3.2056 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{17/2}} \, dx\)

Optimal. Leaf size=366 \[ \frac {3 c^5 d^5 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{128 e^{7/2} \left (c d^2-a e^2\right )^{5/2}}+\frac {3 c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 e^3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}+\frac {c^3 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 e^3 (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac {c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{16 e^3 (d+e x)^{7/2}}-\frac {c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}} \]

[Out]

-1/8*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/e^2/(e*x+d)^(11/2)-1/5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2
)/e/(e*x+d)^(15/2)+3/128*c^5*d^5*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))/e^(7/2)/(-a*e^2+c*d^2)^(5/2)-1/16*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/e^3/(e*x+d)^(7
/2)+1/64*c^3*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/e^3/(-a*e^2+c*d^2)/(e*x+d)^(5/2)+3/128*c^4*d^4*(a*d*e
+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/e^3/(-a*e^2+c*d^2)^2/(e*x+d)^(3/2)

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Rubi [A]  time = 0.28, antiderivative size = 366, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {662, 672, 660, 205} \[ \frac {3 c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 e^3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}+\frac {c^3 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 e^3 (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac {c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{16 e^3 (d+e x)^{7/2}}+\frac {3 c^5 d^5 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{128 e^{7/2} \left (c d^2-a e^2\right )^{5/2}}-\frac {c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(17/2),x]

[Out]

-(c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(16*e^3*(d + e*x)^(7/2)) + (c^3*d^3*Sqrt[a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2])/(64*e^3*(c*d^2 - a*e^2)*(d + e*x)^(5/2)) + (3*c^4*d^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2])/(128*e^3*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2)) - (c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2
))/(8*e^2*(d + e*x)^(11/2)) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(5*e*(d + e*x)^(15/2)) + (3*c^5*d^
5*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*e^(7
/2)*(c*d^2 - a*e^2)^(5/2))

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]

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 662

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

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]

Rubi steps

\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}+\frac {(c d) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{13/2}} \, dx}{2 e}\\ &=-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}+\frac {\left (3 c^2 d^2\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{9/2}} \, dx}{16 e^2}\\ &=-\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 e^3 (d+e x)^{7/2}}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}+\frac {\left (c^3 d^3\right ) \int \frac {1}{(d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{32 e^3}\\ &=-\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 e^3 (d+e x)^{7/2}}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 e^3 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}+\frac {\left (3 c^4 d^4\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{128 e^3 \left (c d^2-a e^2\right )}\\ &=-\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 e^3 (d+e x)^{7/2}}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 e^3 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {3 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 e^3 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}+\frac {\left (3 c^5 d^5\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 e^3 \left (c d^2-a e^2\right )^2}\\ &=-\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 e^3 (d+e x)^{7/2}}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 e^3 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {3 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 e^3 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}+\frac {\left (3 c^5 d^5\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 e^2 \left (c d^2-a e^2\right )^2}\\ &=-\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 e^3 (d+e x)^{7/2}}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 e^3 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {3 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 e^3 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}+\frac {3 c^5 d^5 \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 e^{7/2} \left (c d^2-a e^2\right )^{5/2}}\\ \end {align*}

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Mathematica [C]  time = 0.07, size = 83, normalized size = 0.23 \[ \frac {2 c^5 d^5 ((d+e x) (a e+c d x))^{7/2} \, _2F_1\left (\frac {7}{2},6;\frac {9}{2};\frac {e (a e+c d x)}{a e^2-c d^2}\right )}{7 (d+e x)^{7/2} \left (c d^2-a e^2\right )^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(17/2),x]

[Out]

(2*c^5*d^5*((a*e + c*d*x)*(d + e*x))^(7/2)*Hypergeometric2F1[7/2, 6, 9/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)
])/(7*(c*d^2 - a*e^2)^6*(d + e*x)^(7/2))

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fricas [B]  time = 1.22, size = 1687, normalized size = 4.61 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(17/2),x, algorithm="fricas")

[Out]

