3.2079 \(\int \frac{1}{(d+e x)^{5/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\)

Optimal. Leaf size=457 \[ \frac{1155 c^4 d^4 e \sqrt{d+e x}}{64 \left (c d^2-a e^2\right )^6 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{385 c^3 d^3 e}{64 \sqrt{d+e x} \left (c d^2-a e^2\right )^5 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{77 c^3 d^3 \sqrt{d+e x}}{32 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{33 c^2 d^2}{32 \sqrt{d+e x} \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{1155 c^4 d^4 e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{64 \left (c d^2-a e^2\right )^{13/2}}+\frac{11 c d}{24 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{1}{4 (d+e x)^{5/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

[Out]

1/(4*(c*d^2 - a*e^2)*(d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (11*c*d)/(24*(c*d^2 - a*
e^2)^2*(d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (33*c^2*d^2)/(32*(c*d^2 - a*e^2)^3*Sqr
t[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (77*c^3*d^3*Sqrt[d + e*x])/(32*(c*d^2 - a*e^2)^4*(
a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (385*c^3*d^3*e)/(64*(c*d^2 - a*e^2)^5*Sqrt[d + e*x]*Sqrt[a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (1155*c^4*d^4*e*Sqrt[d + e*x])/(64*(c*d^2 - a*e^2)^6*Sqrt[a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2]) + (1155*c^4*d^4*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])])/(64*(c*d^2 - a*e^2)^(13/2))

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Rubi [A]  time = 0.501198, antiderivative size = 457, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {672, 666, 660, 205} \[ \frac{1155 c^4 d^4 e \sqrt{d+e x}}{64 \left (c d^2-a e^2\right )^6 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{385 c^3 d^3 e}{64 \sqrt{d+e x} \left (c d^2-a e^2\right )^5 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{77 c^3 d^3 \sqrt{d+e x}}{32 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{33 c^2 d^2}{32 \sqrt{d+e x} \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{1155 c^4 d^4 e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{64 \left (c d^2-a e^2\right )^{13/2}}+\frac{11 c d}{24 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac{1}{4 (d+e x)^{5/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1/(4*(c*d^2 - a*e^2)*(d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (11*c*d)/(24*(c*d^2 - a*
e^2)^2*(d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (33*c^2*d^2)/(32*(c*d^2 - a*e^2)^3*Sqr
t[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (77*c^3*d^3*Sqrt[d + e*x])/(32*(c*d^2 - a*e^2)^4*(
a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (385*c^3*d^3*e)/(64*(c*d^2 - a*e^2)^5*Sqrt[d + e*x]*Sqrt[a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (1155*c^4*d^4*e*Sqrt[d + e*x])/(64*(c*d^2 - a*e^2)^6*Sqrt[a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2]) + (1155*c^4*d^4*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])])/(64*(c*d^2 - a*e^2)^(13/2))

