3.2080 \(\int \frac{1}{(d+e x)^{7/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\)

Optimal. Leaf size=519 \[ \frac{3003 c^5 d^5 e \sqrt{d+e x}}{128 \left (c d^2-a e^2\right )^7 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{1001 c^4 d^4 e}{128 \sqrt{d+e x} \left (c d^2-a e^2\right )^6 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{1001 c^4 d^4 \sqrt{d+e x}}{320 \left (c d^2-a e^2\right )^5 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{429 c^3 d^3}{320 \sqrt{d+e x} \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{143 c^2 d^2}{240 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{3003 c^5 d^5 e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{128 \left (c d^2-a e^2\right )^{15/2}}+\frac{13 c d}{40 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{1}{5 (d+e x)^{7/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

[Out]

1/(5*(c*d^2 - a*e^2)*(d + e*x)^(7/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (13*c*d)/(40*(c*d^2 - a*
e^2)^2*(d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (143*c^2*d^2)/(240*(c*d^2 - a*e^2)^3*(
d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (429*c^3*d^3)/(320*(c*d^2 - a*e^2)^4*Sqrt[d +
e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (1001*c^4*d^4*Sqrt[d + e*x])/(320*(c*d^2 - a*e^2)^5*(a*d
*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (1001*c^4*d^4*e)/(128*(c*d^2 - a*e^2)^6*Sqrt[d + e*x]*Sqrt[a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (3003*c^5*d^5*e*Sqrt[d + e*x])/(128*(c*d^2 - a*e^2)^7*Sqrt[a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2]) + (3003*c^5*d^5*e^(3/2)*ArcTan[(Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/
(Sqrt[c*d^2 - a*e^2]*Sqrt[d + e*x])])/(128*(c*d^2 - a*e^2)^(15/2))

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Rubi [A]  time = 0.598339, antiderivative size = 519, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {672, 666, 660, 205} \[ \frac{3003 c^5 d^5 e \sqrt{d+e x}}{128 \left (c d^2-a e^2\right )^7 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{1001 c^4 d^4 e}{128 \sqrt{d+e x} \left (c d^2-a e^2\right )^6 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{1001 c^4 d^4 \sqrt{d+e x}}{320 \left (c d^2-a e^2\right )^5 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{429 c^3 d^3}{320 \sqrt{d+e x} \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{143 c^2 d^2}{240 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{3003 c^5 d^5 e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{128 \left (c d^2-a e^2\right )^{15/2}}+\frac{13 c d}{40 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{1}{5 (d+e x)^{7/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(7/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]

[Out]

1/(5*(c*d^2 - a*e^2)*(d + e*x)^(7/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (13*c*d)/(40*(c*d^2 - a*
e^2)^2*(d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (143*c^2*d^2)/(240*(c*d^2 - a*e^2)^3*(
d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (429*c^3*d^3)/(320*(c*d^2 - a*e^2)^4*Sqrt[d +
e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (1001*c^4*d^4*Sqrt[d + e*x])/(320*(c*d^2 - a*e^2)^5*(a*d
*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (1001*c^4*d^4*e)/(128*(c*d^2 - a*e^2)^6*Sqrt[d + e*x]*Sqrt[a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (3003*c^5*d^5*e*Sqrt[d + e*x])/(128*(c*d^2 - a*e^2)^7*Sqrt[a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2]) + (3003*c^5*d^5*e^(3/2)*ArcTan[(Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/
(Sqrt[c*d^2 - a*e^2]*Sqrt[d + e*x])])/(128*(c*d^2 - a*e^2)^(15/2))

