3.377 \(\int \frac{1}{(d+e x)^{7/2} (b x+c x^2)^2} \, dx\)

Optimal. Leaf size=349 \[ -\frac{e \left (26 b^2 c^2 d^2 e^2-24 b^3 c d e^3+7 b^4 e^4-4 b c^3 d^3 e+2 c^4 d^4\right )}{b^2 d^4 \sqrt{d+e x} (c d-b e)^4}-\frac{e (2 c d-b e) \left (7 b^2 e^2-3 b c d e+3 c^2 d^2\right )}{3 b^2 d^3 (d+e x)^{3/2} (c d-b e)^3}-\frac{e \left (7 b^2 e^2-10 b c d e+10 c^2 d^2\right )}{5 b^2 d^2 (d+e x)^{5/2} (c d-b e)^2}-\frac{c^{9/2} (4 c d-11 b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 (c d-b e)^{9/2}}+\frac{(7 b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3 d^{9/2}}-\frac{c x (2 c d-b e)+b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{5/2} (c d-b e)} \]

[Out]

-(e*(10*c^2*d^2 - 10*b*c*d*e + 7*b^2*e^2))/(5*b^2*d^2*(c*d - b*e)^2*(d + e*x)^(5/2)) - (e*(2*c*d - b*e)*(3*c^2
*d^2 - 3*b*c*d*e + 7*b^2*e^2))/(3*b^2*d^3*(c*d - b*e)^3*(d + e*x)^(3/2)) - (e*(2*c^4*d^4 - 4*b*c^3*d^3*e + 26*
b^2*c^2*d^2*e^2 - 24*b^3*c*d*e^3 + 7*b^4*e^4))/(b^2*d^4*(c*d - b*e)^4*Sqrt[d + e*x]) - (b*(c*d - b*e) + c*(2*c
*d - b*e)*x)/(b^2*d*(c*d - b*e)*(d + e*x)^(5/2)*(b*x + c*x^2)) + ((4*c*d + 7*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d
]])/(b^3*d^(9/2)) - (c^(9/2)*(4*c*d - 11*b*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*(c*d - b*
e)^(9/2))

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Rubi [A]  time = 0.738855, antiderivative size = 349, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {740, 828, 826, 1166, 208} \[ -\frac{e \left (26 b^2 c^2 d^2 e^2-24 b^3 c d e^3+7 b^4 e^4-4 b c^3 d^3 e+2 c^4 d^4\right )}{b^2 d^4 \sqrt{d+e x} (c d-b e)^4}-\frac{e (2 c d-b e) \left (7 b^2 e^2-3 b c d e+3 c^2 d^2\right )}{3 b^2 d^3 (d+e x)^{3/2} (c d-b e)^3}-\frac{e \left (7 b^2 e^2-10 b c d e+10 c^2 d^2\right )}{5 b^2 d^2 (d+e x)^{5/2} (c d-b e)^2}-\frac{c^{9/2} (4 c d-11 b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 (c d-b e)^{9/2}}+\frac{(7 b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3 d^{9/2}}-\frac{c x (2 c d-b e)+b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{5/2} (c d-b e)} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(7/2)*(b*x + c*x^2)^2),x]

[Out]

-(e*(10*c^2*d^2 - 10*b*c*d*e + 7*b^2*e^2))/(5*b^2*d^2*(c*d - b*e)^2*(d + e*x)^(5/2)) - (e*(2*c*d - b*e)*(3*c^2
*d^2 - 3*b*c*d*e + 7*b^2*e^2))/(3*b^2*d^3*(c*d - b*e)^3*(d + e*x)^(3/2)) - (e*(2*c^4*d^4 - 4*b*c^3*d^3*e + 26*
b^2*c^2*d^2*e^2 - 24*b^3*c*d*e^3 + 7*b^4*e^4))/(b^2*d^4*(c*d - b*e)^4*Sqrt[d + e*x]) - (b*(c*d - b*e) + c*(2*c
*d - b*e)*x)/(b^2*d*(c*d - b*e)*(d + e*x)^(5/2)*(b*x + c*x^2)) + ((4*c*d + 7*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d
]])/(b^3*d^(9/2)) - (c^(9/2)*(4*c*d - 11*b*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*(c*d - b*
e)^(9/2))

