\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{x^7 (d+e x)} \, dx\) [467]

Optimal result

Integrand size = 40, antiderivative size = 386 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^7 (d+e x)} \, dx=-\frac {\left (c d^2-a e^2\right )^3 \left (5 c d^2+7 a e^2\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 a^3 d^4 e^3 x^2}+\frac {\left (c d^2-a e^2\right ) \left (5 c d^2+7 a e^2\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 a^2 d^3 e^2 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 d x^6}-\frac {\left (\frac {5 c}{a e}-\frac {7 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 x^5}+\frac {\left (c d^2-a e^2\right )^5 \left (5 c d^2+7 a e^2\right ) \text {arctanh}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{1024 a^{7/2} d^{9/2} e^{7/2}} \]

[Out]

1/192*(-a*e^2+c*d^2)*(7*a*e^2+5*c*d^2)*(2*a*d*e+(a*e^2+c*d^2)*x)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/a^2/d
^3/e^2/x^4-1/6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/d/x^6-1/60*(5*c/a/e-7*e/d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d
*e*x^2)^(5/2)/x^5+1/1024*(-a*e^2+c*d^2)^5*(7*a*e^2+5*c*d^2)*arctanh(1/2*(2*a*d*e+(a*e^2+c*d^2)*x)/a^(1/2)/d^(1
/2)/e^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/a^(7/2)/d^(9/2)/e^(7/2)-1/512*(-a*e^2+c*d^2)^3*(7*a*e^2+5
*c*d^2)*(2*a*d*e+(a*e^2+c*d^2)*x)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a^3/d^4/e^3/x^2

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {863, 848, 820, 734, 738, 212} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^7 (d+e x)} \, dx=\frac {\left (7 a e^2+5 c d^2\right ) \left (c d^2-a e^2\right )^5 \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 a^{7/2} d^{9/2} e^{7/2}}-\frac {\left (7 a e^2+5 c d^2\right ) \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 a^3 d^4 e^3 x^2}+\frac {\left (7 a e^2+5 c d^2\right ) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{192 a^2 d^3 e^2 x^4}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 d x^6}-\frac {\left (\frac {5 c}{a e}-\frac {7 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{60 x^5} \]

[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]

-1/512*((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])/(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 212

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

Rule 734

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] /; Free
Q[{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 738

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 820

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)/(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[S
implify[m + 2*p + 3], 0]

Rule 848

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 863

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*} \text {integral}& = \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 ) \text {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] (verified)

Time = 0.91 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^7 (d+e x)} \, dx=\frac {\left (-c d^2+a e^2\right )^5 ((a e+c d x) (d+e x))^{3/2} \left (\frac {\sqrt {a} \sqrt {d} \sqrt {e} \left (75 c^5 d^{10} x^5-5 a c^4 d^8 e x^4 (10 d+49 e x)+10 a^2 c^3 d^6 e^2 x^3 \left (4 d^2+16 d e x+15 e^2 x^2\right )+6 a^3 c^2 d^4 e^3 x^2 \left (360 d^3+564 d^2 e x+58 d e^2 x^2-91 e^3 x^3\right )+a^4 c d^2 e^4 x \left (3200 d^4+4448 d^3 e x+216 d^2 e^2 x^2-272 d e^3 x^3+415 e^4 x^4\right )+a^5 e^5 \left (1280 d^5+1664 d^4 e x+48 d^3 e^2 x^2-56 d^2 e^3 x^3+70 d e^4 x^4-105 e^5 x^5\right )\right )}{\left (c d^2-a e^2\right )^5 x^6 (a e+c d x) (d+e x)}-\frac {15 \left (5 c d^2+7 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{(a e+c d x)^{3/2} (d+e x)^{3/2}}\right )}{7680 a^{7/2} d^{9/2} e^{7/2}} \]

[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]

((-(c*d^2) + a*e^2)^5*((a*e + c*d*x)*(d + e*x))^(3/2)*((Sqrt[a]*Sqrt[d]*Sqrt[e]*(75*c^5*d^10*x^5 - 5*a*c^4*d^8
*e*x^4*(10*d + 49*e*x) + 10*a^2*c^3*d^6*e^2*x^3*(4*d^2 + 16*d*e*x + 15*e^2*x^2) + 6*a^3*c^2*d^4*e^3*x^2*(360*d
^3 + 564*d^2*e*x + 58*d*e^2*x^2 - 91*e^3*x^3) + a^4*c*d^2*e^4*x*(3200*d^4 + 4448*d^3*e*x + 216*d^2*e^2*x^2 - 2
72*d*e^3*x^3 + 415*e^4*x^4) + a^5*e^5*(1280*d^5 + 1664*d^4*e*x + 48*d^3*e^2*x^2 - 56*d^2*e^3*x^3 + 70*d*e^4*x^
4 - 105*e^5*x^5)))/((c*d^2 - a*e^2)^5*x^6*(a*e + c*d*x)*(d + e*x)) - (15*(5*c*d^2 + 7*a*e^2)*ArcTanh[(Sqrt[d]*
Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/((a*e + c*d*x)^(3/2)*(d + e*x)^(3/2))))/(7680*a^(7/2)*d^(
9/2)*e^(7/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(32290\) vs. \(2(352)=704\).

Time = 1.97 (sec) , antiderivative size = 32291, normalized size of antiderivative = 83.66

method	result	size
default	\(\text {Expression too large to display}\)	\(32291\)

[In]

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

[Out]

result too large to display

Fricas [A] (verification not implemented)

none

Time = 41.44 (sec) , antiderivative size = 1072, normalized size of antiderivative = 2.78 \[ \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=\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 ] \]

[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)]

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^7 (d+e x)} \, dx=\text {Timed out} \]

[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

Maxima [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^7 (d+e x)} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )} x^{7}} \,d x } \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3388 vs. \(2 (352) = 704\).

