3.21.80 \(\int \frac {1}{(d+e x)^{7/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [2080]

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

[Out]

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

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Rubi [A]
time = 0.40, antiderivative size = 519, normalized size of antiderivative = 1.00, 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 = {686, 680, 674, 211} \begin {gather*} \frac {3003 c^5 d^5 e^{3/2} \text {ArcTan}\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 {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 {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}} \end {gather*}

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 211

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

Rule 674

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 680

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 686

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))),
Int[(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] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, 0] && NeQ[m + p + 1, 0] && IntegerQ[2*p]

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 ) \text {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*}

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Mathematica [A]
time = 2.23, size = 437, normalized size = 0.84 \begin {gather*} \frac {c^5 d^5 (d+e x)^{5/2} \left (\frac {(a e+c d x) \left (384 a^6 e^{12}-48 a^5 c d e^{10} (61 d+13 e x)+8 a^4 c^2 d^2 e^8 \left (1253 d^2+676 d e x+143 e^2 x^2\right )-2 a^3 c^3 d^3 e^6 \left (10535 d^3+11557 d^2 e x+6149 d e^2 x^2+1287 e^3 x^3\right )+3 a^2 c^4 d^4 e^4 \left (11865 d^4+26390 d^3 e x+28028 d^2 e^2 x^2+14586 d e^3 x^3+3003 e^4 x^4\right )+2 a c^5 d^5 e^2 \left (12160 d^5+96395 d^4 e x+232375 d^3 e^2 x^2+260403 d^2 e^3 x^3+141141 d e^4 x^4+30030 e^5 x^5\right )+c^6 d^6 \left (-1280 d^6+16640 d^5 e x+137995 d^4 e^2 x^2+338910 d^3 e^3 x^3+384384 d^2 e^4 x^4+210210 d e^5 x^5+45045 e^6 x^6\right )\right )}{c^5 d^5 \left (c d^2-a e^2\right )^7 (d+e x)^5}+\frac {45045 e^{3/2} (a e+c d x)^{5/2} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{15/2}}\right )}{1920 ((a e+c d x) (d+e x))^{5/2}} \end {gather*}

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]

(c^5*d^5*(d + e*x)^(5/2)*(((a*e + c*d*x)*(384*a^6*e^12 - 48*a^5*c*d*e^10*(61*d + 13*e*x) + 8*a^4*c^2*d^2*e^8*(
1253*d^2 + 676*d*e*x + 143*e^2*x^2) - 2*a^3*c^3*d^3*e^6*(10535*d^3 + 11557*d^2*e*x + 6149*d*e^2*x^2 + 1287*e^3
*x^3) + 3*a^2*c^4*d^4*e^4*(11865*d^4 + 26390*d^3*e*x + 28028*d^2*e^2*x^2 + 14586*d*e^3*x^3 + 3003*e^4*x^4) + 2
*a*c^5*d^5*e^2*(12160*d^5 + 96395*d^4*e*x + 232375*d^3*e^2*x^2 + 260403*d^2*e^3*x^3 + 141141*d*e^4*x^4 + 30030
*e^5*x^5) + c^6*d^6*(-1280*d^6 + 16640*d^5*e*x + 137995*d^4*e^2*x^2 + 338910*d^3*e^3*x^3 + 384384*d^2*e^4*x^4
+ 210210*d*e^5*x^5 + 45045*e^6*x^6)))/(c^5*d^5*(c*d^2 - a*e^2)^7*(d + e*x)^5) + (45045*e^(3/2)*(a*e + c*d*x)^(
5/2)*ArcTan[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[c*d^2 - a*e^2]])/(c*d^2 - a*e^2)^(15/2)))/(1920*((a*e + c*d*x)*(d
+ e*x))^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1542\) vs. \(2(463)=926\).
time = 0.75, size = 1543, normalized size = 2.97

method	result	size
default	\(\text {Expression too large to display}\)	\(1543\)

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,method=_RETURNVERBOSE)

[Out]