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 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)^{7/2} \left (b x+c x^2\right )^2} \, dx &=-\frac{b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) (d+e x)^{5/2} \left (b x+c x^2\right )}-\frac{\int \frac{\frac{1}{2} (c d-b e) (4 c d+7 b e)+\frac{7}{2} c e (2 c d-b e) x}{(d+e x)^{7/2} \left (b x+c x^2\right )} \, dx}{b^2 d (c d-b e)}\\ &=-\frac{e \left (10 c^2 d^2-10 b c d e+7 b^2 e^2\right )}{5 b^2 d^2 (c d-b e)^2 (d+e x)^{5/2}}-\frac{b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) (d+e x)^{5/2} \left (b x+c x^2\right )}-\frac{\int \frac{\frac{1}{2} (c d-b e)^2 (4 c d+7 b e)+\frac{1}{2} c e \left (10 c^2 d^2-10 b c d e+7 b^2 e^2\right ) x}{(d+e x)^{5/2} \left (b x+c x^2\right )} \, dx}{b^2 d^2 (c d-b e)^2}\\ &=-\frac{e \left (10 c^2 d^2-10 b c d e+7 b^2 e^2\right )}{5 b^2 d^2 (c d-b e)^2 (d+e x)^{5/2}}-\frac{e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+7 b^2 e^2\right )}{3 b^2 d^3 (c d-b e)^3 (d+e x)^{3/2}}-\frac{b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) (d+e x)^{5/2} \left (b x+c x^2\right )}-\frac{\int \frac{\frac{1}{2} (c d-b e)^3 (4 c d+7 b e)+\frac{1}{2} c e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+7 b^2 e^2\right ) x}{(d+e x)^{3/2} \left (b x+c x^2\right )} \, dx}{b^2 d^3 (c d-b e)^3}\\ &=-\frac{e \left (10 c^2 d^2-10 b c d e+7 b^2 e^2\right )}{5 b^2 d^2 (c d-b e)^2 (d+e x)^{5/2}}-\frac{e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+7 b^2 e^2\right )}{3 b^2 d^3 (c d-b e)^3 (d+e x)^{3/2}}-\frac{e \left (2 c^4 d^4-4 b c^3 d^3 e+26 b^2 c^2 d^2 e^2-24 b^3 c d e^3+7 b^4 e^4\right )}{b^2 d^4 (c d-b e)^4 \sqrt{d+e x}}-\frac{b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) (d+e x)^{5/2} \left (b x+c x^2\right )}-\frac{\int \frac{\frac{1}{2} (c d-b e)^4 (4 c d+7 b e)+\frac{1}{2} c e \left (2 c^4 d^4-4 b c^3 d^3 e+26 b^2 c^2 d^2 e^2-24 b^3 c d e^3+7 b^4 e^4\right ) x}{\sqrt{d+e x} \left (b x+c x^2\right )} \, dx}{b^2 d^4 (c d-b e)^4}\\ &=-\frac{e \left (10 c^2 d^2-10 b c d e+7 b^2 e^2\right )}{5 b^2 d^2 (c d-b e)^2 (d+e x)^{5/2}}-\frac{e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+7 b^2 e^2\right )}{3 b^2 d^3 (c d-b e)^3 (d+e x)^{3/2}}-\frac{e \left (2 c^4 d^4-4 b c^3 d^3 e+26 b^2 c^2 d^2 e^2-24 b^3 c d e^3+7 b^4 e^4\right )}{b^2 d^4 (c d-b e)^4 \sqrt{d+e x}}-\frac{b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) (d+e x)^{5/2} \left (b x+c x^2\right )}-\frac{2 \operatorname{Subst}\left (\int \frac{\frac{1}{2} e (c d-b e)^4 (4 c d+7 b e)-\frac{1}{2} c d e \left (2 c^4 d^4-4 b c^3 d^3 e+26 b^2 c^2 d^2 e^2-24 b^3 c d e^3+7 b^4 e^4\right )+\frac{1}{2} c e \left (2 c^4 d^4-4 b c^3 d^3 e+26 b^2 c^2 d^2 e^2-24 b^3 c d e^3+7 b^4 e^4\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^2 d^4 (c d-b e)^4}\\ &=-\frac{e \left (10 c^2 d^2-10 b c d e+7 b^2 e^2\right )}{5 b^2 d^2 (c d-b e)^2 (d+e x)^{5/2}}-\frac{e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+7 b^2 e^2\right )}{3 b^2 d^3 (c d-b e)^3 (d+e x)^{3/2}}-\frac{e \left (2 c^4 d^4-4 b c^3 d^3 e+26 b^2 c^2 d^2 e^2-24 b^3 c d e^3+7 b^4 e^4\right )}{b^2 d^4 (c d-b e)^4 \sqrt{d+e x}}-\frac{b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) (d+e x)^{5/2} \left (b x+c x^2\right )}+\frac{\left (c^5 (4 c d-11 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)^4}-\frac{(c (4 c d+7 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^4}\\ &=-\frac{e \left (10 c^2 d^2-10 b c d e+7 b^2 e^2\right )}{5 b^2 d^2 (c d-b e)^2 (d+e x)^{5/2}}-\frac{e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+7 b^2 e^2\right )}{3 b^2 d^3 (c d-b e)^3 (d+e x)^{3/2}}-\frac{e \left (2 c^4 d^4-4 b c^3 d^3 e+26 b^2 c^2 d^2 e^2-24 b^3 c d e^3+7 b^4 e^4\right )}{b^2 d^4 (c d-b e)^4 \sqrt{d+e x}}-\frac{b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) (d+e x)^{5/2} \left (b x+c x^2\right )}+\frac{(4 c d+7 b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3 d^{9/2}}-\frac{c^{9/2} (4 c d-11 b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 (c d-b e)^{9/2}}\\ \end{align*}