Rule 672

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

Rule 666

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

Rule 660

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2*e, Subst[Int[1/(
2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^
2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{1}{(d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=\frac{1}{5 \left (c d^2-a e^2\right ) (d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{(13 c d) \int \frac{1}{(d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx}{10 \left (c d^2-a e^2\right )}\\ &=\frac{1}{5 \left (c d^2-a e^2\right ) (d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{13 c d}{40 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{\left (143 c^2 d^2\right ) \int \frac{1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx}{80 \left (c d^2-a e^2\right )^2}\\ &=\frac{1}{5 \left (c d^2-a e^2\right ) (d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{13 c d}{40 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{143 c^2 d^2}{240 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{\left (429 c^3 d^3\right ) \int \frac{1}{\sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx}{160 \left (c d^2-a e^2\right )^3}\\ &=\frac{1}{5 \left (c d^2-a e^2\right ) (d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{13 c d}{40 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{143 c^2 d^2}{240 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{429 c^3 d^3}{320 \left (c d^2-a e^2\right )^4 \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{\left (3003 c^4 d^4\right ) \int \frac{\sqrt{d+e x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx}{640 \left (c d^2-a e^2\right )^4}\\ &=\frac{1}{5 \left (c d^2-a e^2\right ) (d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{13 c d}{40 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{143 c^2 d^2}{240 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{429 c^3 d^3}{320 \left (c d^2-a e^2\right )^4 \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{1001 c^4 d^4 \sqrt{d+e x}}{320 \left (c d^2-a e^2\right )^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{\left (1001 c^4 d^4 e\right ) \int \frac{1}{\sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{128 \left (c d^2-a e^2\right )^5}\\ &=\frac{1}{5 \left (c d^2-a e^2\right ) (d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{13 c d}{40 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{143 c^2 d^2}{240 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{429 c^3 d^3}{320 \left (c d^2-a e^2\right )^4 \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{1001 c^4 d^4 \sqrt{d+e x}}{320 \left (c d^2-a e^2\right )^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{1001 c^4 d^4 e}{128 \left (c d^2-a e^2\right )^6 \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{\left (3003 c^5 d^5 e\right ) \int \frac{\sqrt{d+e x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{256 \left (c d^2-a e^2\right )^6}\\ &=\frac{1}{5 \left (c d^2-a e^2\right ) (d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{13 c d}{40 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{143 c^2 d^2}{240 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{429 c^3 d^3}{320 \left (c d^2-a e^2\right )^4 \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{1001 c^4 d^4 \sqrt{d+e x}}{320 \left (c d^2-a e^2\right )^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{1001 c^4 d^4 e}{128 \left (c d^2-a e^2\right )^6 \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{3003 c^5 d^5 e \sqrt{d+e x}}{128 \left (c d^2-a e^2\right )^7 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{\left (3003 c^5 d^5 e^2\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{256 \left (c d^2-a e^2\right )^7}\\ &=\frac{1}{5 \left (c d^2-a e^2\right ) (d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{13 c d}{40 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{143 c^2 d^2}{240 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{429 c^3 d^3}{320 \left (c d^2-a e^2\right )^4 \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{1001 c^4 d^4 \sqrt{d+e x}}{320 \left (c d^2-a e^2\right )^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{1001 c^4 d^4 e}{128 \left (c d^2-a e^2\right )^6 \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{3003 c^5 d^5 e \sqrt{d+e x}}{128 \left (c d^2-a e^2\right )^7 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{\left (3003 c^5 d^5 e^3\right ) \operatorname{Subst}\left (\int \frac{1}{2 c d^2 e-e \left (c d^2+a e^2\right )+e^2 x^2} \, dx,x,\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}}\right )}{128 \left (c d^2-a e^2\right )^7}\\ &=\frac{1}{5 \left (c d^2-a e^2\right ) (d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{13 c d}{40 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{143 c^2 d^2}{240 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{429 c^3 d^3}{320 \left (c d^2-a e^2\right )^4 \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{1001 c^4 d^4 \sqrt{d+e x}}{320 \left (c d^2-a e^2\right )^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{1001 c^4 d^4 e}{128 \left (c d^2-a e^2\right )^6 \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{3003 c^5 d^5 e \sqrt{d+e x}}{128 \left (c d^2-a e^2\right )^7 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{3003 c^5 d^5 e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{c d^2-a e^2} \sqrt{d+e x}}\right )}{128 \left (c d^2-a e^2\right )^{15/2}}\\ \end{align*}

