 - b^10*d^5*e^3)*x^2), 1/8*(6*((16*c^7*d^5*e - 36*b*c^6*d^4*e^2 + 17*b^
2*c^5*d^3*e^3 + 5*b^3*c^4*d^2*e^4 + 3*b^4*c^3*d*e^5 - 5*b^5*c^2*e^6)*x^5 + (16*c^7*d^6 - 4*b*c^6*d^5*e - 55*b^
2*c^5*d^4*e^2 + 39*b^3*c^4*d^3*e^3 + 13*b^4*c^3*d^2*e^4 + b^5*c^2*d*e^5 - 10*b^6*c*e^6)*x^4 + (32*b*c^6*d^6 -
56*b^2*c^5*d^5*e - 2*b^3*c^4*d^4*e^2 + 27*b^4*c^3*d^3*e^3 + 11*b^5*c^2*d^2*e^4 - 7*b^6*c*d*e^5 - 5*b^7*e^6)*x^
3 + (16*b^2*c^5*d^6 - 36*b^3*c^4*d^5*e + 17*b^4*c^3*d^4*e^2 + 5*b^5*c^2*d^3*e^3 + 3*b^6*c*d^2*e^4 - 5*b^7*d*e^
5)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) - 3*((16*c^7*d^6*e - 44*b*c^6*d^5*e^2 + 33*b^2*c^5*d^4*e^3)*
x^5 + (16*c^7*d^7 - 12*b*c^6*d^6*e - 55*b^2*c^5*d^5*e^2 + 66*b^3*c^4*d^4*e^3)*x^4 + (32*b*c^6*d^7 - 72*b^2*c^5
*d^6*e + 22*b^3*c^4*d^5*e^2 + 33*b^4*c^3*d^4*e^3)*x^3 + (16*b^2*c^5*d^7 - 44*b^3*c^4*d^6*e + 33*b^4*c^3*d^5*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^3*d^6 - 6*b^5*c^2*d^5*e + 6*b^6*c*d^4*e^2 - 2*b^7*d^3*e^3 - 3*(8*b*c^6*d^5*e - 16*b^2*c^5*d^
4*e^2 + 5*b^3*c^4*d^3*e^3 + 3*b^4*c^3*d^2*e^4 - 5*b^5*c^2*d*e^5)*x^4 - (24*b*c^6*d^6 - 12*b^2*c^5*d^5*e - 58*b
^3*c^4*d^4*e^2 + 33*b^4*c^3*d^3*e^3 + 13*b^5*c^2*d^2*e^4 - 30*b^6*c*d*e^5)*x^3 - (36*b^2*c^5*d^6 - 65*b^3*c^4*
d^5*e + 7*b^4*c^3*d^4*e^2 + 23*b^5*c^2*d^3*e^3 - b^6*c*d^2*e^4 - 15*b^7*d*e^5)*x^2 - (8*b^3*c^4*d^6 - 19*b^4*c
^3*d^5*e + 9*b^5*c^2*d^4*e^2 + 7*b^6*c*d^3*e^3 - 5*b^7*d^2*e^4)*x)*sqrt(e*x + d))/((b^5*c^5*d^7*e - 3*b^6*c^4*
d^6*e^2 + 3*b^7*c^3*d^5*e^3 - b^8*c^2*d^4*e^4)*x^5 + (b^5*c^5*d^8 - b^6*c^4*d^7*e - 3*b^7*c^3*d^6*e^2 + 5*b^8*
c^2*d^5*e^3 - 2*b^9*c*d^4*e^4)*x^4 + (2*b^6*c^4*d^8 - 5*b^7*c^3*d^7*e + 3*b^8*c^2*d^6*e^2 + b^9*c*d^5*e^3 - b^
10*d^4*e^4)*x^3 + (b^7*c^3*d^8 - 3*b^8*c^2*d^7*e + 3*b^9*c*d^6*e^2 - b^10*d^5*e^3)*x^2), 1/4*(3*((16*c^7*d^6*e
- 44*b*c^6*d^5*e^2 + 33*b^2*c^5*d^4*e^3)*x^5 + (16*c^7*d^7 - 12*b*c^6*d^6*e - 55*b^2*c^5*d^5*e^2 + 66*b^3*c^4
*d^4*e^3)*x^4 + (32*b*c^6*d^7 - 72*b^2*c^5*d^6*e + 22*b^3*c^4*d^5*e^2 + 33*b^4*c^3*d^4*e^3)*x^3 + (16*b^2*c^5*
d^7 - 44*b^3*c^4*d^6*e + 33*b^4*c^3*d^5*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^7*d^5*e - 36*b*c^6*d^4*e^2 + 17*b^2*c^5*d^3*e^3 + 5*b^3*c^4*d^2*e^4
+ 3*b^4*c^3*d*e^5 - 5*b^5*c^2*e^6)*x^5 + (16*c^7*d^6 - 4*b*c^6*d^5*e - 55*b^2*c^5*d^4*e^2 + 39*b^3*c^4*d^3*e^3
+ 13*b^4*c^3*d^2*e^4 + b^5*c^2*d*e^5 - 10*b^6*c*e^6)*x^4 + (32*b*c^6*d^6 - 56*b^2*c^5*d^5*e - 2*b^3*c^4*d^4*e
^2 + 27*b^4*c^3*d^3*e^3 + 11*b^5*c^2*d^2*e^4 - 7*b^6*c*d*e^5 - 5*b^7*e^6)*x^3 + (16*b^2*c^5*d^6 - 36*b^3*c^4*d
^5*e + 17*b^4*c^3*d^4*e^2 + 5*b^5*c^2*d^3*e^3 + 3*b^6*c*d^2*e^4 - 5*b^7*d*e^5)*x^2)*sqrt(-d)*arctan(sqrt(e*x +