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)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=\frac{1}{4 \left (c d^2-a e^2\right ) (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{(11 c d) \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}{8 \left (c d^2-a e^2\right )}\\ &=\frac{1}{4 \left (c d^2-a e^2\right ) (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{11 c d}{24 \left (c d^2-a e^2\right )^2 (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 (33 c^2 d^2\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}{16 \left (c d^2-a e^2\right )^2}\\ &=\frac{1}{4 \left (c d^2-a e^2\right ) (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{11 c d}{24 \left (c d^2-a e^2\right )^2 (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{33 c^2 d^2}{32 \left (c d^2-a e^2\right )^3 \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 (231 c^3 d^3\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}{64 \left (c d^2-a e^2\right )^3}\\ &=\frac{1}{4 \left (c d^2-a e^2\right ) (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{11 c d}{24 \left (c d^2-a e^2\right )^2 (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{33 c^2 d^2}{32 \left (c d^2-a e^2\right )^3 \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{77 c^3 d^3 \sqrt{d+e x}}{32 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{\left (385 c^3 d^3 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}{64 \left (c d^2-a e^2\right )^4}\\ &=\frac{1}{4 \left (c d^2-a e^2\right ) (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{11 c d}{24 \left (c d^2-a e^2\right )^2 (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{33 c^2 d^2}{32 \left (c d^2-a e^2\right )^3 \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{77 c^3 d^3 \sqrt{d+e x}}{32 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{385 c^3 d^3 e}{64 \left (c d^2-a e^2\right )^5 \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{\left (1155 c^4 d^4 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}{128 \left (c d^2-a e^2\right )^5}\\ &=\frac{1}{4 \left (c d^2-a e^2\right ) (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{11 c d}{24 \left (c d^2-a e^2\right )^2 (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{33 c^2 d^2}{32 \left (c d^2-a e^2\right )^3 \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{77 c^3 d^3 \sqrt{d+e x}}{32 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{385 c^3 d^3 e}{64 \left (c d^2-a e^2\right )^5 \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{1155 c^4 d^4 e \sqrt{d+e x}}{64 \left (c d^2-a e^2\right )^6 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{\left (1155 c^4 d^4 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}{128 \left (c d^2-a e^2\right )^6}\\ &=\frac{1}{4 \left (c d^2-a e^2\right ) (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{11 c d}{24 \left (c d^2-a e^2\right )^2 (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{33 c^2 d^2}{32 \left (c d^2-a e^2\right )^3 \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{77 c^3 d^3 \sqrt{d+e x}}{32 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{385 c^3 d^3 e}{64 \left (c d^2-a e^2\right )^5 \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{1155 c^4 d^4 e \sqrt{d+e x}}{64 \left (c d^2-a e^2\right )^6 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{\left (1155 c^4 d^4 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 )}{64 \left (c d^2-a e^2\right )^6}\\ &=\frac{1}{4 \left (c d^2-a e^2\right ) (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{11 c d}{24 \left (c d^2-a e^2\right )^2 (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{33 c^2 d^2}{32 \left (c d^2-a e^2\right )^3 \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{77 c^3 d^3 \sqrt{d+e x}}{32 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{385 c^3 d^3 e}{64 \left (c d^2-a e^2\right )^5 \sqrt{d+e x} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{1155 c^4 d^4 e \sqrt{d+e x}}{64 \left (c d^2-a e^2\right )^6 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{1155 c^4 d^4 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 )}{64 \left (c d^2-a e^2\right )^{13/2}}\\ \end{align*}

Mathematica [C]  time = 0.0411061, size = 83, normalized size = 0.18 \[ -\frac{2 c^4 d^4 (d+e x)^{3/2} \, _2F_1\left (-\frac{3}{2},5;-\frac{1}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right )}{3 \left (c d^2-a e^2\right )^5 ((d+e x) (a e+c d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*c^4*d^4*(d + e*x)^(3/2)*Hypergeometric2F1[-3/2, 5, -1/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(3*(c*d^2
- a*e^2)^5*((a*e + c*d*x)*(d + e*x))^(3/2))

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Maple [B]  time = 0.227, size = 1225, normalized size = 2.7 \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)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)

[Out]