Mathematica [C]  time = 0.0379719, size = 83, normalized size = 0.16 \[ -\frac{2 c^5 d^5 (d+e x)^{3/2} \, _2F_1\left (-\frac{3}{2},6;-\frac{1}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right )}{3 \left (c d^2-a e^2\right )^6 ((d+e x) (a e+c d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(7/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]

[Out]

(-2*c^5*d^5*(d + e*x)^(3/2)*Hypergeometric2F1[-3/2, 6, -1/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(3*(c*d^2
- a*e^2)^6*((a*e + c*d*x)*(d + e*x))^(3/2))

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Maple [B]  time = 0.259, size = 1553, normalized size = 3. \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)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)

[Out]

1/1920*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(-45045*((a*e^2-c*d^2)*e)^(1/2)*x^6*c^6*d^6*e^6-210210*((a*e^2-
c*d^2)*e)^(1/2)*x^5*c^6*d^7*e^5-384384*((a*e^2-c*d^2)*e)^(1/2)*x^4*c^6*d^8*e^4-338910*((a*e^2-c*d^2)*e)^(1/2)*
x^3*c^6*d^9*e^3-137995*((a*e^2-c*d^2)*e)^(1/2)*x^2*c^6*d^10*e^2-24320*((a*e^2-c*d^2)*e)^(1/2)*a*c^5*d^10*e^2+2
928*((a*e^2-c*d^2)*e)^(1/2)*a^5*c*d^2*e^10-10024*((a*e^2-c*d^2)*e)^(1/2)*a^4*c^2*d^4*e^8+21070*((a*e^2-c*d^2)*
e)^(1/2)*a^3*c^3*d^6*e^6-35595*((a*e^2-c*d^2)*e)^(1/2)*a^2*c^4*d^8*e^4-16640*((a*e^2-c*d^2)*e)^(1/2)*x*c^6*d^1
1*e+1280*((a*e^2-c*d^2)*e)^(1/2)*c^6*d^12-384*((a*e^2-c*d^2)*e)^(1/2)*a^6*e^12+225225*arctanh(e*(c*d*x+a*e)^(1
/2)/((a*e^2-c*d^2)*e)^(1/2))*x*a*c^5*d^9*e^4*(c*d*x+a*e)^(1/2)+45045*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2
)*e)^(1/2))*x^5*a*c^5*d^5*e^8*(c*d*x+a*e)^(1/2)+225225*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x^
4*a*c^5*d^6*e^7*(c*d*x+a*e)^(1/2)+450450*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x^3*a*c^5*d^7*e^
6*(c*d*x+a*e)^(1/2)+450450*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x^2*a*c^5*d^8*e^5*(c*d*x+a*e)^
(1/2)+225225*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x^5*c^6*d^7*e^6*(c*d*x+a*e)^(1/2)+450450*arc
tanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x^4*c^6*d^8*e^5*(c*d*x+a*e)^(1/2)+450450*arctanh(e*(c*d*x+a*
e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x^3*c^6*d^9*e^4*(c*d*x+a*e)^(1/2)+225225*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2
-c*d^2)*e)^(1/2))*x^2*c^6*d^10*e^3*(c*d*x+a*e)^(1/2)+45045*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2)
)*x*c^6*d^11*e^2*(c*d*x+a*e)^(1/2)-60060*((a*e^2-c*d^2)*e)^(1/2)*x^5*a*c^5*d^5*e^7-282282*((a*e^2-c*d^2)*e)^(1
/2)*x^4*a*c^5*d^6*e^6-520806*((a*e^2-c*d^2)*e)^(1/2)*x^3*a*c^5*d^7*e^5-464750*((a*e^2-c*d^2)*e)^(1/2)*x^2*a*c^
5*d^8*e^4-192790*((a*e^2-c*d^2)*e)^(1/2)*x*a*c^5*d^9*e^3-9009*((a*e^2-c*d^2)*e)^(1/2)*x^4*a^2*c^4*d^4*e^8+2574
*((a*e^2-c*d^2)*e)^(1/2)*x^3*a^3*c^3*d^3*e^9-43758*((a*e^2-c*d^2)*e)^(1/2)*x^3*a^2*c^4*d^5*e^7-1144*((a*e^2-c*
d^2)*e)^(1/2)*x^2*a^4*c^2*d^2*e^10+12298*((a*e^2-c*d^2)*e)^(1/2)*x^2*a^3*c^3*d^4*e^8-84084*((a*e^2-c*d^2)*e)^(
1/2)*x^2*a^2*c^4*d^6*e^6+624*((a*e^2-c*d^2)*e)^(1/2)*x*a^5*c*d*e^11-5408*((a*e^2-c*d^2)*e)^(1/2)*x*a^4*c^2*d^3
*e^9+23114*((a*e^2-c*d^2)*e)^(1/2)*x*a^3*c^3*d^5*e^7-79170*((a*e^2-c*d^2)*e)^(1/2)*x*a^2*c^4*d^7*e^5+45045*arc
tanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x^6*c^6*d^6*e^7*(c*d*x+a*e)^(1/2)+45045*arctanh(e*(c*d*x+a*e
)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^5*d^10*e^3*(c*d*x+a*e)^(1/2))/(e*x+d)^(11/2)/(c*d*x+a*e)^2/(a*e^2-c*d^2)^
7/((a*e^2-c*d^2)*e)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{5}{2}}{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="maxima")