1/1920*((c*d*x+a*e)*(e*x+d))^(1/2)*(-520806*((a*e^2-c*d^2)*e)^(1/2)*a*c^5*d^7*e^5*x^3-464750*((a*e^2-c*d^2)*e)
^(1/2)*a*c^5*d^8*e^4*x^2-9009*((a*e^2-c*d^2)*e)^(1/2)*a^2*c^4*d^4*e^8*x^4+2574*((a*e^2-c*d^2)*e)^(1/2)*a^3*c^3
*d^3*e^9*x^3-43758*((a*e^2-c*d^2)*e)^(1/2)*a^2*c^4*d^5*e^7*x^3-1144*((a*e^2-c*d^2)*e)^(1/2)*a^4*c^2*d^2*e^10*x
^2+12298*((a*e^2-c*d^2)*e)^(1/2)*a^3*c^3*d^4*e^8*x^2-84084*((a*e^2-c*d^2)*e)^(1/2)*a^2*c^4*d^6*e^6*x^2+624*((a
*e^2-c*d^2)*e)^(1/2)*a^5*c*d*e^11*x-5408*((a*e^2-c*d^2)*e)^(1/2)*a^4*c^2*d^3*e^9*x+23114*((a*e^2-c*d^2)*e)^(1/
2)*a^3*c^3*d^5*e^7*x-79170*((a*e^2-c*d^2)*e)^(1/2)*a^2*c^4*d^7*e^5*x+2928*((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+45045*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^5*d^5*e^8*x^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))*a*c^5*d^6*e^7*x^4*(c*d*x+a*e)^(1/2
)-60060*((a*e^2-c*d^2)*e)^(1/2)*a*c^5*d^5*e^7*x^5+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^12-24320*((a*e^2-c*d^2)*e)^(1/2)*a*c^5*d^10*e^2-45045*((a*e^2-c*d^2)*e)^(1/2)*c^6*d^6*e^6*x^6-2822
82*((a*e^2-c*d^2)*e)^(1/2)*a*c^5*d^6*e^6*x^4-192790*((a*e^2-c*d^2)*e)^(1/2)*a*c^5*d^9*e^3*x-210210*((a*e^2-c*d
^2)*e)^(1/2)*c^6*d^7*e^5*x^5-384384*((a*e^2-c*d^2)*e)^(1/2)*c^6*d^8*e^4*x^4-338910*((a*e^2-c*d^2)*e)^(1/2)*c^6
*d^9*e^3*x^3-137995*((a*e^2-c*d^2)*e)^(1/2)*c^6*d^10*e^2*x^2-16640*((a*e^2-c*d^2)*e)^(1/2)*c^6*d^11*e*x+450450
*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^5*d^7*e^6*x^3*(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))*a*c^5*d^8*e^5*x^2*(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))*a*c^5*d^9*e^4*x*(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))*c^6*d^6*e^7*x^6*(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))*c^6*d^7
*e^6*x^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))*c^6*d^8*e^5*x^4*(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))*c^6*d^9*e^4*x^3*(c*d*x+a*e)^(1/2)+22522
5*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^6*d^10*e^3*x^2*(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))*c^6*d^11*e^2*x*(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1911 vs. \(2 (470) = 940\).
time = 6.36, size = 3860, normalized size = 7.44 \begin {gather*} \text {Too large to display} \end {gather*}

