### 3.2009 $$\int \frac{(d+e x)^{13/2}}{(a d e+(c d^2+a e^2) x+c d e x^2)^2} \, dx$$

Optimal. Leaf size=210 $\frac{9 e (d+e x)^{5/2} \left (c d^2-a e^2\right )}{5 c^3 d^3}+\frac{3 e (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}{c^4 d^4}+\frac{9 e \sqrt{d+e x} \left (c d^2-a e^2\right )^3}{c^5 d^5}-\frac{9 e \left (c d^2-a e^2\right )^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{11/2} d^{11/2}}-\frac{(d+e x)^{9/2}}{c d (a e+c d x)}+\frac{9 e (d+e x)^{7/2}}{7 c^2 d^2}$

[Out]

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

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Rubi [A]  time = 0.176113, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 37, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.135, Rules used = {626, 47, 50, 63, 208} $\frac{9 e (d+e x)^{5/2} \left (c d^2-a e^2\right )}{5 c^3 d^3}+\frac{3 e (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}{c^4 d^4}+\frac{9 e \sqrt{d+e x} \left (c d^2-a e^2\right )^3}{c^5 d^5}-\frac{9 e \left (c d^2-a e^2\right )^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{11/2} d^{11/2}}-\frac{(d+e x)^{9/2}}{c d (a e+c d x)}+\frac{9 e (d+e x)^{7/2}}{7 c^2 d^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^(13/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]

[Out]

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

Rule 626

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

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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{(d+e x)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx &=\int \frac{(d+e x)^{9/2}}{(a e+c d x)^2} \, dx\\ &=-\frac{(d+e x)^{9/2}}{c d (a e+c d x)}+\frac{(9 e) \int \frac{(d+e x)^{7/2}}{a e+c d x} \, dx}{2 c d}\\ &=\frac{9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac{(d+e x)^{9/2}}{c d (a e+c d x)}+\frac{\left (9 e \left (c d^2-a e^2\right )\right ) \int \frac{(d+e x)^{5/2}}{a e+c d x} \, dx}{2 c^2 d^2}\\ &=\frac{9 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^3 d^3}+\frac{9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac{(d+e x)^{9/2}}{c d (a e+c d x)}+\frac{\left (9 e \left (c d^2-a e^2\right )^2\right ) \int \frac{(d+e x)^{3/2}}{a e+c d x} \, dx}{2 c^3 d^3}\\ &=\frac{3 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{c^4 d^4}+\frac{9 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^3 d^3}+\frac{9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac{(d+e x)^{9/2}}{c d (a e+c d x)}+\frac{\left (9 e \left (c d^2-a e^2\right )^3\right ) \int \frac{\sqrt{d+e x}}{a e+c d x} \, dx}{2 c^4 d^4}\\ &=\frac{9 e \left (c d^2-a e^2\right )^3 \sqrt{d+e x}}{c^5 d^5}+\frac{3 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{c^4 d^4}+\frac{9 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^3 d^3}+\frac{9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac{(d+e x)^{9/2}}{c d (a e+c d x)}+\frac{\left (9 e \left (c d^2-a e^2\right )^4\right ) \int \frac{1}{(a e+c d x) \sqrt{d+e x}} \, dx}{2 c^5 d^5}\\ &=\frac{9 e \left (c d^2-a e^2\right )^3 \sqrt{d+e x}}{c^5 d^5}+\frac{3 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{c^4 d^4}+\frac{9 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^3 d^3}+\frac{9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac{(d+e x)^{9/2}}{c d (a e+c d x)}+\frac{\left (9 \left (c d^2-a e^2\right )^4\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{c d^2}{e}+a e+\frac{c d x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{c^5 d^5}\\ &=\frac{9 e \left (c d^2-a e^2\right )^3 \sqrt{d+e x}}{c^5 d^5}+\frac{3 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{c^4 d^4}+\frac{9 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^3 d^3}+\frac{9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac{(d+e x)^{9/2}}{c d (a e+c d x)}-\frac{9 e \left (c d^2-a e^2\right )^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{11/2} d^{11/2}}\\ \end{align*}