[Out]

integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(e*x + d)^(7/2)), x)

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Fricas [B]  time = 2.80526, size = 8296, normalized size = 15.98 \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)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fricas")

[Out]

[1/3840*(45045*(c^7*d^7*e^7*x^8 + a^2*c^5*d^11*e^3 + 2*(3*c^7*d^8*e^6 + a*c^6*d^6*e^8)*x^7 + (15*c^7*d^9*e^5 +
12*a*c^6*d^7*e^7 + a^2*c^5*d^5*e^9)*x^6 + 2*(10*c^7*d^10*e^4 + 15*a*c^6*d^8*e^6 + 3*a^2*c^5*d^6*e^8)*x^5 + 5*
(3*c^7*d^11*e^3 + 8*a*c^6*d^9*e^5 + 3*a^2*c^5*d^7*e^7)*x^4 + 2*(3*c^7*d^12*e^2 + 15*a*c^6*d^10*e^4 + 10*a^2*c^
5*d^8*e^6)*x^3 + (c^7*d^13*e + 12*a*c^6*d^11*e^3 + 15*a^2*c^5*d^9*e^5)*x^2 + 2*(a*c^6*d^12*e^2 + 3*a^2*c^5*d^1
0*e^4)*x)*sqrt(-e/(c*d^2 - a*e^2))*log(-(c*d*e^2*x^2 + 2*a*e^3*x - c*d^3 + 2*a*d*e^2 + 2*sqrt(c*d*e*x^2 + a*d*
e + (c*d^2 + a*e^2)*x)*(c*d^2 - a*e^2)*sqrt(e*x + d)*sqrt(-e/(c*d^2 - a*e^2)))/(e^2*x^2 + 2*d*e*x + d^2)) + 2*
(45045*c^6*d^6*e^6*x^6 - 1280*c^6*d^12 + 24320*a*c^5*d^10*e^2 + 35595*a^2*c^4*d^8*e^4 - 21070*a^3*c^3*d^6*e^6
+ 10024*a^4*c^2*d^4*e^8 - 2928*a^5*c*d^2*e^10 + 384*a^6*e^12 + 30030*(7*c^6*d^7*e^5 + 2*a*c^5*d^5*e^7)*x^5 + 3
003*(128*c^6*d^8*e^4 + 94*a*c^5*d^6*e^6 + 3*a^2*c^4*d^4*e^8)*x^4 + 858*(395*c^6*d^9*e^3 + 607*a*c^5*d^7*e^5 +
51*a^2*c^4*d^5*e^7 - 3*a^3*c^3*d^3*e^9)*x^3 + 143*(965*c^6*d^10*e^2 + 3250*a*c^5*d^8*e^4 + 588*a^2*c^4*d^6*e^6
- 86*a^3*c^3*d^4*e^8 + 8*a^4*c^2*d^2*e^10)*x^2 + 26*(640*c^6*d^11*e + 7415*a*c^5*d^9*e^3 + 3045*a^2*c^4*d^7*e
^5 - 889*a^3*c^3*d^5*e^7 + 208*a^4*c^2*d^3*e^9 - 24*a^5*c*d*e^11)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*
