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

Optimal. Leaf size=386 \[ \frac {\left (7 a e^2+5 c d^2\right ) \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 a^{7/2} d^{9/2} e^{7/2}}-\frac {\left (7 a e^2+5 c d^2\right ) \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 a^3 d^4 e^3 x^2}+\frac {\left (7 a e^2+5 c d^2\right ) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{192 a^2 d^3 e^2 x^4}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 d x^6}-\frac {\left (\frac {5 c}{a e}-\frac {7 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{60 x^5} \]

________________________________________________________________________________________

Rubi [A]  time = 0.49, antiderivative size = 386, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {849, 834, 806, 720, 724, 206} \begin {gather*} -\frac {\left (7 a e^2+5 c d^2\right ) \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 a^3 d^4 e^3 x^2}+\frac {\left (7 a e^2+5 c d^2\right ) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{192 a^2 d^3 e^2 x^4}+\frac {\left (7 a e^2+5 c d^2\right ) \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 a^{7/2} d^{9/2} e^{7/2}}-\frac {\left (\frac {5 c}{a e}-\frac {7 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{60 x^5}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 d x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((c*d^2 - a*e^2)^3*(5*c*d^2 + 7*a*e^2)*(2*a*d*e + (c*d^2 + a*e^2)*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x
^2])/(512*a^3*d^4*e^3*x^2) + ((c*d^2 - a*e^2)*(5*c*d^2 + 7*a*e^2)*(2*a*d*e + (c*d^2 + a*e^2)*x)*(a*d*e + (c*d^
2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(192*a^2*d^3*e^2*x^4) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(6*d*x^
6) - (((5*c)/(a*e) - (7*e)/d^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(60*x^5) + ((c*d^2 - a*e^2)^5*(
5*c*d^2 + 7*a*e^2)*ArcTanh[(2*a*d*e + (c*d^2 + a*e^2)*x)/(2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^
2)*x + c*d*e*x^2])])/(1024*a^(7/2)*d^(9/2)*e^(7/2))

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 720

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

Rule 724

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

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rule 849

Int[((x_)^(n_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Int[x^n*(a/d + (c*
x)/e)*(a + b*x + c*x^2)^(p - 1), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b
*d*e + a*e^2, 0] &&  !IntegerQ[p] && ( !IntegerQ[n] ||  !IntegerQ[2*p] || IGtQ[n, 2] || (GtQ[p, 0] && NeQ[n, 2
]))