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^13*x^2*e + a^2*c^5*d^5*x^6*e^9 + 2*(a*c^6*d^6*x^7 + 3*a^2*c^5*d^6*x^5)*e^8 + (c^7*d^7*x^
8 + 12*a*c^6*d^7*x^6 + 15*a^2*c^5*d^7*x^4)*e^7 + 2*(3*c^7*d^8*x^7 + 15*a*c^6*d^8*x^5 + 10*a^2*c^5*d^8*x^3)*e^6
+ 5*(3*c^7*d^9*x^6 + 8*a*c^6*d^9*x^4 + 3*a^2*c^5*d^9*x^2)*e^5 + 2*(10*c^7*d^10*x^5 + 15*a*c^6*d^10*x^3 + 3*a^
2*c^5*d^10*x)*e^4 + (15*c^7*d^11*x^4 + 12*a*c^6*d^11*x^2 + a^2*c^5*d^11)*e^3 + 2*(3*c^7*d^12*x^3 + a*c^6*d^12*
x)*e^2)*sqrt(-e/(c*d^2 - a*e^2))*log((c*d^3 - 2*a*x*e^3 - 2*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*(c*d^2
- a*e^2)*sqrt(x*e + d)*sqrt(-e/(c*d^2 - a*e^2)) - (c*d*x^2 + 2*a*d)*e^2)/(x^2*e^2 + 2*d*x*e + d^2)) + 2*(1664
0*c^6*d^11*x*e - 1280*c^6*d^12 - 624*a^5*c*d*x*e^11 + 384*a^6*e^12 + 8*(143*a^4*c^2*d^2*x^2 - 366*a^5*c*d^2)*e
^10 - 26*(99*a^3*c^3*d^3*x^3 - 208*a^4*c^2*d^3*x)*e^9 + (9009*a^2*c^4*d^4*x^4 - 12298*a^3*c^3*d^4*x^2 + 10024*
a^4*c^2*d^4)*e^8 + 26*(2310*a*c^5*d^5*x^5 + 1683*a^2*c^4*d^5*x^3 - 889*a^3*c^3*d^5*x)*e^7 + 7*(6435*c^6*d^6*x^
6 + 40326*a*c^5*d^6*x^4 + 12012*a^2*c^4*d^6*x^2 - 3010*a^3*c^3*d^6)*e^6 + 78*(2695*c^6*d^7*x^5 + 6677*a*c^5*d^
7*x^3 + 1015*a^2*c^4*d^7*x)*e^5 + (384384*c^6*d^8*x^4 + 464750*a*c^5*d^8*x^2 + 35595*a^2*c^4*d^8)*e^4 + 130*(2
607*c^6*d^9*x^3 + 1483*a*c^5*d^9*x)*e^3 + 5*(27599*c^6*d^10*x^2 + 4864*a*c^5*d^10)*e^2)*sqrt(c*d^2*x + a*x*e^2
+ (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(c^9*d^22*x^2 - a^9*x^6*e^22 - 2*(a^8*c*d*x^7 + 3*a^9*d*x^5)*e^21 - (a^7*
c^2*d^2*x^8 + 5*a^8*c*d^2*x^6 + 15*a^9*d^2*x^4)*e^20 + 4*(2*a^7*c^2*d^3*x^7 + 3*a^8*c*d^3*x^5 - 5*a^9*d^3*x^3)
*e^19 + (7*a^6*c^3*d^4*x^8 + 48*a^7*c^2*d^4*x^6 + 65*a^8*c*d^4*x^4 - 15*a^9*d^4*x^2)*e^18 + 2*(32*a^7*c^2*d^5*
x^5 + 55*a^8*c*d^5*x^3 - 3*a^9*d^5*x)*e^17 - (21*a^5*c^4*d^6*x^8 + 112*a^6*c^3*d^6*x^6 + 50*a^7*c^2*d^6*x^4 -
93*a^8*c*d^6*x^2 + a^9*d^6)*e^16 - 8*(7*a^5*c^4*d^7*x^7 + 35*a^6*c^3*d^7*x^5 + 27*a^7*c^2*d^7*x^3 - 5*a^8*c*d^
7*x)*e^15 + (35*a^4*c^5*d^8*x^8 + 70*a^5*c^4*d^8*x^6 - 210*a^6*c^3*d^8*x^4 - 232*a^7*c^2*d^8*x^2 + 7*a^8*c*d^8
)*e^14 + 28*(5*a^4*c^5*d^9*x^7 + 15*a^5*c^4*d^9*x^5 + 4*a^6*c^3*d^9*x^3 - 4*a^7*c^2*d^9*x)*e^13 - 7*(5*a^3*c^6
*d^10*x^8 - 18*a^4*c^5*d^10*x^6 - 80*a^5*c^4*d^10*x^4 - 40*a^6*c^3*d^10*x^2 + 3*a^7*c^2*d^10)*e^12 - 56*(3*a^3
*c^6*d^11*x^7 + 4*a^4*c^5*d^11*x^5 - 4*a^5*c^4*d^11*x^3 - 3*a^6*c^3*d^11*x)*e^11 + 7*(3*a^2*c^7*d^12*x^8 - 40*
a^3*c^6*d^12*x^6 - 80*a^4*c^5*d^12*x^4 - 18*a^5*c^4*d^12*x^2 + 5*a^6*c^3*d^12)*e^10 + 28*(4*a^2*c^7*d^13*x^7 -
4*a^3*c^6*d^13*x^5 - 15*a^4*c^5*d^13*x^3 - 5*a^5*c^4*d^13*x)*e^9 - (7*a*c^8*d^14*x^8 - 232*a^2*c^7*d^14*x^6 -