-1/192*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(-12705*((a*e^2-c*d^2)*e)^(1/2)*x^4*c^5*d^6*e^4-16863*((a*e^2-c
*d^2)*e)^(1/2)*x^3*c^5*d^7*e^3-9207*((a*e^2-c*d^2)*e)^(1/2)*x^2*c^5*d^8*e^2-1408*((a*e^2-c*d^2)*e)^(1/2)*x*c^5
*d^9*e-328*((a*e^2-c*d^2)*e)^(1/2)*a^4*c*d^2*e^8+1030*((a*e^2-c*d^2)*e)^(1/2)*a^3*c^2*d^4*e^6-2295*((a*e^2-c*d
^2)*e)^(1/2)*a^2*c^3*d^6*e^4-3465*((a*e^2-c*d^2)*e)^(1/2)*x^5*c^5*d^5*e^5+3465*arctanh(e*(c*d*x+a*e)^(1/2)/((a
*e^2-c*d^2)*e)^(1/2))*x^4*a*c^4*d^4*e^7*(c*d*x+a*e)^(1/2)+13860*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^
(1/2))*x^3*a*c^4*d^5*e^6*(c*d*x+a*e)^(1/2)+20790*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x^2*a*c^
4*d^6*e^5*(c*d*x+a*e)^(1/2)+13860*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x*a*c^4*d^7*e^4*(c*d*x+
a*e)^(1/2)-2048*((a*e^2-c*d^2)*e)^(1/2)*a*c^4*d^8*e^2+128*((a*e^2-c*d^2)*e)^(1/2)*c^5*d^10+48*((a*e^2-c*d^2)*e
)^(1/2)*a^5*e^10-4620*((a*e^2-c*d^2)*e)^(1/2)*x^4*a*c^4*d^4*e^6-693*((a*e^2-c*d^2)*e)^(1/2)*x^3*a^2*c^3*d^3*e^
7+198*((a*e^2-c*d^2)*e)^(1/2)*x^2*a^3*c^2*d^2*e^8-2673*((a*e^2-c*d^2)*e)^(1/2)*x^2*a^2*c^3*d^4*e^6-88*((a*e^2-
c*d^2)*e)^(1/2)*x*a^4*c*d*e^9+748*((a*e^2-c*d^2)*e)^(1/2)*x*a^3*c^2*d^3*e^7-17094*((a*e^2-c*d^2)*e)^(1/2)*x^3*
a*c^4*d^5*e^5-22968*((a*e^2-c*d^2)*e)^(1/2)*x^2*a*c^4*d^6*e^4-12782*((a*e^2-c*d^2)*e)^(1/2)*x*a*c^4*d^7*e^3-37
95*((a*e^2-c*d^2)*e)^(1/2)*x*a^2*c^3*d^5*e^5+3465*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x^5*c^5
*d^5*e^6*(c*d*x+a*e)^(1/2)+13860*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x^4*c^5*d^6*e^5*(c*d*x+a
*e)^(1/2)+20790*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x^3*c^5*d^7*e^4*(c*d*x+a*e)^(1/2)+13860*a
rctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x^2*c^5*d^8*e^3*(c*d*x+a*e)^(1/2)+3465*arctanh(e*(c*d*x+a*
e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*x*c^5*d^9*e^2*(c*d*x+a*e)^(1/2)+3465*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d
^2)*e)^(1/2))*a*c^4*d^8*e^3*(c*d*x+a*e)^(1/2))/(e*x+d)^(9/2)/(c*d*x+a*e)^2/(a*e^2-c*d^2)^6/((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{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [B]  time = 2.45268, size = 6433, normalized size = 14.08 \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)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fricas")

[Out]