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}}{x^7 (d+e x)} \, dx &=\int \frac {(a e+c d x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^7} \, dx\\ &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 d x^6}-\frac {\int \frac {\left (-\frac {1}{2} a e \left (5 c d^2-7 a e^2\right )+a c d e^2 x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^6} \, dx}{6 a d e}\\ &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 d x^6}-\frac {\left (\frac {5 c}{a e}-\frac {7 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 x^5}-\frac {\left (\frac {5 c^2 d^2}{a}+2 c e^2-\frac {7 a e^4}{d^2}\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^5} \, dx}{24 e}\\ &=\frac {\left (c d^2-a e^2\right ) \left (5 c d^2+7 a e^2\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 a^2 d^3 e^2 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 d x^6}-\frac {\left (\frac {5 c}{a e}-\frac {7 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 x^5}+\frac {\left (\left (c d^2-a e^2\right )^3 \left (5 c d^2+7 a e^2\right )\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^3} \, dx}{128 a^2 d^3 e^2}\\ &=-\frac {\left (c d^2-a e^2\right )^3 \left (5 c d^2+7 a e^2\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 a^3 d^4 e^3 x^2}+\frac {\left (c d^2-a e^2\right ) \left (5 c d^2+7 a e^2\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 a^2 d^3 e^2 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 d x^6}-\frac {\left (\frac {5 c}{a e}-\frac {7 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 x^5}-\frac {\left (\left (c d^2-a e^2\right )^5 \left (5 c d^2+7 a e^2\right )\right ) \int \frac {1}{x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{1024 a^3 d^4 e^3}\\ &=-\frac {\left (c d^2-a e^2\right )^3 \left (5 c d^2+7 a e^2\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 a^3 d^4 e^3 x^2}+\frac {\left (c d^2-a e^2\right ) \left (5 c d^2+7 a e^2\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 a^2 d^3 e^2 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 d x^6}-\frac {\left (\frac {5 c}{a e}-\frac {7 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 x^5}+\frac {\left (\left (c d^2-a e^2\right )^5 \left (5 c d^2+7 a e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 a d e-x^2} \, dx,x,\frac {2 a d e-\left (-c d^2-a e^2\right ) x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{512 a^3 d^4 e^3}\\ &=-\frac {\left (c d^2-a e^2\right )^3 \left (5 c d^2+7 a e^2\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 a^3 d^4 e^3 x^2}+\frac {\left (c d^2-a e^2\right ) \left (5 c d^2+7 a e^2\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 a^2 d^3 e^2 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 d x^6}-\frac {\left (\frac {5 c}{a e}-\frac {7 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 x^5}+\frac {\left (c d^2-a e^2\right )^5 \left (5 c d^2+7 a e^2\right ) \tanh ^{-1}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{1024 a^{7/2} d^{9/2} e^{7/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.99, size = 344, normalized size = 0.89 \begin {gather*} \frac {((d+e x) (a e+c d x))^{3/2} \left (-\frac {\left (7 a e^2+5 c d^2\right ) \left (5 x \left (c d^2-a e^2\right ) \left (\frac {x \left (c d^2-a e^2\right ) \left (\frac {x \left (a e^2-c d^2\right ) \left (3 x^2 \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )+\sqrt {a} \sqrt {d} \sqrt {e} \sqrt {d+e x} \sqrt {a e+c d x} \left (a e (2 d+5 e x)-3 c d^2 x\right )\right )}{a^{5/2} \sqrt {d} e^{5/2}}-8 (d+e x)^{5/2} \sqrt {a e+c d x}\right )}{d}-16 (d+e x)^{5/2} (a e+c d x)^{3/2}\right )-128 d (d+e x)^{5/2} (a e+c d x)^{5/2}\right )}{1280 d^2 x^5 (d+e x)^{3/2} (a e+c d x)^{3/2}}-\frac {(d+e x) (a e+c d x)^2}{x^6}\right )}{6 a d e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(((a*e + c*d*x)*(d + e*x))^(3/2)*(-(((a*e + c*d*x)^2*(d + e*x))/x^6) - ((5*c*d^2 + 7*a*e^2)*(-128*d*(a*e + c*d
*x)^(5/2)*(d + e*x)^(5/2) + 5*(c*d^2 - a*e^2)*x*(-16*(a*e + c*d*x)^(3/2)*(d + e*x)^(5/2) + ((c*d^2 - a*e^2)*x*
(-8*Sqrt[a*e + c*d*x]*(d + e*x)^(5/2) + ((-(c*d^2) + a*e^2)*x*(Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[
d + e*x]*(-3*c*d^2*x + a*e*(2*d + 5*e*x)) + 3*(c*d^2 - a*e^2)^2*x^2*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[
a]*Sqrt[e]*Sqrt[d + e*x])]))/(a^(5/2)*Sqrt[d]*e^(5/2))))/d)))/(1280*d^2*x^5*(a*e + c*d*x)^(3/2)*(d + e*x)^(3/2
))))/(6*a*d*e)

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 180.31, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