210*a^3*c^6*d^14*x^4 + 70*a^4*c^5*d^14*x^2 + 35*a^5*c^4*d^14)*e^8 - 8*(5*a*c^8*d^15*x^7 - 27*a^2*c^7*d^15*x^5
- 35*a^3*c^6*d^15*x^3 - 7*a^4*c^5*d^15*x)*e^7 + (c^9*d^16*x^8 - 93*a*c^8*d^16*x^6 + 50*a^2*c^7*d^16*x^4 + 112
*a^3*c^6*d^16*x^2 + 21*a^4*c^5*d^16)*e^6 + 2*(3*c^9*d^17*x^7 - 55*a*c^8*d^17*x^5 - 32*a^2*c^7*d^17*x^3)*e^5 +
(15*c^9*d^18*x^6 - 65*a*c^8*d^18*x^4 - 48*a^2*c^7*d^18*x^2 - 7*a^3*c^6*d^18)*e^4 + 4*(5*c^9*d^19*x^5 - 3*a*c^8
*d^19*x^3 - 2*a^2*c^7*d^19*x)*e^3 + (15*c^9*d^20*x^4 + 5*a*c^8*d^20*x^2 + a^2*c^7*d^20)*e^2 + 2*(3*c^9*d^21*x^
3 + a*c^8*d^21*x)*e), 1/1920*(45045*(c^7*d^13*x^2*e + a^2*c^5*d^5*x^6*e^9 + 2*(a*c^6*d^6*x^7 + 3*a^2*c^5*d^6*x
^5)*e^8 + (c^7*d^7*x^8 + 12*a*c^6*d^7*x^6 + 15*a^2*c^5*d^7*x^4)*e^7 + 2*(3*c^7*d^8*x^7 + 15*a*c^6*d^8*x^5 + 10
*a^2*c^5*d^8*x^3)*e^6 + 5*(3*c^7*d^9*x^6 + 8*a*c^6*d^9*x^4 + 3*a^2*c^5*d^9*x^2)*e^5 + 2*(10*c^7*d^10*x^5 + 15*
a*c^6*d^10*x^3 + 3*a^2*c^5*d^10*x)*e^4 + (15*c^7*d^11*x^4 + 12*a*c^6*d^11*x^2 + a^2*c^5*d^11)*e^3 + 2*(3*c^7*d
^12*x^3 + a*c^6*d^12*x)*e^2)*arctan(-sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(c*d^2 - a*e^2)*sqrt(x*e
+ d)*e^(1/2)/(c*d^2*x*e + a*x*e^3 + (c*d*x^2 + a*d)*e^2))*e^(1/2)/sqrt(c*d^2 - a*e^2) + (16640*c^6*d^11*x*e -
1280*c^6*d^12 - 624*a^5*c*d*x*e^11 + 384*a^6*e^12 + 8*(143*a^4*c^2*d^2*x^2 - 366*a^5*c*d^2)*e^10 - 26*(99*a^3*
c^3*d^3*x^3 - 208*a^4*c^2*d^3*x)*e^9 + (9009*a^2*c^4*d^4*x^4 - 12298*a^3*c^3*d^4*x^2 + 10024*a^4*c^2*d^4)*e^8
+ 26*(2310*a*c^5*d^5*x^5 + 1683*a^2*c^4*d^5*x^3 - 889*a^3*c^3*d^5*x)*e^7 + 7*(6435*c^6*d^6*x^6 + 40326*a*c^5*d
^6*x^4 + 12012*a^2*c^4*d^6*x^2 - 3010*a^3*c^3*d^6)*e^6 + 78*(2695*c^6*d^7*x^5 + 6677*a*c^5*d^7*x^3 + 1015*a^2*
c^4*d^7*x)*e^5 + (384384*c^6*d^8*x^4 + 464750*a*c^5*d^8*x^2 + 35595*a^2*c^4*d^8)*e^4 + 130*(2607*c^6*d^9*x^3 +
1483*a*c^5*d^9*x)*e^3 + 5*(27599*c^6*d^10*x^2 + 4864*a*c^5*d^10)*e^2)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d
)*e)*sqrt(x*e + d))/(c^9*d^22*x^2 - a^9*x^6*e^22 - 2*(a^8*c*d*x^7 + 3*a^9*d*x^5)*e^21 - (a^7*c^2*d^2*x^8 + 5*a
^8*c*d^2*x^6 + 15*a^9*d^2*x^4)*e^20 + 4*(2*a^7*c^2*d^3*x^7 + 3*a^8*c*d^3*x^5 - 5*a^9*d^3*x^3)*e^19 + (7*a^6*c^
3*d^4*x^8 + 48*a^7*c^2*d^4*x^6 + 65*a^8*c*d^4*x^4 - 15*a^9*d^4*x^2)*e^18 + 2*(32*a^7*c^2*d^5*x^5 + 55*a^8*c*d^
5*x^3 - 3*a^9*d^5*x)*e^17 - (21*a^5*c^4*d^6*x^8 + 112*a^6*c^3*d^6*x^6 + 50*a^7*c^2*d^6*x^4 - 93*a^8*c*d^6*x^2
+ a^9*d^6)*e^16 - 8*(7*a^5*c^4*d^7*x^7 + 35*a^6*c^3*d^7*x^5 + 27*a^7*c^2*d^7*x^3 - 5*a^8*c*d^7*x)*e^15 + (35*a
^4*c^5*d^8*x^8 + 70*a^5*c^4*d^8*x^6 - 210*a^6*c...

