3.5.55 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{x^6 (d+e x)} \, dx\) [455]

Optimal. Leaf size=395 \[ -\frac {\left (c d^2-a e^2\right ) \left (3 c^2 d^4+6 a c d^2 e^2+7 a^2 e^4\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 a^3 d^4 e^3 x^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 d x^5}-\frac {\left (\frac {3 c}{a e}-\frac {7 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{40 x^4}+\frac {\left (15 c^2 d^4+12 a c d^2 e^2-35 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{240 a^2 d^3 e^2 x^3}+\frac {\left (c d^2-a e^2\right )^3 \left (3 c^2 d^4+6 a c d^2 e^2+7 a^2 e^4\right ) \tanh ^{-1}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{256 a^{7/2} d^{9/2} e^{7/2}} \]

[Out]

-1/5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/d/x^5-1/40*(3*c/a/e-7*e/d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3
/2)/x^4+1/240*(-35*a^2*e^4+12*a*c*d^2*e^2+15*c^2*d^4)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/a^2/d^3/e^2/x^3+
1/256*(-a*e^2+c*d^2)^3*(7*a^2*e^4+6*a*c*d^2*e^2+3*c^2*d^4)*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)-1/128*(-a*e^2+c*d^2)*(7*a^2*e^4+6*
a*c*d^2*e^2+3*c^2*d^4)*(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

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Rubi [A]
time = 0.32, antiderivative size = 395, 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 = {863, 848, 820, 734, 738, 212} \begin {gather*} \frac {\left (-35 a^2 e^4+12 a c d^2 e^2+15 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{240 a^2 d^3 e^2 x^3}+\frac {\left (7 a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\right ) \left (c d^2-a e^2\right )^3 \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 )}{256 a^{7/2} d^{9/2} e^{7/2}}-\frac {\left (7 a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\right ) \left (c d^2-a e^2\right ) \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}}{128 a^3 d^4 e^3 x^2}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 d x^5}-\frac {\left (\frac {3 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 )^{3/2}}{40 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^6*(d + e*x)),x]

[Out]