$Aborted

________________________________________________________________________________________

fricas [A]  time = 60.75, size = 1072, normalized size = 2.78 \begin {gather*} \left [-\frac {15 \, {\left (5 \, c^{6} d^{12} - 18 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} + 20 \, a^{3} c^{3} d^{6} e^{6} - 45 \, a^{4} c^{2} d^{4} e^{8} + 30 \, a^{5} c d^{2} e^{10} - 7 \, a^{6} e^{12}\right )} \sqrt {a d e} x^{6} \log \left (\frac {8 \, a^{2} d^{2} e^{2} + {\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {a d e} + 8 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x}{x^{2}}\right ) + 4 \, {\left (1280 \, a^{6} d^{6} e^{6} + {\left (75 \, a c^{5} d^{11} e - 245 \, a^{2} c^{4} d^{9} e^{3} + 150 \, a^{3} c^{3} d^{7} e^{5} - 546 \, a^{4} c^{2} d^{5} e^{7} + 415 \, a^{5} c d^{3} e^{9} - 105 \, a^{6} d e^{11}\right )} x^{5} - 2 \, {\left (25 \, a^{2} c^{4} d^{10} e^{2} - 80 \, a^{3} c^{3} d^{8} e^{4} - 174 \, a^{4} c^{2} d^{6} e^{6} + 136 \, a^{5} c d^{4} e^{8} - 35 \, a^{6} d^{2} e^{10}\right )} x^{4} + 8 \, {\left (5 \, a^{3} c^{3} d^{9} e^{3} + 423 \, a^{4} c^{2} d^{7} e^{5} + 27 \, a^{5} c d^{5} e^{7} - 7 \, a^{6} d^{3} e^{9}\right )} x^{3} + 16 \, {\left (135 \, a^{4} c^{2} d^{8} e^{4} + 278 \, a^{5} c d^{6} e^{6} + 3 \, a^{6} d^{4} e^{8}\right )} x^{2} + 128 \, {\left (25 \, a^{5} c d^{7} e^{5} + 13 \, a^{6} d^{5} e^{7}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{30720 \, a^{4} d^{5} e^{4} x^{6}}, -\frac {15 \, {\left (5 \, c^{6} d^{12} - 18 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} + 20 \, a^{3} c^{3} d^{6} e^{6} - 45 \, a^{4} c^{2} d^{4} e^{8} + 30 \, a^{5} c d^{2} e^{10} - 7 \, a^{6} e^{12}\right )} \sqrt {-a d e} x^{6} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {-a d e}}{2 \, {\left (a c d^{2} e^{2} x^{2} + a^{2} d^{2} e^{2} + {\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )}}\right ) + 2 \, {\left (1280 \, a^{6} d^{6} e^{6} + {\left (75 \, a c^{5} d^{11} e - 245 \, a^{2} c^{4} d^{9} e^{3} + 150 \, a^{3} c^{3} d^{7} e^{5} - 546 \, a^{4} c^{2} d^{5} e^{7} + 415 \, a^{5} c d^{3} e^{9} - 105 \, a^{6} d e^{11}\right )} x^{5} - 2 \, {\left (25 \, a^{2} c^{4} d^{10} e^{2} - 80 \, a^{3} c^{3} d^{8} e^{4} - 174 \, a^{4} c^{2} d^{6} e^{6} + 136 \, a^{5} c d^{4} e^{8} - 35 \, a^{6} d^{2} e^{10}\right )} x^{4} + 8 \, {\left (5 \, a^{3} c^{3} d^{9} e^{3} + 423 \, a^{4} c^{2} d^{7} e^{5} + 27 \, a^{5} c d^{5} e^{7} - 7 \, a^{6} d^{3} e^{9}\right )} x^{3} + 16 \, {\left (135 \, a^{4} c^{2} d^{8} e^{4} + 278 \, a^{5} c d^{6} e^{6} + 3 \, a^{6} d^{4} e^{8}\right )} x^{2} + 128 \, {\left (25 \, a^{5} c d^{7} e^{5} + 13 \, a^{6} d^{5} e^{7}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{15360 \, a^{4} d^{5} e^{4} x^{6}}\right ] \end {gather*}

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)/x^7/(e*x+d),x, algorithm="fricas")

[Out]