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 4847 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1009 vs. \(2 (470) = 940\).
time = 1.95, size = 1009, normalized size = 1.94 \begin {gather*} \frac {1}{1920} \, {\left (\frac {45045 \, c^{5} d^{5} \arctan \left (\frac {\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right ) e}{{\left (c^{7} d^{14} - 7 \, a c^{6} d^{12} e^{2} + 21 \, a^{2} c^{5} d^{10} e^{4} - 35 \, a^{3} c^{4} d^{8} e^{6} + 35 \, a^{4} c^{3} d^{6} e^{8} - 21 \, a^{5} c^{2} d^{4} e^{10} + 7 \, a^{6} c d^{2} e^{12} - a^{7} e^{14}\right )} \sqrt {c d^{2} e - a e^{3}}} - \frac {1280 \, {\left (c^{6} d^{7} e^{2} - a c^{5} d^{5} e^{4} - 18 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )} c^{5} d^{5} e\right )}}{{\left (c^{7} d^{14} - 7 \, a c^{6} d^{12} e^{2} + 21 \, a^{2} c^{5} d^{10} e^{4} - 35 \, a^{3} c^{4} d^{8} e^{6} + 35 \, a^{4} c^{3} d^{6} e^{8} - 21 \, a^{5} c^{2} d^{4} e^{10} + 7 \, a^{6} c d^{2} e^{12} - a^{7} e^{14}\right )} {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}}} + \frac {{\left (35595 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{9} d^{13} e^{5} - 142380 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a c^{8} d^{11} e^{7} + 121310 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{8} d^{11} e^{4} + 213570 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c^{7} d^{9} e^{9} - 363930 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{7} d^{9} e^{6} + 160384 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c^{7} d^{9} e^{3} - 142380 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a^{3} c^{6} d^{7} e^{11} + 363930 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} c^{6} d^{7} e^{8} - 320768 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a c^{6} d^{7} e^{5} + 96290 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} c^{6} d^{7} e^{2} + 35595 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a^{4} c^{5} d^{5} e^{13} - 121310 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{3} c^{5} d^{5} e^{10} + 160384 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a^{2} c^{5} d^{5} e^{7} - 96290 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} a c^{5} d^{5} e^{4} + 22005 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {9}{2}} c^{5} d^{5} e\right )} e^{\left (-5\right )}}{{\left (c^{7} d^{14} - 7 \, a c^{6} d^{12} e^{2} + 21 \, a^{2} c^{5} d^{10} e^{4} - 35 \, a^{3} c^{4} d^{8} e^{6} + 35 \, a^{4} c^{3} d^{6} e^{8} - 21 \, a^{5} c^{2} d^{4} e^{10} + 7 \, a^{6} c d^{2} e^{12} - a^{7} e^{14}\right )} {\left (x e + d\right )}^{5} c^{5} d^{5}}\right )} e \end {gather*}