-1/128*((c*d^2 - a*e^2)*(3*c^2*d^4 + 6*a*c*d^2*e^2 + 7*a^2*e^4)*(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) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(5*d*x^5) - (((
3*c)/(a*e) - (7*e)/d^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(40*x^4) + ((15*c^2*d^4 + 12*a*c*d^2*e^
2 - 35*a^2*e^4)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(240*a^2*d^3*e^2*x^3) + ((c*d^2 - a*e^2)^3*(3*c
^2*d^4 + 6*a*c*d^2*e^2 + 7*a^2*e^4)*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])])/(256*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*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^6 (d+e x)} \, dx &=\int \frac {(a e+c d x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^6} \, dx\\ &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 d x^5}-\frac {\int \frac {\left (-\frac {1}{2} a e \left (3 c d^2-7 a e^2\right )+2 a c d e^2 x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^5} \, dx}{5 a d e}\\ &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 d x^5}-\frac {\left (\frac {3 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 )^{3/2}}{40 x^4}+\frac {\int \frac {\left (-\frac {1}{4} a e \left (15 c^2 d^4+12 a c d^2 e^2-35 a^2 e^4\right )-\frac {1}{2} a c d e^2 \left (3 c d^2-7 a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^4} \, dx}{20 a^2 d^2 e^2}\\ &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 d x^5}-\frac {\left (\frac {3 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 )^{3/2}}{40 x^4}+\frac {\left (15 c^2 d^4+12 a c d^2 e^2-35 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{240 a^2 d^3 e^2 x^3}+\frac {\left (\left (c d^2-a e^2\right ) \left (3 c^2 d^4+6 a c d^2 e^2+7 a^2 e^4\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}{32 a^2 d^3 e^2}\\ &=-\frac {\left (c d^2-a e^2\right ) \left (3 c^2 d^4+6 a c d^2 e^2+7 a^2 e^4\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}}{128 a^3 d^4 e^3 x^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 d x^5}-\frac {\left (\frac {3 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 )^{3/2}}{40 x^4}+\frac {\left (15 c^2 d^4+12 a c d^2 e^2-35 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{240 a^2 d^3 e^2 x^3}-\frac {\left (\left (c d^2-a e^2\right )^3 \left (3 c^2 d^4+6 a c d^2 e^2+7 a^2 e^4\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}{256 a^3 d^4 e^3}\\ &=-\frac {\left (c d^2-a e^2\right ) \left (3 c^2 d^4+6 a c d^2 e^2+7 a^2 e^4\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}}{128 a^3 d^4 e^3 x^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 d x^5}-\frac {\left (\frac {3 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 )^{3/2}}{40 x^4}+\frac {\left (15 c^2 d^4+12 a c d^2 e^2-35 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{240 a^2 d^3 e^2 x^3}+\frac {\left (\left (c d^2-a e^2\right )^3 \left (3 c^2 d^4+6 a c d^2 e^2+7 a^2 e^4\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 )}{128 a^3 d^4 e^3}\\ &=-\frac {\left (c d^2-a e^2\right ) \left (3 c^2 d^4+6 a c d^2 e^2+7 a^2 e^4\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}}{128 a^3 d^4 e^3 x^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 d x^5}-\frac {\left (\frac {3 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 )^{3/2}}{40 x^4}+\frac {\left (15 c^2 d^4+12 a c d^2 e^2-35 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{240 a^2 d^3 e^2 x^3}+\frac {\left (c d^2-a e^2\right )^3 \left (3 c^2 d^4+6 a c d^2 e^2+7 a^2 e^4\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 )}{256 a^{7/2} d^{9/2} e^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.84, size = 317, normalized size = 0.80 \begin {gather*} \frac {\sqrt {(a e+c d x) (d+e x)} \left (-\frac {\sqrt {a} \sqrt {d} \sqrt {e} \left (45 c^4 d^8 x^4-30 a c^3 d^6 e x^3 (d+e x)+6 a^2 c^2 d^4 e^2 x^2 \left (4 d^2+3 d e x-6 e^2 x^2\right )+2 a^3 c d^2 e^3 x \left (264 d^3+48 d^2 e x-61 d e^2 x^2+95 e^3 x^3\right )+a^4 e^4 \left (384 d^4+48 d^3 e x-56 d^2 e^2 x^2+70 d e^3 x^3-105 e^4 x^4\right )\right )}{x^5}+\frac {15 \left (c d^2-a e^2\right )^3 \left (3 c^2 d^4+6 a c d^2 e^2+7 a^2 e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {e} \sqrt {d+e x}}{\sqrt {d} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{1920 a^{7/2} d^{9/2} e^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^6*(d + e*x)),x]

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-((Sqrt[a]*Sqrt[d]*Sqrt[e]*(45*c^4*d^8*x^4 - 30*a*c^3*d^6*e*x^3*(d + e*x) + 6*
a^2*c^2*d^4*e^2*x^2*(4*d^2 + 3*d*e*x - 6*e^2*x^2) + 2*a^3*c*d^2*e^3*x*(264*d^3 + 48*d^2*e*x - 61*d*e^2*x^2 + 9
5*e^3*x^3) + a^4*e^4*(384*d^4 + 48*d^3*e*x - 56*d^2*e^2*x^2 + 70*d*e^3*x^3 - 105*e^4*x^4)))/x^5) + (15*(c*d^2
- a*e^2)^3*(3*c^2*d^4 + 6*a*c*d^2*e^2 + 7*a^2*e^4)*ArcTanh[(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])/(Sqrt[d]*Sqrt[a*e +
c*d*x])])/(Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(1920*a^(7/2)*d^(9/2)*e^(7/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(10574\) vs. \(2(361)=722\).
time = 0.08, size = 10575, normalized size = 26.77

method	result	size
default	\(\text {Expression too large to display}\)	\(10575\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^6/(e*x+d),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^6/(e*x+d),x, algorithm="maxima")

[Out]

integrate((c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/((x*e + d)*x^6), x)