Mathematica [C]  time = 0.117751, size = 171, normalized size = 0.49 \[ \frac{c^2 d^2 x (b+c x) (4 c d-11 b e) \, _2F_1\left (-\frac{5}{2},1;-\frac{3}{2};\frac{c (d+e x)}{c d-b e}\right )-(c d-b e) \left (x (b+c x) \left (-7 b^2 e^2+3 b c d e+4 c^2 d^2\right ) \, _2F_1\left (-\frac{5}{2},1;-\frac{3}{2};\frac{e x}{d}+1\right )-5 b d \left (b^2 e+b c (e x-d)-2 c^2 d x\right )\right )}{5 b^3 d^2 x (b+c x) (d+e x)^{5/2} (c d-b e)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(7/2)*(b*x + c*x^2)^2),x]

[Out]

(c^2*d^2*(4*c*d - 11*b*e)*x*(b + c*x)*Hypergeometric2F1[-5/2, 1, -3/2, (c*(d + e*x))/(c*d - b*e)] - (c*d - b*e
)*(-5*b*d*(b^2*e - 2*c^2*d*x + b*c*(-d + e*x)) + (4*c^2*d^2 + 3*b*c*d*e - 7*b^2*e^2)*x*(b + c*x)*Hypergeometri
c2F1[-5/2, 1, -3/2, 1 + (e*x)/d]))/(5*b^3*d^2*(c*d - b*e)^2*x*(b + c*x)*(d + e*x)^(5/2))

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Maple [A]  time = 0.237, size = 364, normalized size = 1. \begin{align*} -{\frac{2\,{e}^{3}}{5\,{d}^{2} \left ( be-cd \right ) ^{2}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}}-{\frac{4\,{e}^{4}b}{3\,{d}^{3} \left ( be-cd \right ) ^{3}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}+{\frac{8\,{e}^{3}c}{3\,{d}^{2} \left ( be-cd \right ) ^{3}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}-6\,{\frac{{e}^{5}{b}^{2}}{{d}^{4} \left ( be-cd \right ) ^{4}\sqrt{ex+d}}}+20\,{\frac{{e}^{4}bc}{{d}^{3} \left ( be-cd \right ) ^{4}\sqrt{ex+d}}}-20\,{\frac{{e}^{3}{c}^{2}}{{d}^{2} \left ( be-cd \right ) ^{4}\sqrt{ex+d}}}-{\frac{e{c}^{5}}{{b}^{2} \left ( be-cd \right ) ^{4} \left ( cex+be \right ) }\sqrt{ex+d}}-11\,{\frac{e{c}^{5}}{{b}^{2} \left ( be-cd \right ) ^{4}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+4\,{\frac{{c}^{6}d}{{b}^{3} \left ( be-cd \right ) ^{4}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-{\frac{1}{{b}^{2}{d}^{4}x}\sqrt{ex+d}}+7\,{\frac{e}{{b}^{2}{d}^{9/2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }+4\,{\frac{c}{{b}^{3}{d}^{7/2}}{\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)^(7/2)/(c*x^2+b*x)^2,x)

[Out]

-2/5*e^3/d^2/(b*e-c*d)^2/(e*x+d)^(5/2)-4/3*e^4/d^3/(b*e-c*d)^3/(e*x+d)^(3/2)*b+8/3*e^3/d^2/(b*e-c*d)^3/(e*x+d)
^(3/2)*c-6*e^5/d^4/(b*e-c*d)^4/(e*x+d)^(1/2)*b^2+20*e^4/d^3/(b*e-c*d)^4/(e*x+d)^(1/2)*b*c-20*e^3/d^2/(b*e-c*d)
^4/(e*x+d)^(1/2)*c^2-e*c^5/b^2/(b*e-c*d)^4*(e*x+d)^(1/2)/(c*e*x+b*e)-11*e*c^5/b^2/(b*e-c*d)^4/((b*e-c*d)*c)^(1
/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))+4*c^6/b^3/(b*e-c*d)^4/((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^4*(e*x+d)^(1/2)/x+7*e/b^2/d^(9/2)*arctanh((e*x+d)^(1/2)/d^(1/2))+4/b^3/d^(7/
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)^(7/2)/(c*x^2+b*x)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 45.9547, size = 11756, normalized size = 33.68 \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)^(7/2)/(c*x^2+b*x)^2,x, algorithm="fricas")

[Out]