Mathematica [C]  time = 0.0251819, size = 59, normalized size = 0.28 $\frac{2 e (d+e x)^{11/2} \, _2F_1\left (2,\frac{11}{2};\frac{13}{2};-\frac{c d (d+e x)}{a e^2-c d^2}\right )}{11 \left (a e^2-c d^2\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^(13/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]

[Out]

(2*e*(d + e*x)^(11/2)*Hypergeometric2F1[2, 11/2, 13/2, -((c*d*(d + e*x))/(-(c*d^2) + a*e^2))])/(11*(-(c*d^2) +
a*e^2)^2)

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Maple [B]  time = 0.203, size = 628, normalized size = 3. \begin{align*}{\frac{2\,e}{7\,{c}^{2}{d}^{2}} \left ( ex+d \right ) ^{{\frac{7}{2}}}}-{\frac{4\,a{e}^{3}}{5\,{c}^{3}{d}^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+{\frac{4\,e}{5\,{c}^{2}d} \left ( ex+d \right ) ^{{\frac{5}{2}}}}+2\,{\frac{ \left ( ex+d \right ) ^{3/2}{a}^{2}{e}^{5}}{{c}^{4}{d}^{4}}}-4\,{\frac{ \left ( ex+d \right ) ^{3/2}a{e}^{3}}{{c}^{3}{d}^{2}}}+2\,{\frac{e \left ( ex+d \right ) ^{3/2}}{{c}^{2}}}-8\,{\frac{{a}^{3}{e}^{7}\sqrt{ex+d}}{{c}^{5}{d}^{5}}}+24\,{\frac{{a}^{2}{e}^{5}\sqrt{ex+d}}{{c}^{4}{d}^{3}}}-24\,{\frac{a{e}^{3}\sqrt{ex+d}}{{c}^{3}d}}+8\,{\frac{de\sqrt{ex+d}}{{c}^{2}}}-{\frac{{a}^{4}{e}^{9}}{{c}^{5}{d}^{5} \left ( cdex+a{e}^{2} \right ) }\sqrt{ex+d}}+4\,{\frac{{a}^{3}{e}^{7}\sqrt{ex+d}}{{c}^{4}{d}^{3} \left ( cdex+a{e}^{2} \right ) }}-6\,{\frac{{a}^{2}{e}^{5}\sqrt{ex+d}}{{c}^{3}d \left ( cdex+a{e}^{2} \right ) }}+4\,{\frac{d\sqrt{ex+d}a{e}^{3}}{{c}^{2} \left ( cdex+a{e}^{2} \right ) }}-{\frac{e{d}^{3}}{c \left ( cdex+a{e}^{2} \right ) }\sqrt{ex+d}}+9\,{\frac{{a}^{4}{e}^{9}}{{c}^{5}{d}^{5}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{\sqrt{ex+d}cd}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }-36\,{\frac{{a}^{3}{e}^{7}}{{c}^{4}{d}^{3}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{\sqrt{ex+d}cd}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }+54\,{\frac{{a}^{2}{e}^{5}}{{c}^{3}d\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{\sqrt{ex+d}cd}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }-36\,{\frac{ad{e}^{3}}{{c}^{2}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{\sqrt{ex+d}cd}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }+9\,{\frac{e{d}^{3}}{c\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{\sqrt{ex+d}cd}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(13/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x)

[Out]