[-1/1280*(15*(c^5*d^5*e^6*x^6 + 6*c^5*d^6*e^5*x^5 + 15*c^5*d^7*e^4*x^4 + 20*c^5*d^8*e^3*x^3 + 15*c^5*d^9*e^2*x
^2 + 6*c^5*d^10*e*x + c^5*d^11)*sqrt(-c*d^2*e + a*e^3)*log(-(c*d*e^2*x^2 + 2*a*e^3*x - c*d^3 + 2*a*d*e^2 - 2*s
qrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d^2*e + a*e^3)*sqrt(e*x + d))/(e^2*x^2 + 2*d*e*x + d^2)) +
2*(15*c^5*d^10*e - 5*a*c^4*d^8*e^3 - 2*a^2*c^3*d^6*e^5 - 184*a^3*c^2*d^4*e^7 + 304*a^4*c*d^2*e^9 - 128*a^5*e^1
1 - 15*(c^5*d^6*e^5 - a*c^4*d^4*e^7)*x^4 - 10*(7*c^5*d^7*e^4 - 8*a*c^4*d^5*e^6 + a^2*c^3*d^3*e^8)*x^3 + 2*(64*
c^5*d^8*e^3 - 297*a*c^4*d^6*e^5 + 357*a^2*c^3*d^4*e^7 - 124*a^3*c^2*d^2*e^9)*x^2 + 2*(35*c^5*d^9*e^2 - 12*a*c^
4*d^7*e^4 - 279*a^2*c^3*d^5*e^6 + 424*a^3*c^2*d^3*e^8 - 168*a^4*c*d*e^10)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 +
a*e^2)*x)*sqrt(e*x + d))/(c^3*d^12*e^4 - 3*a*c^2*d^10*e^6 + 3*a^2*c*d^8*e^8 - a^3*d^6*e^10 + (c^3*d^6*e^10 -
3*a*c^2*d^4*e^12 + 3*a^2*c*d^2*e^14 - a^3*e^16)*x^6 + 6*(c^3*d^7*e^9 - 3*a*c^2*d^5*e^11 + 3*a^2*c*d^3*e^13 - a
^3*d*e^15)*x^5 + 15*(c^3*d^8*e^8 - 3*a*c^2*d^6*e^10 + 3*a^2*c*d^4*e^12 - a^3*d^2*e^14)*x^4 + 20*(c^3*d^9*e^7 -
3*a*c^2*d^7*e^9 + 3*a^2*c*d^5*e^11 - a^3*d^3*e^13)*x^3 + 15*(c^3*d^10*e^6 - 3*a*c^2*d^8*e^8 + 3*a^2*c*d^6*e^1
0 - a^3*d^4*e^12)*x^2 + 6*(c^3*d^11*e^5 - 3*a*c^2*d^9*e^7 + 3*a^2*c*d^7*e^9 - a^3*d^5*e^11)*x), -1/640*(15*(c^
5*d^5*e^6*x^6 + 6*c^5*d^6*e^5*x^5 + 15*c^5*d^7*e^4*x^4 + 20*c^5*d^8*e^3*x^3 + 15*c^5*d^9*e^2*x^2 + 6*c^5*d^10*
e*x + c^5*d^11)*sqrt(c*d^2*e - a*e^3)*arctan(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d^2*e - a*e^3)
*sqrt(e*x + d)/(c*d*e^2*x^2 + a*d*e^2 + (c*d^2*e + a*e^3)*x)) + (15*c^5*d^10*e - 5*a*c^4*d^8*e^3 - 2*a^2*c^3*d
^6*e^5 - 184*a^3*c^2*d^4*e^7 + 304*a^4*c*d^2*e^9 - 128*a^5*e^11 - 15*(c^5*d^6*e^5 - a*c^4*d^4*e^7)*x^4 - 10*(7
*c^5*d^7*e^4 - 8*a*c^4*d^5*e^6 + a^2*c^3*d^3*e^8)*x^3 + 2*(64*c^5*d^8*e^3 - 297*a*c^4*d^6*e^5 + 357*a^2*c^3*d^
4*e^7 - 124*a^3*c^2*d^2*e^9)*x^2 + 2*(35*c^5*d^9*e^2 - 12*a*c^4*d^7*e^4 - 279*a^2*c^3*d^5*e^6 + 424*a^3*c^2*d^
3*e^8 - 168*a^4*c*d*e^10)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^3*d^12*e^4 - 3*a*c^
2*d^10*e^6 + 3*a^2*c*d^8*e^8 - a^3*d^6*e^10 + (c^3*d^6*e^10 - 3*a*c^2*d^4*e^12 + 3*a^2*c*d^2*e^14 - a^3*e^16)*
x^6 + 6*(c^3*d^7*e^9 - 3*a*c^2*d^5*e^11 + 3*a^2*c*d^3*e^13 - a^3*d*e^15)*x^5 + 15*(c^3*d^8*e^8 - 3*a*c^2*d^6*e
^10 + 3*a^2*c*d^4*e^12 - a^3*d^2*e^14)*x^4 + 20*(c^3*d^9*e^7 - 3*a*c^2*d^7*e^9 + 3*a^2*c*d^5*e^11 - a^3*d^3*e^
13)*x^3 + 15*(c^3*d^10*e^6 - 3*a*c^2*d^8*e^8 + 3*a^2*c*d^6*e^10 - a^3*d^4*e^12)*x^2 + 6*(c^3*d^11*e^5 - 3*a*c^
2*d^9*e^7 + 3*a^2*c*d^7*e^9 - a^3*d^5*e^11)*x)]