[-1/30*(15*((4*c^6*d^6*e^3 - 11*b*c^5*d^5*e^4)*x^5 + (12*c^6*d^7*e^2 - 29*b*c^5*d^6*e^3 - 11*b^2*c^4*d^5*e^4)*
x^4 + 3*(4*c^6*d^8*e - 7*b*c^5*d^7*e^2 - 11*b^2*c^4*d^6*e^3)*x^3 + (4*c^6*d^9 + b*c^5*d^8*e - 33*b^2*c^4*d^7*e
^2)*x^2 + (4*b*c^5*d^9 - 11*b^2*c^4*d^8*e)*x)*sqrt(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e + 2*(c*d - b*e)*sqr
t(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x + b)) - 15*((4*c^6*d^5*e^3 - 9*b*c^5*d^4*e^4 - 4*b^2*c^4*d^3*e^5 + 26*b^3
*c^3*d^2*e^6 - 24*b^4*c^2*d*e^7 + 7*b^5*c*e^8)*x^5 + (12*c^6*d^6*e^2 - 23*b*c^5*d^5*e^3 - 21*b^2*c^4*d^4*e^4 +
74*b^3*c^3*d^3*e^5 - 46*b^4*c^2*d^2*e^6 - 3*b^5*c*d*e^7 + 7*b^6*e^8)*x^4 + 3*(4*c^6*d^7*e - 5*b*c^5*d^6*e^2 -
13*b^2*c^4*d^5*e^3 + 22*b^3*c^3*d^4*e^4 + 2*b^4*c^2*d^3*e^5 - 17*b^5*c*d^2*e^6 + 7*b^6*d*e^7)*x^3 + (4*c^6*d^
8 + 3*b*c^5*d^7*e - 31*b^2*c^4*d^6*e^2 + 14*b^3*c^3*d^5*e^3 + 54*b^4*c^2*d^4*e^4 - 65*b^5*c*d^3*e^5 + 21*b^6*d
^2*e^6)*x^2 + (4*b*c^5*d^8 - 9*b^2*c^4*d^7*e - 4*b^3*c^3*d^6*e^2 + 26*b^4*c^2*d^5*e^3 - 24*b^5*c*d^4*e^4 + 7*b
^6*d^3*e^5)*x)*sqrt(d)*log((e*x + 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(15*b^2*c^4*d^8 - 60*b^3*c^3*d^7*e + 9
0*b^4*c^2*d^6*e^2 - 60*b^5*c*d^5*e^3 + 15*b^6*d^4*e^4 + 15*(2*b*c^5*d^5*e^3 - 4*b^2*c^4*d^4*e^4 + 26*b^3*c^3*d
^3*e^5 - 24*b^4*c^2*d^2*e^6 + 7*b^5*c*d*e^7)*x^4 + 5*(18*b*c^5*d^6*e^2 - 33*b^2*c^4*d^5*e^3 + 170*b^3*c^3*d^4*
e^4 - 90*b^4*c^2*d^3*e^5 - 23*b^5*c*d^2*e^6 + 21*b^6*d*e^7)*x^3 + (90*b*c^5*d^7*e - 135*b^2*c^4*d^6*e^2 + 436*
b^3*c^3*d^5*e^3 + 358*b^4*c^2*d^4*e^4 - 679*b^5*c*d^3*e^5 + 245*b^6*d^2*e^6)*x^2 + (30*b*c^5*d^8 - 15*b^2*c^4*
d^7*e - 90*b^3*c^3*d^6*e^2 + 556*b^4*c^2*d^5*e^3 - 537*b^5*c*d^4*e^4 + 161*b^6*d^3*e^5)*x)*sqrt(e*x + d))/((b^
3*c^5*d^9*e^3 - 4*b^4*c^4*d^8*e^4 + 6*b^5*c^3*d^7*e^5 - 4*b^6*c^2*d^6*e^6 + b^7*c*d^5*e^7)*x^5 + (3*b^3*c^5*d^
10*e^2 - 11*b^4*c^4*d^9*e^3 + 14*b^5*c^3*d^8*e^4 - 6*b^6*c^2*d^7*e^5 - b^7*c*d^6*e^6 + b^8*d^5*e^7)*x^4 + 3*(b
^3*c^5*d^11*e - 3*b^4*c^4*d^10*e^2 + 2*b^5*c^3*d^9*e^3 + 2*b^6*c^2*d^8*e^4 - 3*b^7*c*d^7*e^5 + b^8*d^6*e^6)*x^
3 + (b^3*c^5*d^12 - b^4*c^4*d^11*e - 6*b^5*c^3*d^10*e^2 + 14*b^6*c^2*d^9*e^3 - 11*b^7*c*d^8*e^4 + 3*b^8*d^7*e^