x)*sqrt(e*x + d))/(a^2*c^7*d^20*e^2 - 7*a^3*c^6*d^18*e^4 + 21*a^4*c^5*d^16*e^6 - 35*a^5*c^4*d^14*e^8 + 35*a^6*
c^3*d^12*e^10 - 21*a^7*c^2*d^10*e^12 + 7*a^8*c*d^8*e^14 - a^9*d^6*e^16 + (c^9*d^16*e^6 - 7*a*c^8*d^14*e^8 + 21
*a^2*c^7*d^12*e^10 - 35*a^3*c^6*d^10*e^12 + 35*a^4*c^5*d^8*e^14 - 21*a^5*c^4*d^6*e^16 + 7*a^6*c^3*d^4*e^18 - a
^7*c^2*d^2*e^20)*x^8 + 2*(3*c^9*d^17*e^5 - 20*a*c^8*d^15*e^7 + 56*a^2*c^7*d^13*e^9 - 84*a^3*c^6*d^11*e^11 + 70
*a^4*c^5*d^9*e^13 - 28*a^5*c^4*d^7*e^15 + 4*a^7*c^2*d^3*e^19 - a^8*c*d*e^21)*x^7 + (15*c^9*d^18*e^4 - 93*a*c^8
*d^16*e^6 + 232*a^2*c^7*d^14*e^8 - 280*a^3*c^6*d^12*e^10 + 126*a^4*c^5*d^10*e^12 + 70*a^5*c^4*d^8*e^14 - 112*a
^6*c^3*d^6*e^16 + 48*a^7*c^2*d^4*e^18 - 5*a^8*c*d^2*e^20 - a^9*e^22)*x^6 + 2*(10*c^9*d^19*e^3 - 55*a*c^8*d^17*
e^5 + 108*a^2*c^7*d^15*e^7 - 56*a^3*c^6*d^13*e^9 - 112*a^4*c^5*d^11*e^11 + 210*a^5*c^4*d^9*e^13 - 140*a^6*c^3*
d^7*e^15 + 32*a^7*c^2*d^5*e^17 + 6*a^8*c*d^3*e^19 - 3*a^9*d*e^21)*x^5 + 5*(3*c^9*d^20*e^2 - 13*a*c^8*d^18*e^4
+ 10*a^2*c^7*d^16*e^6 + 42*a^3*c^6*d^14*e^8 - 112*a^4*c^5*d^12*e^10 + 112*a^5*c^4*d^10*e^12 - 42*a^6*c^3*d^8*e
^14 - 10*a^7*c^2*d^6*e^16 + 13*a^8*c*d^4*e^18 - 3*a^9*d^2*e^20)*x^4 + 2*(3*c^9*d^21*e - 6*a*c^8*d^19*e^3 - 32*
a^2*c^7*d^17*e^5 + 140*a^3*c^6*d^15*e^7 - 210*a^4*c^5*d^13*e^9 + 112*a^5*c^4*d^11*e^11 + 56*a^6*c^3*d^9*e^13 -
108*a^7*c^2*d^7*e^15 + 55*a^8*c*d^5*e^17 - 10*a^9*d^3*e^19)*x^3 + (c^9*d^22 + 5*a*c^8*d^20*e^2 - 48*a^2*c^7*d
^18*e^4 + 112*a^3*c^6*d^16*e^6 - 70*a^4*c^5*d^14*e^8 - 126*a^5*c^4*d^12*e^10 + 280*a^6*c^3*d^10*e^12 - 232*a^7
*c^2*d^8*e^14 + 93*a^8*c*d^6*e^16 - 15*a^9*d^4*e^18)*x^2 + 2*(a*c^8*d^21*e - 4*a^2*c^7*d^19*e^3 + 28*a^4*c^5*d
^15*e^7 - 70*a^5*c^4*d^13*e^9 + 84*a^6*c^3*d^11*e^11 - 56*a^7*c^2*d^9*e^13 + 20*a^8*c*d^7*e^15 - 3*a^9*d^5*e^1