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(17/2),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.09, size = 910, normalized size = 2.49 \[ -\frac {\sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, \left (15 c^{5} d^{5} e^{5} x^{5} \arctanh \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right )+75 c^{5} d^{6} e^{4} x^{4} \arctanh \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right )+150 c^{5} d^{7} e^{3} x^{3} \arctanh \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right )+150 c^{5} d^{8} e^{2} x^{2} \arctanh \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right )+75 c^{5} d^{9} e x \arctanh \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right )+15 c^{5} d^{10} \arctanh \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right )-15 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c^{4} d^{4} e^{4} x^{4}+10 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a \,c^{3} d^{3} e^{5} x^{3}-70 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c^{4} d^{5} e^{3} x^{3}+248 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a^{2} c^{2} d^{2} e^{6} x^{2}-466 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a \,c^{3} d^{4} e^{4} x^{2}+128 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c^{4} d^{6} e^{2} x^{2}+336 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a^{3} c d \,e^{7} x -512 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a^{2} c^{2} d^{3} e^{5} x +46 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a \,c^{3} d^{5} e^{3} x +70 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c^{4} d^{7} e x +128 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a^{4} e^{8}-176 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a^{3} c \,d^{2} e^{6}+8 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a^{2} c^{2} d^{4} e^{4}+10 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a \,c^{3} d^{6} e^{2}+15 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c^{4} d^{8}\right )}{640 \left (e x +d \right )^{\frac {11}{2}} \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {c d x +a e}\, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)/(e*x+d)^(17/2),x)

[Out]

-1/640*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(15*arctanh((c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2)*e)*x^5*c^
5*d^5*e^5+75*arctanh((c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2)*e)*x^4*c^5*d^6*e^4+150*arctanh((c*d*x+a*e)^(1/2
)/((a*e^2-c*d^2)*e)^(1/2)*e)*x^3*c^5*d^7*e^3+150*arctanh((c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2)*e)*x^2*c^5*
d^8*e^2-15*x^4*c^4*d^4*e^4*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+75*arctanh((c*d*x+a*e)^(1/2)/((a*e^2-c*d^
2)*e)^(1/2)*e)*x*c^5*d^9*e+10*x^3*a*c^3*d^3*e^5*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)-70*x^3*c^4*d^5*e^3*(
c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+15*arctanh((c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2)*e)*c^5*d^10+248*
x^2*a^2*c^2*d^2*e^6*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)-466*x^2*a*c^3*d^4*e^4*(c*d*x+a*e)^(1/2)*((a*e^2-
c*d^2)*e)^(1/2)+128*x^2*c^4*d^6*e^2*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+336*x*a^3*c*d*e^7*(c*d*x+a*e)^(1
/2)*((a*e^2-c*d^2)*e)^(1/2)-512*x*a^2*c^2*d^3*e^5*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+46*x*a*c^3*d^5*e^3
*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+70*x*c^4*d^7*e*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+128*(c*d*x
+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)*a^4*e^8-176*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)*a^3*c*d^2*e^6+8*(c*d
*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)*a^2*c^2*d^4*e^4+10*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)*a*c^3*d^6*e
^2+15*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)*c^4*d^8)/(e*x+d)^(11/2)/((a*e^2-c*d^2)*e)^(1/2)/e^3/(a*e^2-c*d
^2)^2/(c*d*x+a*e)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {17}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(17/2),x, algorithm="maxima")

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/(e*x + d)^(17/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{{\left (d+e\,x\right )}^{17/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(d + e*x)^(17/2),x)

[Out]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(d + e*x)^(17/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**(17/2),x)

[Out]

Timed out

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