d)*sqrt(-d)/d) - (2*b^4*c^3*d^6 - 6*b^5*c^2*d^5*e + 6*b^6*c*d^4*e^2 - 2*b^7*d^3*e^3 - 3*(8*b*c^6*d^5*e - 16*b
^2*c^5*d^4*e^2 + 5*b^3*c^4*d^3*e^3 + 3*b^4*c^3*d^2*e^4 - 5*b^5*c^2*d*e^5)*x^4 - (24*b*c^6*d^6 - 12*b^2*c^5*d^5
*e - 58*b^3*c^4*d^4*e^2 + 33*b^4*c^3*d^3*e^3 + 13*b^5*c^2*d^2*e^4 - 30*b^6*c*d*e^5)*x^3 - (36*b^2*c^5*d^6 - 65
*b^3*c^4*d^5*e + 7*b^4*c^3*d^4*e^2 + 23*b^5*c^2*d^3*e^3 - b^6*c*d^2*e^4 - 15*b^7*d*e^5)*x^2 - (8*b^3*c^4*d^6 -
19*b^4*c^3*d^5*e + 9*b^5*c^2*d^4*e^2 + 7*b^6*c*d^3*e^3 - 5*b^7*d^2*e^4)*x)*sqrt(e*x + d))/((b^5*c^5*d^7*e - 3
*b^6*c^4*d^6*e^2 + 3*b^7*c^3*d^5*e^3 - b^8*c^2*d^4*e^4)*x^5 + (b^5*c^5*d^8 - b^6*c^4*d^7*e - 3*b^7*c^3*d^6*e^2
+ 5*b^8*c^2*d^5*e^3 - 2*b^9*c*d^4*e^4)*x^4 + (2*b^6*c^4*d^8 - 5*b^7*c^3*d^7*e + 3*b^8*c^2*d^6*e^2 + b^9*c*d^5
*e^3 - b^10*d^4*e^4)*x^3 + (b^7*c^3*d^8 - 3*b^8*c^2*d^7*e + 3*b^9*c*d^6*e^2 - b^10*d^5*e^3)*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)**(3/2)/(c*x**2+b*x)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.44012, size = 1062, normalized size = 2.87 \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)^(3/2)/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

-3/4*(16*c^6*d^2 - 44*b*c^5*d*e + 33*b^2*c^4*e^2)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b^5*c^3*d^3 -
3*b^6*c^2*d^2*e + 3*b^7*c*d*e^2 - b^8*e^3)*sqrt(-c^2*d + b*c*e)) - 2*e^5/((c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*
d^4*e^2 - b^3*d^3*e^3)*sqrt(x*e + d)) + 1/4*(24*(x*e + d)^(7/2)*c^6*d^4*e - 72*(x*e + d)^(5/2)*c^6*d^5*e + 72*
(x*e + d)^(3/2)*c^6*d^6*e - 24*sqrt(x*e + d)*c^6*d^7*e - 48*(x*e + d)^(7/2)*b*c^5*d^3*e^2 + 180*(x*e + d)^(5/2
)*b*c^5*d^4*e^2 - 216*(x*e + d)^(3/2)*b*c^5*d^5*e^2 + 84*sqrt(x*e + d)*b*c^5*d^6*e^2 + 15*(x*e + d)^(7/2)*b^2*
c^4*d^2*e^3 - 118*(x*e + d)^(5/2)*b^2*c^4*d^3*e^3 + 199*(x*e + d)^(3/2)*b^2*c^4*d^4*e^3 - 96*sqrt(x*e + d)*b^2
*c^4*d^5*e^3 + 9*(x*e + d)^(7/2)*b^3*c^3*d*e^4 - 3*(x*e + d)^(5/2)*b^3*c^3*d^2*e^4 - 38*(x*e + d)^(3/2)*b^3*c^
3*d^3*e^4 + 30*sqrt(x*e + d)*b^3*c^3*d^4*e^4 - 7*(x*e + d)^(7/2)*b^4*c^2*e^5 + 41*(x*e + d)^(5/2)*b^4*c^2*d*e^
5 - 58*(x*e + d)^(3/2)*b^4*c^2*d^2*e^5 + 30*sqrt(x*e + d)*b^4*c^2*d^3*e^5 - 14*(x*e + d)^(5/2)*b^5*c*e^6 + 41*
(x*e + d)^(3/2)*b^5*c*d*e^6 - 33*sqrt(x*e + d)*b^5*c*d^2*e^6 - 7*(x*e + d)^(3/2)*b^6*e^7 + 9*sqrt(x*e + d)*b^6
*d*e^7)/((b^4*c^3*d^6 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4*e^2 - b^7*d^3*e^3)*((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 + 12*b*c*d*e + 5*b^2*e^2)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^
5*sqrt(-d)*d^3)