2/7*e*(e*x+d)^(7/2)/c^2/d^2-4/5/c^3/d^3*(e*x+d)^(5/2)*a*e^3+4/5*e/c^2/d*(e*x+d)^(5/2)+2/c^4/d^4*(e*x+d)^(3/2)*
a^2*e^5-4/c^3/d^2*(e*x+d)^(3/2)*a*e^3+2*e/c^2*(e*x+d)^(3/2)-8/c^5/d^5*a^3*e^7*(e*x+d)^(1/2)+24/c^4/d^3*a^2*e^5
*(e*x+d)^(1/2)-24/c^3/d*a*e^3*(e*x+d)^(1/2)+8*e/c^2*d*(e*x+d)^(1/2)-1/c^5/d^5*(e*x+d)^(1/2)/(c*d*e*x+a*e^2)*a^
4*e^9+4/c^4/d^3*(e*x+d)^(1/2)/(c*d*e*x+a*e^2)*a^3*e^7-6/c^3/d*(e*x+d)^(1/2)/(c*d*e*x+a*e^2)*a^2*e^5+4/c^2*d*(e
*x+d)^(1/2)/(c*d*e*x+a*e^2)*a*e^3-e/c*d^3*(e*x+d)^(1/2)/(c*d*e*x+a*e^2)+9/c^5/d^5/((a*e^2-c*d^2)*c*d)^(1/2)*ar
ctan((e*x+d)^(1/2)*c*d/((a*e^2-c*d^2)*c*d)^(1/2))*a^4*e^9-36/c^4/d^3/((a*e^2-c*d^2)*c*d)^(1/2)*arctan((e*x+d)^
(1/2)*c*d/((a*e^2-c*d^2)*c*d)^(1/2))*a^3*e^7+54/c^3/d/((a*e^2-c*d^2)*c*d)^(1/2)*arctan((e*x+d)^(1/2)*c*d/((a*e
^2-c*d^2)*c*d)^(1/2))*a^2*e^5-36/c^2*d/((a*e^2-c*d^2)*c*d)^(1/2)*arctan((e*x+d)^(1/2)*c*d/((a*e^2-c*d^2)*c*d)^
(1/2))*a*e^3+9*e/c*d^3/((a*e^2-c*d^2)*c*d)^(1/2)*arctan((e*x+d)^(1/2)*c*d/((a*e^2-c*d^2)*c*d)^(1/2))

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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((e*x+d)^(13/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.03539, size = 1629, normalized size = 7.76 \begin{align*} \left [\frac{315 \,{\left (a c^{3} d^{6} e^{2} - 3 \, a^{2} c^{2} d^{4} e^{4} + 3 \, a^{3} c d^{2} e^{6} - a^{4} e^{8} +{\left (c^{4} d^{7} e - 3 \, a c^{3} d^{5} e^{3} + 3 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x\right )} \sqrt{\frac{c d^{2} - a e^{2}}{c d}} \log \left (\frac{c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt{e x + d} c d \sqrt{\frac{c d^{2} - a e^{2}}{c d}}}{c d x + a e}\right ) + 2 \,{\left (10 \, c^{4} d^{4} e^{4} x^{4} - 35 \, c^{4} d^{8} + 528 \, a c^{3} d^{6} e^{2} - 1218 \, a^{2} c^{2} d^{4} e^{4} + 1050 \, a^{3} c d^{2} e^{6} - 315 \, a^{4} e^{8} + 2 \,{\left (29 \, c^{4} d^{5} e^{3} - 9 \, a c^{3} d^{3} e^{5}\right )} x^{3} + 6 \,{\left (26 \, c^{4} d^{6} e^{2} - 23 \, a c^{3} d^{4} e^{4} + 7 \, a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 2 \,{\left (194 \, c^{4} d^{7} e - 426 \, a c^{3} d^{5} e^{3} + 357 \, a^{2} c^{2} d^{3} e^{5} - 105 \, a^{3} c d e^{7}\right )} x\right )} \sqrt{e x + d}}{70 \,{\left (c^{6} d^{6} x + a c^{5} d^{5} e\right )}}, -\frac{315 \,{\left (a c^{3} d^{6} e^{2} - 3 \, a^{2} c^{2} d^{4} e^{4} + 3 \, a^{3} c d^{2} e^{6} - a^{4} e^{8} +{\left (c^{4} d^{7} e - 3 \, a c^{3} d^{5} e^{3} + 3 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x\right )} \sqrt{-\frac{c d^{2} - a e^{2}}{c d}} \arctan \left (-\frac{\sqrt{e x + d} c d \sqrt{-\frac{c d^{2} - a e^{2}}{c d}}}{c d^{2} - a e^{2}}\right ) -{\left (10 \, c^{4} d^{4} e^{4} x^{4} - 35 \, c^{4} d^{8} + 528 \, a c^{3} d^{6} e^{2} - 1218 \, a^{2} c^{2} d^{4} e^{4} + 1050 \, a^{3} c d^{2} e^{6} - 315 \, a^{4} e^{8} + 2 \,{\left (29 \, c^{4} d^{5} e^{3} - 9 \, a c^{3} d^{3} e^{5}\right )} x^{3} + 6 \,{\left (26 \, c^{4} d^{6} e^{2} - 23 \, a c^{3} d^{4} e^{4} + 7 \, a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 2 \,{\left (194 \, c^{4} d^{7} e - 426 \, a c^{3} d^{5} e^{3} + 357 \, a^{2} c^{2} d^{3} e^{5} - 105 \, a^{3} c d e^{7}\right )} x\right )} \sqrt{e x + d}}{35 \,{\left (c^{6} d^{6} x + a c^{5} d^{5} e\right )}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(13/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="fricas")