5)*x^2 + (b^4*c^4*d^12 - 4*b^5*c^3*d^11*e + 6*b^6*c^2*d^10*e^2 - 4*b^7*c*d^9*e^3 + b^8*d^8*e^4)*x), -1/30*(30*
((4*c^6*d^6*e^3 - 11*b*c^5*d^5*e^4)*x^5 + (12*c^6*d^7*e^2 - 29*b*c^5*d^6*e^3 - 11*b^2*c^4*d^5*e^4)*x^4 + 3*(4*
c^6*d^8*e - 7*b*c^5*d^7*e^2 - 11*b^2*c^4*d^6*e^3)*x^3 + (4*c^6*d^9 + b*c^5*d^8*e - 33*b^2*c^4*d^7*e^2)*x^2 + (
4*b*c^5*d^9 - 11*b^2*c^4*d^8*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)) - 15*((4*c^6*d^5*e^3 - 9*b*c^5*d^4*e^4 - 4*b^2*c^4*d^3*e^5 + 26*b^3*c^3*d^2*e^6 - 24*b^4*c^2*d
*e^7 + 7*b^5*c*e^8)*x^5 + (12*c^6*d^6*e^2 - 23*b*c^5*d^5*e^3 - 21*b^2*c^4*d^4*e^4 + 74*b^3*c^3*d^3*e^5 - 46*b^
4*c^2*d^2*e^6 - 3*b^5*c*d*e^7 + 7*b^6*e^8)*x^4 + 3*(4*c^6*d^7*e - 5*b*c^5*d^6*e^2 - 13*b^2*c^4*d^5*e^3 + 22*b^
3*c^3*d^4*e^4 + 2*b^4*c^2*d^3*e^5 - 17*b^5*c*d^2*e^6 + 7*b^6*d*e^7)*x^3 + (4*c^6*d^8 + 3*b*c^5*d^7*e - 31*b^2*
c^4*d^6*e^2 + 14*b^3*c^3*d^5*e^3 + 54*b^4*c^2*d^4*e^4 - 65*b^5*c*d^3*e^5 + 21*b^6*d^2*e^6)*x^2 + (4*b*c^5*d^8
- 9*b^2*c^4*d^7*e - 4*b^3*c^3*d^6*e^2 + 26*b^4*c^2*d^5*e^3 - 24*b^5*c*d^4*e^4 + 7*b^6*d^3*e^5)*x)*sqrt(d)*log(
(e*x + 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(15*b^2*c^4*d^8 - 60*b^3*c^3*d^7*e + 90*b^4*c^2*d^6*e^2 - 60*b^5*
c*d^5*e^3 + 15*b^6*d^4*e^4 + 15*(2*b*c^5*d^5*e^3 - 4*b^2*c^4*d^4*e^4 + 26*b^3*c^3*d^3*e^5 - 24*b^4*c^2*d^2*e^6
+ 7*b^5*c*d*e^7)*x^4 + 5*(18*b*c^5*d^6*e^2 - 33*b^2*c^4*d^5*e^3 + 170*b^3*c^3*d^4*e^4 - 90*b^4*c^2*d^3*e^5 -
23*b^5*c*d^2*e^6 + 21*b^6*d*e^7)*x^3 + (90*b*c^5*d^7*e - 135*b^2*c^4*d^6*e^2 + 436*b^3*c^3*d^5*e^3 + 358*b^4*c
^2*d^4*e^4 - 679*b^5*c*d^3*e^5 + 245*b^6*d^2*e^6)*x^2 + (30*b*c^5*d^8 - 15*b^2*c^4*d^7*e - 90*b^3*c^3*d^6*e^2
+ 556*b^4*c^2*d^5*e^3 - 537*b^5*c*d^4*e^4 + 161*b^6*d^3*e^5)*x)*sqrt(e*x + d))/((b^3*c^5*d^9*e^3 - 4*b^4*c^4*d
^8*e^4 + 6*b^5*c^3*d^7*e^5 - 4*b^6*c^2*d^6*e^6 + b^7*c*d^5*e^7)*x^5 + (3*b^3*c^5*d^10*e^2 - 11*b^4*c^4*d^9*e^3
+ 14*b^5*c^3*d^8*e^4 - 6*b^6*c^2*d^7*e^5 - b^7*c*d^6*e^6 + b^8*d^5*e^7)*x^4 + 3*(b^3*c^5*d^11*e - 3*b^4*c^4*d
^10*e^2 + 2*b^5*c^3*d^9*e^3 + 2*b^6*c^2*d^8*e^4 - 3*b^7*c*d^7*e^5 + b^8*d^6*e^6)*x^3 + (b^3*c^5*d^12 - b^4*c^4
*d^11*e - 6*b^5*c^3*d^10*e^2 + 14*b^6*c^2*d^9*e^3 - 11*b^7*c*d^8*e^4 + 3*b^8*d^7*e^5)*x^2 + (b^4*c^4*d^12 - 4*