[1/384*(3465*(c^6*d^6*e^6*x^7 + a^2*c^4*d^9*e^3 + (5*c^6*d^7*e^5 + 2*a*c^5*d^5*e^7)*x^6 + (10*c^6*d^8*e^4 + 10
*a*c^5*d^6*e^6 + a^2*c^4*d^4*e^8)*x^5 + 5*(2*c^6*d^9*e^3 + 4*a*c^5*d^7*e^5 + a^2*c^4*d^5*e^7)*x^4 + 5*(c^6*d^1
0*e^2 + 4*a*c^5*d^8*e^4 + 2*a^2*c^4*d^6*e^6)*x^3 + (c^6*d^11*e + 10*a*c^5*d^9*e^3 + 10*a^2*c^4*d^7*e^5)*x^2 +
(2*a*c^5*d^10*e^2 + 5*a^2*c^4*d^8*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*(3465*c^5*d^5*e^5*x^5 - 128*c^5*d^10 + 2048*a*c^4*d^8*e^2 + 2295*a^2*c^3*d^6*e
^4 - 1030*a^3*c^2*d^4*e^6 + 328*a^4*c*d^2*e^8 - 48*a^5*e^10 + 1155*(11*c^5*d^6*e^4 + 4*a*c^4*d^4*e^6)*x^4 + 23
1*(73*c^5*d^7*e^3 + 74*a*c^4*d^5*e^5 + 3*a^2*c^3*d^3*e^7)*x^3 + 99*(93*c^5*d^8*e^2 + 232*a*c^4*d^6*e^4 + 27*a^
2*c^3*d^4*e^6 - 2*a^3*c^2*d^2*e^8)*x^2 + 11*(128*c^5*d^9*e + 1162*a*c^4*d^7*e^3 + 345*a^2*c^3*d^5*e^5 - 68*a^3
*c^2*d^3*e^7 + 8*a^4*c*d*e^9)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(a^2*c^6*d^17*e^2
- 6*a^3*c^5*d^15*e^4 + 15*a^4*c^4*d^13*e^6 - 20*a^5*c^3*d^11*e^8 + 15*a^6*c^2*d^9*e^10 - 6*a^7*c*d^7*e^12 + a^
8*d^5*e^14 + (c^8*d^14*e^5 - 6*a*c^7*d^12*e^7 + 15*a^2*c^6*d^10*e^9 - 20*a^3*c^5*d^8*e^11 + 15*a^4*c^4*d^6*e^1
3 - 6*a^5*c^3*d^4*e^15 + a^6*c^2*d^2*e^17)*x^7 + (5*c^8*d^15*e^4 - 28*a*c^7*d^13*e^6 + 63*a^2*c^6*d^11*e^8 - 7
0*a^3*c^5*d^9*e^10 + 35*a^4*c^4*d^7*e^12 - 7*a^6*c^2*d^3*e^16 + 2*a^7*c*d*e^18)*x^6 + (10*c^8*d^16*e^3 - 50*a*
c^7*d^14*e^5 + 91*a^2*c^6*d^12*e^7 - 56*a^3*c^5*d^10*e^9 - 35*a^4*c^4*d^8*e^11 + 70*a^5*c^3*d^6*e^13 - 35*a^6*
c^2*d^4*e^15 + 4*a^7*c*d^2*e^17 + a^8*e^19)*x^5 + 5*(2*c^8*d^17*e^2 - 8*a*c^7*d^15*e^4 + 7*a^2*c^6*d^13*e^6 +
14*a^3*c^5*d^11*e^8 - 35*a^4*c^4*d^9*e^10 + 28*a^5*c^3*d^7*e^12 - 7*a^6*c^2*d^5*e^14 - 2*a^7*c*d^3*e^16 + a^8*
d*e^18)*x^4 + 5*(c^8*d^18*e - 2*a*c^7*d^16*e^3 - 7*a^2*c^6*d^14*e^5 + 28*a^3*c^5*d^12*e^7 - 35*a^4*c^4*d^10*e^
9 + 14*a^5*c^3*d^8*e^11 + 7*a^6*c^2*d^6*e^13 - 8*a^7*c*d^4*e^15 + 2*a^8*d^2*e^17)*x^3 + (c^8*d^19 + 4*a*c^7*d^
17*e^2 - 35*a^2*c^6*d^15*e^4 + 70*a^3*c^5*d^13*e^6 - 35*a^4*c^4*d^11*e^8 - 56*a^5*c^3*d^9*e^10 + 91*a^6*c^2*d^
7*e^12 - 50*a^7*c*d^5*e^14 + 10*a^8*d^3*e^16)*x^2 + (2*a*c^7*d^18*e - 7*a^2*c^6*d^16*e^3 + 35*a^4*c^4*d^12*e^7
- 70*a^5*c^3*d^10*e^9 + 63*a^6*c^2*d^8*e^11 - 28*a^7*c*d^6*e^13 + 5*a^8*d^4*e^15)*x), 1/192*(3465*(c^6*d^6*e^
6*x^7 + a^2*c^4*d^9*e^3 + (5*c^6*d^7*e^5 + 2*a*c^5*d^5*e^7)*x^6 + (10*c^6*d^8*e^4 + 10*a*c^5*d^6*e^6 + a^2*c^4