[Out]

[1/70*(315*(a*c^3*d^6*e^2 - 3*a^2*c^2*d^4*e^4 + 3*a^3*c*d^2*e^6 - a^4*e^8 + (c^4*d^7*e - 3*a*c^3*d^5*e^3 + 3*a
^2*c^2*d^3*e^5 - a^3*c*d*e^7)*x)*sqrt((c*d^2 - a*e^2)/(c*d))*log((c*d*e*x + 2*c*d^2 - a*e^2 - 2*sqrt(e*x + d)*
c*d*sqrt((c*d^2 - a*e^2)/(c*d)))/(c*d*x + a*e)) + 2*(10*c^4*d^4*e^4*x^4 - 35*c^4*d^8 + 528*a*c^3*d^6*e^2 - 121
8*a^2*c^2*d^4*e^4 + 1050*a^3*c*d^2*e^6 - 315*a^4*e^8 + 2*(29*c^4*d^5*e^3 - 9*a*c^3*d^3*e^5)*x^3 + 6*(26*c^4*d^
6*e^2 - 23*a*c^3*d^4*e^4 + 7*a^2*c^2*d^2*e^6)*x^2 + 2*(194*c^4*d^7*e - 426*a*c^3*d^5*e^3 + 357*a^2*c^2*d^3*e^5
- 105*a^3*c*d*e^7)*x)*sqrt(e*x + d))/(c^6*d^6*x + a*c^5*d^5*e), -1/35*(315*(a*c^3*d^6*e^2 - 3*a^2*c^2*d^4*e^4
+ 3*a^3*c*d^2*e^6 - a^4*e^8 + (c^4*d^7*e - 3*a*c^3*d^5*e^3 + 3*a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*x)*sqrt(-(c*d^2
- a*e^2)/(c*d))*arctan(-sqrt(e*x + d)*c*d*sqrt(-(c*d^2 - a*e^2)/(c*d))/(c*d^2 - a*e^2)) - (10*c^4*d^4*e^4*x^4
- 35*c^4*d^8 + 528*a*c^3*d^6*e^2 - 1218*a^2*c^2*d^4*e^4 + 1050*a^3*c*d^2*e^6 - 315*a^4*e^8 + 2*(29*c^4*d^5*e^
3 - 9*a*c^3*d^3*e^5)*x^3 + 6*(26*c^4*d^6*e^2 - 23*a*c^3*d^4*e^4 + 7*a^2*c^2*d^2*e^6)*x^2 + 2*(194*c^4*d^7*e -
426*a*c^3*d^5*e^3 + 357*a^2*c^2*d^3*e^5 - 105*a^3*c*d*e^7)*x)*sqrt(e*x + d))/(c^6*d^6*x + a*c^5*d^5*e)]

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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((e*x+d)**(13/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)

[Out]

Timed out

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Giac [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((e*x+d)^(13/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="giac")

[Out]

Timed out