b^5*c^3*d^11*e + 6*b^6*c^2*d^10*e^2 - 4*b^7*c*d^9*e^3 + b^8*d^8*e^4)*x), -1/30*(30*((4*c^6*d^5*e^3 - 9*b*c^5*d
^4*e^4 - 4*b^2*c^4*d^3*e^5 + 26*b^3*c^3*d^2*e^6 - 24*b^4*c^2*d*e^7 + 7*b^5*c*e^8)*x^5 + (12*c^6*d^6*e^2 - 23*b
*c^5*d^5*e^3 - 21*b^2*c^4*d^4*e^4 + 74*b^3*c^3*d^3*e^5 - 46*b^4*c^2*d^2*e^6 - 3*b^5*c*d*e^7 + 7*b^6*e^8)*x^4 +
3*(4*c^6*d^7*e - 5*b*c^5*d^6*e^2 - 13*b^2*c^4*d^5*e^3 + 22*b^3*c^3*d^4*e^4 + 2*b^4*c^2*d^3*e^5 - 17*b^5*c*d^2
*e^6 + 7*b^6*d*e^7)*x^3 + (4*c^6*d^8 + 3*b*c^5*d^7*e - 31*b^2*c^4*d^6*e^2 + 14*b^3*c^3*d^5*e^3 + 54*b^4*c^2*d^
4*e^4 - 65*b^5*c*d^3*e^5 + 21*b^6*d^2*e^6)*x^2 + (4*b*c^5*d^8 - 9*b^2*c^4*d^7*e - 4*b^3*c^3*d^6*e^2 + 26*b^4*c
^2*d^5*e^3 - 24*b^5*c*d^4*e^4 + 7*b^6*d^3*e^5)*x)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + 15*((4*c^6*d^6*e
^3 - 11*b*c^5*d^5*e^4)*x^5 + (12*c^6*d^7*e^2 - 29*b*c^5*d^6*e^3 - 11*b^2*c^4*d^5*e^4)*x^4 + 3*(4*c^6*d^8*e - 7
*b*c^5*d^7*e^2 - 11*b^2*c^4*d^6*e^3)*x^3 + (4*c^6*d^9 + b*c^5*d^8*e - 33*b^2*c^4*d^7*e^2)*x^2 + (4*b*c^5*d^9 -
11*b^2*c^4*d^8*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*(15*b^2*c^4*d^8 - 60*b^3*c^3*d^7*e + 90*b^4*c^2*d^6*e^2 - 60*b^5*c*d^5*e^3 + 15*b^6*d^4
*e^4 + 15*(2*b*c^5*d^5*e^3 - 4*b^2*c^4*d^4*e^4 + 26*b^3*c^3*d^3*e^5 - 24*b^4*c^2*d^2*e^6 + 7*b^5*c*d*e^7)*x^4
+ 5*(18*b*c^5*d^6*e^2 - 33*b^2*c^4*d^5*e^3 + 170*b^3*c^3*d^4*e^4 - 90*b^4*c^2*d^3*e^5 - 23*b^5*c*d^2*e^6 + 21*
b^6*d*e^7)*x^3 + (90*b*c^5*d^7*e - 135*b^2*c^4*d^6*e^2 + 436*b^3*c^3*d^5*e^3 + 358*b^4*c^2*d^4*e^4 - 679*b^5*c
*d^3*e^5 + 245*b^6*d^2*e^6)*x^2 + (30*b*c^5*d^8 - 15*b^2*c^4*d^7*e - 90*b^3*c^3*d^6*e^2 + 556*b^4*c^2*d^5*e^3
- 537*b^5*c*d^4*e^4 + 161*b^6*d^3*e^5)*x)*sqrt(e*x + d))/((b^3*c^5*d^9*e^3 - 4*b^4*c^4*d^8*e^4 + 6*b^5*c^3*d^7
*e^5 - 4*b^6*c^2*d^6*e^6 + b^7*c*d^5*e^7)*x^5 + (3*b^3*c^5*d^10*e^2 - 11*b^4*c^4*d^9*e^3 + 14*b^5*c^3*d^8*e^4
- 6*b^6*c^2*d^7*e^5 - b^7*c*d^6*e^6 + b^8*d^5*e^7)*x^4 + 3*(b^3*c^5*d^11*e - 3*b^4*c^4*d^10*e^2 + 2*b^5*c^3*d^
9*e^3 + 2*b^6*c^2*d^8*e^4 - 3*b^7*c*d^7*e^5 + b^8*d^6*e^6)*x^3 + (b^3*c^5*d^12 - b^4*c^4*d^11*e - 6*b^5*c^3*d^
10*e^2 + 14*b^6*c^2*d^9*e^3 - 11*b^7*c*d^8*e^4 + 3*b^8*d^7*e^5)*x^2 + (b^4*c^4*d^12 - 4*b^5*c^3*d^11*e + 6*b^6