[-1/30720*(15*(5*c^6*d^12 - 18*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 + 20*a^3*c^3*d^6*e^6 - 45*a^4*c^2*d^4*e^8 +
30*a^5*c*d^2*e^10 - 7*a^6*e^12)*sqrt(a*d*e)*x^6*log((8*a^2*d^2*e^2 + (c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4)*x^2
- 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*a*d*e + (c*d^2 + a*e^2)*x)*sqrt(a*d*e) + 8*(a*c*d^3*e + a^2
*d*e^3)*x)/x^2) + 4*(1280*a^6*d^6*e^6 + (75*a*c^5*d^11*e - 245*a^2*c^4*d^9*e^3 + 150*a^3*c^3*d^7*e^5 - 546*a^4
*c^2*d^5*e^7 + 415*a^5*c*d^3*e^9 - 105*a^6*d*e^11)*x^5 - 2*(25*a^2*c^4*d^10*e^2 - 80*a^3*c^3*d^8*e^4 - 174*a^4
*c^2*d^6*e^6 + 136*a^5*c*d^4*e^8 - 35*a^6*d^2*e^10)*x^4 + 8*(5*a^3*c^3*d^9*e^3 + 423*a^4*c^2*d^7*e^5 + 27*a^5*
c*d^5*e^7 - 7*a^6*d^3*e^9)*x^3 + 16*(135*a^4*c^2*d^8*e^4 + 278*a^5*c*d^6*e^6 + 3*a^6*d^4*e^8)*x^2 + 128*(25*a^
5*c*d^7*e^5 + 13*a^6*d^5*e^7)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^4*d^5*e^4*x^6), -1/15360*(15*
(5*c^6*d^12 - 18*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 + 20*a^3*c^3*d^6*e^6 - 45*a^4*c^2*d^4*e^8 + 30*a^5*c*d^2*
e^10 - 7*a^6*e^12)*sqrt(-a*d*e)*x^6*arctan(1/2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*a*d*e + (c*d^2 +
a*e^2)*x)*sqrt(-a*d*e)/(a*c*d^2*e^2*x^2 + a^2*d^2*e^2 + (a*c*d^3*e + a^2*d*e^3)*x)) + 2*(1280*a^6*d^6*e^6 + (
75*a*c^5*d^11*e - 245*a^2*c^4*d^9*e^3 + 150*a^3*c^3*d^7*e^5 - 546*a^4*c^2*d^5*e^7 + 415*a^5*c*d^3*e^9 - 105*a^
6*d*e^11)*x^5 - 2*(25*a^2*c^4*d^10*e^2 - 80*a^3*c^3*d^8*e^4 - 174*a^4*c^2*d^6*e^6 + 136*a^5*c*d^4*e^8 - 35*a^6
*d^2*e^10)*x^4 + 8*(5*a^3*c^3*d^9*e^3 + 423*a^4*c^2*d^7*e^5 + 27*a^5*c*d^5*e^7 - 7*a^6*d^3*e^9)*x^3 + 16*(135*
a^4*c^2*d^8*e^4 + 278*a^5*c*d^6*e^6 + 3*a^6*d^4*e^8)*x^2 + 128*(25*a^5*c*d^7*e^5 + 13*a^6*d^5*e^7)*x)*sqrt(c*d
*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^4*d^5*e^4*x^6)]

________________________________________________________________________________________

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

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)/x^7/(e*x+d),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [B]  time = 0.06, size = 4735, normalized size = 12.27 \begin {gather*} \text {output too large to display} \end {gather*}

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)/x^7/(e*x+d),x)

[Out]