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Fricas [A]
time = 27.84, size = 907, normalized size = 2.30 \begin {gather*} \left [-\frac {{\left (15 \, {\left (3 \, c^{5} d^{10} x^{5} - 3 \, a c^{4} d^{8} x^{5} e^{2} - 2 \, a^{2} c^{3} d^{6} x^{5} e^{4} - 6 \, a^{3} c^{2} d^{4} x^{5} e^{6} + 15 \, a^{4} c d^{2} x^{5} e^{8} - 7 \, a^{5} x^{5} e^{10}\right )} \sqrt {a d} e^{\frac {1}{2}} \log \left (\frac {c^{2} d^{4} x^{2} + 8 \, a c d^{3} x e + a^{2} x^{2} e^{4} + 8 \, a^{2} d x e^{3} - 4 \, {\left (c d^{2} x + a x e^{2} + 2 \, a d e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {a d} e^{\frac {1}{2}} + 2 \, {\left (3 \, a c d^{2} x^{2} + 4 \, a^{2} d^{2}\right )} e^{2}}{x^{2}}\right ) + 4 \, {\left (45 \, a c^{4} d^{9} x^{4} e - 30 \, a^{2} c^{3} d^{8} x^{3} e^{2} - 105 \, a^{5} d x^{4} e^{9} + 70 \, a^{5} d^{2} x^{3} e^{8} + 2 \, {\left (95 \, a^{4} c d^{3} x^{4} - 28 \, a^{5} d^{3} x^{2}\right )} e^{7} - 2 \, {\left (61 \, a^{4} c d^{4} x^{3} - 24 \, a^{5} d^{4} x\right )} e^{6} - 12 \, {\left (3 \, a^{3} c^{2} d^{5} x^{4} - 8 \, a^{4} c d^{5} x^{2} - 32 \, a^{5} d^{5}\right )} e^{5} + 6 \, {\left (3 \, a^{3} c^{2} d^{6} x^{3} + 88 \, a^{4} c d^{6} x\right )} e^{4} - 6 \, {\left (5 \, a^{2} c^{3} d^{7} x^{4} - 4 \, a^{3} c^{2} d^{7} x^{2}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-4\right )}}{7680 \, a^{4} d^{5} x^{5}}, -\frac {{\left (15 \, {\left (3 \, c^{5} d^{10} x^{5} - 3 \, a c^{4} d^{8} x^{5} e^{2} - 2 \, a^{2} c^{3} d^{6} x^{5} e^{4} - 6 \, a^{3} c^{2} d^{4} x^{5} e^{6} + 15 \, a^{4} c d^{2} x^{5} e^{8} - 7 \, a^{5} x^{5} e^{10}\right )} \sqrt {-a d e} \arctan \left (\frac {{\left (c d^{2} x + a x e^{2} + 2 \, a d e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {-a d e}}{2 \, {\left (a c d^{3} x e + a^{2} d x e^{3} + {\left (a c d^{2} x^{2} + a^{2} d^{2}\right )} e^{2}\right )}}\right ) + 2 \, {\left (45 \, a c^{4} d^{9} x^{4} e - 30 \, a^{2} c^{3} d^{8} x^{3} e^{2} - 105 \, a^{5} d x^{4} e^{9} + 70 \, a^{5} d^{2} x^{3} e^{8} + 2 \, {\left (95 \, a^{4} c d^{3} x^{4} - 28 \, a^{5} d^{3} x^{2}\right )} e^{7} - 2 \, {\left (61 \, a^{4} c d^{4} x^{3} - 24 \, a^{5} d^{4} x\right )} e^{6} - 12 \, {\left (3 \, a^{3} c^{2} d^{5} x^{4} - 8 \, a^{4} c d^{5} x^{2} - 32 \, a^{5} d^{5}\right )} e^{5} + 6 \, {\left (3 \, a^{3} c^{2} d^{6} x^{3} + 88 \, a^{4} c d^{6} x\right )} e^{4} - 6 \, {\left (5 \, a^{2} c^{3} d^{7} x^{4} - 4 \, a^{3} c^{2} d^{7} x^{2}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-4\right )}}{3840 \, a^{4} d^{5} x^{5}}\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)^(3/2)/x^6/(e*x+d),x, algorithm="fricas")