*c^2*d^10*e^2 - 4*b^7*c*d^9*e^3 + b^8*d^8*e^4)*x), -1/15*(15*((4*c^6*d^6*e^3 - 11*b*c^5*d^5*e^4)*x^5 + (12*c^6
*d^7*e^2 - 29*b*c^5*d^6*e^3 - 11*b^2*c^4*d^5*e^4)*x^4 + 3*(4*c^6*d^8*e - 7*b*c^5*d^7*e^2 - 11*b^2*c^4*d^6*e^3)
*x^3 + (4*c^6*d^9 + b*c^5*d^8*e - 33*b^2*c^4*d^7*e^2)*x^2 + (4*b*c^5*d^9 - 11*b^2*c^4*d^8*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)) + 15*((4*c^6*d^5*e^3 - 9*b*c^5*d^
4*e^4 - 4*b^2*c^4*d^3*e^5 + 26*b^3*c^3*d^2*e^6 - 24*b^4*c^2*d*e^7 + 7*b^5*c*e^8)*x^5 + (12*c^6*d^6*e^2 - 23*b*
c^5*d^5*e^3 - 21*b^2*c^4*d^4*e^4 + 74*b^3*c^3*d^3*e^5 - 46*b^4*c^2*d^2*e^6 - 3*b^5*c*d*e^7 + 7*b^6*e^8)*x^4 +
3*(4*c^6*d^7*e - 5*b*c^5*d^6*e^2 - 13*b^2*c^4*d^5*e^3 + 22*b^3*c^3*d^4*e^4 + 2*b^4*c^2*d^3*e^5 - 17*b^5*c*d^2*
e^6 + 7*b^6*d*e^7)*x^3 + (4*c^6*d^8 + 3*b*c^5*d^7*e - 31*b^2*c^4*d^6*e^2 + 14*b^3*c^3*d^5*e^3 + 54*b^4*c^2*d^4
*e^4 - 65*b^5*c*d^3*e^5 + 21*b^6*d^2*e^6)*x^2 + (4*b*c^5*d^8 - 9*b^2*c^4*d^7*e - 4*b^3*c^3*d^6*e^2 + 26*b^4*c^
2*d^5*e^3 - 24*b^5*c*d^4*e^4 + 7*b^6*d^3*e^5)*x)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + (15*b^2*c^4*d^8 -
60*b^3*c^3*d^7*e + 90*b^4*c^2*d^6*e^2 - 60*b^5*c*d^5*e^3 + 15*b^6*d^4*e^4 + 15*(2*b*c^5*d^5*e^3 - 4*b^2*c^4*d
^4*e^4 + 26*b^3*c^3*d^3*e^5 - 24*b^4*c^2*d^2*e^6 + 7*b^5*c*d*e^7)*x^4 + 5*(18*b*c^5*d^6*e^2 - 33*b^2*c^4*d^5*e
^3 + 170*b^3*c^3*d^4*e^4 - 90*b^4*c^2*d^3*e^5 - 23*b^5*c*d^2*e^6 + 21*b^6*d*e^7)*x^3 + (90*b*c^5*d^7*e - 135*b
^2*c^4*d^6*e^2 + 436*b^3*c^3*d^5*e^3 + 358*b^4*c^2*d^4*e^4 - 679*b^5*c*d^3*e^5 + 245*b^6*d^2*e^6)*x^2 + (30*b*
c^5*d^8 - 15*b^2*c^4*d^7*e - 90*b^3*c^3*d^6*e^2 + 556*b^4*c^2*d^5*e^3 - 537*b^5*c*d^4*e^4 + 161*b^6*d^3*e^5)*x
)*sqrt(e*x + d))/((b^3*c^5*d^9*e^3 - 4*b^4*c^4*d^8*e^4 + 6*b^5*c^3*d^7*e^5 - 4*b^6*c^2*d^6*e^6 + b^7*c*d^5*e^7
)*x^5 + (3*b^3*c^5*d^10*e^2 - 11*b^4*c^4*d^9*e^3 + 14*b^5*c^3*d^8*e^4 - 6*b^6*c^2*d^7*e^5 - b^7*c*d^6*e^6 + b^
8*d^5*e^7)*x^4 + 3*(b^3*c^5*d^11*e - 3*b^4*c^4*d^10*e^2 + 2*b^5*c^3*d^9*e^3 + 2*b^6*c^2*d^8*e^4 - 3*b^7*c*d^7*
e^5 + b^8*d^6*e^6)*x^3 + (b^3*c^5*d^12 - b^4*c^4*d^11*e - 6*b^5*c^3*d^10*e^2 + 14*b^6*c^2*d^9*e^3 - 11*b^7*c*d
^8*e^4 + 3*b^8*d^7*e^5)*x^2 + (b^4*c^4*d^12 - 4*b^5*c^3*d^11*e + 6*b^6*c^2*d^10*e^2 - 4*b^7*c*d^9*e^3 + b^8*d^
8*e^4)*x)]