7)*x), 1/1920*(45045*(c^7*d^7*e^7*x^8 + a^2*c^5*d^11*e^3 + 2*(3*c^7*d^8*e^6 + a*c^6*d^6*e^8)*x^7 + (15*c^7*d^9
*e^5 + 12*a*c^6*d^7*e^7 + a^2*c^5*d^5*e^9)*x^6 + 2*(10*c^7*d^10*e^4 + 15*a*c^6*d^8*e^6 + 3*a^2*c^5*d^6*e^8)*x^
5 + 5*(3*c^7*d^11*e^3 + 8*a*c^6*d^9*e^5 + 3*a^2*c^5*d^7*e^7)*x^4 + 2*(3*c^7*d^12*e^2 + 15*a*c^6*d^10*e^4 + 10*
a^2*c^5*d^8*e^6)*x^3 + (c^7*d^13*e + 12*a*c^6*d^11*e^3 + 15*a^2*c^5*d^9*e^5)*x^2 + 2*(a*c^6*d^12*e^2 + 3*a^2*c
^5*d^10*e^4)*x)*sqrt(e/(c*d^2 - a*e^2))*arctan(-sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d^2 - a*e^2)*sq
rt(e*x + d)*sqrt(e/(c*d^2 - a*e^2))/(c*d*e^2*x^2 + a*d*e^2 + (c*d^2*e + a*e^3)*x)) + (45045*c^6*d^6*e^6*x^6 -
1280*c^6*d^12 + 24320*a*c^5*d^10*e^2 + 35595*a^2*c^4*d^8*e^4 - 21070*a^3*c^3*d^6*e^6 + 10024*a^4*c^2*d^4*e^8 -
2928*a^5*c*d^2*e^10 + 384*a^6*e^12 + 30030*(7*c^6*d^7*e^5 + 2*a*c^5*d^5*e^7)*x^5 + 3003*(128*c^6*d^8*e^4 + 94
*a*c^5*d^6*e^6 + 3*a^2*c^4*d^4*e^8)*x^4 + 858*(395*c^6*d^9*e^3 + 607*a*c^5*d^7*e^5 + 51*a^2*c^4*d^5*e^7 - 3*a^
3*c^3*d^3*e^9)*x^3 + 143*(965*c^6*d^10*e^2 + 3250*a*c^5*d^8*e^4 + 588*a^2*c^4*d^6*e^6 - 86*a^3*c^3*d^4*e^8 + 8
*a^4*c^2*d^2*e^10)*x^2 + 26*(640*c^6*d^11*e + 7415*a*c^5*d^9*e^3 + 3045*a^2*c^4*d^7*e^5 - 889*a^3*c^3*d^5*e^7
+ 208*a^4*c^2*d^3*e^9 - 24*a^5*c*d*e^11)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(a^2*c^
7*d^20*e^2 - 7*a^3*c^6*d^18*e^4 + 21*a^4*c^5*d^16*e^6 - 35*a^5*c^4*d^14*e^8 + 35*a^6*c^3*d^12*e^10 - 21*a^7*c^
2*d^10*e^12 + 7*a^8*c*d^8*e^14 - a^9*d^6*e^16 + (c^9*d^16*e^6 - 7*a*c^8*d^14*e^8 + 21*a^2*c^7*d^12*e^10 - 35*a
^3*c^6*d^10*e^12 + 35*a^4*c^5*d^8*e^14 - 21*a^5*c^4*d^6*e^16 + 7*a^6*c^3*d^4*e^18 - a^7*c^2*d^2*e^20)*x^8 + 2*