Time = 0.51 (sec) , antiderivative size = 3388, normalized size of antiderivative = 8.78 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^7 (d+e x)} \, dx=\text {Too large to display} \]

[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]

-1/512*(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)*arctan(-(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))/sqrt(-a*d*e))/
(sqrt(-a*d*e)*a^3*d^4*e^3) + 1/7680*(75*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^5*c^6*
d^17*e^5 - 270*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^6*c^5*d^15*e^7 + 225*(sqrt(c*d*
e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^7*c^4*d^13*e^9 + 15660*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 +
c*d^2*x + a*e^2*x + a*d*e))*a^8*c^3*d^11*e^11 + 14685*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a
*d*e))*a^9*c^2*d^9*e^13 + 450*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^10*c*d^7*e^15 -
105*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^11*d^5*e^17 + 3072*sqrt(c*d*e)*a^9*c^2*d^1
0*e^12 - 425*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^4*c^6*d^16*e^4 + 1530*(sqrt(c*d
*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^5*c^5*d^14*e^6 + 75525*(sqrt(c*d*e)*x - sqrt(c*d*e*x^
2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^6*c^4*d^12*e^8 + 203100*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*
x + a*d*e))^3*a^7*c^3*d^10*e^10 + 142065*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^8*c
^2*d^8*e^12 + 28170*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^9*c*d^6*e^14 + 595*(sqrt
(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^10*d^4*e^16 + 46080*sqrt(c*d*e)*(sqrt(c*d*e)*x -
sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^2*a^7*c^3*d^11*e^9 + 73728*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*
e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^2*a^8*c^2*d^9*e^11 + 30720*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c
*d^2*x + a*e^2*x + a*d*e))^2*a^9*c*d^7*e^13 + 990*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)
)^5*a^3*c^6*d^15*e^3 + 42516*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^5*a^4*c^5*d^13*e^5
+ 279450*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^5*a^5*c^4*d^11*e^7 + 526200*(sqrt(c*d*e
)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^5*a^6*c^3*d^9*e^9 + 313650*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2
+ c*d^2*x + a*e^2*x + a*d*e))^5*a^7*c^2*d^7*e^11 + 52020*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x +
a*d*e))^5*a^8*c*d^5*e^13 + 1686*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^5*a^9*d^3*e^15
+ 76800*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^4*a^5*c^4*d^12*e^6 + 307200*
sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^4*a^6*c^3*d^10*e^8 + 368640*sqrt(c*d
*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^4*a^7*c^2*d^8*e^10 + 122880*sqrt(c*d*e)*(sqr
t(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^4*a^8*c*d^6*e^12 + 15360*sqrt(c*d*e)*(sqrt(c*d*e)*x
- sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^4*a^9*d^4*e^14 + 990*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x
+ a*e^2*x + a*d*e))^7*a^2*c^6*d^14*e^2 + 39444*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^7
*a^3*c^5*d^12*e^4 + 233370*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^7*a^4*c^4*d^10*e^6 +
372600*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^7*a^5*c^3*d^8*e^8 + 160050*(sqrt(c*d*e)*x
- sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^7*a^6*c^2*d^6*e^10 + 5940*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*
d^2*x + a*e^2*x + a*d*e))^7*a^7*c*d^4*e^12 - 1386*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)
)^7*a^8*d^2*e^14 + 15360*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^6*a^3*c^5*d
^13*e^3 + 143360*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^6*a^4*c^4*d^11*e^5
+ 460800*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^6*a^5*c^3*d^9*e^7 + 430080*
sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^6*a^6*c^2*d^7*e^9 + 128000*sqrt(c*d*
e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^6*a^7*c*d^5*e^11 - 425*(sqrt(c*d*e)*x - sqrt(
c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^9*a*c^6*d^13*e + 1530*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^
2*x + a*d*e))^9*a^2*c^5*d^11*e^3 + 44805*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^9*a^3*c
^4*d^9*e^5 + 64860*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^9*a^4*c^3*d^7*e^7 + 3825*(sqr
t(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^9*a^5*c^2*d^5*e^9 - 2550*(sqrt(c*d*e)*x - sqrt(c*d*e
*x^2 + c*d^2*x + a*e^2*x + a*d*e))^9*a^6*c*d^3*e^11 + 595*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x
+ a*d*e))^9*a^7*d*e^13 + 61440*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^8*a^3
*c^4*d^10*e^4 + 184320*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^8*a^4*c^3*d^8
*e^6 + 138240*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^8*a^5*c^2*d^6*e^8 + 75
*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^11*c^6*d^12 - 270*(sqrt(c*d*e)*x - sqrt(c*d*e*x
^2 + c*d^2*x + a*e^2*x + a*d*e))^11*a*c^5*d^10*e^2 + 225*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x +
a*d*e))^11*a^2*c^4*d^8*e^4 + 300*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^11*a^3*c^3*d^6
*e^6 - 675*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^11*a^4*c^2*d^4*e^8 + 450*(sqrt(c*d*e)
*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^11*a^5*c*d^2*e^10 - 105*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*
d^2*x + a*e^2*x + a*d*e))^11*a^6*e^12 + 15360*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x
+ a*d*e))^10*a^3*c^3*d^7*e^5)/((a*d*e - (sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^2)^6*a^3
*d^4*e^3)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^7 (d+e x)} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{x^7\,\left (d+e\,x\right )} \,d x \]

[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)