*d^4*e^8)*x^5 + 5*(2*c^6*d^9*e^3 + 4*a*c^5*d^7*e^5 + a^2*c^4*d^5*e^7)*x^4 + 5*(c^6*d^10*e^2 + 4*a*c^5*d^8*e^4
+ 2*a^2*c^4*d^6*e^6)*x^3 + (c^6*d^11*e + 10*a*c^5*d^9*e^3 + 10*a^2*c^4*d^7*e^5)*x^2 + (2*a*c^5*d^10*e^2 + 5*a^
2*c^4*d^8*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)*
sqrt(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)) + (3465*c^5*d^5*e^5*x^5 -
128*c^5*d^10 + 2048*a*c^4*d^8*e^2 + 2295*a^2*c^3*d^6*e^4 - 1030*a^3*c^2*d^4*e^6 + 328*a^4*c*d^2*e^8 - 48*a^5*
e^10 + 1155*(11*c^5*d^6*e^4 + 4*a*c^4*d^4*e^6)*x^4 + 231*(73*c^5*d^7*e^3 + 74*a*c^4*d^5*e^5 + 3*a^2*c^3*d^3*e^
7)*x^3 + 99*(93*c^5*d^8*e^2 + 232*a*c^4*d^6*e^4 + 27*a^2*c^3*d^4*e^6 - 2*a^3*c^2*d^2*e^8)*x^2 + 11*(128*c^5*d^
9*e + 1162*a*c^4*d^7*e^3 + 345*a^2*c^3*d^5*e^5 - 68*a^3*c^2*d^3*e^7 + 8*a^4*c*d*e^9)*x)*sqrt(c*d*e*x^2 + a*d*e
+ (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(a^2*c^6*d^17*e^2 - 6*a^3*c^5*d^15*e^4 + 15*a^4*c^4*d^13*e^6 - 20*a^5*c^3
*d^11*e^8 + 15*a^6*c^2*d^9*e^10 - 6*a^7*c*d^7*e^12 + a^8*d^5*e^14 + (c^8*d^14*e^5 - 6*a*c^7*d^12*e^7 + 15*a^2*
c^6*d^10*e^9 - 20*a^3*c^5*d^8*e^11 + 15*a^4*c^4*d^6*e^13 - 6*a^5*c^3*d^4*e^15 + a^6*c^2*d^2*e^17)*x^7 + (5*c^8
*d^15*e^4 - 28*a*c^7*d^13*e^6 + 63*a^2*c^6*d^11*e^8 - 70*a^3*c^5*d^9*e^10 + 35*a^4*c^4*d^7*e^12 - 7*a^6*c^2*d^
3*e^16 + 2*a^7*c*d*e^18)*x^6 + (10*c^8*d^16*e^3 - 50*a*c^7*d^14*e^5 + 91*a^2*c^6*d^12*e^7 - 56*a^3*c^5*d^10*e^
9 - 35*a^4*c^4*d^8*e^11 + 70*a^5*c^3*d^6*e^13 - 35*a^6*c^2*d^4*e^15 + 4*a^7*c*d^2*e^17 + a^8*e^19)*x^5 + 5*(2*
c^8*d^17*e^2 - 8*a*c^7*d^15*e^4 + 7*a^2*c^6*d^13*e^6 + 14*a^3*c^5*d^11*e^8 - 35*a^4*c^4*d^9*e^10 + 28*a^5*c^3*
d^7*e^12 - 7*a^6*c^2*d^5*e^14 - 2*a^7*c*d^3*e^16 + a^8*d*e^18)*x^4 + 5*(c^8*d^18*e - 2*a*c^7*d^16*e^3 - 7*a^2*
c^6*d^14*e^5 + 28*a^3*c^5*d^12*e^7 - 35*a^4*c^4*d^10*e^9 + 14*a^5*c^3*d^8*e^11 + 7*a^6*c^2*d^6*e^13 - 8*a^7*c*
d^4*e^15 + 2*a^8*d^2*e^17)*x^3 + (c^8*d^19 + 4*a*c^7*d^17*e^2 - 35*a^2*c^6*d^15*e^4 + 70*a^3*c^5*d^13*e^6 - 35
*a^4*c^4*d^11*e^8 - 56*a^5*c^3*d^9*e^10 + 91*a^6*c^2*d^7*e^12 - 50*a^7*c*d^5*e^14 + 10*a^8*d^3*e^16)*x^2 + (2*
a*c^7*d^18*e - 7*a^2*c^6*d^16*e^3 + 35*a^4*c^4*d^12*e^7 - 70*a^5*c^3*d^10*e^9 + 63*a^6*c^2*d^8*e^11 - 28*a^7*c
*d^6*e^13 + 5*a^8*d^4*e^15)*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)**(5/2)/(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)^(5/2)/(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]