(3*c^9*d^17*e^5 - 20*a*c^8*d^15*e^7 + 56*a^2*c^7*d^13*e^9 - 84*a^3*c^6*d^11*e^11 + 70*a^4*c^5*d^9*e^13 - 28*a^
5*c^4*d^7*e^15 + 4*a^7*c^2*d^3*e^19 - a^8*c*d*e^21)*x^7 + (15*c^9*d^18*e^4 - 93*a*c^8*d^16*e^6 + 232*a^2*c^7*d
^14*e^8 - 280*a^3*c^6*d^12*e^10 + 126*a^4*c^5*d^10*e^12 + 70*a^5*c^4*d^8*e^14 - 112*a^6*c^3*d^6*e^16 + 48*a^7*
c^2*d^4*e^18 - 5*a^8*c*d^2*e^20 - a^9*e^22)*x^6 + 2*(10*c^9*d^19*e^3 - 55*a*c^8*d^17*e^5 + 108*a^2*c^7*d^15*e^
7 - 56*a^3*c^6*d^13*e^9 - 112*a^4*c^5*d^11*e^11 + 210*a^5*c^4*d^9*e^13 - 140*a^6*c^3*d^7*e^15 + 32*a^7*c^2*d^5
*e^17 + 6*a^8*c*d^3*e^19 - 3*a^9*d*e^21)*x^5 + 5*(3*c^9*d^20*e^2 - 13*a*c^8*d^18*e^4 + 10*a^2*c^7*d^16*e^6 + 4
2*a^3*c^6*d^14*e^8 - 112*a^4*c^5*d^12*e^10 + 112*a^5*c^4*d^10*e^12 - 42*a^6*c^3*d^8*e^14 - 10*a^7*c^2*d^6*e^16
+ 13*a^8*c*d^4*e^18 - 3*a^9*d^2*e^20)*x^4 + 2*(3*c^9*d^21*e - 6*a*c^8*d^19*e^3 - 32*a^2*c^7*d^17*e^5 + 140*a^
3*c^6*d^15*e^7 - 210*a^4*c^5*d^13*e^9 + 112*a^5*c^4*d^11*e^11 + 56*a^6*c^3*d^9*e^13 - 108*a^7*c^2*d^7*e^15 + 5
5*a^8*c*d^5*e^17 - 10*a^9*d^3*e^19)*x^3 + (c^9*d^22 + 5*a*c^8*d^20*e^2 - 48*a^2*c^7*d^18*e^4 + 112*a^3*c^6*d^1
6*e^6 - 70*a^4*c^5*d^14*e^8 - 126*a^5*c^4*d^12*e^10 + 280*a^6*c^3*d^10*e^12 - 232*a^7*c^2*d^8*e^14 + 93*a^8*c*
d^6*e^16 - 15*a^9*d^4*e^18)*x^2 + 2*(a*c^8*d^21*e - 4*a^2*c^7*d^19*e^3 + 28*a^4*c^5*d^15*e^7 - 70*a^5*c^4*d^13
*e^9 + 84*a^6*c^3*d^11*e^11 - 56*a^7*c^2*d^9*e^13 + 20*a^8*c*d^7*e^15 - 3*a^9*d^5*e^17)*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)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 2\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")

[Out]

[undef, undef, undef, undef, undef, undef, undef, undef, undef, undef, undef, 2]