[Out]

[-1/7680*(15*(3*c^5*d^10*x^5 - 3*a*c^4*d^8*x^5*e^2 - 2*a^2*c^3*d^6*x^5*e^4 - 6*a^3*c^2*d^4*x^5*e^6 + 15*a^4*c*
d^2*x^5*e^8 - 7*a^5*x^5*e^10)*sqrt(a*d)*e^(1/2)*log((c^2*d^4*x^2 + 8*a*c*d^3*x*e + a^2*x^2*e^4 + 8*a^2*d*x*e^3
- 4*(c*d^2*x + a*x*e^2 + 2*a*d*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(a*d)*e^(1/2) + 2*(3*a*c*d^
2*x^2 + 4*a^2*d^2)*e^2)/x^2) + 4*(45*a*c^4*d^9*x^4*e - 30*a^2*c^3*d^8*x^3*e^2 - 105*a^5*d*x^4*e^9 + 70*a^5*d^2
*x^3*e^8 + 2*(95*a^4*c*d^3*x^4 - 28*a^5*d^3*x^2)*e^7 - 2*(61*a^4*c*d^4*x^3 - 24*a^5*d^4*x)*e^6 - 12*(3*a^3*c^2
*d^5*x^4 - 8*a^4*c*d^5*x^2 - 32*a^5*d^5)*e^5 + 6*(3*a^3*c^2*d^6*x^3 + 88*a^4*c*d^6*x)*e^4 - 6*(5*a^2*c^3*d^7*x
^4 - 4*a^3*c^2*d^7*x^2)*e^3)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e))*e^(-4)/(a^4*d^5*x^5), -1/3840*(15*(3
*c^5*d^10*x^5 - 3*a*c^4*d^8*x^5*e^2 - 2*a^2*c^3*d^6*x^5*e^4 - 6*a^3*c^2*d^4*x^5*e^6 + 15*a^4*c*d^2*x^5*e^8 - 7
*a^5*x^5*e^10)*sqrt(-a*d*e)*arctan(1/2*(c*d^2*x + a*x*e^2 + 2*a*d*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*
e)*sqrt(-a*d*e)/(a*c*d^3*x*e + a^2*d*x*e^3 + (a*c*d^2*x^2 + a^2*d^2)*e^2)) + 2*(45*a*c^4*d^9*x^4*e - 30*a^2*c^
3*d^8*x^3*e^2 - 105*a^5*d*x^4*e^9 + 70*a^5*d^2*x^3*e^8 + 2*(95*a^4*c*d^3*x^4 - 28*a^5*d^3*x^2)*e^7 - 2*(61*a^4
*c*d^4*x^3 - 24*a^5*d^4*x)*e^6 - 12*(3*a^3*c^2*d^5*x^4 - 8*a^4*c*d^5*x^2 - 32*a^5*d^5)*e^5 + 6*(3*a^3*c^2*d^6*
x^3 + 88*a^4*c*d^6*x)*e^4 - 6*(5*a^2*c^3*d^7*x^4 - 4*a^3*c^2*d^7*x^2)*e^3)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 +
a*d)*e))*e^(-4)/(a^4*d^5*x^5)]

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Sympy [F(-1)] Timed out
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)**(3/2)/x**6/(e*x+d),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2381 vs. \(2 (354) = 708\).
time = 1.52, size = 2381, normalized size = 6.03 \begin {gather*} \text {Too large to display} \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)^(3/2)/x^6/(e*x+d),x, algorithm="giac")

[Out]