1/16/d^4*e^5*c*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(3/2)-3/128/d*e^4*c^2*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x
+d/e))^(1/2)+7/1536/d^6*a*e^7*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)+7/512/d^5*a^2*e^8*(c*d*e*x^2+a*d*e+(a*e^
2+c*d^2)*x)^(1/2)-35/384/d^4*e^5*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*c+17/60/d^3/a/x^5*(c*d*e*x^2+a*d*e+(a
*e^2+c*d^2)*x)^(7/2)+1/512*d^3/a^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*c^4-59/320/d/a^3*(c*d*e*x^2+a*d*e+(
a*e^2+c*d^2)*x)^(5/2)*c^3+25/512/d*e^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*c^2+9/64/d^6*e^9*(c*d*e*x^2+a*d
*e+(a*e^2+c*d^2)*x)^(1/2)*x*a^2+15/128/d^5*e^10*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d*e*x^2+a*d*
e+(a*e^2+c*d^2)*x)^(1/2))/(c*d*e)^(1/2)*a^3-101/512/d^7*e^6*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)-1/5/d^7*e^
6*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(5/2)+3/64/d^7*e^10/c*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^3-3/
256*d*e^4*c^3*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/(c*d*e)^
(1/2)+1017/2560/d^7/a*e^4/x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)-1/512*d^5/a^6/e^6*(c*d*e*x^2+a*d*e+(a*e^2+
c*d^2)*x)^(5/2)*c^6-5/512*d^7/a^4/e^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*c^6-397/960/d^5/a*e^4*(c*d*e*x^2
+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*c-35/1536/d^2/a*e^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*c^2+49/1536*d^2/a^3/
e*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*c^4-5/64/d^3*a*e^6*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*c-2681/76
80/d^3/a^2*e^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*c^2+1/64*d^5/a^3/e^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^
(1/2)*c^5-221/7680*d/a^4/e^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*c^4-57/160/d^4/a*e/x^4*(c*d*e*x^2+a*d*e+(
a*e^2+c*d^2)*x)^(7/2)+7/384*d^4/a^4/e^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*c^5-1/64*d/a*e^2*(c*d*e*x^2+a*
d*e+(a*e^2+c*d^2)*x)^(1/2)*c^3-5/256/a*e^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*c^3-11/480/a^4/e^3/x^2*(c
*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)*c^3-45/1024*a*e^5/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1
/2)*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/x)*c^2-1/32/a^3/e^3/x^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)*c
^2-81/1280/a^4/e*c^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*x-1/6/d^2/a/e/x^6*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*
x)^(7/2)+35/768*d/a^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x*c^4+5/256*d^2*e^3/(a*d*e)^(1/2)*ln((2*a*d*e+(a
*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/x)*c^3-185/1536/d^5*e^6*(c*d*e*x^2+a*d*
e+(a*e^2+c*d^2)*x)^(3/2)*x*c+381/1280/d^3/a^3/x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)*c^2+15/512/d^2*e^5*(c*
d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*c^2+89/320/d^3/a^2/x^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)*c-7/1024
/d^4*a^3*e^9/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)
)/x)+1/120*d^3/a^5/e^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*c^5-1543/3840/d^6/a*e^3/x^2*(c*d*e*x^2+a*d*e+(a
*e^2+c*d^2)*x)^(7/2)-5/1536*d^6/a^5/e^5*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*c^6+377/960/d^5/a*e^2/x^3*(c*d
*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)+1/8/d^5*e^6*c*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(3/2)*x-1/8/d^7*e^8*
a*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(3/2)*x-3/64/d^7*e^10*a^3/c*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^
(1/2)-1/16/d^8*e^9*a^2/c*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(3/2)-9/64/d^6*e^9*a^2*((x+d/e)^2*c*d*e+(a*e^
2-c*d^2)*(x+d/e))^(1/2)*x+3/128/d^9*e^12*a^4/c^2*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)+3/64/d^3*e^6*a*
c*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)-15/128/d^5*e^10*a^3*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*
d*e)^(1/2)+((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)-3/64/d^2*e^5*c^2*((x+d/e)^2*c*d*e+(a*e
^2-c*d^2)*(x+d/e))^(1/2)*x+3/256*d*e^4*c^3*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+((x+d/e)^2*c*d
*e+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)+1/8/d^7*e^8*a*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x+1/16/d^