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]

1/1920*(45045*c^5*d^5*arctan(sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)/sqrt(c*d^2*e - a*e^3))*e/((c^7*d^14 - 7*a
*c^6*d^12*e^2 + 21*a^2*c^5*d^10*e^4 - 35*a^3*c^4*d^8*e^6 + 35*a^4*c^3*d^6*e^8 - 21*a^5*c^2*d^4*e^10 + 7*a^6*c*
d^2*e^12 - a^7*e^14)*sqrt(c*d^2*e - a*e^3)) - 1280*(c^6*d^7*e^2 - a*c^5*d^5*e^4 - 18*((x*e + d)*c*d*e - c*d^2*
e + a*e^3)*c^5*d^5*e)/((c^7*d^14 - 7*a*c^6*d^12*e^2 + 21*a^2*c^5*d^10*e^4 - 35*a^3*c^4*d^8*e^6 + 35*a^4*c^3*d^
6*e^8 - 21*a^5*c^2*d^4*e^10 + 7*a^6*c*d^2*e^12 - a^7*e^14)*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)) + (35595
*sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*c^9*d^13*e^5 - 142380*sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*a*c^8*d
^11*e^7 + 121310*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^8*d^11*e^4 + 213570*sqrt((x*e + d)*c*d*e - c*d^2*
e + a*e^3)*a^2*c^7*d^9*e^9 - 363930*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*c^7*d^9*e^6 + 160384*((x*e + d
)*c*d*e - c*d^2*e + a*e^3)^(5/2)*c^7*d^9*e^3 - 142380*sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*a^3*c^6*d^7*e^11
+ 363930*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*c^6*d^7*e^8 - 320768*((x*e + d)*c*d*e - c*d^2*e + a*e^
3)^(5/2)*a*c^6*d^7*e^5 + 96290*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*c^6*d^7*e^2 + 35595*sqrt((x*e + d)*c*
d*e - c*d^2*e + a*e^3)*a^4*c^5*d^5*e^13 - 121310*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^3*c^5*d^5*e^10 +
160384*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*c^5*d^5*e^7 - 96290*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(
7/2)*a*c^5*d^5*e^4 + 22005*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(9/2)*c^5*d^5*e)*e^(-5)/((c^7*d^14 - 7*a*c^6*d^
12*e^2 + 21*a^2*c^5*d^10*e^4 - 35*a^3*c^4*d^8*e^6 + 35*a^4*c^3*d^6*e^8 - 21*a^5*c^2*d^4*e^10 + 7*a^6*c*d^2*e^1
2 - a^7*e^14)*(x*e + d)^5*c^5*d^5))*e

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (d+e\,x\right )}^{7/2}\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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