-1/128*(3*c^5*d^10 - 3*a*c^4*d^8*e^2 - 2*a^2*c^3*d^6*e^4 - 6*a^3*c^2*d^4*e^6 + 15*a^4*c*d^2*e^8 - 7*a^5*e^10)*
arctan(-(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))/sqrt(-a*d*e))*e^(-3)/(sqrt(-a*d*e)
*a^3*d^4) + 1/1920*(45*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))*a^4*c^5*d^14*e^4 -
210*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^3*a^3*c^5*d^13*e^3 - 384*(sqrt(c*d)*x*
e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^5*a^2*c^5*d^12*e^2 + 210*(sqrt(c*d)*x*e^(1/2) - sqrt(c*
d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^7*a*c^5*d^11*e - 45*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*
x*e^2 + a*d*e))^9*c^5*d^10 - 3840*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^4*sqrt(c
*d)*a^3*c^4*d^11*e^(7/2) - 45*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))*a^5*c^4*d^12
*e^6 - 7470*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^3*a^4*c^4*d^11*e^5 - 9600*(sqr
t(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^5*a^3*c^4*d^10*e^4 - 210*(sqrt(c*d)*x*e^(1/2)
- sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^7*a^2*c^4*d^9*e^3 + 45*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e +
c*d^2*x + a*x*e^2 + a*d*e))^9*a*c^4*d^8*e^2 - 7680*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 +
a*d*e))^2*sqrt(c*d)*a^5*c^3*d^10*e^(13/2) - 26880*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 +
a*d*e))^4*sqrt(c*d)*a^4*c^3*d^9*e^(11/2) - 11520*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 +
a*d*e))^6*sqrt(c*d)*a^3*c^3*d^8*e^(9/2) - 3870*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d
*e))*a^6*c^3*d^10*e^8 - 34420*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^3*a^5*c^3*d^
9*e^7 - 37120*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^5*a^4*c^3*d^8*e^6 - 5260*(sq
rt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^7*a^3*c^3*d^7*e^5 + 30*(sqrt(c*d)*x*e^(1/2) -
sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^9*a^2*c^3*d^6*e^4 - 768*sqrt(c*d)*a^7*c^2*d^9*e^(19/2) - 23040*(
sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^2*sqrt(c*d)*a^6*c^2*d^8*e^(17/2) - 53760*(s
qrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^4*sqrt(c*d)*a^5*c^2*d^7*e^(15/2) - 19200*(sq
rt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^6*sqrt(c*d)*a^4*c^2*d^6*e^(13/2) - 7770*(sqrt
(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))*a^7*c^2*d^8*e^10 - 41820*(sqrt(c*d)*x*e^(1/2) -
sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^3*a^6*c^2*d^7*e^9 - 30720*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e
+ c*d^2*x + a*x*e^2 + a*d*e))^5*a^5*c^2*d^6*e^8 - 420*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^
2 + a*d*e))^7*a^4*c^2*d^5*e^7 + 90*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^9*a^3*c
^2*d^4*e^6 - 1280*sqrt(c*d)*a^8*c*d^7*e^(23/2) - 16640*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e
^2 + a*d*e))^2*sqrt(c*d)*a^7*c*d^6*e^(21/2) - 28160*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2
+ a*d*e))^4*sqrt(c*d)*a^6*c*d^5*e^(19/2) - 3615*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*
d*e))*a^8*c*d^6*e^12 - 12570*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^3*a^7*c*d^5*e
^11 - 1920*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^5*a^6*c*d^4*e^10 + 1050*(sqrt(c
*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^7*a^5*c*d^3*e^9 - 225*(sqrt(c*d)*x*e^(1/2) - sqrt
(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^9*a^4*c*d^2*e^8 - 3840*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*
x + a*x*e^2 + a*d*e))^2*sqrt(c*d)*a^8*d^4*e^(25/2) - 105*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x
*e^2 + a*d*e))*a^9*d^4*e^14 - 790*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^3*a^8*d^
3*e^13 + 896*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^5*a^7*d^2*e^12 - 490*(sqrt(c*
d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^7*a^6*d*e^11 + 105*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d
*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^9*a^5*e^10)*e^(-3)/((a*d*e - (sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^
2*x + a*x*e^2 + a*d*e))^2)^5*a^3*d^4)

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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 )}^{3/2}}{x^6\,\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)^(3/2)/(x^6*(d + e*x)),x)

[Out]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)/(x^6*(d + e*x)), x)

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