8*e^9*a^2/c*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)-3/128/d^9*e^12*a^4/c^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(
1/2)+3/64/d^8*e^11*a^3/c*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x+9/64/d^4*e^7*a*c*((x+d/e)^2*c*d*e+(a*
e^2-c*d^2)*(x+d/e))^(1/2)*x-3/256/d^9*e^14*a^5/c^2*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+((x+d/
e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)+15/128/d^3*e^8*a^2*c*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*
d*e)/(c*d*e)^(1/2)+((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)-15/256/d*e^6*a*c^2*ln((1/2*a*e
^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)+15/256/
d^7*e^12*a^4/c*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1
/2))/(c*d*e)^(1/2)-15/128/d^3*e^8*c*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d*e*x^2+a*d*e+(a*e^2+c*d
^2)*x)^(1/2))/(c*d*e)^(1/2)*a^2-3/64/d^8*e^11*a^3/c*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x+3/256/d^9*e^14*a
^5/c^2*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/(c*d*e)^(1/2)+1
5/256/d*e^6*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/(c*d*e)^(1
/2)*a*c^2-15/256/d^7*e^12/c*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(
1/2))/(c*d*e)^(1/2)*a^4+15/512/d^2*a^2*e^7/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(c*d*e*x^
2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/x)*c+3/512*d^4/a^3/e*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*c^5-5/512*d^6/a
^4/e^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*c^6-25/768/d/a^2*e^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*
x*c^3+89/7680*d^2/a^5/e^3*c^5*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*x-1/512*d^4/a^6/e^5*c^6*(c*d*e*x^2+a*d*e
+(a*e^2+c*d^2)*x)^(5/2)*x+1/768*d^2/a^5/e^5/x^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)*c^4+43/1536*d^3/a^4/e^
2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x*c^5-5/1536*d^5/a^5/e^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x*c
^6+29/320/d/a^3/e^2/x^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)*c^2+1/192*d/a^4/e^4/x^3*(c*d*e*x^2+a*d*e+(a*e^
2+c*d^2)*x)^(7/2)*c^3-9/512*d^6/a^2/e/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(c*d*e*x^2+a*d
*e+(a*e^2+c*d^2)*x)^(1/2))/x)*c^5+5/1024*d^8/a^3/e^3/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)
*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/x)*c^6+81/1280/d/a^4/e^2/x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)*c
^3-89/7680*d/a^5/e^4/x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)*c^4+1/512*d^3/a^6/e^6/x*(c*d*e*x^2+a*d*e+(a*e^2
+c*d^2)*x)^(7/2)*c^5-11/30/d^4/a^2*e/x^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)*c+3211/7680/d^5/a^2*e^2/x*(c*
d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)*c-1017/2560/d^6/a*e^5*c*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*x-3211/76
80/d^4/a^2*e^3*c^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*x-381/1280/d^2/a^3*e*c^3*(c*d*e*x^2+a*d*e+(a*e^2+c*
d^2)*x)^(5/2)*x-65/512/d^4*a*e^7*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*c+7/256*d^2/a^2*e*(c*d*e*x^2+a*d*e+
(a*e^2+c*d^2)*x)^(1/2)*x*c^4+1/12/d/a^2/e^2/x^5*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)*c+15/1024*d^4/a*e/(a*d
*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/x)*c^4-65/1536/
d^3/a*e^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x*c^2-43/240/d^2/a^2/e/x^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)
^(7/2)*c-113/640/d^2/a^3/e/x^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)*c^2

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 )} x^{7}}\,{d x} \end {gather*}

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)/x^7/(e*x+d),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)*x^7), x)

________________________________________________________________________________________

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

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)/(x^7*(d + e*x)),x)

[Out]

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

________________________________________________________________________________________

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

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)/x**7/(e*x+d),x)

[Out]

Timed out

________________________________________________________________________________________