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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)**(7/2)/(c*x**2+b*x)**2,x)

[Out]

Timed out

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Giac [B]  time = 1.40457, size = 868, normalized size = 2.49 \begin{align*} \frac{{\left (4 \, c^{6} d - 11 \, b c^{5} e\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{{\left (b^{3} c^{4} d^{4} - 4 \, b^{4} c^{3} d^{3} e + 6 \, b^{5} c^{2} d^{2} e^{2} - 4 \, b^{6} c d e^{3} + b^{7} e^{4}\right )} \sqrt{-c^{2} d + b c e}} - \frac{2 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{5} d^{4} e - 2 \, \sqrt{x e + d} c^{5} d^{5} e - 4 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{4} d^{3} e^{2} + 5 \, \sqrt{x e + d} b c^{4} d^{4} e^{2} + 6 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} c^{3} d^{2} e^{3} - 10 \, \sqrt{x e + d} b^{2} c^{3} d^{3} e^{3} - 4 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} c^{2} d e^{4} + 10 \, \sqrt{x e + d} b^{3} c^{2} d^{2} e^{4} +{\left (x e + d\right )}^{\frac{3}{2}} b^{4} c e^{5} - 5 \, \sqrt{x e + d} b^{4} c d e^{5} + \sqrt{x e + d} b^{5} e^{6}}{{\left (b^{2} c^{4} d^{8} - 4 \, b^{3} c^{3} d^{7} e + 6 \, b^{4} c^{2} d^{6} e^{2} - 4 \, b^{5} c d^{5} e^{3} + b^{6} d^{4} e^{4}\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{2 \,{\left (150 \,{\left (x e + d\right )}^{2} c^{2} d^{2} e^{3} + 20 \,{\left (x e + d\right )} c^{2} d^{3} e^{3} + 3 \, c^{2} d^{4} e^{3} - 150 \,{\left (x e + d\right )}^{2} b c d e^{4} - 30 \,{\left (x e + d\right )} b c d^{2} e^{4} - 6 \, b c d^{3} e^{4} + 45 \,{\left (x e + d\right )}^{2} b^{2} e^{5} + 10 \,{\left (x e + d\right )} b^{2} d e^{5} + 3 \, b^{2} d^{2} e^{5}\right )}}{15 \,{\left (c^{4} d^{8} - 4 \, b c^{3} d^{7} e + 6 \, b^{2} c^{2} d^{6} e^{2} - 4 \, b^{3} c d^{5} e^{3} + b^{4} d^{4} e^{4}\right )}{\left (x e + d\right )}^{\frac{5}{2}}} - \frac{{\left (4 \, c d + 7 \, b e\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b^{3} \sqrt{-d} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(7/2)/(c*x^2+b*x)^2,x, algorithm="giac")

[Out]

(4*c^6*d - 11*b*c^5*e)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b^3*c^4*d^4 - 4*b^4*c^3*d^3*e + 6*b^5*c^
2*d^2*e^2 - 4*b^6*c*d*e^3 + b^7*e^4)*sqrt(-c^2*d + b*c*e)) - (2*(x*e + d)^(3/2)*c^5*d^4*e - 2*sqrt(x*e + d)*c^
5*d^5*e - 4*(x*e + d)^(3/2)*b*c^4*d^3*e^2 + 5*sqrt(x*e + d)*b*c^4*d^4*e^2 + 6*(x*e + d)^(3/2)*b^2*c^3*d^2*e^3
- 10*sqrt(x*e + d)*b^2*c^3*d^3*e^3 - 4*(x*e + d)^(3/2)*b^3*c^2*d*e^4 + 10*sqrt(x*e + d)*b^3*c^2*d^2*e^4 + (x*e
+ d)^(3/2)*b^4*c*e^5 - 5*sqrt(x*e + d)*b^4*c*d*e^5 + sqrt(x*e + d)*b^5*e^6)/((b^2*c^4*d^8 - 4*b^3*c^3*d^7*e +
6*b^4*c^2*d^6*e^2 - 4*b^5*c*d^5*e^3 + b^6*d^4*e^4)*((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e -
b*d*e)) - 2/15*(150*(x*e + d)^2*c^2*d^2*e^3 + 20*(x*e + d)*c^2*d^3*e^3 + 3*c^2*d^4*e^3 - 150*(x*e + d)^2*b*c*
d*e^4 - 30*(x*e + d)*b*c*d^2*e^4 - 6*b*c*d^3*e^4 + 45*(x*e + d)^2*b^2*e^5 + 10*(x*e + d)*b^2*d*e^5 + 3*b^2*d^2
*e^5)/((c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 - 4*b^3*c*d^5*e^3 + b^4*d^4*e^4)*(x*e + d)^(5/2)) - (4*c*d
+ 7*b*e)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^3*sqrt(-d)*d^4)